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Connection matrices in combinatorial topological dynamics

Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of the boundary operator in the Morse complex in Morse theory. Their computability has recently been addressed by Harker, Mischaikow, and Spendlove in the context of lattice filtered chain complexes. In the current paper, we extend the recently introduced Conley theory for combinatorial vector and multivector fields on Lefschetz complexes by transferring the concept of connection matrix to this setting. This is accomplished by the notion of connection matrix for arbitrary poset filtered chain complexes, as well as an associated equivalence, which allows for changes in the underlying posets. We show that for the special case of gradient combinatorial vector fields in the sense of Forman \cite{Fo98a}, connection matrices are necessarily unique. Thus, the classical results of Reineck have a natural analogue in the combinatorial setting.


Introduction
Classical Morse theory concerns a compact smooth manifold together with a smooth, real-valued function with non-degenerate critical points. An important part of the theory concerns the Morse complex which is a chain complex whose ith chain group is a free abelian group spanned by critical points of Morse index i and whose boundary homomorphism is defined by counting the (oriented) flow lines between critical points in the gradient flow induced by the Morse function (see [12,Section 4.2]). The fundamental result of classical Morse theory states that the homology of the manifold is isomorphic to the homology of the Morse complex.
The stationary points of the gradient flow in Morse theory, which are precisely the critical points of the Morse function, provide the simplest example of an isolated invariant set, a key concept of Conley theory [3]. For every isolated invariant set there is a homology module associated with it. It is called the homology Conley index. A (minimal) Morse decomposition is a decomposition of space into a partially ordered collection of isolated invariant sets, called Morse sets, such that every recurrent trajectory (in particular every stationary or periodic trajectory) is located in a Morse set and every nonrecurrent trajectory is a heteroclinic connection between Morse sets from a higher Morse set to a lower Morse set in the poset structure of the Morse decomposition. The collection of stationary points of the gradient flow of a Morse function provides the simplest example of a Morse decomposition in which the Morse sets are just the stationary points and the Conley index of a stationary point coincides with the homology of a pointed k-dimensional sphere with k equal to the Morse index of the point.
Conley theory in its simplest form may be viewed as a twofold generalization of Morse theory. On one hand it substantially weakens the general assumptions by replacing the smooth manifold by a compact metric space and the gradient flow of the Morse function by an arbitrary (semi)flow. On the other hand it replaces the collection of critical points of the Morse function by the more general Morse decomposition in which the counterpart of the Morse complex takes the form of the direct sum of the Conley indexes of all Morse sets. The homology of this generalized complex, as in the Morse theory, is isomorphic to the homology of the space. The boundary operator in this setting is called the connection matrix [7]. As in the Morse theory the connection matrix is defined in terms of the flow. The construction of connection matrix is technically complicated, in part because the generalized setting captures the situations of bifurcations when, unlike the Morse theory, the connection matrix need not be uniquely determined by the flow. Since a few years ago, connection matrices, despite their complicated construction, may be computed by algorithmic means due to an efficient algorithm recently proposed by Harker, Mischaikow, and Spendlove [8]. The algorithm utilizes a clear separation of dynamics and algebra which facilitates viewing a connection matrix as a purely algebraic object associated with every lattice filtered chain complex and considering a connection matrix of a Morse decomposition as a connection matrix of a filtered chain complex constructed from a lattice of attractors of the Morse decomposition.
Conley theory, in particular via connection matrices, is a very useful tool in the qualitative study of dynamical systems. However, to apply it one requires a well-defined dynamical system on a compact metric space. This is not the case when the dynamical system is exposed only via a finite set of samples as in the case of time series collected from observations or experiments. The study of dynamical systems known only from samples becomes an important part of the rapidly growing field of data science. In this context a generalization of Morse theory presented by Robin Forman [6] turned out to be very fruitful. In this generalization the smooth manifold is replaced by a finite CW complex and the gradient vector field of the Morse function by the concept of a combinatorial vector field. These structures may be easily constructed from data and analyzed by means of the combinatorial, also called discrete, Morse theory by Forman. Recently, the concepts of isolated invariant set and Conley index have been carried over to this combinatorial setting [13,17,11]. In this paper we extend the ideas introduced there by constructing connection matrices of a Morse decomposition of a combinatorial multivector field. A problem here is that the dynamics of combinatorial vector and multivector fields is, by its very nature, multivalued. One can give examples which demonstrate that index filtrations, that is certain lattice homomorphisms needed in the construction of the connection matrices, fail to exist in the case of multivalued dynamics. Actually, such examples exist also in the setting of multivalued flows in compact metric spaces. We overcome this difficulty by looking into the whole family of Morse decompositions with varying poset structure. By adding some Morse sets with trivial Conley index we reduce the problem to a special case when the index filtrations do exist and their construction is straightforward. The added Morse sets may then be removed by introducing a certain equivalence in the category of filtered chain complexes with a varying poset structure. This lets us consider connection matrices in the setting of combinatorial vector and multivector fields and prove the uniqueness of connection matrices for gradient combinatorial vector fields.

Main results
The main result of the paper is the following theorem.
Theorem 2.1. Assume V is a gradient combinatorial vector field on a Lefschetz complex X. Then the Morse decomposition consisting of all the critical cells of V has precisely one connection matrix. It coincides with the matrix of the boundary operator of the associated Morse complex.
The connection matrix need not be unique in the case of a general combinatorial multivector field or in the case of a non-gradient combinatorial The matrix C 1 is the unique matrix of the combinatorial multivector field shown in the middle of Figure 1, while C 2 is the unique matrix for the combinatorial multivector field shown on the right.

Preliminaries
3.1. Sets and maps. We denote the sets of reals, integers, positive integers, and non-negative integers by R, Z, N, and N 0 , respectively. Given a set X, we write card X for the number of elements of X and we denote the family of all subsets of X by P(X). By a partition of a set X we mean a family E ⊂ P(X) of mutually disjoint non-empty subsets of X such that E = X. For a subfamily E ⊂ E of a partition E of X we will use the compact notation |E | to denote the union of sets in E , that is, |E | = E . We say that a subset E ⊂ X is E-compatible with respect to a partition E of X if we have E = |E | for a subfamily E ⊂ E.
A K-indexed family of subsets of a set X is an injective map K k → X k ∈ P(X). We denote such a family by (X k ) k∈K . Note that every family E ⊂ P(X) may be considered as an E-indexed family E E → E ∈ P(X). In this case we say that E is a self-indexing family. By a K-gradation of a set X we mean a K-indexed family (X k ) k∈K such that {X k } k∈K is a partition of X. Note that every surjective map f : X → K provides a K-gradation (f −1 (k)) k∈K of X. We refer to this K-gradation as the f -gradation of X. Given A ⊂ X and a gradation (X k ) k∈K of X, by the induced gradation we mean the gradation ( We write f : X Y for a partial map from X to Y , that is, a map defined on a subset dom f ⊂ X, called the domain of f , and such that the set of values of f , denoted im f , is contained in Y . Partial maps are composed in the obvious way: If f : X Y and g : Y Z are partial maps, then the domain of their composition g • f : X Z is given by dom(g • f ) = f −1 (dom g), and on this domain we have (g • f )(x) = g(f (x)).
In the sequel, we work with the category of finite sets with a distinguished subset, denoted DSet, and defined as follows. The objects of DSet are pairs (X, X ), where X is a finite set and X ⊂ X is a distinguished subset. The morphisms from (X, X ) to (Y, Y ) in DSet are subset preserving partial maps, that is, partial maps f : X Y such that X ⊂ dom f and f (X ) ⊂ Y . One easily verifies that DSet with the composition of morphisms defined as the composition of partial maps and the identity morphism defined as the identity map is indeed a category.

