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Complexity of PL-manifolds

We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant). Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with non-positive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed 4-manifolds of complexity zero (manifolds without 3-handles, doubles of 2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group).


Introduction
The complexity c(M ) of a compact 3-manifold M (possibly with boundary) was defined in a nice paper of Matveev [27] as the minimum number of vertices of an almost simple spine of M . In that paper he proved the following properties: Thanks to the combinatorial nature of spines, it is not hard to classify all manifolds having increasing complexity 0, 1, 2, . . . Tables have been produced in various contexts, see [8,9,13,23,24,28,29] and the references therein (and Table 1 below). Some of these classifications were actually done using the dual viewpoint of singular triangulations, which turns out to be equivalent to Matveev's for the most interesting 3-manifolds. We extend here Matveev's complexity from dimension 3 to arbitrary dimension. To do this, we need to choose an appropriate notion of spine. In another paper [25] written in 1973, Matveev defined and studied simple spines of manifolds in arbitrary dimension. A simple spine of a compact manifold is a (locally flat) codimension-1 subpolyhedron with generic singularities, onto which the manifold collapses. If the manifold is closed there cannot be any collapse at all and we therefore need to priorly remove one ball.
Simple spines are actually not flexible enough for defining a good complexity. In dimension 3, as an example, any simple spine for S 3 (or, equivalently, D 3 ) is a too complicated and unnatural object, such as Bing's house or the abalone. Every simple spine of D 3 has at least one vertex: however, a reasonable complexity must be zero on discs and spheres.
To gain more flexibility, Matveev defined in 1988 the more general class of almost simple spines [26,27] of 3-manifolds. An almost simple polyhedron is a compact polyhedron that can be locally embedded in a simple one. This more general definition allows one to use very natural objects as spines, such as a point for D 3 or a circle for D 2 × S 1 : a point is not a vertex by definition, and hence c(D 3 ) = 0, as required. We show here that the notion of almost simple spine extends naturally to all dimensions. This is in fact not the only way to gain more flexibility. A different possibility consists in enlarging the notion of spine by admitting an arbitrary number of open balls in its complement. Following that road, a 2-sphere is a simple spine of S 3 (or D 3 ) without vertices, and hence c(D 3 ) = 0 again. One might also allow simultaneously almost simple polyhedra and more balls in their complement.
In our attempt to define a suitable complexity in any dimension n, we are apparently forced to choose among three different definitions of complexity, and the choice seems only a matter of taste: as a spine for S n , do we allow a point, an equator (n − 1)-sphere, or both?
Luckily, these three definitions are actually equivalent and lead to the same complexity c(M n ), in every dimension n. This non-trivial fact shows that c(M n ) is indeed a very natural quantity to associate to a compact manifold M n . For the sake of clarity, we choose in Section 3 the simplest definition, which takes simple polyhedra and admits more balls in their complement. The other definitions and the proof of their equivalence is deferred to Section 7.
Having settled the problem of defining c, we turn to studying its properties. Three-dimensional complexity is already widely studied, and 1-and 2-dimensional ones are quite boring, so in this introduction we focus mainly on dimension 4.
We start by studying how c varies when a handle is added. Quite surprisingly, complexity can always be controlled. If a 4-manifold N is obtained from M by adding a handle of index i > 0, we have c(N ) c(M ), except when i = 3: in that case we get the opposite inequality c(N ) c(M ). When i = 4 we actually have c(N ) = c(M ).
These simple inequalities already allow to prove many things, and namely that plenty of 4-manifolds have complexity zero, in contrast with the 3-dimensional case. These include all 4-manifolds (with or without boundary) having a handledecomposition without 3-handles, and all the doubles of 2-handlebodies (i.e. manifolds decomposing without 3-and 4-handles). The first set includes many simply connected manifolds (maybe all), the second set contains closed manifolds with arbitrary (finitely presented) fundamental group.
We can find more. It is easy to see that a non-trivial product M k × N n−k with boundary has a spine without vertices. Therefore every closed 4-manifold obtained from a non-trivial product by adding handles of index = 3 has complexity zero. Among manifolds that may constructed in this way, we find the infinitely many exotic K3 surfaces discovered by Fintushel and Stern in [12] and the closed symplectic manifolds with arbitrary fundamental group exhibited by Gompf in [15] (both types of manifolds are built by attaching handles of index = 3 to a product M 3 × S 1 ).
As we have seen, there are plenty of 4-manifolds of complexity zero, although in many cases describing explicitly their spines is not obvious. One could guess that complexity is just zero on all 4-manifolds. Luckily, this is not the case. Various non-triviality results (in all dimensions n) are proved in this paper.
A closed n-manifold M with complexity zero must indeed fulfill some strict requirements. First of all, it cannot be aspherical. Moreover, its Gromov norm ||M || vanishes. If π 1 (M ) is infinite and (virtually) torsion-free, some other geometric invariants of M also vanish: the spherical volume T (M ) defined by Besson, Courtois, and Gallot in [7], the volume entropy λ(M ), and the systolic constant σ(M ), defined by Gromov in [18].
Concerning Gromov norm, we actually have c(M ) ||M || for every closed aspherical manifold. This shows in particular that there are closed manifolds of arbitrarily high complexity in all dimensions. It also implies that there are finitely many closed hyperbolic n-manifolds of bounded complexity: this is a mild extension of the 3-dimensional finiteness property, proved by Matveev for all closed irreducible 3-manifolds.
The triviality and non-triviality results just stated suggest that c(M ) is a wellbalanced quantity which could reasonably measure how "complicate" a manifold is. We hope that this new invariant will help to understand better the enormous set of PL (equivalently, smooth) compact 4-manifolds.
Structure of the paper. In Section 1 we list all the properties of c that are proved in this paper. Some basic notions of piecewise-linear topology are collected in Section 2. Simple spines and complexity are then introduced in Section 3. Some of our definitions are somehow different from the ones given by Matveev: in Sections 4 and 7 we prove that they are equivalent.
In Section 5 we construct simple spines as objects dual to triangulations. In Section 6 we show how to modify correspondingly a spine when the manifold is drilled along some sbpolyhedron. This basic operation will be used at many stages in the rest of the paper.
In Section 8 we study how complexity changes under handle addition, sphere drilling, and connected sum. In Section 9 we study the complexity of products and of finite coverings. In Section 10 we introduce a generalization of normal surfaces to arbitrary dimension and show how to "cut" a simple spine along a normal hypersurface.
In Section 11 we study the nerve of a simple spine P : the nerve is a simplicial complex determined by the stratification of P , which contains many informations on the topology of the manifold. The nerve is the key tool to prove various nontriviality results for c. The relations between complexity and homotopy invariants, Gromov norm, and riemannian geometry are then studied in Sections 12, 13, and 14. Finally, Section 15 is devoted to four-manifolds.