3.2.
Relations, multivalued maps and digraphs. Given a set X and a binary relation R ⊂ X × X, we use the shorthand xRy for (x, y) ∈ R. The inverse of R, denoted R −1 , is the relation R −1 := { (y, x) | xRy }. By the transitive closure of R we mean the relationR ⊂ X × X given by xRy if there exists a sequence x = x 0 , x 1 , . . . , x n = y such that n ≥ 1 and x i−1 Rx i for i = 1, 2, . . . , n. Note thatR is transitive but need not be reflexive. The relationR ∪ id X , where id X stands for the identity relation on X, is reflexive and transitive. Hence, it is a preorder, called the preorder induced by R. An element y ∈ X covers an x ∈ X in the relation R if xRy and x = y, but there is no z ∈ X such that x = z = y and both xRz and zRy are satisfied.
A multivalued map F : X Y is a map F : X → P(Y ). For A ⊂ X we define the image of A by F (A) := { F (x) | x ∈ A } and for B ⊂ Y we define the preimage of B by F −1 (B) : Given a relation R, we associate with it a multivalued map F R : X X, by F R (x) := R(x), where R(x) := { y ∈ X | xRy } is the image of x ∈ X in R. Obviously, R → F R is a one-to-one correspondence between binary relations in X and multivalued maps from X to X. Often, it will be convenient to interpret the relation R as a directed graph whose set of vertices is X and a directed arrow goes from x to y whenever xRy. The three concepts relation, multivalued map and directed graph are equivalent on the formal level and the distinction is used only to emphasize different viewpoints. However, in this paper it will be convenient to use all these concepts interchangeably.
3.3. Partial orders. Recall that a preorder in a set P is a reflexive and transitive relation in P . If not stated otherwise we denote a preorder by ≤ and its inverse by ≥. We also write < and > for the associated strict preorders, that is, the relations ≤ and ≥ excluding equality. A partial order is a preorder which is antisymmetric. We also recall that a preordered set (a poset) is a pair (P, ≤) where P is a set and ≤ is a preorder (a partial order) in P . Whenever the preorder (the partial order) is clear from the context, we shorten the notation (P, ≤) to P .
Given a preordered set P we say that a q ∈ P covers a p ∈ P if p < q and there is no r ∈ P such that p < r < q. We say that p is a predecessor of q if q covers p. Recall that a subset A ⊂ P is convex if the conditions x, z ∈ A and x ≤ y ≤ z imply y ∈ A. A set A ⊂ P is a down set or a lower set if the conditions z ∈ A and y ≤ z yield y ∈ A. Dually, the subset A ⊂ P is an upper set if z ∈ A and z ≤ y guarantee y ∈ A.
We denote the family of all down sets in P by Down(P ). For A ⊂ P we further write A ≤ := { x ∈ P | ∃ a∈A x ≤ a } and A < := A ≤ \ A.
Proposition 3.1. If I is convex, then both I ≤ and I < are down sets.
Proof: The verification that I ≤ is a down set is straightforward. To see that I < is a down set take an x ∈ I < . Then we have x ∈ I and x < z for some z ∈ I. Let y ≤ x. Then y ∈ I ≤ . Since I is convex, we cannot have the inclusion y ∈ I. It follows that y ∈ I < .
Let P and P be posets. We say that a partial map f : P P is order preserving if the conditions x, y ∈ dom f and x ≤ y imply f (x) ≤ f (y). We say that f is an order isomorphism if f is an order preserving bijection such that f −1 is also order preserving.
We define the category DPSet of posets with a distinguished subset as follows. Its objects are pairs (P, P ) where P is a finite poset and P ⊂ P is a distinguished subset. A morphism from (P, P ) to (P , P ) in DPSet is an order preserving partial map f : P P such that P ⊂ dom f and f (P ) ⊂ P . One easily verifies that DPSet with the composition of morphisms defined as the composition of partial maps and the identity morphism defined as the identity map is indeed a category. Moreover, since every set may be considered as a poset partially ordered by the identity and, clearly, every partial map preserves identities, we may consider DSet as a subcategory of DPSet. In this case, every partial map is automatically order preserving.
To simplify notation, in the sequel we denote objects of DSet and DPSet with a single capital letter and the distinguished subset by the same letter with subscript .
3.4. Distributive lattices. For the terminology of this paper concerning lattices we refer to [23]. Here we briefly recall that a lattice is a triple (L, ∨, ∧) where L is a set and ∨, ∧ : L × L → L are binary operations on L which are called join and meet, respectively, and which are idempotent, commutative, associative and satisfy the absorption rule, that is, a∨(a∧b) = a∧(a∨b) = a for all a, b ∈ L. A lattice is distributive if the operations ∨ and ∧ are mutually distributive. If there exists an element a ∈ L such that a = a ∧ b for every b ∈ L, it is called a zero element. It is straightforward to observe that if a zero element exists, it is unique. We denote it by 0. Similarly, if there exists an element a ∈ L such that a = a ∨ b for every b ∈ L, it is called a unit element. If a unit element exists, it is unique. We denote it by 1. A bounded lattice is a quintuple (L, 0, 1, ∨, ∧) such that (L, ∨, ∧) is a lattice with 0 ∈ L as zero element and 1 ∈ L as unit element.
In the sequel we assume that every considered lattice is a finite, distributive, and bounded lattice. By saying that L is a lattice, we actually mean the lattice (L, 0, 1, ∨, ∧) in general notation -unless we specify precisely or it is clear from context what the zero, the unit, and the operators ∨ and ∧ are.
A subset L ⊂ L is a sublattice of a lattice L if the zero and unit elements of L belong to L and the binary operations ∨ and ∧ in L are closed with respect to L , that is, we have both ∨(L × L ) ⊂ L and ∧(L × L ) ⊂ L . Then L with 0, 1, and the restricted operations ∨ |L ×L , ∧ |L ×L is itself a lattice.
Given two lattices L and L , a map h : L → L is called a lattice homomorphism (or briefly, a homomorphism) if it preserves the binary operations ∨ and ∧, and it maps the zero and unit elements in L to the zero and unit elements in L , respectively.
A non-zero element a ∈ L is called join-irreducible, if a = a 1 ∨ a 2 implies that a = a 1 or a = a 2 . We denote the set of join-irreducible elements of L by JIrr(L). Every lattice induces a partial order ≤ L on L given by a ≤ L b if one has a = a ∧ b. We note that zero is the only element in the resulting poset (X, ≤ L ) which has no predecessor. In fact, we have the following proposition.
Proposition 3.2. Assume that L is a finite, bounded, distributive lattice and that a ∈ L. Then a is join-irreducible if and only if a has exactly one predecessor with respect to the lattice induced partial order ≤ L .
Proof: Assume a ∈ L is join-irreducible. Then, a = 0. Hence, a has a predecessor. Suppose b i for i ∈ {1, 2} are two predecessors of a.
In consequence b 1 = b 2 , which proves the uniqueness of the predecessor.
To prove the opposite implication, assume to the contrary that a has exactly one predecessor and that a is not join-irreducible. Then a = 0 and a = a 1 ∨ a 2 for some a 1 , a 2 ∈ L such that a 1 = a = a 2 . By the absorption rule a i ≤ L a for i ∈ {1, 2}. Letā i be a predecessor of a such that a i ≤ Lāi . It suffices to prove thatā 1 =ā 2 , because then a has more than one predecessor. Thus, assumeā 1 =ā 2 and setā :=ā 1 =ā 2 . Then (a 1 ∨ a 2 ) ∧ā = (a 1 ∧ā) ∨ (a 2 ∧ā) = a 1 ∨ a 2 . It immediately follows that one then has a = a 1 ∨ a 2 ≤ Lā < L a, a contradiction proving thatā 1 =ā 2 . Therefore, a has at least two predecessors -which in turn contradicts the assumption that a has exactly one predecessor.
If a ∈ L is join-irreducible, we denote the unique predecessor of a with respect to ≤ L by a .
We recall that by P(X) we denote the family of all subsets of a set X. It is straightforward to observe that given an arbitrary finite set X, the quintuple (P(X), ∅, X, ∪, ∩) is a lattice. Moreover, the lattice induced partial order on P(X) coincides with the inclusion relation ⊂. If P is a finite poset, then Down(P ) is a sublattice of the lattice P(P ). The Birkhoff representation theorem [2] states that every finite, bounded, distributive lattice is isomorphic to a lattice of down sets. More precisely, we have the following theorem.
Theorem 3.3 (Birkhoff, 1937). Every finite, bounded, distributive lattice L is isomorphic to the lattice of down sets of the partially ordered set of joinirreducible elements of L.
3.5. Topological spaces. For our terminology concerning topological spaces we refer the reader to [4,19]. Recall that a topology on a set X is a family T of subsets of X which is closed under finite intersections and arbitrary unions, and which satisfies ∅, We denote by Clsd(X) the family of closed subsets of X. Note that Clsd(X) is a sublattice of the lattice P(X) of subsets of X. The closure of A, denoted cl T A, or by cl A if T is clear from context, is the intersection of all closed supersets of A.
Given two topological spaces (X, T ) and (X , T ) we say that f : X → X is continuous if U ∈ T implies f −1 (U ) ∈ T . If we want to emphasize the involved topologies we say that f : (X, T ) → (X , T ) is continuous.
Recall that a subset A of a topological space X is called locally closed, if every point x ∈ A admits a neighborhood U in X such that A ∩ U is closed in U . Locally closed sets as well as mo A := cl A \ A, which we refer to as the mouth of A, are important in the sequel. Proposition 3.4 (see Problem 2.7.1 in [4]). Assume that A is a subset of a topological space X. Then the following conditions are pairwise equivalent.
(i) A is locally closed, (ii) the mouth of A is closed in X, (iii) A is a difference of two closed subsets of X, (iv) A is an intersection of an open set in X and a closed set in X.
Proof: It is obvious that (ii) implies (iii), (iii) implies (iv), and (iv) implies (i). Thus, it suffices to prove that if (ii) fails, then so does (i). Hence, assume that the mouth mo A = cl A\A is not closed in X. Then there exists a point x ∈ cl(cl A \ A) \ (cl A \ A). It follows that x ∈ A. Let U be an arbitrary neighborhood of x in X. We will prove that A ∩ U is not closed in U . Since x ∈ cl(cl A \ A), we can find a point y ∈ (cl A \ A) ∩ U . Let V be an open neighborhood of y in X. Then also V ∩ U is a neighborhood of y in X.
However, y ∈ A, hence, y ∈ U ∩ A. It follows that U ∩ A is not closed in U . This holds for every neighborhood U of x in X. Therefore, A is not locally closed.
The topology T is T 2 or Hausdorff if for any two different points x, y ∈ X there exist disjoint sets U, V ∈ T such that x ∈ U and y ∈ V . It is T 0 or Kolmogorov if for any two different points x, y ∈ X there exists a U ∈ T such that U ∩ {x, y} is a singleton.
A topological space is a pair (X, T ) where T is a topology on X. We often refer to X as a topological space assuming that the topology T on X is clear from context. Given Y ⊂ X, the family 3.6. Finite topological spaces. We say that a topological space X is a finite topological space if X is finite. Finite topological spaces differ from general topological spaces because the only Hausdorff topology on a finite topological space X is the discrete topology consisting of all subsets of X. In consequence, unlike in Hausdorff topological spaces, the closure of a singleton need not be a singleton. Moreover, for any A ⊂ X we have where we abbreviate cl{x} to the more convenient notation cl x. A remarkable feature of finite topological spaces is the following theorem.
Theorem 3.5 (P. Alexandrov [1]). For a preorder ≤ on a finite set X, there exists a topology T ≤ on X whose open sets are precisely the upper sets with respect to ≤. Conversely, if for a given topology T on a finite set X we define then ≤ T is a preorder. The correspondences T → ≤ T and ≤ → T ≤ are mutually inverse. Under these correspondences, continuous maps are transformed into order-preserving maps, and vice versa. Moreover, the topology T is T 0 if and only if the preorder ≤ T is a partial order.
Locally closed sets have a special characterization in finite topological spaces.
Proposition 3.6. Assume X is a finite topological space and A ⊂ X. Then the set A is locally closed if and only if for all x, y, z ∈ X one has (2) x, z ∈ A, x ∈ cl y, y ∈ cl z ⇒ y ∈ A.
Proof: Assume A ⊂ X is locally closed, x, z ∈ A, x ∈ cl y, y ∈ cl z, as well as y ∈ A. Then y ∈ mo A. Since x ∈ cl y, we get from Proposition 3.4 that x ∈ mo A. In particular, one obtains x ∈ A, which is a contradiction proving (2). To prove the opposite implication, assume that a subset A ⊂ X satisfies assumption (2). By Proposition 3.4 it suffices to prove that mo A is closed. For this, take an x ∈ cl mo A and assume x ∈ mo A. Since cl mo A ⊂ cl A, we get x ∈ cl A. Since x ∈ mo A, one has x ∈ A. Thus, we get from (1) that x ∈ cl y for a y ∈ mo A. In particular, y ∈ cl A and by (1) y ∈ cl z for a z ∈ A. Hence, it follows from (2) that y ∈ A. But, y ∈ mo A implies that y ∈ A, a contradiction proving that mo A is closed.

3.7.
Modules. For the terminology used in the paper concerning modules we refer the reader to [14]. Here, we briefly recall that a module over a ring R is a triple (M, +, ·) such that (M, +) is an abelian group and · : R × M → M is a scalar multiplication which is distributive with respect to the additions in M and R, and which satisfies both a · (b · x) = (ab) · x and 1 R · x = x for x ∈ M and a, b ∈ R. We shorten the notation a · x to ax. We often say that M is a module assuming that the operations + and · are clear from context. Recall that a submodule of M is a subset N ⊂ M such that N with the operations + and · restricted to N is itself again a module. Finally, for submodules N 1 , N 2 , . . . , N k of a given module M , their algebraic sum is easily seen to be a submodule of M . We call it a direct sum and denote it by the symbol N 1 ⊕ N 2 ⊕ . . . ⊕ N k , if for each x ∈ N 1 + N 2 + . . . + N k the representation x = x 1 + x 2 + . . . + x k with x i ∈ N i is unique. We note that for a module M the family of submodules of M , denoted by Sub(M ), forms a lattice (Sub(M ), 0, M, ∩, +).
A subset Z ⊂ M generates M if for every element x ∈ M \ {0} there exist elements x 1 , x 2 , . . . , x n ∈ Z and a 1 , a 2 , . . . , a n ∈ R such that x = n i=1 a i x i . A subset Z ⊂ M is linearly independent if for any choice of x 1 , x 2 , . . . , x n ∈ Z and a 1 , a 2 , . . . , a n ∈ R the equality n i=1 a i x i = 0 implies a 1 = a 2 = . . . = 0. A linearly independent subset B ⊂ M which generates M is called a basis of M . A module may not have a basis. If it does have a basis, the module is called free. We note that ∅ is the unique basis of the trivial module {0}.
Assume that M is a free module and that B ⊂ M is a fixed basis. Then we have the associated bilinear map ·, · B : M × M → R, called scalar product, and defined on basis elements b, b ∈ B by b, b B := 0 for b = b , as well as b, b B := 1 for b = b . For an element x ∈ M we define its support with respect to B as |x| B := { b ∈ B | x, b B = 0 }. One easily verifies that for all x, y ∈ M Assume that X is a finite set. Then the collection R X := { f : X → R } of all functions forms a module with respect to pointwise addition and scalar multiplication. We note that R ∅ is the zero module. For every x ∈ X we have a functionx : M → R which sends x to 1 and any other element of X to 0. One easily verifies thatX := {x | x ∈ X } is a basis of R X . Therefore, R X is a free module, and we call R X the free module spanned by X. In the sequel we identifyx with x.
Let M and M be two modules over R.

Poset graded and filtered modules
4.1. Gradation of modules. Assume R is a fixed ring and M is a module over R. Let K be an arbitrary set. Then a K-gradation of M is a collection of submodules {M k } k∈K of M indexed by K, and such that M = k∈K M k is the direct sum of the submodules M k . This definition requires that K = ∅, but it is convenient to assume that the direct sum over the empty set is always the zero module. Thus, the only module admitting the ∅-gradation is the zero module. By a K-graded module over R we mean a module M over R together with a fixed, implicitly given, K-gradation. Note that the zero module admits not only the ∅-gradation, but also a unique K-gradation for each non-empty set K. Thus, for each set K the zero module is a Kgraded module. We denote it by 0 K .
If B is a basis of a free module M and A is a partition of B, then M is A-graded with the gradation where the expression on the right-hand side is defined as Note that in the case that A is finite, one can identify R · A with R A . As a special case, consider the situation when the partition of B consists only of singletons. Then the gradation becomes We refer to this gradation as the B-basis gradation of M .
We say that a submodule M of M is a K-graded submodule if M is also K-graded with the decomposition Proposition and Definition 4.1. Assume K and K are finite. Then In particular, f is uniquely determined by the matrix [f ij ] i∈K ,j∈K of homomorphisms f ij : M j → M i . We refer to this matrix as the (K, K )-matrix of the homomorphism f .
Notice that in the case when the K-and K -gradations are basis gradations given by bases B = {b 1 , b 2 , . . . , b n } in M and B = {b 1 , b 2 , . . . , b m } in M , respectively, then the homomorphisms f ij take the form which means that the (K, K )-matrix of f may be identified in this case with the matrix of coefficients f (b j ), b i .
As in the case of classical matrices one easily verifies the following proposition.