Main results
We define the complexity c(M ) of any compact PL manifold M in Section 3. The definition is of course also applicable to every smooth compact manifold by taking its unique PL structure [37]. In this section we collect all the properties of c proved in this paper.
Topological operations. Simple spines are flexible. Most topological operations on manifolds can be translated into some corresponding modifications of their spines. Various estimates on the complexity are therefore proved by examining how the number of vertices may vary along these modifications.
We collect here some estimates. We start with products. In contrast with Gromov norm, this inequality is far from being an equality in general. For instance, lens spaces have arbitrarily high complexity while their universal covering S 3 has complexity zero.
We investigate the effect of adding a i-handle to a n-manifold M . Quite surprisingly, we always get a one-side estimate when n 4, which depends only on the codimension n − i.
handles: Let N be obtained from M by adding a handle of index i. We have: connected sum: We have c(M n #N n ) c(M n ) + c(N n ) for every (boundary) connected sum in dimension n 3. Matveev proved that an equality holds in dimension three [27]. We do not know if it still holds in dimension n 4.
We turn to sphere drilling. If S ⊂ M is a submanifold, we denote by M S the manifold obtained by removing from M an open regular neighborhood of S. As for handle addition, if S is a k-sphere and n 4 we get a one-side estimate which depends only on the dimension k.
sphere drilling: Let S ⊂ M be a k-sphere. We have The (PL-)sphere S does not need to be locally flat. If S has a product regular neighborhood D n−k ×S we can perform a surgery by substituting this neighborhood with S n−k−1 × D k+1 along some map. If k = 1, the previous result implies the following.
surgery: If N is obtained from M by surgery along a simple closed curve and n 4, then c(N ) c(M ). A strict inequality holds in some cases.
strict inequality: If M is closed with c(M ) > 0 and n 4, there is a simple closed curve γ ⊂ M such that c(N ) < c(M ) if N is obtained by drilling or surgery along γ. This implies the following result. Very often in dimension 4 a complicate manifold becomes "simpler" after summing it with S 2 × S 2 . The complexity might estimate this phenomenon as follows.
stabilization: If M is a simply connected closed 4-manifold with c(M ) > 0 then c M #(S 2 × S 2 ) < c(M ). However, we do not know if there exists any simply connected 4-manifold of positive complexity! Finally, an important result of 3-dimensional complexity, due to Matveev [27], says that c(M S ) c(M ) whenever S is an incompressible surface. Unfortunately, the notion of incompressibility does not extend appropriately to higher dimensions. Having in mind that every class in H 2 (M 3 , Z) is represented by an incompressible surface, we extend a weaker version of Matveev's result as follows.
hypersurfaces: Every class in H n−1 (M, Z 2 ) is represented by a hypersurface S such that c(M S ) c(M ). This result is proved by extending the 3-dimensional notion of normal surface to any dimension: this extension might be of independent interest.
The results just stated are proved in Sections 8, 9, and 10. Note that if M is aspherical then π 1 (M ) is torsion-free and hence the inequalities hold for any aspherical manifold M . Actually, only the left inequality requires this hypothesis on π 1 (M ), and we do not know if it is really necessary. Both inequalities might be justified informally by saying that simples spines are more flexible than triangulations, but not as flexible as real homology cycles. The above result can be strengthened in complexity zero, by dropping the hypothesis on π 1 (M ) and admitting amenable boundary. The boundary ∂M is amenable if the image of every connected component of ∂M in π 1 (M ) is an amenable group.
The results just stated are proved in Sections 5 and 13.
Homotopy type. It might be that every simply connected manifold has complexity zero. This question is open only in dimension 4. simply connected: Every simply connected manifold of dimension = 4 has complexity zero.
On the other hand, we have the following. arbitrary fundamental group: Every finitely presented group is the fundamental group of a closed 4-manifold with complexity zero. Complexity detects aspherical manifolds, in some sense.
aspherical manifolds: If M is closed aspherical, then c(M ) > 0. This shows in particular that complexity behaves quite differently from Gromov norm. For instance, complexity detects non-positive curvature, while Gromov norm detects negative curvature: the n-torus T n has c(T ) > 0 and ||T || = 0.
We also note that complexity is not a homotopy invariant, since it distinguishes some homotopically equivalent lens spaces: we have c(L 7,1 ) = 4 and c(L 7,2 ) = 2, see [27]. We do not know if it distinguishes different PL manifolds sharing the same topological structure.
The results just stated are proved in Section 12.
Riemannian geometry. We compare the complexity of a smooth manifold M with other invariants coming from riemannian geometry. A relation between the volume of a riemannian manifold and its complexity can be given by bounding both the sectional curvature and the injectivity radius. The second inequality in the following result is due to Gromov [17]. volume: Let M n be a riemannian manifold with everywhere bounded sectional curvature |K(M )| 1. Then Here inj(M ) is the injectivity radius, inj * (M ) = min{inj(M ), 1}, and const n is a constant depending on n. The same formula holds for Gromov norm ||M ||: in that case however the factor inj * (M ) −n can be removed when π 1 (M ) is residually finite [17]. It is not possible to remove this factor here, since there are infinitely many hyperbolic 3-manifolds with bounded volume, while only finitely many can have bounded complexity. This holds in fact in all dimensions. finiteness: There are finitely many closed hyperbolic n-manifolds of bounded complexity, for every n. We do not know if the finiteness property can be extended to manifolds of nonnegative curvature, or more generally to aspherical manifolds. As far as we know, it might also hold for elliptic manifolds. The results on Gromov norm allow to prove also the following.
cusped hyperbolic manifolds: Let M be a compact manifold whose interior admits a complete hyperbolic metric of finite volume. Then c(M ) > 0. This result is sharp since the Gieseking 3-manifold has complexity 1, see [9]. Complexity is also related to other geometric invariants. A nice chain of inequalities, taken from [20,33], holds for every closed orientable manifold M : From left to right, we find Gromov norm ||M ||, the spherical volume T (M ) defined by Besson, Courtois, and Gallot in [7], the volume entropy λ(M ), the topological entropy h(M ), and the minimum volume MinVol(M ) defined by Gromov in [17].  Table 1. The number of irreducible orientable 3-manifolds of complexity c 11 in each geometry. The non-geometric manifolds decompose into geometric pieces according to their JSJ decomposition along tori (they are all graph manifolds when c 10).
Another interesting invariant is the systolic constant σ(M ), defined by Gromov in [18]. We have the following. thin manifolds: There are some constants a 2 < a 3 < . . . such that if a riemannian manifold M n with boundary has (curvature-normalized) inradius less than a n , then c(M n ) = 0. The results just stated are proved in Section 14.
Low dimensions. The complexity of manifolds of dimension 1 and 2 is easily calculated. Concerning 1-manifolds, we have c(S 1 ) = c(D 1 ) = 1. Turning to dimension 2, the complexity of a (compact) surface Σ turns out to be as follows: The compact surfaces having complexity zero are S 2 , RP 2 , the annulus, and the Möbius strip. The torus and the pair-of-pants have complexity 2. The complexity of 3-manifolds has been widely studied. Manifolds of low complexity have been listed via computer by various authors [8,9,13,23,24,28,29]: the closed orientable irreducible ones are collected in Table 1 according to their geometry. The closed irreducible manifolds having complexity zero are S 3 , RP 3 , and L 3,1 .
We now devote our attention to dimension 4. We start by studying the set of 4-manifolds of complexity zero. We describe here some interesting classes of such manifolds. These classes seem however far to exhaust the set of all 4-manifolds with complexity zero.
The various results stated above show that the set of all 4-manifolds of complexity zero contains all products N × N with non-empty boundary or N ∈ {S 2 , S 3 }, and is closed under connected sums, finite coverings, addition of handles of index = 3, and drilling (or surgery) along simple closed curves. All the examples presented here are of this kind. We concentrate on closed manifolds for simplicity. no 3-handles: Every closed 4-manifold that has a handle decomposition without 3-handles has complexity zero.
Every such manifold is necessarily simply connected. However, for many simply connected manifolds a decomposition without 3-handles does not seem to be known. Among these, we find the exotic K3 surfaces constructed by Fintushel and Stern in [12]. In fact, these manifolds are constructed by attaching handles of index = 3 to a product M 3 × S 1 . Therefore we have the following. exotic K3: The (infinitely many) exotic K3 surfaces X K constructed via Fintushel and Stern's knot construction [12] from a knot K ⊂ S 3 have complexity zero.
We now introduce two different classes of closed 4-manifolds with arbitrary (finitely presented) fundamental group. Let a 2-handlebody be a 4-manifold which has a decomposition with 0-, 1-, and 2-handles.
doubles of 2-handlebodies: The double of any orientable 2-handlebody has complexity zero.
These manifolds have complexity zero because they are obtained by surgerying (S 1 × S 3 )# . . . #(S 1 × S 3 ) along some curves. Every finitely presented group is the fundamental group of a 2-handlebody, which is in turn isomorphic to the fundamental group of its double. It is not true that any double has complexity zero, because a double can be aspherical (for instance, a product of surfaces). Another class was constructed by Gompf in [15], in order to show that symplectic 4-manifolds may have arbitrary fundamental group. As above, these manifolds are constructed by attaching handles of index = 3 to a product M 3 × S 1 , so we have the following.
symplectic manifolds: The closed symplectic manifolds with arbitrary fundamental group constructed by Gompf in [15] have complexity zero.
The results just stated are proved in Section 15.