4.2.
Graded and filtered homomorphisms. Consider a K-graded module M and a K -graded module M . Let α : K K be a partial map. We say that a homomorphism h : M → M is α-graded if for every pair of indices j ∈ K and i ∈ K we have which is equivalent to the implication If in addition the sets K and K are partially ordered sets and α is order preserving, then we say that a homomorphism h : M → M is α-filtered if for every j ∈ K and i ∈ K These two definitions are illustrated in Figures 2 and 3. The left image in Figure 2 shows two partially ordered sets together with a partial map α between them. In the right image of the same figure arrows indicate which of the homomorphisms h ij can be nontrivial if h is α-graded. In contrast, Figure 3 depicts all possible nontrivial homomorphisms h ij if h is α-filtered. The more involved implication (7) is illustrated in the top left image, which indicates the sets i ≤ and j ≤ in light blue. The remaining three panels in the figure depict all possible nontrivial homomorphisms h ij . Clearly, αfiltered homomorphisms are far less restricted than α-graded ones. In fact, an α-filtered homomorphism can have a nontrivial h ij even if i ∈ dom α or if j ∈ im α. Conditions (6) and (7) for α-graded and α-filtered homomorphisms, respectively, may be simplified if dom α = K , i.e., if α is actually a function. This is the subject of the following straightforward proposition.
and h is α-filtered if and only if for every i ∈ K and j ∈ K.
Obviously, if α is order preserving and h is an α-graded homomorphism then h is an α-filtered homomorphism. However, the converse is not true in general. This can readily be seen from Figures 2 and 3.
We say that a homomorphism h : M → M is filtered (respectively graded) if K = K and h is id K -filtered (respectively id K -graded). As an immediate consequence of Proposition 4.3 we get the following corollary.
and h is filtered if and only if one has Consider now the special case when K = K = Z and α : Z i → i−k ∈ Z is the left-shift by k on the integers. If in this situation h is α-graded, we say that it is a Z-graded homomorphism of degree k. When k is clear from the context, we shorten the notation h j+k,j to h j . Proposition 4.5. Assume that P and P are posets, α : P P is order preserving, M is a P -graded module, and M is a P -graded module. A module homomorphism h : M → M is an α-filtered homomorphism if and only if for every L ∈ Down(P ).
Proof: Assume that h : M → M is a module homomorphism such that (10) is satisfied for every L ∈ Down(P ). Moreover, assume that h pq = 0 for an element q ∈ P and a p ∈ P . Let x ∈ M q be such that h pq (x) = 0. If we consider L := q ≤ ∈ Down(P ), then (10) Thus, h p q (x) = π p (h(x)) = 0 for p ∈ α −1 (L) ≤ , which in turn implies that p ∈ α −1 (L) ≤ , in view of h pq (x) =  π p (h(x)) = 0. In consequence, p ≤p for ap with α(p) ∈ L = q ≤ . Therefore, p ∈ α −1 (q ≤ ) ≤ which implies (7) and proves that h is α-filtered. To prove the opposite implication, assume that h : M → M is a module homomorphism such that property (7) holds. Let L ∈ Down(P ) and consider x ∈ M L . Without loss of generality we may assume that x ∈ M q for a q ∈ L. Then an application of (7) yields This establishes the inclusion in (10).
In the special case when P = P and α = id P we have the following corollary of Proposition 4.5.
Corollary 4.6. Assume that P is a poset and that M is a P -graded module. Then h : M → M is a filtered homomorphism if and only if (11) h(M L ) ⊂ M L for every L ∈ Down(P ).
Proof: By applying our assumptions P = P and α = id P to the setting of Proposition 4.5, one obtains that α −1 (L) ≤ = L ≤ = L. Therefore, the condition in (10) reduces to the one in (11). As an immediate consequence of Corollary 4.6 one obtains the following proposition.

4.3.
The category of graded and filtered moduli. We define the two categories GMod of graded moduli and FMod of filtered moduli as follows. The objects of GMod (respectively FMod) are pairs (P, M ) where P is an object of DSet (respectively DPSet) and M is a P -graded module; see also Sections 3.1 and 3.3. The morphisms in GMod (respectively FMod) are pairs (α, h) : (P, M ) → (P , M ) such that α : P P is a morphism in DSet (respectively DPSet) and h : M → M is an α-graded (respectively α-filtered) module homomorphism. In the following, we will briefly refer to morphisms in GMod (respectively FMod) as graded (respectively filtered) morphisms (α, h) : (P, M ) → (P , M ). In the special case when (P, M ) = (P , M ) and α = id P we simplify the terminology be referring to h as a filtered (respectively graded) morphism meaning that h is id P -filtered (respectively id P -graded).
One can easily verify that id (  Proof: The only nontrivial part of the proof is the verification that the composition of two graded morphisms is again a graded morphism. For this, let (α , h ) := (α , h ) • (α, h) be given as above. Obviously h h is a module homomorphism. We will show that h pr = 0 for p ∈ P and r ∈ P implies (αα )(p) = r. Hence, assume that h pr = 0 for p ∈ P and r ∈ P . It follows from Proposition 4.2 that there exists a q ∈ P such that both h pq = 0 and h qr = 0 are satisfied. Therefore, one has to have p ∈ dom α and α (p) = q as well as q ∈ dom α and α(q) = r. It follows that p ∈ dom(αα ) and (αα )(p) = r. This proves that (α , h ) is a graded morphism.
If f and f are graded-conjugate, then we say that their respective (P, P ) and (P , P ) matrices are graded-similar. It is easily seen that the graded similarity of matrices is an equivalence relation. Proof: The only nontrivial property to verify is that the composition of filtered morphisms is again a filtered morphism. For this, consider a composition (α , h ) := (α , h ) • (α, h) of filtered morphisms as above. Obviously, the map αα is order preserving and h h is a module homomorphism. Thus, we only need to verify that that h pr = 0 for p ∈ P and r ∈ P implies the inclusion p ∈ (αα ) −1 (r ≤ ) ≤ . Hence, assume that h pr = 0 for p ∈ P and r ∈ P . It follows from Proposition 4.2 that there exists a q ∈ P such that h pq = 0 and h qr = 0. Therefore, there existp ≥ p andq ≥ q such that α (p) ≤ q and α(q) ≤ r. Since α is order preserving, it follows that (αα )(p) ≤ α(q) ≤ r. Hence,p ∈ (αα ) −1 (r ≤ ) and p ∈ (αα ) −1 (r ≤ ) ≤ . This proves that (α , h ) is a filtered morphism. If p ∈ P and p ∈ P are such that αα (p ) = p, then Proof: We get from Proposition 4.2 that It follows from (9) that the index p of a non-zero term on the right-hand side of (13) must satisfy α (p ) ≤ p and α(p ) ≤ p. Hence, It follows that α(p ) = p and, since α is injective, we get p = α (p ). This proves (12).
As an immediate consequence of Proposition 4.11 we get the following corollary.  Proof: First assume that (α, h) is an isomorphism in the category FMod. Let (α , h ) : (P , M ) → (P, M ) be the inverse of (α, h). Then αα = id P and α α = id P . Since α and α are order preserving, we see that α : P → P is an order isomorphism. It follows from Corollary 4.12 that (14) id But, α and α are mutually inverse bijections. Hence, substituting p := α (p) in (14) we get p = α(p ) and It follows that h α(p )p is the inverse of h p α(p ) . Therefore, h p α(p ) is a module isomorphism for every p ∈ P . To see the opposite implication, assume that (α, h) : (P, M ) → (P , M ) is a filtered homomorphism such that α : P → P is an order isomorphism and h p α(p ) : M α(p ) → M p is a module isomorphism for all p ∈ P . Since P and P as objects of DPSet are finite sets, we may proceed by induction on their cardinality n := card P = card P . If n = 1, then P = {p}, P = {p } and h p α(p ) = h p p = h is a module isomorphism. Clearly, its inverse is a filtered homomorphism. Hence, h is an isomorphism in FMod. Thus, we now assume n > 1. Letp be a maximal element in P . Setp := α(p ). Since α is an order isomorphism, we see thatp is a maximal element in P .  (16) hp p = 0 for p ∈P .
Arguing by contradiction, assume that hp p = 0 for a p ∈P . Since h is an α-filtered homomorphism, we get from (7) that hp p = 0 impliesp ≤p for somep ∈ dom α such that α(p ) ≤ p. Sincep is maximal in P , we obtain bothp =p andp = α(p ) ≤ p. Sincep is maximal in P , this further implies the equality p =p, a contradiction proving (16). Hence, the identity in (4) yields hp P = p∈P hp p • π p = 0. Therefore, the (Q, Q )-matrix of h is By induction assumption (α |P , hP P ) : MP → MP is an isomorphism in FMod. Set α := α −1 and let h : M → M be the homomorphism given by One easily verifies that h is α -filtered and a straightforward computation Proof: Due to Lemma 4.13 the map α : P → P is an order isomorphism and h p α(p ) : M α(p ) → M p is a module isomorphism. Let g : M → M be the α −1 -graded homomorphism with (P , P )-matrix given by It is straightforward to verify that (α −1 , g) is the inverse of (α, h) in GMod.

Chain complexes.
Recall that a chain complex is a pair (C, d) consisting of a Z-graded R-module C and a Z-graded homomorphism d : C → C of degree −1, satisfying d 2 = 0. The homomorphism d is called the boundary homomorphism of the chain complex (C, d). Let (C , d ) be another chain complex. A chain map ϕ : The following proposition is straightforward.
is a chain map and ϕ is an isomorphism of Z-graded modules then ϕ −1 is also a chain map.
We denote by Cc the category whose object are chain complexes and whose morphisms are chain maps. One easily verifies that this is indeed a category.
A subset C ⊂ C is a chain subcomplex of C if C is a Z-graded submodule of C such that d(C ) ⊂ C . Recall that given a subcomplex C ⊂ C we have a well-defined quotient complex (C/C , d ) where d : C/C → C/C is the boundary homomorphism induced by d.

5.2.
Homotopy category of chain complexes. A pair of chain maps ϕ, ϕ : (C, d) → (C , d ) are called chain homotopic if there exists a chain homotopy joining ϕ and ϕ , i.e., a module homomorphism S : C → C of degree +1 such that ϕ −ϕ = d S+Sd. The existence of a chain homotopy between two chain maps is easily seen to be an equivalence relation in the set of chain maps Cc((C, d), (C , d )). Given a chain map ϕ ∈ Cc((C, d), (C , d )) we denote by [ϕ] the equivalence class of ϕ with respect to this equivalence relation. We define the homotopy category ChCc of chain complexes by taking chain complexes as objects, equivalence classes of morphisms in Cc as morphisms in ChCc, and the formula (17) [ψ] for ψ ∈ Cc((C , d ), (C , d )) as the definition of composition of morphisms in ChCc. Note that then the equivalence classes of identities in Cc are the identities in ChCc.
Proposition 5.2. The category ChCc is well-defined.
Proof: The only non-obvious part of the argument is the verification that the composition given by (17) is well-defined. Thus, we need to prove that if ϕ, ϕ : (C, d) → (C , d ) and ψ, ψ : (C , d ) → (C , d ) are chain homotopic, then ψϕ and ψ ϕ are chain homotopic. Let S : C → C and S : C → C be chain homotopies between ϕ, ϕ and ψ, ψ , respectively. Moreover, consider the map S := ψ S + S ϕ. Then S is clearly a degree +1 homomorphism and we have which proves that ψϕ and ψ ϕ are indeed chain homotopic.
We have a covariant functor Ch : Cc → ChCc which fixes objects and sends a chain map to its chain homotopy equivalence class. Moreover, we say that two chain complexes (C, d) and (C , d ) are chain homotopic if they are isomorphic in ChCc.
Note that 0 Z , the Z-graded zero module, together with zero homomorphism as the boundary map, is a chain complex. We call it the zero chain complex. We say that a chain complex (C, d) is homotopically essential if it is not chain homotopic to the zero chain complex. Otherwise we say that the chain complex (C, d) is homotopically trivial or homotopically inessential. Finally, we call a chain complex (C, d) boundaryless if d = 0. Proof: Suppose that the chain complexes (C, d) and (C , d ) are boundaryless. First assume that C and C are isomorphic as Z-graded modules.
Let ϕ : C → C and ϕ : C → C be mutually inverse isomorphisms of Zgraded modules. Since both boundary homomorphisms d and d are zero, we have ϕd = 0 = d ϕ and ϕ d = 0 = dϕ . Hence, ϕ and ϕ are chain maps and from ϕ ϕ = id C and ϕϕ = id This shows that [ϕ] and [ϕ ] are mutually inverse isomorphisms in ChCc.
To prove the opposite implication assume that (C, d) and (C , d ) are chain homotopic. Choose ϕ : Let S : C → C and S : C → C be the chain homotopies between ϕ ϕ and id C , and between ϕϕ and id C , respectively. Then id C −ϕ ϕ = Sd + dS = 0 and id C −ϕϕ = S d + d S = 0, which proves that ϕ and ϕ are mutually inverse isomorphisms in Cc.
Corollary 5.4. The only chain complex which is both boundaryless and homotopically trivial is the zero chain complex.
Recall that the homology module of a chain complex ( In the sequel, we will consider the homology module as a boundaryless chain complex, that is, as a chain complex with zero boundary homomorphism. By a homology decomposition of a chain complex (C, d) we mean a di- Note that then also d |B = 0. By a homology complex of a chain complex (C, d) we mean a boundaryless chain complex which is chain homotopic to (C, d). Then one has the following proposition.
Proposition 5.5. Assume that R is a field and that the pair (C, d) is a chain complex over R.
(i) There exists a homology decomposition of (C, d).
is a homology decomposition of the chain complex (C, d), then (H, 0) is chain homotopic to (C, d). Therefore, it is a homology complex of (C, d). Moreover, the modules H and H(C) are isomorphic as Z-graded modules, where H(C) denotes the homology module of C.
Proof: To prove statement (i), let Z := ker d and To see that it is also onto, take a y ∈ B. Since B = im d, we have y = dx for an x ∈ C. But, x = v + z for a v ∈ V and a z ∈ Z. It follows that dv = dv + dz = dx = y, which establishes d |V as an epimorphism, and proves (i).
To prove (ii), assume that C = V ⊕ H ⊕ B is a homology decomposition of (C, d). Let ι : H → C denote inclusion, and let π : C → H be the One can easily see that ι : (H, 0) → (C, d) and π : (C, d) → (H, 0) are chain maps, and that πι = id H . We will show that ιπ is chain homotopic to id C . For this, define the degree +1 homomorphism S : which proves that S is a chain homotopy joining ιπ and id C . Finally, directly from the homology decomposition definition one obtains ker d = H ⊕ B and im d = B. Therefore, In the case of field coefficients the concepts of homology module and homology complex are essentially the same as the following theorem shows.
Theorem 5.6. Assume that R is a field. Then the following hold.
(i) Every chain complex admits a homology complex.
(ii) Two chain complexes are chain homotopic if and only if the associated homology complexes are isomorphic as Z-graded modules. (iii) The homology module of a chain complex is its homology complex.
(iv) Two chain complexes are chain homotopic if and only if the associated homology modules are isomorphic as Z-graded modules.
Proof: Property (i) follows immediately from Proposition 5.5. In order to prove (ii), observe that two chain complexes are chain homotopic if and only if the associated homology complexes are chain homotopic. Therefore, property (ii) follows immediately from Proposition 5.3. To prove (iii), take a chain complex (C, d). By (i) we may consider a homology complex (H, 0) of (C, d). Since (C, d) and (H, 0) are chain homotopic, by a standard theorem of homology theory [9, §13] the homology modules of (C, d) and (H, 0) are isomorphic as Z-graded modules. Hence, it follows from Proposition 5.5(ii) that the homology module of (C, d) is its homology complex. Finally, property (iv) follows immediately from (ii) and (iii).