Piecewise-linear topology
We collect here the informations on piecewise-linear topology that we will need. The basic definitions and tools are listed in Sections 2.1 and 2.2. More material can be found in [34]. The notion of intrinsic stratification is taken from [3,5,30,36] and described in Section 2.3. Stein factorization (which we take from [10]) is introduced in Section 2.4. Finally, in Section 2.5 we define the nerve of a pair (X, Y ) of polyhedra: this definition is original and might be of independent interest.
The material contained in Sections 2.4 and 2.5 is used in Section 11 to define the nerve of a pair (M, P ) when P is a simple spine of M .

Basic definitions.
2.1.1. Simplicial complexes. A (finite and abstract) simplicial complex K is a set of nonempty subsets of a given finite set V (K) (the vertices of K), such that {v} ∈ K for all v ∈ V (K) and if σ ∈ K and τ ⊂ σ then τ ∈ K. An element of K is a face. A subcomplex is a subset of K which is a complex. If K and L are simplicial complexes, a simplicial map f : 2.1.2. Triangulations. A (finite) simplicial complex K induces a compact topological space |K|, defined by taking a standard euclidean simplex for each element of K and identifying them according to the face relations. A triangulation of a compact topological space X is a simplicial complex K and a homeomorphism f : |K| → X. Another triangulation (L, g) of X is a subdivision of (K, f ) if the image of every simplex of L is contained as a straight simplex in some simplex of K. Two triangulations of X are related if they have a common subdivision.
We will use the letter T to indicate a triangulation, i.e. a pair (K, f ).

Polyhedra.
A compact polyhedron is a compact topological space X equipped with a maximal family of related triangulations. A subpolyhedron X ⊂ X is a subset which is the image of a subcomplex of some triangulation of X. If X is a polyhedron containing compact polyhedra X 1 , . . . , X k , a triangulation K of (X, X 1 , . . . , X k ) is a triangulation of X where each X i is represented by some subcomplex; such a triangulation can be found by taking a common subdivision of the triangulations realizing X i as a subcomplex. The standard n-simplex ∆ n is a polyhedron. We define the n-disc D n and (n − 1)-sphere S n−1 respectively as ∆ n and ∂∆ n .

Manifolds and maps.
A simplicial map f : K → L induces a continuous map f : |K| → |L|. A map between polyhedra is piecewise-linear (shortly, PL) if it is induced by a simplicial map on some triangulations. A polyhedron is a PL-manifold (with boundary) if it is locally PL-homeomorphic to some point in S n (D n ). Every manifold and map mentioned in this paper is tacitly assumed to be PL.