5.3.
Poset filtered chain complexes. Let P be an arbitrary finite set and (C, d) a chain complex. We call (C, d) a P -graded chain complex, if C is a P -graded module in which each C p ⊂ C is a Z-graded submodule of C.
Since the chain complex C is P -graded, its boundary homomorphism d has a (P, P )-matrix. A partial order ≤ in P is called (C, d)-admissible, or briefly d-admissible, if d is a filtered homomorphism with respect to (P, ≤), i.e., if (18) d pq = 0 ⇒ p ≤ q for all p, q ∈ P . Note that a (C, d)-admissible partial order on P may not always exist. However, we have the following straightforward proposition and definition.
Proposition and Definition 5.7. If a given P -graded chain complex (C, d) admits a (C, d)-admissible partial order on P , then the intersection of all such (C, d)-admissible partial orders on P is again a (C, d)-admissible partial order on P . We call it the native partial order of d.
Definition 5.8. We say that the triple (P, C, d) is a poset filtered chain complex if (C, d) is a P -graded chain complex and the partial order in P is d-admissible.
Note that for a poset filtered chain complex (P, C, d) the module C is not only P -graded but also Z-graded where the nth summand of the Z-gradation is the direct sum over p ∈ P of the nth summands in the Z-gradation of C p . and the free module C := Q X with basis X and coefficients in the field Q of rational numbers. For a word x ∈ X define its dimension as one less than the number of characters in x and denote the set of words of dimension i by X i . Setting we obtain a Z-gradation of C given by . and a homomorphism d : C → C defined on the basis X by the matrix One can check that the triple (A, C, d) with the Z-gradation (19) and A partially ordered as in the Hasse diagram (20) is a well-defined poset filtered chain complex.
Note that if J ⊂ P , then (C J , d JJ ) need not be a chain complex in general. However, we have the following proposition.
Proposition 5.10. Assume that J is a convex subset of P . Then we have: Notice that the quotient complex is well-defined since J < is a down set due to the convexity of J.
Proof: In order to prove (i) we need to verify that d 2 JJ = 0. Assume first that J is a down set. Let x ∈ C J . Then Corollary 4.6 immediately implies that dx ∈ C J . Hence, one obtains both If J is just convex, we consider the down sets I := J < and K := J ≤ , which clearly satisfy the identity K = I ∪ J. Since d is a filtered homomorphism, we have d pq = 0 for p ∈ J and q ∈ I.
Therefore, the matrix of d KK takes the form.
Since K is a down set, we already have verified that d 2 KK = 0. It follows that To establish (ii), we consider again the down sets I := J < and K := J ≤ , as well as the homomorphism κ : I denotes the equivalence class of x in the quotient module C K /C I . Let x ∈ C J and letd KK denote the boundary homomorphism induced by d KK on the quotient module This proves that κ is a monomorphism. Finally, given the class [y] I ∈ C K /C I generated by y ∈ C K , we have y = y I + y J for a y I ∈ C I and a y J ∈ C J . It follows that [y] I = [y J ] I = κy J , proving that κ is an epimorphism. Hence, κ is an isomorphism.
Given a poset filtered chain complex (P, C, d) we consider the poset P as an object of DPSet with the distinguished subset given by Recall that C p is homotopically essential if it is not chain homotopic to the zero chain complex. We would like to point out that the definition of P depends on C. In the rare cases when we simultaneously consider two poset filtered chain complexes with the same underlying poset P , we write P C instead of P to indicate the dependence of P on C.
Given two poset filtered chain complexes (P, C, d) and (P , C , d ) we say that the map (α, h) : We define the category PfCc of filtered chain complexes by taking poset filtered chain complexes as objects and filtered chain morphisms as morphisms. One easily verifies that this is indeed a category. We also define the subcategory PgCc of graded chain complexes by taking the same objects as in PfCc and graded chain morphisms as morphisms.
The following proposition is straightforward.
Proposition 5.12. The relation ∼ in the set of morphisms from (P, C, d) to (P , C , d ) in PfCc is an equivalence relation.
We say that a poset filtered chain complex (P, C, d) is homotopically essential if (C p , d pp ) is homotopically essential for each p ∈ P , i.e., if P = P .
We refer to the equivalence classes of ∼ as the homotopy equivalence classes. We define the homotopy category of poset filtered chain complexes, denoted ChPfCc, by taking poset filtered chain complexes as objects, homotopy equivalence classes of filtered chain morphisms in PfCc as morphisms in ChPfCc, and use the formula for the two filtered morphisms (α, h) ∈ PfCc((P, C, d), (P , C , d )) and (β, g) ∈ PfCc((P , C , d ), (P , C , d )) as the definition of composition of morphisms in ChPfCc. Finally, equivalence classes of identities in PfCc are identities in ChPfCc.
Definition 5.15. Let (P, C, d) be a poset filtered chain complex. By a representation of (P, C, d) we mean an object (P , C , d ) in PfCc which is filtered chain homotopic to (P, C, d), together with mutually inverse filtered chain equivalences (α, ϕ) : (P, C, d) → (P , C , d ) and (β, ψ) : (P , C , d ) → (P, C, d). More formally speaking, a representation of (P, C, d) is a triple ((P , C , d ), (α, ϕ), (β, ψ)). To simplify the terminology, when speaking about a representation we always assume that the associated mutually inverse chain equivalences are implicitly given.
Definition 5.16. We define the transfer morphism from the filtered chain complex (P , C , d ) to the filtered chain complex (P , C , d ) as the filtered chain morphism (βα , ϕ ψ).
We finish this section with the following auxiliary proposition.
Proposition 5.17. Assume that (P, C, d) is a poset filtered chain complex with the P -gradation (C p ) p∈P , and let (W p ) p∈P denote another P -gradation of C. If (C p ) p∈P and (W p ) p∈P are filtered equivalent in the sense of Definition 4.7, then the triple (P, W, d) with W := p∈P W p and P -gradation given by (W p ) p∈P is also a poset filtered chain complex. Moreover, (P, W, d) and (P, C, d) are isomorphic in ChPfCc.
Proof: Note that W and C are in fact the same modules, but the gradations (W p ) p∈P and (C p ) p∈P need not be the same. We know that d is a filtered homomorphism with respect to the (C p ) p∈P gradation of C. Since the (W p ) p∈P gradation of W = C is filtered equivalent to the (C p ) p∈P gradation, we see from Proposition 4.8 that d is a filtered homomorphism with respect to the (W p ) p∈P gradation of C as well. Hence, (P, W, d) is a poset filtered chain complex. We will prove that are mutually inverse isomorphisms in PfCc. Let p ∈ P . Since both (C p ) p∈P and (W p ) p∈P are filtered equivalent, using Proposition 5.10(ii) we get Hence, C p is essential if and only if W p is essential. It follows that P C = P W . In consequence, id P : (P, P C ) → (P, P W ) and id P : (P, P W ) → (P, P C ) are well-defined morphisms in DPSet. Since (C p ) p∈P and (W p ) p∈P are filtered equivalent, we get from Corollary 4.6 that (id P , id C ) : (P, C) → (P, W ) and (id P , id W ) : (P, W ) → (P, C) are filtered morphisms. Clearly, both are chain maps. It follows that the morphisms (24) and (25) are well-defined. Obviously, they are mutually inverse. Thus, the conclusion follows.
6. Algebraic connection matrices 6.1. Reduced filtered chain complexes. Given a poset filtered chain complex (P, C, d), a singleton {p} ⊂ P is always convex. Therefore, by Proposition 5.10(ii) the pair (C p , d pp ) is a chain complex.  (20), is a well-defined poset filtered chain complex. It is reduced, because in each case exactly one chain group is nontrivial and the boundary operator is always zero within each C p .
As an immediate consequence of Corollary 5.4 we get the following proposition. Proposition 6.4. Assume that (P, C, d) and (P , C , d ) are two reduced poset filtered chain complexes and that (α, h), (β, g) : (P, C, d) → (P , C , d ) are filtered chain homotopic morphisms. If α is injective, then for p ∈ P and q ∈ P we have α(q) = p ⇒ h qp = g qp .
Proof: Since (P , C , d ) is reduced, it is homotopically essential. Thus, we get from Proposition 5.13 that α = β : P → P and there exists an α-filtered degree +1 homomorphism Γ : C → C such that g − h = d Γ + Γd. Since (P, C, d) and (P , C , d ) are reduced, we get from Proposition 4.11 the identities For this, recall also that d and d are id P -filtered, and that id P is injective. This completes the proof.
If filtered chain complexes (P, C, d) and (P , C , d ) are filtered chain homotopic, then the equivalence class of every filtered chain equivalence (α, ϕ) : (P, C, d) → (P , C , d ) is automatically an isomorphism in ChPfCc. However, in the category PfCc one has to prove that fact. This is addressed in the following result. Theorem 6.5. Assume that the two poset filtered chain complexes (P, C, d) and (P , C , d ) are reduced. If they are filtered chain homotopic, then every filtered chain equivalence (α, ϕ) : (P, C, d) → (P , C , d ) is an isomorphism in PfCc. In particular, (P, C, d) and (P , C , d ) are isomorphic in PfCc.
The following proposition is an immediate consequence of Proposition 5.14 and the fact that the composition of graded chain morphisms is again a graded chain morphism. The later statement in turn follows from the fact that the composition of graded morphisms is a graded morphism, and that the composition of chain maps is again a chain map. Thus, we have the following result. It is straightforward to observe that every identity morphism in PfCc is essentially graded, and that it is its own graded representation. Thus, in view of Proposition 6.8 we have a well-defined wide subcategory EgPfCc of PfCc whose morphisms are essentially graded morphisms in PfCc. Proposition 6.9. Assume that both (P, C, d) and (P , C , d ) are reduced poset filtered chain complexes, and that both (α, h) : (P, C, d) → (P , C , d ) and (α , h ) : (P , C , d ) → (P, C, d) are mutually inverse chain equivalences. Suppose further that (α, h) is essentially graded and has the graded representation (ᾱ,h). Then (α , h ) is essentially graded as well, (ᾱ,h) is an isomorphism in PgCc, and (ᾱ,h) −1 is the graded representation of (α , h ).
Proof: By Definition 6.6 we have (α, h) ∼ (ᾱ,h). Hence, we get from Proposition 5.14 the equivalences which imply that (ᾱ,h) is a filtered chain equivalence. Now Theorem 6.5 shows that (ᾱ,h) is an isomorphism in PfCc, and it follows that (ᾱ,h) is an isomorphism in FMod. Thus, we get from Proposition 5.13 thatᾱ = α and from Lemma 4.13 that α is an order isomorphism. In consequence, the inverse of (ᾱ,h) = (α,h) in PfCc takes the form (α −1 ,h −1 ) and, clearly, the inverseh −1 is α −1 -graded. Moreover, from (28) we get which proves that (α , h ) is essentially graded with (α,h) −1 = (ᾱ,h) −1 as its graded representation. 6.3. Conley complex and connection matrices. Throughout this section, let (P, C, d) denote a poset filtered chain complex. The following definition is the central concept of this paper. Definition 6.10. By a Conley complex of (P, C, d) we mean every reduced representation of (P, C, d), that is, a poset filtered chain complex (P ,C,d) which is reduced and filtered chain homotopic to (P, C, d). The (P ,P )matrix of the boundary homomorphismd is then called a connection matrix of the poset filtered chain complex (P, C, d).
As an immediate consequence of Theorem 6.5, we obtain the following two corollaries.
Corollary 6.11. The Conley complex of a poset filtered chain complex is unique up to an isomorphism in PfCc. In particular, the transfer homomorphism between two Conley complexes of a given poset filtered chain complex is an isomorphism in PfCc. Corollary 6.12. If two poset filtered chain complexes are filtered chain homotopic, that is, isomorphic in ChPfCc, then their Conley complexes are isomorphic in PfCc.
Example 6.13. One can verify that the reduced filtered chain complex in Example 6.2 is a Conley complex of the filtered chain complex in Example 5.9 with the associated connection matrix (27). More precisely, consider the map α : A → A given by and homomorphisms h : C → C and g : C → C given by the matrices One can verify that with the partial map (α ) −1 : A A defined as the inverse relation of α , are mutually inverse filtered chain equivalences. This immediately implies that a Conley complex of the filtered chain complex (A, C, d) in Example 5.9 is given by the representation ((A , C , d ), (α , h ), ((α ) −1 , g )).
6.4. Equivalence of Conley complexes. Definition 6.14. We say that two Conley complexes of a poset filtered chain complex are equivalent if the associated transfer homomorphism is essentially graded.
Note that the transfer homomorphism is always an isomorphism in PfCc. For equivalence, one needs in addition that it is essentially graded, i.e., that it is filtered chain homotopic to a graded chain isomorphism.
It follows from Proposition 6.8 that the equivalence of Conley complexes of a given poset filtered chain complex is indeed an equivalence relation. Two equivalent Conley complexes, as well as the associated connection matrices, are basically the same. This justifies the following definition. Note that two non-equivalent Conley complexes of a filtered chain complex may be graded-conjugate and the associated connection matrices may be graded similar. Thus, graded similarity of connection matrices is not sufficient for their uniqueness.
Then one can verify that ((A , C , d ), (α , h ), ((α ) −1 , g )) is another reduced representation, that is, a Conley complex of the filtered chain complex (A, C, d) presented in Example 5.9. We claim that this Conley complex is not equivalent to the Conley complex (A , C , d ) presented in Example 6.13. To see this, assume to the contrary that the transfer homomorphism from (A , C , d ) to (A , C , d ) is essentially graded. Observe that this transfer homomorphism is (γ, h g ), where γ = (α ) −1 α : A → A is given by and h g has the matrix Let (γ , f ) be a graded representation of (γ, h g ). Then (γ , f ) ∼ (γ, h g ) and from Proposition 5.13 we get γ = γ, as well as for some γ-filtered degree +1 homomorphism Γ : C → C . Evaluating both sides of (32) at AD we get  (27) and (31), respectively. Note that despite the fact that these matrices are not equivalent as connection matrices of (A, C, d), they are graded similar. To see this define χ : A → A by and h : C → C by the matrix Then one easily verifies that χ is an order preserving bijection, that (χ, h) is a graded chain isomorphism, and that we have This readily establishes the graded similarity.
6.5. Existence of Conley complexes. Although in our setting the poset in a poset filtered chain complex is not fixed as in [22] and [8], the existence proof of a Conley complex for a poset filtered chain complex in our sense can be adapted from the argument in [ (i) C = p∈P W p and (W p ) p∈P is a P -gradation of C which is filtered equivalent to the gradation (C p ) p∈P , Proof: Assume that (P, C, d) is a poset filtered chain complex. We proceed by induction on n := card P . First assume that n = 1. Let p * be the unique element of P . Then C = C p * . We set W p * := C. It follows that the P -gradations (C p ) p∈P and (W p ) p∈P are identical, and therefore they are filtered equivalent, i.e., (i) is satisfied. The existence of V p * , B p * , H p * satisfying properties (ii)-(v) follows from Proposition 5.5. Now, assume that n > 1. Let r ∈ P be a maximal element in P and consider the down set P := P \ {r} ∈ Down(P ). Let C := p∈P C p . Since d is a filtered homomorphism, we have d(C ) ⊂ C . Thus, (P , C , d |C ) is a P -filtered chain complex. Since card P = n − 1, by our induction hypothesis, there exist families {W p } p∈P , {V p } p∈P , {B p } p∈P , {H p } p∈P of submodules such that properties (i)-(iv) hold for (P , C , d |C ). In order to have respective families for (P, C, d) we will extend the families over P to families over P by constructing in turn the modules V r , B r , H r and W r .
To begin with, for a family {M p } p∈P of submodules of C which satisfies the identity M p ∩M q = {0} for all p = q, and an L ∈ Down(P ), we introduce We will show that Indeed, if x ∈ V and x = x 1 + dx 2 for an x 1 ∈ H and an x 2 ∈ C, then one obtains dx = dx 1 . By (v) of the induction assumption, (P , H , d |H ) is a poset filtered chain complex. Hence, dx = dx 1 = d |H x 1 ∈ H . Also, by (iii) of the induction assumption we have dx ∈ B , and therefore dx ∈ H ∩B = 0. Since x ∈ V one then obtains x = 0. This proves (35).
where we also use the fact that the modules have field coefficients. We will prove that To see that the right-hand side of (37) is contained in the left-hand side observe that the right-hand side is obviously contained in C r ≤ . Since C is a filtered chain complex and r < is a down set, we have d(C r < ) ⊂ C r < and, in consequence, C r < ⊂ d −1 (C r < ). We also have d −1 (H r < ) ⊂ d −1 (C r < ), because H r < ⊂ W r < by (ii) and W r < = C r < by (i). To prove the opposite inclusion take an x ∈ C r ≤ ∩ d −1 (C r < ). Then dx ∈ C r < . Hence, by (34), we can find We also have x ∈ C r ≤ and y V ∈ V r < ⊂ C r < ⊂ C r ≤ . Therefore, one finally obtains . This completes the proof of (37). Now set B r := d(V r ). We will prove that (38) B r ∩ C r < = 0.
Let x ∈ B r ∩ C r < . Then x = dy for some y ∈ V r ⊂ C r ≤ . Since x ∈ C r < , we get y ∈ d −1 (C r < ). Therefore, y ∈ V r ∩ C r ≤ ∩ d −1 (C r < ). It follows from (36) that y = 0. Hence, x = d0 = 0, which proves (38). Since (P, C, d) is a poset filtered chain complex, the boundary homomorphism d is a filtered homomorphism. Therefore, we have C r < ⊂ d −1 (C r < ).
Thus, by (38), we have a direct sum of Z-graded submodules Hence, we can choose a Z-graded submodule H r such that Thus, it follows from (37) and Proposition 6.17 that Therefore, setting W r := V r ⊕ B r ⊕ H r we get from (36) that We now have well-defined families {W p } p∈P , {V p } p∈P , {B p } p∈P , {H p } p∈P of submodules of C. We will prove that they indeed satisfy properties (i)-(v) for (P, C, d). To prove (i) observe that by (42) We claim that C = C ⊕ W r . Indeed, by (42) we have W r ⊂ C r ≤ . Therefore, This together with the induction assumption shows that To show that (W p ) p∈P is filtered equivalent to (C p ) p∈P , one needs to verify that C L = W L for all L ∈ Down(P ). Note that by the induction assumption (43) C L = W L for L ∈ Down(P ).
Thus, we only need to consider the case when r ∈ L. Let L := L \ {r}. Then L = L ∪ r ≤ , and (43) and (42) yield This proves property (i). By induction assumption properties (ii)-(v) need to be verified only for p = r. Property (ii) for p = r follows from the definition of W r . To see (iii) for p = r take an x ∈ V r such that dx = 0 ∈ C r < . Since the inclusion V r ⊂ C r ≤ holds, it follows that x ∈ V r ∩ C r ≤ ∩ d −1 (C r < ) and we get from (36) that x = 0. Thus, d |Vr is a monomorphism. By the definition of B r it is an epimorphism, which proves (iii). Finally, by (40) we have H r ⊂ d −1 (H r < ) which implies Therefore, d rr (H r ) = 0 which proves (iv) for p = r. To show that (P, H, d |H ) is a poset filtered chain complex, we will prove that (45) d(H L ) ⊂ H L for every L ∈ Down(P ).
Property (45) holds by induction assumption if r ∈ L. Thus, assume r ∈ L and set L := L\{r}. Then H L = H L +H r < +H r , and by (44) and induction assumption one has which proves (45). Since P ∈ Down(P ) and H P = H, we get from (44)   is a chain complex. Let L ∈ Down(Q) and let L := { p ∈ P | ∃ q∈L p ≤ q }. Then L ∈ Down(P ) and H p = 0 for p ∈ L \ L. Therefore, H L = H L . Thus, since d |H is P -filtered, it follows from Corollary 4.6 that d |H is also Q-filtered. Hence, (Q, H, d |H ) is a filtered chain complex. It follows from Lemma 6.18(iv) that (Q, H, d |H ) is boundaryless. Hence, we conclude from Corollary 5.4 and the definition of Q that (Q, H, d |H ) is reduced. We will prove that (Q, H, d |H ) is filtered chain homotopic to (P, W, d). Let ι : H → W and π : W → H be respectively the inclusion and the projection homomorphisms. It follows from Lemma 6.18(v) that for x ∈ H we have dιx = dx = ιdx, that is, ι is a chain map. Also, for x ∈ W we have proving that π is a chain map. Let α : Q → P denote the inclusion map. Clearly, α is order preserving. Since α is injective, the inverse relation β := α −1 : P Q is a well-defined partial map which is also order preserving. Recall that by Proposition 5.10(i) for every p ∈ P we have a chain complex (W p , d pp ). By Lemma 6.18 it has homology decomposition W p = V p ⊕ H p ⊕ B p . Thus, by Proposition 5.5(ii), the chain complexes (W p , d pp ) and (H p , 0) are chain homotopic. It follows that (W p , d pp ) is essential if and only if (H p , 0) is essential. Moreover, by Corollary 5.4, (H p , 0) is essential if and only if H p = 0. Therefore, P = Q = Q which implies that α : (Q, Q ) → (P, P ) and β : (P, P ) → (Q, Q ) are well-defined morphisms in DPSet. Obviously, the inclusion homomorphism ι : H → W is β-graded and, in consequence, β-filtered. Similarly, the projection homomorphism π : W → H is α-filtered. Therefore, we have well-defined filtered morphisms (β, ι) : (Q, H, d |H ) → (P, W, d) and (α, π) : (P, W, d) → (Q, H, d |H ). Obviously, (α, π) • (β, ι) = (βα, πι) = (id Q , id H ) = id (Q,H) . We will show that (β, ι) • (α, π) = (αβ, ιπ) is filtered chain homotopic to id (P,W ) . Let be the projection map and let ν : V x → x ∈ W be the inclusion map. Clearly, µ and ν are graded and, in consequence, filtered homomorphisms. By Lemma 6.18(iii) we have a well-defined P -graded degree To see (46), take an arbitrary |V (x B ) and also dΓx = x B . It follows that (Γd + dΓ)(x) = x V + x B which proves the identity (46). Clearly, the identity (βα) |P = id P = id P |P holds. Hence, (id P , Γ) is an elementary filtered chain homotopy between (αβ, ιπ) and id (P,W ) . Thus, (Q, H, d |H ) and (P, W, d) are filtered chain homotopic. Since Q = Q , we deduce from Lemma 6.18(iv) and Proposition 6.3 that (Q, H, d |H ) is reduced. It follows that the poset filtered chain complex (Q, H, d |H ) is in fact a Conley complex for (P, W, d) and that the (Q, Q)-matrix of d |H is a connection matrix of the poset filtered chain complex (P, W, d).

Connection matrices in Lefschetz complexes
7.1. Lefschetz complexes. With this section we turn our attention to a more specialized situation. Rather than continuing to study connection matrices for general poset filtered chain complexes, we now consider the setting of Lefschetz complexes. The following definition goes back to S. Lefschetz, see [15, Chapter III, Section 1, Definition 1.1].
Definition 7.1. We say that (X, κ) is a Lefschetz complex over a ring R if X = (X q ) q∈N 0 is a finite set with N 0 -gradation, κ : X × X → R is a map such that (47) κ(x, y) = 0 ⇒ x ∈ X q , y ∈ X q−1 , and for any x, z ∈ X we have (48) y∈X κ(x, y)κ(y, z) = 0.
We refer to the elements of X as cells, to κ(x, y) as the incidence coefficient of the cells x and y, and to κ as the incidence coefficient map. We define the dimension of a cell x ∈ X q as q, and denote it by dim x. Whenever the incidence coefficient map is clear from the context we often refer just to X as Lefschetz complex. We say that (X, κ) is regular if for any x, y ∈ X the incidence coefficient κ(x, y) is either zero or it is invertible in R.
Let (X, κ) be a given Lefschetz complex. We denote by C k (X) := R X k the free R-module spanned by the set X k of cells of dimension k for k ∈ N 0 , and let C k (X) denote the zero module for k < 0. Then it is clear that the sum C(X) := k∈Z C k (X) is a free Z-graded R-module generated by X. Finally, define the module homomorphism ∂ κ : C(X) → C(X) on generators by Proposition and Definition 7.2. The pair (C(X), ∂ κ ) is a chain complex. We call it the chain complex of (X, κ) and we refer to the homology of this chain complex as the Lefschetz homology of (X, κ).
Note that every finitely generated free chain complex is the chain complex of a Lefschetz complex obtained by selecting a basis. More precisely, assume that (C, ∂) is a finitely generated free chain complex over a ring R and U ⊂ C is a fixed basis of C. Suppose further that C k = 0 for all k < 0. Then for every v ∈ U there are uniquely determined coefficients a vu ∈ R such that ∂v = u∈U a vu u.
The following proposition is straightforward.
The family of cells of a simplicial complex [10, Definition 11.8] and the family of elementary cubes of a cubical set [10, Definition 2.9] provide simple but important examples of Lefschetz complexes. In these two cases the respective formulas for the incident coefficients are explicit and elementary, see for example [18]. Also a general regular cellular complex, or a regular finite CW complex as considered in [16,Section IX.3], is an example of a Lefschetz complex. In this case the incident coefficients may be obtained from a system of equations as described in [16,Section IX.5]. Note that a Lefschetz complex over a field is always regular. Given x, y ∈ X we say that y is a facet of x, and we write y ≺ κ x, if we have κ(x, y) = 0. It is easily seen that the reflexive and transitive closure of the relation ≺ κ is a partial order. We denote this partial order by ≤ κ and the associated strict order by < κ . As an immediate consequence of (47) we have the following proposition.
Proposition 7.4. The map dim : (X, ≤ κ ) → (Z, ≤) which assigns any cell x ∈ X its dimension dim x ∈ Z is order preserving. Moreover, if the inequality x ≺ κ y holds, then one has dim y = dim x + 1.
We say that y is a face of x if y ≤ κ x. The T 0 topology defined via Theorem 3.5 by the partial order ≤ κ will be called the Lefschetz topology of the Lefschetz complex (X, κ). Observe that the closure of a set A ⊂ X in this topology consists of all faces of all cells in A.
We say that the subset A ⊂ X is a κ-subcomplex or Lefschetz subcomplex of X, if (A, κ |A×A ) with the Z-gradation induced from X is a Lefschetz complex in its own right. The following proposition provides sufficient conditions for a subset of a Lefschetz complex to be a Lefschetz subcomplex. For more details see [17,Proposition 5.3] and [18, Theorem 3.1].
Proposition 7.5. If A ⊂ X is locally closed in the Lefschetz topology, then A is a Lefschetz subcomplex of (X, κ). In particular, every open and every closed subset of X is a Lefschetz subcomplex.
The following result was established in [17,Proposition 5.2], and it computes the homology of some simple Lefschetz complexes. Proposition 7.6. For every x ∈ X the singleton {x} is a Lefschetz subcomplex of X and For every x, y ∈ X such that x is a facet of y the doubleton {x, y} is a Lefschetz subcomplex of X and for all q ∈ Z one has Note that in general the chain complex of a Lefschetz subcomplex A of a Lefschetz complex X is not a chain subcomplex of the chain complex of X. However, we have the following proposition. In particular, the chain complex (C(A), ∂ κ |A×A ) is a chain subcomplex of the chain complex (C(X), ∂ κ ).
Proof: To see (50) it suffices to verify the equality on basis elements. Thus, take an x ∈ A. Then because κ(x, y) = 0 implies y ∈ cl x ⊂ cl A = A. Property (51) and the remaining assertion are obvious.
Given a closed subset A ⊂ X in the Lefschetz topology we define the relative Lefschetz homology H(X, A) as the homology of the quotient chain complex (C(X, A),∂), where C(X, A) := C(X)/C(A) and∂ stands for the induced boundary map.
Example 7.8. The easiest way to visualize a Lefschetz complex is by presenting it as a κ-subcomplex of a simplicial complex. A sample Lefschetz complex is presented in Figure 4 as a locally closed collection of simplices of a simplicial complex. Note, however, that not all Lefschetz complexes can be written in this form.
7.2. Acyclic partitions. Let X be a finite topological space and let E be a partition of X into locally closed sets. Consider the relation E in E defined by (52) We say that E is an acyclic partition of X if E may be extended to a partial order on E. Every such extension is called E-admissible. Note that the smallest E-admissible partial order on an acyclic partition E is the transitive closure of E . We call it the inherent partial order of E and denote it by ≤ E .
Proposition 7.9. Assume that E is an acyclic partition of a finite topological space X, that ≤ is an E-admissible partial order, and that D ⊂ E.
(i) If D is a down set with respect to ≤ E , then |D| is closed.
(ii) If D is convex with respect to ≤ E , then |D| is locally closed.
Proof: In order to prove (i), we assume that D ⊂ E is a down set and let x ∈ cl |D|. Then there exists an E ∈ D such that x ∈ cl E . Since E is a partition of X, there exists an E ∈ E such that x ∈ E. It follows that E ∩ cl E = ∅ and, since ≤ is an E-admissible partial order, we get E ≤ E . Since D is a down set, this in turn implies E ∈ D. Thus, x ∈ E ⊂ |D|, and therefore |D| is closed. This completes the proof of (i). To prove (ii), we now assume that D is convex. It follows from Proposition 3.1 that both D ≤ and D < are down sets, and due to (i) the sets |D ≤ | and |D < | are closed. Moreover, since D = D ≤ \D < and E is a partition, we have |D| = |D ≤ |\|D < |. Hence, |D| is locally closed by Proposition 3.4.