Basic tools.
2.2.1. Derived complexes. A simplicial complex K defines a partially ordered set (briefly, a poset) i(K) = (K, ⊆), the set of faces with their face relations. Conversely, a poset (A, ) defines a simplicial complex η(A, ), whose vertices are the elements of A, and whose faces are all finite subsets {a 0 , . . . , a i } such that a 0 < . . . < a i . The simplicial complex η(A, ) is the nerve of (A, ).
The simplicial complex K = η • i(K) is the derived simplicial complex of K. Vertices of K correspond to faces of K. A simplicial map f : K → L induces an order-preserving map i(K) → i(L) and hence a derived simplicial map f : K → L .
A triangulation T = (K, f ) of a space X determines a barycentric subdivision T = (K , f ) of X, obtained by composing f with the homeomorphism |K | → |K| which sends every vertex of K to the barycenter of the corresponding face of K (and is extended linearly on the rest of |K |) 2.2.2. Join, cone, and suspension. The join K * L of two simplicial complexes K and L (with disjoint vertices) is the complex with vertices V (K * L) = V (K) ∪ V (L) and with faces K ∪ L ∪ {σ ∪ τ |σ ∈ K, τ ∈ L}. The polyhedron |K * L| depends only on |K| and |L| (up to homeomorphism) and can thus be denoted by |K| * |L|.
The cone and suspension of a polyhedron P are respectively C(P ) = P * D 0 and Σ(P ) = P * S 0 . We have Σ k (P ) ≡ P * S k−1 .
2.2.3. Link, star, and regular neighborhood. Let K be a simplicial complex and L ⊂ K a subcomplex. The star st(L, K) of L in K is the minimal subcomplex of K containing all faces that intersect some face of L. The link lk(L, K) is the subcomplex of st(L, K) consisting of all faces not intersecting any face of L.
When Y ⊂ X are polyhedra and T is a triangulation of (X, Y ), we indicate by lk(Y, T ) and st(Y, T ) the corresponding subpolyhedra of X. When Y = {y} is a point, these polyhedra depend (up to homeomorphism) only on y and not on T .
In general, if T is sufficiently subdivided, the star st(Y, T ) does not depend on T up to an isotopy in X keeping Y fixed: for instance, this holds after two barycentric subdivisions. In that case, the polyhedron st(Y, T ) is the regular neighbhorhood of Y in X, which we denote by R(Y ).
When X is a manifold, the regular neighborhood R(Y ) is a manifold with boundary.

2.2.4.
Collapse. Let K be a simplicial complex. Let σ ∈ K be a face which is properly contained in a unique face η. The subcomplex L = K \ {σ, η} is obtained from L by an elementary collapse.
Let Y ⊂ X be any polyhedra. The polyhedron Y is obtained from X via a elementary collapse if it is so on some triangulation. More generally, a collapse of X onto a subpolyhedron Z is a combination of finitely many simplicial collapses.
Let Y ⊂ X be any polyhedra and x ∈ X a point. The intrinsic dimension d(x; X, Y ) of the pair (X, Y ) at x is the maximum number t such that (1) there is a triangulation of (X, Y ) with x contained in the interior of a t-simplex. If x ∈ Y , this is equivalent to each of the following conditions: The intrinsic dimension can be easily calculated using the following nice result of Armstrong and Morton [5,32]: This easily implies the following.
The Stein factorization of a map.
The intrinsic dimension induces an intrinsic stratification of any pair (X, Y ). The points of intrinisic dimension k in Y form the k-stratum of Y . The k-stratum is an (open) k-dimensional manifold made of finitely many connected components, called k-components (or simply components). Points in a k-component are all homogeneous, i.e. there is an ambient isotopy of Y sending a point to any other. In particular, they have the same link.
The union of all points of intrinsic dimension k is the k-skeleton: it is a k-dimensional polyhedron.

Stein factorization. A Stein factorization of a (piecewise-linear) map
such that h has connected fibers and g is finite-to-one. Every f has a unique Stein factorization: the map h is the quotient onto the space Z of connected components of the fibers of f , see Fig 1. We learned about this notion from [10].
We define the Stein factorization in the category of simplicial complexes. Let f : K → L be a simplicial map. Let f : K → L be its derived map. We define an intermediate simplicial complex H as follows. Consider the map f : |K | → |L |. The vertices of H are the connected components of (f ) −1 (v) when v varies among the vertices of L . The map f : V (K ) → V (L ) naturally splits along two maps We now define a simplex in H to be the image of any simplex in K along h. The resulting maps Since we used the derived map f , the map h : |K | → |H| has indeed connected fibers everywhere (not only at the vertices of H). The map g : |H| → |L | is finite-to-one: this is equivalent to the condition that dim g(σ) = dim σ for every simplex σ of H.