Connection matrices of acyclic partitions.
Assume that E is an acyclic partition of a Lefschetz complex X. Then the module C(X) admits an E-gradation We have the following proposition.
Proposition 7.10. The triple (E, C(X), ∂ κ ) with an E-admissible partial order ≤ on E and the E-gradation (53) is a poset filtered chain complex.
Proof: We need to prove that ∂ κ is a filtered module homomorphism, that is, ∂ κ is an α-filtered module homomorphism with α = id E . For this, we will verify (11) of Corollary 4.6 for M = C(X) and L = D ∈ Down(E). Then one immediately obtains M L = C(X) D = C(|D|), and we need to verify that the inclusion ∂ κ (C(|D|)) ⊂ C(|D|) holds. But, this follows from Proposition 7.7, because, by Proposition 7.9(i), the set |D| is closed. Thus, by Corollary 4.6, we see that the boundary homomorphism is id E -filtered. Therefore, (E, C(X), ∂ κ ) is a poset filtered chain complex. Proposition 7.10 lets us define the Conley complex and connection matrices of acyclic partitions. Definition 7.11. By a filtration of an acyclic partition E of a Lefschetz complex X we mean a poset filtered chain complex (E, C(X), ∂ κ ) with E ordered by an E-admissible partial order. The Conley complex and the associated connection matrices of E are defined as the Conley complex and the associated connection matrices of the filtration of E, that is, the Conley complex and connection matrices of (E, C(X), ∂ κ ). 7.4. Singleton partition. We have the following proposition and definition.
Proposition and Definition 7.12. Assume (X, κ) is a Lefschetz complex. Then the following hold: (i) The family X := { {x} | x ∈ X } is a partition of X into locally closed sets.
(ii) The map sing : X x → {x} ∈ X is a bijection which preserves in both directions the relation ≤ κ on X and the relation X on X given by (52). (iii) The relation X is a partial order on X which coincides with the inherent partial order ≤ X on X . (iv) The family X is an acyclic partition of X. We call X the singleton partition of X.
Proof: Clearly X is a partition of X and every singleton {x} is locally closed, because we have {x} = x ≤κ \ x <κ and the sets x ≤κ and x <κ are both down sets, hence closed. This proves (i).
Now let x, y ∈ X. Then one easily verifies the sequence of equivalences which proves (ii). Hence, X is a partial order on X . In particular, X is transitively closed, and this implies that X coincides with the inherent partial order on X . This proves (iii). It follows that X is an acyclic partition of X, which finally establishes (iv).
Proposition 7.12 shows that we have a well-defined poset filtered chain complex (X , C(X), ∂ κ ). Using the order isomorphism sing : X → X we identify it with (X, C(X), ∂ κ ). We note that via this identification closed sets in X are in one-to-one correspondence with down sets in X .
Proof: According to Definition 5.7 we have to prove that for every admissible partial order ≤ on X and x, y ∈ X the inequality x ≤ κ y implies the inequality x ≤ y. Clearly, it suffices to prove that x ≺ κ y implies x ≤ y, because ≤ κ is the transitive closure of ≺ κ . But, x ≺ κ y by definition means that κ(y, x) = 0 which implies ∂ κ xy = 0, and by the admissibility of ≤ finally gives the inequality x ≤ y.
Note that the bijection sing lets us identify partial orders in X with partial orders in the associated singleton partition X of X. As an immediate consequence of Proposition 7.12 and Proposition 7.13 we get the following corollary.
Corollary 7.14. Let (X, κ) be a Lefschetz complex and let X denote the associated singleton partition. A partial order is ∂ κ -admissible if and only if it is X -admissible.
We say that a ∂ κ -admissible partial order ≤ in X is natural if (54) x ≤ y and dim x = dim y ⇒ x = y.
Note that the native partial order of ∂ κ is natural, but that a ∂ κ -admissible partial order ≤ in X does not need to be natural in general.
Definition 7.15. A filtration of a Lefschetz complex (X, κ) is defined as a filtration of the singleton partition of X, that is, the poset filtered chain complex (X, C(X), ∂ κ ) with X ordered by a ∂ κ -admissible partial order. When we consider X as a poset ordered by a natural partial order in X we refer to the filtration (X, C(X), ∂ κ ) as a natural filtration of X. When we consider X as a poset ordered by the native partial order of X we refer to the filtration (X, C(X), ∂ κ ) as the native filtration of X.
Theorem 7.16. Let (X, κ) be a Lefschetz complex. Then the following hold: (i) Every filtration of the Lefschetz complex (X, κ) is reduced, that is, it is a Conley complex of itself. In particular, X = X. (ii) Every natural filtration of a Lefschetz complex (X, κ) has a uniquely determined Conley complex and connection matrix. The connection matrix coincides with the (X, X)-matrix of the boundary homomorphism ∂ κ .
In order to prove (ii), it suffices to verify that every transfer morphism from (X, C(X), ∂ κ ) to a Conley complex (P, C, d) of (X, C(X), ∂ κ ) is essentially graded. Actually, we will prove the stronger fact that every such transfer morphism is graded. Thus, assume that (X, C(X), ∂ κ ) is a natural filtration of X, that (P, C, d) is another Conley complex of X, and that the map (α, ϕ) : (X, C(X), ∂ κ ) → (P, C, d) is a filtered chain isomorphism.
Since in view of (i) the filtration (X, C(X), ∂ κ ) is a Conley complex of itself, the transfer morphism from (X, C(X), ∂ κ ) to (P, C, d) is just (α, ϕ).
Since both (X, C(X), ∂ κ ) and (P, C, d), as Conley complexes, are reduced, we see from Proposition 6.9 that (α, ϕ) is also an isomorphism in FMod. Thus, it follows from Lemma 4.13 that α : P → X is an order isomorphism and ϕ pα(p) : C({α(p)}) → C p is an isomorphism of Z-graded moduli for every p ∈ P . But, Proposition 7.6 provides the structure of C({α(p)}). Hence, we can choose a non-zero c p ∈ C such that In other words, C p is a one-dimensional Z-graded module generated by c p such that In order to prove that (α, ϕ) is an isomorphism in GMod we will first show that ϕ is α-graded by verifying (6). Assume therefore that we have ϕ py = 0 for p ∈ P and y ∈ X. Furthermore, let x = α(p). Since (α, ϕ) is a filtered module homomorphism, we get from (7) that p ∈ α −1 (y ≤ ) ≤ . This implies that there exists a p ∈ P such that p ≤ p and α(p ) ≤ y. Since α is an order isomorphism, one further obtains x = α(p) ≤ α(p ) ≤ y. Note that C p is a one-dimensional Z-graded module which is non-zero only in dimension n = dim α(p) = dim x. Since ϕ py : C({y}) → C p is Z-graded homomorphism of degree zero, both C({y}) and C p are one-dimensional Zgraded modules, and we have ϕ py = 0, one immediately obtains that ϕ py maps Ry isomorphically onto Rc p . It follows that dim c p = dim y. In view of x = α(p) and (55) we therefore have dim x = dim y. Since x ≤ y and the partial order is natural, this implies x = y, as well as α(p) = x = y. This in turn proves that ϕ is α-graded, that is, (α, ϕ) is a module homomorphism in GMod. Since it is an isomorphism in FMod, one obtains from Corollary 4.14 that it is also an isomorphism in GMod. Since it is a chain map, Proposition 5.1 shows that it is an isomorphism in PgCc. This completes the proof of (ii).
8. Dynamics of combinatorial multivector fields 8.1. Combinatorial multivector fields. The concept of a combinatorial multivector field was proposed in [17, Definition 5.10] as a dynamically oriented extension of the notion of combinatorial vector field which has been introduced by Forman [6,5]. The definition of combinatorial multivector field we consider here is based on [13] restricted to the special setting of Lefschetz complexes. Let X be a Lefschetz complex. By a combinatorial multivector in X we mean a non-empty subset of X which is locally closed with respect to the Lefschetz topology. A combinatorial multivector field is a partition V of X into combinatorial multivectors. Since in this paper we do not use any other concepts of vector fields, in the sequel we simplify the terminology by dropping the adjective combinatorial in combinatorial multivector and combinatorial multivector field. Vectors and multivectors are marked with an arrow from simplex x to simplex y whenever y ∈ Π V (x) and y ∈ cl x. The case when y ∈ cl x is not marked to keep the image readable.
We say that a multivector V is critical, if the relative Lefschetz homology H(cl V, mo V ) is non-zero. A multivector V which is not critical is called regular. For each x ∈ X we denote by [x] V the unique multivector in V which contains x. If the multivector field V is clear from context, we abbreviate the notation by writing [x] := [x] V . We say that x ∈ X is critical (respectively regular) with respect to the multivector field V, if the multivector [x] V is critical (respectively regular). We say that a subset We associate with every multivector field a multivalued map Π V : X X given by The multivalued map Π V may be interpreted as a digraph with vertices in X and an arrow from x ∈ X to y ∈ X if y ∈ Π V (x). Clearly, every multivector V ∈ V forms a clique in this digraph. By collapsing all vertices in a multivector to a point we obtain an induced digraph with vertices in V and an arrow from V ∈ V to W ∈ W if there exist an x ∈ V and a y ∈ W such that y ∈ Π V (x). We refer to this digraph as the V-digraph.
Since V is a partition of X into locally closed subsets, we can consider the relation V in V introduced in Section 7.2 . The following proposition shows that the V-digraph and the relation V are the same concepts. Proof: Assume there is an arrow from V to W in the V-digraph of V, and let x, y ∈ X be such that x ∈ V , y ∈ W , and y ∈ Π then we can take a y ∈ W ∩ cl V and an x ∈ V such that y ∈ cl x. It follows that y ∈ Π V (x). Hence, there is an arrow in the V-digraph from V to W .
Example 8.2. Figure 5 presents three different combinatorial multivector fields V 0 , V 1 , and V 2 on the Lefschetz complex X introduced in Example 7.8. The V 0 -digraph coincides with the Hasse diagram (20). Notice that by using the Hasse diagram representation of the digraph, we implicitly assume that arrows always point downwards, i.e., we represent them without arrow heads. Moreover, to keep the diagrams as simple as possible, we do not indicate the loops which are present at every node. Similarly, the V 1 -digraph is A solution of a multivector field V in A ⊂ X is a partial map : Z A whose domain, denoted by dom , is a Z-interval and for any i, i + 1 ∈ dom the inclusion (i + 1) ∈ Π V ( (i)) holds. The solution passes through x ∈ X if 0 ∈ dom and x = (0) for some i ∈ dom . The solution is full if dom = Z. It is a backward solution if dom is left infinite. It is a forward solution if dom is right infinite. It is a partial solution or simply a path if dom is bounded. We refer to the cardinality of the domain of a path as the length of the path. If the maximum of dom exists, we call the value of at this maximum the right endpoint of . If the minimum of dom exists, we call the value of at this minimum the left endpoint of . We denote the left and right endpoints of by and , respectively. Given a full solution through x ∈ X, we denote by + := |Z + 0 the forward solution through x, and by − := |Z − 0 the backward solution through x. By a shift of a solution we mean the composition • τ n , where τ n : Z m → m + n ∈ Z is the translation map. Given two solutions ϕ and ψ such that ψ and ϕ exist and ψ ∈ Π V (ϕ ), there is a unique shift τ n such that ϕ ∪ ψ • τ n is a solution. We call this union of paths the concatenation of ϕ and ψ and denote it ϕ · ψ. We also identify each x ∈ X with the path of length one whose image is {x}. The following proposition is straightforward.