2.5.
Nerve. The nerve of a polyhedron is a simplicial complex which encodes the incidences between its components, see Section 2.3. We define it for pairs (X, Y ).
Let Y ⊂ X be any polyhedra. The components of (X, Y ) form a partially ordered set (C, ): we set C C if C ⊂ C . If C < C then dim C < dim C . The pre-nerve of (X, Y ) is the nerve N 0 = η(C, ) of this partially ordered set, see Section 2.2.1 above.
Let T be a triangulation of (X, Y ). If T is sufficiently subdivided, by sending every vertex of T to the component to which it belongs we get a surjective simplicial map ϕ 0 : T → N 0 , called the pre-nerve map. It induces a surjective continuous The pre-nerve map does not necessarily have connected fibers, so we prefer to consider its Stein factorization, see Section 2.4. The nerve of (X, Y ) is the complex N obtained via the Stein factorization The map ϕ : T → N is the nerve map. More generally, a nerve map is a map ϕ : X → |N | induced by some (sufficiently subdivided) triangulation of (X, Y ). Exercise 2.3. The pre-nerve of (X, Y ) = (S 1 , {pt}) is a segment, while the nerve is a circle.

Simple polyhedra
The definition of simple polyhedra in arbitrary dimensions is due to Matveev [25]. We use it in Section 3.3 to define the complexity of a manifold. This definition extends Matveev's complexity of 3-manifolds [27].
3.1. The local model. Let ∆ = ∆ n+1 be the (n + 1)-simplex. Let Π n be the cone over the (n − 1)-skeleton of ∆. The base of the cone is its boundary ∂Π n , while int(Π n ) = Π n \ ∂Π n is its interior. Some examples are shown in Fig. 2.
There are two representations of Π n inside ∆, shown in Fig 3: the standard and dual representation. They both describe Π n as a subcomplex of the barycentric subdivision ∆ . See also Fig. 4. Both representations induce the same pair (D n+1 , Π n ) up to homeomorphism. The dual description is investigated below in Section 5.1.
We define Π n k as Π n k = Π n−k ×D k . The pair (D n+1 , Π n k ) = D k ×(D n−k+1 , Π n−k ) is well-defined up to homeomorphism. The boundary ∂Π n k = Π n k ∩ S n is homeomorphic to the k-th suspension Σ k (∂Π n−k ). Following Matveev, a point x in a   polyhedron P is of type k if its link is homeomorphic to ∂Π n k (and hence its star is homeomorphic to Π n k ) 1 . See Fig. 5. The polyhedron Π n has a natural triangulation induced by that of ∆.
Proof. Since ∂Π n−k is not a suspension, a point of type k has intrinsic dimension k by Proposition 2.1.
The polyhedron Π n may be constructed recursively. In the following, we see both Π n−1 and S n−1 inside D n . See Fig. 6.
1 Actually, our Π n k corresponds to Matveev's Π n n−k : we prefer to define the type of a point coherently with Armstrong's general notion of intrinsic dimension. = + Figure 6. We have ∂Π n ∼ = Π n−1 ∪ S n−1 . Here n = 2.
n = 1: n = 2: Figure 8. Simple polyhedra of dimension n = 1 (a circle and a trivalent graph) and n = 2 (a sphere and a torus with two discs attached).
Proof. Take a vertex v and the opposite face f in ∆ n+1 . The (n − 1)-skeleton of ∆ n+1 is the union of ∂f and a cone over the (n − 2)-skeleton of f with base v.
The following is an easy corollary of Proposition 2.1.
Definition 3.5. A compact polyhedron P n is simple if every point of P is of some type k (that is, its link is homeomorphic to ∂Π n k ). See some examples in Fig. 8. A point of type 0 is called a vertex. In this paper, every simple polyhedron P ⊂ int(M n ) contained in some manifold M n will be tacitly assumed to have codimension 1 and to be locally unknotted, see Section 2.3. This is equivalent to require that P is properly embedded in Matveev's sense [25]: the equivalence is proved in Section 4.1. Local unknottedness is actually automatic in dimension n 4, see Remark 4.3 below.
Exercise 3.6. The polyhedron ∂Π n is simple with n + 2 vertices. The exercise is also proved as Corollary 5.7 below.
3.3. Complexity. A spine of a manifold is usually defined as a subpolyhedron onto which the manifold collapses. This definition however applies only to manifolds with boundary: in order to extend it to closed manifolds, we allow the removal of an arbitrary number of open balls.
See some examples in Fig. 9. We are now ready to define the complexity of a manifold. Every compact manifold admits a simple spine (see [25] or Corollary 5.5 below) and hence this quantity is indeed finite. A simple spine P ⊂ M is minimal if it has c(M ) vertices.