Isolating neighborhoods and isolated invariant sets. A full solution
: Z → X is called left-essential (respectively right-essential), if for every regular x ∈ im the set { t ∈ Z | (t) ∈ [x] V } is left-infinite (respectively right-infinite). We say that is essential if it is both left-and right-essential. We say that S ⊂ X is V-invariant, or briefly invariant, if for every x ∈ S there exists an essential solution through x in S. A closed set N ⊂ X is an isolating set for a V-invariant subset S ⊂ N if Π V (S) ⊂ N and any path in N with endpoints in S is a path in S. We say that S is an isolated invariant set if S admits an isolating set. 8.4. Recurrent solutions and gradient-like multivector fields. An essential solution : Z → X is recurrent if for every x ∈ im the set −1 (x) is both left-and right-infinite. A multivector field V is called gradient-like if for every recurrent solution : Z → X there exists a multivector V ∈ V such that im ⊂ V . A multivector field V is called gradient if the only recurrent solutions are constant solutions with its value in a singleton of V. We would like to point out that this rules out multivectors of size at least two which have the index of a hyperbolic equilibrium. Moreover, recall that a singleton in V is always critical, see Proposition 7.6. We then have the following proposition.
Proposition 8.4. Let V be a combinatorial multivector field on a Lefschetz complex X. The following properties are pairwise equivalent.
(i) V is gradient-like. (ii) After discarding self-loops, the V-digraph is acyclic.
(iii) V is an acyclic partition in X.
is clearly a partition of X. We call it the partition induced by the Morse decomposition M. Each E ∈ E M either is a Morse set, or it is a multivector not contained in any of the Morse sets. For E, E ∈ E M we write E E if and only if there is a path σ such that σ ∈ E and σ ∈ E.
Proposition 8.6. The relation is a partial order in E M .
Proof: Clearly, the relation is reflexive and transitive. To see that it is antisymmetric, take E, E ∈ E M such that E E and E E. We will prove that the assumption E = E leads to a contradiction. Let σ be a path such that σ ∈ E and σ ∈ E. In addition, let τ denote a path which satisfies both τ ∈ E and τ ∈ E .
Consider first the case when E and E are multivectors disjoint from Morse sets. Then [σ ] = E = [τ ] and [σ ] = E = [τ ]. It follows that the paths σ and τ can be concatenated to a full, periodic solution . Since we assumed that E = E , this solution is essential. Consequently, uim − ( ) and uim + ( ) must be contained in Morse sets. However, it follows from Proposition 8.5 that uim − ( ) = im = uim + ( ), and therefore the point σ is contained in both E and a Morse set, which contradicts our assumption that both E and E are multivectors disjoint from Morse sets. Thus, we have E = E in this case.
Consider now the case when both E and E are Morse sets. In particular, this implies that they are invariant. Let be an essential solution through σ in E , and let π be an essential solution through σ in E. Thenσ := − ·σ·π + is an essential solution with uim − (σ) ⊂ E and uim + (σ) ⊂ E, and therefore it follows from the definition of Morse decomposition that E > E. Similarly, we can extend τ to an essential solutionτ which satisfies both uim − (τ ) ⊂ E and uim + (τ ) ⊂ E . Hence, one obtains E > E , another contradiction.
Finally, consider the case when one of the sets E, E is a Morse set and the other is a multivector disjoint from Morse sets. Without loss of generality we may assume that E is a Morse set and E is a multivector disjoint from Morse sets. Then [σ ] = E = [τ ]. Let π be an essential solution through σ in the Morse set E, and let be an essential solution through τ in E. Then the concatenation ξ := − · τ · σ · π + is an essential solution which satisfies both uim − (ξ) ⊂ E and uim + (ξ) ⊂ E. Thus, it follows from the definition of Morse decomposition that im ξ ⊂ E, a contradiction.
Proposition 8.7. The relations and ≤ E M coincide. In particular, the family E M is an acyclic partition of X.
Proof: We need to prove that for all E, First, we will prove that Let x ∈ E ∩ cl E . Then x ∈ cl y for a y ∈ E . Thus, σ := y · x is a path. Moreover, σ = y ∈ E and σ = x ∈ E, proving (59). Since, by Proposition 8.6, the relation is transitive, it follows from property (59) that the left-hand side of (58) implies the right-hand side of (58).
For the reverse implication assume that E E . Let = x 0 ·x 1 ·. . .·x n be a path from E to E.
Thus, (58) is proved. Hence, it follows from Proposition 8.6 that ≤ E M is a partial order on E M , and this finally implies that E M is an acyclic partition.  The filtered chain complex (E M 0 , C(X), ∂ κ ) coincides with the filtered chain complex considered in Example 5.9. Therefore, the connection matrices given by (27) and (31) are the connection matrices of M 0 . We know that these matrices are not equivalent.