3.4.
Examples. The equator (n − 1)-sphere is a simple spine of S n : the n-sphere collapses to it after removing two small balls centered at the poles. Analogously, a hyperplane is a simple spine of RP n without vertices (RP n collapses to it after removing one ball). When n 2 these spines have no vertices and therefore c(S n ) = c(RP n ) = 0. When n = 1, the circle S 1 has a point as a simple spine, which is indeed a vertex, and hence c(S 1 ) = 1. Figure 9 shows a spine with 2 vertices of the 2-torus T : hence c(T ) 2. It is easy to see that T has no spine with lower number of vertices, and hence c(T ) = 2. A similar argument shows the following.
The surfaces having complexity zero are S 2 , RP 2 , the annulus, and the Möbius strip. They all have a circle as a spine without vertices.
Many examples in dimension 3 can be found in the literature [8,24,23,27,29], so we turn to higher-dimensional manifolds. A nice spine for CP n can be described by using a technique which was inspired to us by tropical geometry [31]. Consider the projection Consider Π n−1 dually embedded in ∆ n . The counterimage p −1 (Π n−1 ) is a simple spine of CP n without vertices (its complement consists of n+1 open balls "centered" at the points [0, . . . , 0, 1, 0, . . . , 0]). Therefore c(CP n ) = 0.
The spine of CP 2 fibers over Π 1 . It consists of three solid tori attached to one 2-torus. We find such a spine also from a different construction. Let M 4 be a closed 4-manifold which decomposes with 0-, 2-, and 4-handles only. The attaching of the 2-handles is encoded by a framed link L ⊂ S 3 . Let P be the union of the boundaries of all the handles involved. It consists of a 3-sphere S 3 plus one solid torus attached to (a regular neighborhood of) each component of L. When L is the 1-framed unknot we find M 4 = CP 2 and we get the same spine as above. In general, we get a simple spine of M without vertices (all points are of type 3 or 2). Therefore c(M 4 ) = 0.

Collars
As proved by Matveev [25], a locally unknotted simple polyhedron P ⊂ M has a kind of collar, similar to a collar of the boundary of a manifold. We introduce the collar by defining the cut map in Section 4.2. To do this, we first need to prove that Matveev's notion of local flatness (which is more useful in the context of simple spines) coincides with the general one introduced in Section 2.3.

4.1.
Matveev's definition. Matveev introduced in [25] a different definition of local unknottedness for simple polyhedra, which is more useful here. We show that it coincides with the general one introduced in Section 2.3. The proof is not strictly necessary (we could use Matveev's notion and forget about the general one), but we include it for completeness.
We defined the pair (D n , Π n−1 k ) in Section 3.1. The following definition is due to Matveev [25].  Proof. It is easy to see that a properly embedded P n−1 ⊂ M n is locally unknotted. We prove the converse by induction on n. The case n = 1 is trivial, so we assume n 2. Let x be a point of P , of some type k. The link of x in (M, P ) is homeomorphic to (S n−1 , Y n−2 ) with Y n−2 ∼ = ∂Π n−1 k . We must show that the homeomorphism extend to pairs, i.e. that (S n−1 , Y n−2 ) ∼ = (S n−1 , ∂Π n−1 k ). Since P is locally unknotted, the pair (S n−1 , Y n−2 ) is also locally unknotted by Exercise 2.2. The polyhedron Y is simple by Exercise 3.6, and is hence properly embedded by our induction hypothesis. Since Y ∼ = ∂Π n−1 k is a special polyhedron (i.e. a simple polyhedron whose (n − 2)-components are discs), [25,Theorem 3]   As noted by Matveev [25], the locally unknotted embedding of a simple polyhedron allows us to define a collar, similar to the collar of a boundary in a manifold.
Let P ⊂ int(M ) be a simple polyhedron in a compact manifold. By cutting M along P as suggested in Fig. 10, we get a manifold M P with boundary and a surjective map f : M P → M .
The set f −1 (P ) ⊂ M P consists of some components of ∂M P , which we denote by ∂ 0 M P . The map f is a local embedding. It is (n − k + 1)-to-1 over a point of type k in (M, P ). In particular, it restricts to a homeomorphism of M P \ ∂ 0 M P onto M \ P .
Regular neighborhoods R(P ) of P in M correspond via f to collars of ∂ 0 M P . The function f , restricted to one such collar, gives a collar ∂R(P ) × [0, 1] → R(P ) of P , as shown in [25]. This discussion implies in particular the following. In other words, a simple polyhedron P ⊂ int(M ) is a spine if and only if M P consists of a collar and some discs.