Connection matrices for Forman's gradient vector fields
Recall that a combinatorial multivector W is a combinatorial vector if card W ≤ 2. A combinatorial multivector field whose multivectors are just vectors is called a combinatorial vector field, a concept introduced by R. Forman [6,5]. In the case of a combinatorial vector field the critical vectors are precisely the singletons and the regular vectors are precisely the doubletons, i.e., vectors of cardinality two. Therefore, every gradient-like combinatorial vector field is a gradient combinatorial vector field.
In this section we assume that V is a gradient combinatorial vector field on a Lefschetz complex X. Let C ⊂ V be the collection of critical vectors in V. Since we assume that V is a combinatorial vector field, it follows from Proposition 7.6 that C is exactly the collection of singletons in V. The following proposition is straightforward.
Proposition 9.1. Assume V is a gradient combinatorial vector field. The collection C is a Morse decomposition of V and the partition of X induced by this Morse decomposition is E C = V.
It follows that the Conley complex of the Morse decomposition C of X is the Conley complex of the filtered chain complex (V, C(X), ∂ κ ). The aim of this section is to prove that the Morse decomposition C has precisely one connection matrix, and that this connection matrix coincides with the matrix of the boundary operator of the associated Morse complex, see also [6,Section 7]. In order to make this statement precise, we first need to recall some concepts.
Unlike a general multivector, a combinatorial vector W ∈ V contains a unique minimal element and a unique maximal element in W with respect to ≤ κ . We denote them by W − and W + , respectively, and we extend the notation to cells by writing x − := [x] − and x + := [x] + . Note that a combinatorial vector is given by W = {W − , W + }, and W is critical if and only if we have W + = W − , which in turn is equivalent to assuming that W is a singleton. As a consequence of Proposition 7.4 one obtains for a vector W the inequality and for all x ∈ W one has We say that a cell x is a tail if x = x − = x + and a head if Clearly, if a cell is neither a tail nor a head, then it is critical. We denote the subsets of critical cells, tails and heads respectively by X c (V), X − (V), and X + (V), and we shorten this notation to X c , X − , and X + whenever V is clear from the context. Note that the collection C of critical vectors of V is precisely { {x} | x ∈ X c }. For a combinatorial vector V ∈ V we set dim V := dim V + . Moreover, recall that V, as a gradient vector field, is an acyclic partition of X according to Proposition 8.4. In particular, V is a poset with the partial order ≤ V . We have the following proposition.
Proposition 9.2. Assume that V is a gradient vector field on a Lefschetz complex X. Then the following hold: topically essential coincides with the set of critical vectors C.
Proof: Since, by definition, the relation ≤ V is the transitive closure of the relation V , to prove (i) it suffices to verify that In particular, this gives dim V = dim W . Thus, consider now case x ∈ W . Since W − ∈ cl W + , we have cl W = cl{W − , W + } = cl W − ∪cl W + = cl W + . It follows that x ∈ cl W + \ {W + }, and therefore dim x < dim W + . Since we assumed x ∈ V , we get from (60) and (61) To see (ii), without loss of generality, we may assume that V ≺ V W , that is, V = W and V ∩cl W = ∅. Observe that since V is critical, both V = {x} and dim V = dim x are satisfied. Since x ∈ W and x ∈ cl W = cl W + , we have dim x < dim W + . Therefore, dim V = dim x < dim W + = dim W .
To see (iii) observe that, by Proposition 7.6, the Lefschetz homology of V is zero if and only if V is not critical. Therefore, property (iii) follows immediately from Theorem 5.6(iv).
Consider now a regular Lefschetz complex X. With a given combinatorial vector field V we can associate the degree +1 map Γ V : C(X) → C(X), which for a generator x ∈ X is defined by In the sequel, we drop the subscript in Γ V whenever V is clear from the context. The following proposition is straightforward.
Proposition 9.3. If X is a regular Lefschetz complex and V is a combinatorial vector field on X, then for every c ∈ C(X) we have |Γc| ⊂ X + .
Proposition 9.4. Suppose that X is a regular Lefschetz complex and V is a combinatorial vector field on X. Then for all x, u ∈ X we have both the implication Proof: The definition of the boundary operator of a Lefschetz complex given in (49) implies κ(x, y)κ(y + , y − ) −1 y, y − y + .
Given a combinatorial vector field V, we define the associated combinatorial flow as in [6, Theorem 6.4] by letting Φ := Φ V : C(X) → C(X) act on a generator x ∈ X through the formula In the sequel, whenever V is clear from the context, we drop the subscript in Φ V . Note that Φ is a degree zero module homomorphism which satisfies the identity Φ∂ = ∂ + ∂Γ∂ = ∂Φ. Hence, Φ is a chain map. We have the following proposition.
Proposition 9.5. Suppose that X is a regular Lefschetz complex and V is a combinatorial vector field on X. Then for all x ∈ X we have
Consider now the case x ∈ X − . Then one has x = x − ∈ X + and we get from (63) that x, Γ∂x = 0. Therefore, the definition (62) yields Finally, we consider the remaining case x ∈ X + . Then x ∈ X − , and (64) shows that x, ∂Γx = 0 . Therefore, since ∂x = y∈X κ(x, y)y, we get κ(x, y)κ(y + , y − ) −1 y, y − x, y + because the only non-zero term in the last sum occurs for the point y ∈ X which satisfies both y + = x = x + and y = y − = x − . Proposition 9.6. Suppose that X is a regular Lefschetz complex and V is a combinatorial vector field on X. Then we have Φ(C(X + )) ⊂ C(X + ).
Proof: Since Φ is linear, it suffices to show that the inclusion x ∈ X + implies |Φx| ⊂ X + . Thus, take x ∈ X + and a u ∈ |Φx|. Then u, Φx = 0. From (64) we get u, ∂Γx = 0. Therefore, 0 = u, Φx = u, x + u, Γ∂x , which implies u, x = 0 or u, Γ∂x = 0. If u, x = 0, then u = x ∈ X + . If the inequality u, Γ∂x = 0 holds, then we get from (63) that u ∈ X + . Hence, |Φx| ⊂ X + . Proposition 9.7. Suppose that X is a regular Lefschetz complex and V is a combinatorial vector field on X. Assume further that A ∈ Down(V, ≤ V ). Proof: Clearly, A is V-compatible, and it follows from Proposition 7.9(i) that A is closed. Take an x ∈ A. We will prove that Φx ∈ C(A). Since A is closed, we have |∂x| ⊂ A. Since A is V-compatible, we have |Γ V x| ⊂ A. Thus, again by closedness and V-compatibility of A, we get both |∂Γ V x| ⊂ A and |Γ V ∂x| ⊂ A. Therefore, This finally implies Φx ∈ C(A) and completes the proof.
Proposition 9.8. Suppose that X is a regular Lefschetz complex and V is a combinatorial vector field on X. Assume further that x, y ∈ X. Then the following hold: (ii) If y ∈ |Φx| and [y] = [x], then y = x .
We will use the fact that Φ stabilizes under iteration, that is, Φ n = Φ n+1 for n sufficiently large. This is established in [6,Theorem 7.2]. One can readily verify that while Forman only considers Z coefficients, his arguments remain valid for the case of general ring coefficients. Let Φ ∞ := Φ n = Φ n+1 denote this stabilized value. Proposition 9.9. Assume that V is a gradient vector field on a regular Lefschetz complex X. If y ∈ |Φ ∞ x|, then [y] ≤ V [x].
Proof: It suffices to prove that y ∈ |Φ n x| implies [y] ≤ V [x] for every positive integer n ∈ N. We proceed by induction on n. For n = 1 the conclusion is the statement of Proposition 9.8. Thus, fix a k ∈ N and assume that the conclusion holds for all n ∈ {1, 2, . . . k}. Let y ∈ |Φ k+1 x| and let c := Φ k x. We have c = w∈|c| c, w w and Φc = w∈|c| c, w Φw. Since y ∈ |Φ k+1 x| = |Φc| ⊂ w∈|c| |Φw|, we get y ∈ |Φw| for some w ∈ |c| = |Φ k x|. Thus, by our induction assumption, we obtain [y] We have the following proposition.
Proposition 9.10. Assume that V is a gradient vector field on a regular Lefschetz complex X. For every x ∈ X c we have Proof: Let x ∈ X c . Since Φ ∞ = Φ n for a large n ∈ N, it suffices to verify that for every positive integer n ∈ N (67) Φ n x = x + r n where r n is a chain satisfying We proceed by induction on n. Assume first that n = 1 and set r 1 := Γ∂x. From Proposition 9.3 we see that |r 1 | ⊂ X + and, since x ∈ X c , one obtains further x, r 1 = 0. It follows from x ∈ X c that Γx = 0. Therefore, the identity Φx = x + Γ∂x = x + r 1 holds. To show that |r 1 | ⊂ | [x] < V |, take an arbitrary y ∈ |r 1 |. Then y, r 1 = 0, and since x, r 1 = 0, we get x = y, that is, y, x = 0. It follows that y, Φx = y, x + y, r 1 = y, r 1 = 0. Hence, y ∈ |Φx|. Thus, Proposition 9.
Since y = x, and in view of the fact that Φ is a degree 0 map, one further has dim y = dim x, and we also get [y] = [x] and [y] ∈ [x] < V . This in turn yields y ∈ | [x] < V |. Therefore, |r 1 | ⊂ | [x] < V | and the proof of (67) for n = 1 is complete. Next, fix a k ∈ N and assume that (67) holds for all n ≤ k with r n satisfying (68). We then have Φ k+1 x = Φ(Φ k x) = Φx + Φr k = x + r 1 + Φr k , and let r k+1 := r 1 + Φr k . It follows from the induction assumption and Proposition 9.6 that |r k+1 | ⊂ |r 1 | ∪ |Φr k | ⊂ X + . Since |r 1 | ⊂ | [x] < V |, it suffices to prove that |Φr k | ⊂ | [x] < V | in order to see that |r k+1 | ⊂ | [x] < V |. For this, take an arbitrary element y ∈ |r k |. Then the induction assumption Thus, we proved that We have r k = y∈|r k | r k , y y and Φr k = y∈|r k | r k , y Φy. By Proposition 9.8(i) and property (69) Thus, which completes the induction argument.
Finally, to prove that Φ ∞ |X c is injective we will show the more general statement that Φ ∞ |C(X c ) is a monomorphism. For this, it suffices to verify that Φ ∞ c = 0 implies c = 0 for c ∈ C(X c ). Let c ∈ C(X c ). Then c = x∈X c a x x, and by (66) Hence, it follows that c = 0.
Recall that X c ⊂ X stands for the collection of critical cells of V. For every x ∈ X c we definex := Φ ∞ x ∈ Fix Φ. Proposition 9.11. Assume that V is a gradient vector field on a regular Lefschetz complex X. The the following hold: (i) The setX := {x | x ∈ X c } is a basis of Fix Φ. (ii) For every x ∈ X c we have for uniquely determined coefficients a xz ∈ R.
(iii) The pair (X,κ), where the Z-gradation ofX is induced by the map dim :X x → dim x ∈ Z andκ(x,z) is the coefficient a xz in (70), is a Lefschetz complex and C(X) = Fix Φ.
Proof: In order to prove (i) consider the projection Π : Fix Φ → C(X c ) given for c ∈ Fix Φ by Πc := x∈X c c, x x. It follows from the proof of [6,Theorem 8.2] that Φ ∞ Π = id Fix Φ . Take a c ∈ Fix Φ. Then proving thatX generates Fix Φ. To see thatX is linearly independent, assume that x∈X c a xx = 0 for some coefficients a x ∈ R. Then 0 = x∈X c a x Φ ∞ x = Φ ∞ ( x∈X c a x x). Since x∈X c a x x ∈ C(X c ), and Φ |C(X c ) : C(X c ) → Fix Φ is an isomorphism due to [6,Theorem 8.2], we conclude that x∈X c a x x = 0. This in turn implies a x = 0 for all x ∈ X c . Thus,X is indeed a basis of Fix Φ.
It follows from Proposition 9.10 that the map X c x →x ∈X is a bijection. Hence, also the map C {x} →x ∈X is a bijection, which lets us carry over the partial order ≤ V from C ⊂ V toX. Thus, forx,ȳ ∈X we writex ≤ Vȳ if {x} ≤ V {y}. Then we have the following result.
Proposition 9.12. Assume that V is a gradient vector field on a regular Lefschetz complex X. Then the partial order ≤ V inX is a natural partial order on the Lefschetz complex (X,κ).
In the sequel we considerX as a poset ordered by the natural partial order ≤ V in V, transferred toX via {x} →x, and the triple (X, C(X), ∂κ) as a natural filtration ofX. Proposition 9.13. Assume that V is a gradient vector field on a regular Lefschetz complex X. Then we have a well-defined filtered morphism (id V , Φ) : (V, C(X), ∂ κ ) → (V, C(X), ∂ κ ).
Moreover, (id V , Φ) n = (id V , Φ n ) is filtered chain homotopic to the identity morphism id (V,C(X)) for every n ∈ N. In particular, (id V , Φ ∞ ) is filtered chain homotopic to id (V,C(X)) .
Proof: Clearly, Φ is a chain map and id V : (V, V ) → (V, V ) is a morphism in DPSet. Hence, to prove that (id V , Φ) is a well-defined filtered morphism we only have to show that Φ is id V -filtered. We will do so by checking property (11) of Corollary 4.6. Consider the down set L ∈ Down(V) and set L := |L|. Then C(X) L = V ∈L C(V ) = C(L), and we need to verify that Φ(C(L)) ⊂ C(L). But this follows from Proposition 9.7, because L is immediately seen to be V-compatible and closed by Proposition 7.9(i).
Since by Proposition 5.14 filtered chain homotopy between morphisms is preserved by composition, in order to prove that (id V , Φ) n = (id V , Φ n ) is filtered homotopic to the identity morphism it suffices to prove that (id V , Φ) is filtered homotopic to the identity morphism. By the definition of Φ we have Φ − id C(X) = ∂Γ + Γ∂. Thus, we only need to check that Γ is id Vfiltered. Hence, assume that Γ V W = 0 for some V, W ∈ V. Then, there exists a w ∈ W such that π V (Γw) = 0. In particular, one has Γw = 0, which implies both w = w − = w + and Γw = −κ(w + , w − ) −1 w + . It follows that w + ∈ V . Thus V = W = id V (W ), which proves that Γ is id V -filtered, in fact, even graded. Finally, since Φ ∞ = Φ n for large n ∈ N, it follows that also (id V , Φ ∞ ) is filtered chain homotopic to id (V,C(X)) . Proposition 9.14. In the situation of the last proposition, we have a welldefined filtered morphism (α, ϕ) : (V, C(X), ∂ κ ) → (X, C(X), ∂κ) with α :X → V defined for x ∈ X c by α(x) := {x} and ϕ : C(X) → C(X) defined for c ∈ C(X) by ϕc := Φ ∞ c.
This verifies (71) and completes the proof. with β : V X defined for x ∈ X c by β({x}) :=x and ψ : C(X) → C(X) defined for c ∈ C(X) by ψc := c.
Proof: As we already mentioned, Proposition 9.2 implies that V = C, and Theorem 7.16(i) givesX =X. Therefore, β is a morphism in DSet. Moreover, since we considerX as ordered by the natural order ≤ V , the map β is trivially order preserving, hence also a morphism in DPSet. Obviously, the map ψ is a chain map. We will show that ψ is β-filtered by checking again property (10) of Proposition 4.5. Consider a down set A ∈ Down(X). Clearly, we have C(X) A = C(A). If we define L := β −1 (A), then in general L is not a down set in V, but L ≤ V is. We therefore consider L := |L ≤ V |, and obtain the identity C(X) β −1 (A) ≤ V = C(X) L ≤ V = C(L). Thus, in our case, condition (10) has to be verified in the form (74) ψ(C(A)) = C(A) ⊂ C(L).
Consider first a chainx ∈ A for an x ∈ X c . Then β({x}) =x ∈ A, which means that {x} ∈ β −1 (A) = L and |x| = {x} = [x] ∈ L ⊂ L ≤ V . It follows from Proposition 9.10 thatx = Φ ∞ x = x + r x where |r x | ⊂ X + ∩ | [x] < V |. Hence, the inclusion |x| ⊂ |x| ∪ |r ∈ L ⊂ L ≤ V and L ≤ V is a down set. This immediately shows thatx ∈ C(L). Since every chain in C(A) is a linear combination of chains in A, the inclusion (74) follows.
Theorem 9.16. Assume that V is a gradient vector field on a regular Lefschetz complex X. Then the filtered morphisms (α, ϕ) and (β, ψ) defined in the last two propositions are mutually inverse filtered chain equivalences. In particular, the filtered chain complexes (V, C(X), ∂ κ ) and (X, C(X), ∂κ) are filtered chain homotopic.
Theorem 9.17. Let V be a gradient combinatorial vector field on a regular Lefschetz complex X. Then C, the collection of the critical vectors of V, is a Morse decomposition of V. It has exactly one connection matrix which coincides with the (X,X)-matrix of ∂ κ | Fix Φ up to a graded similarity. Proof: It follows from Proposition 9.1 that C is a Morse decomposition of V and the partition induced on X by this Morse decomposition is V. Thus, by Definition 8.8, the Conley complex of C is the Conley complex of (V, C(X), ∂ κ ). From Corollary 6.12 and Theorem 9.16 we obtain that the Conley complex of (V, C(X), ∂ κ ) is isomorphic in PfCc to the Conley complex of (X, C(X), ∂κ) withX ordered by a natural partial order. Since, by Proposition 9.11 and Definition 7.15, the latter is the Conley complex of a natural filtration of a Lefschetz complex, the conclusion follows directly from Theorem 7.16(ii). this simple case it coincides with the vector field V 1 . Hence, the Hasse diagram of the partial order ≤ E 1 coincides with diagram (56). It follows that V 1 is a gradient vector field and by Theorem 9.17 M 1 has exactly one connection matrix. One can verify that the morphisms (α , h ) and ((α ) −1 , g ) constructed as in Example 6.13 are also filtered with respect to the filtration (E 1 , C(X), ∂ κ ). Therefore, arguing as in Example 6.13 we conclude that the Conley complex of M 1 is the filtered chain complex in Example 5.9 with the associated connection matrix (27). In fact, the homomorphism h coincides with homomorphism Φ V 1 and Φ V 1 = Φ ∞ V 1 in this case. An analogous argument shows that the Conley complex of M 2 is the filtered chain complex in Example 6.16 with the associated connection matrix (31). Both Conley complexes are visualized in Figure 6 as κ-subcomplexes of simplicial complexes.