Triangulations
We describe here a construction which builds a simple polyhedron P ⊂ M from a triangulation of M and a partition of its vertices. From this we will deduce that c(M ) t(M ) for any compact M .   Figure 11. Models dual to some partitions of the vertices. Here we find Π 2 1 and Π 2 2 .
Definition 5.1. The polyhedron dual to (∆, P) is For the other cases, we have the following.
Proposition 5.2. We have Z ∼ = Π n n+1−k . Proof. If each f i consists of one vertex we get the dual description of Π n and we are done. Otherwise, let ∆ k be a k-dimensional simplex, with vertices w 0 , . . . , w k . By sending V i to w i we get a simplicial map φ : ∆ n+1 → ∆ k . This induces another simplicial map φ : (∆ n+1 ) → (∆ k ) between the derived complexes. Let Z * be the polyhedron dual to (∆ k , {w 0 , . . . , w k }). We have f −1 (Z * ) = Z and Z * ∼ = Π k .
On a small neighborhood of the center C of (∆ n+1 ) the map f is isomorphic to the projection D k × D n−k+1 → D k . Therefore the star of C in Z is homeomorphic to Π k−1 × D n−k+1 ∼ = Π n n−k+1 . Such a star is homeomorphic to Z and we are done. Fig. 11.

5.2.
Simple polyhedra dual to triangulations. Let M be a compact manifold and T a triangulation of M . Let P be a partition of the vertices of T . In each simplex σ of T we have an induced partition of its vertices and hence a dual polyhedron P σ ⊂ σ.
The dual polyhedron is a subcomplex of the barycentric subdivision T . We say that P respects ∂M if for every connected component X of ∂M all the vertices in X belong to the same set of P.
For every set V ∈ P of the partition, we define the submanifold M V ⊂ M as the regular neighborhood in T of the union of all simplexes in T whose vertices lie in V . See Fig. 12. We have the following. Proposition 5.4. Let P be a partition that respects ∂M and P be the polyhedron dual to (T, P). Then P is simple and the following holds.
(1) The vertices of P are the barycenters of the simplexes in T whose vertices lie in n + 1 distinct sets of P.
Proof. The proof is straightforward. Note that by our assumption on P every boundary component lies in some M V and hence P does not intersect ∂M .
Let t(M ) be the minimum number of simplexes arising in a triangulation of M . Remark 5.6. The term "triangulation" is sometimes used for short in dimensions 2 and 3 to indicate a singular triangulation, i.e. the realization of a manifold M n as the union of some n-simplexes whose faces are identified in pairs via some simplicial maps. This is not the case here: in this paper we employ the word "triangulation" only in its original PL meaning.
Corollary 5.7. The polyhedron ∂Π n k is simple. Proof. Represent Π n k as the dual of some partition (∆, P). The boundary is the simple polyhedron dual to the same partition (∂∆, P).
Proposition 5.2 shows that ∂Π n k is homeomorphic to some simple subpolyhedron of ∂Π n . Conversely, we have the following result (which will be used in Section 7).  Figure 13. The simple polyhedron P is obtained from Q by drilling along K.
Proposition 5.8. See ∂Π n dually contained in ∂∆. If n 2, every simple subpolyhedron of ∂Π n is dual to some partition P of the vertices of ∆ and is hence homeomorphic to ∂Π n k for some k. Proof. We prove this result by induction on n. The case n = 2 is easy, so we turn to the case n 3. Let X be a simple subpolyhedron of ∂Π n . Let f 0 , . . . , f n+1 be the facets of ∆. A simple subpolyhedron of a simple polyhedron of the same dimension n − 1 is necessarily the closure of the union of some (n − 1)-components. Therefore each f i ∩ X is a cone over ∂f i ∩ X, which is a simple subpolyhedron of ∂f i ∩ ∂Π n ∼ = ∂Π n−1 . By induction, ∂f i ∩ X is dual to some partition P i of the vertices of f i . Now define a partition P of the vertices of ∆ as follows: two vertices belong to the same set if they do in some P i . Note that if they do belong to the same set in some P i then they do so in any P j , and that the transitivity property of this equivalence relation holds because n 3. The polyhedron X is then dual to (∆, P).

Drilling
Generic soap bubbles in R 3 form a simple polyhedron. Moreover, if a new bubble appears generically somewhere, the polyhedron remains simple. This fact can be generalized to any dimension, as follows.
Let Q ⊂ M be a simple polyhedron in a manifold. Let K ⊂ int(M ) be any compact subpolyhedron. The operation of drilling Q along K consists of removing from Q a small regular neighborhood of K and adding its boundary as in Figg. 13 and 14. More precisely, let T be a triangulation of (M, K, Q). Let R = R(K, T ) be the regular neighborhood of K in the twice subdivided T . The result of this operation is the polyhedron Lemma 6.1. The polyhedron P is simple. If K does not intersect the 1-skeleton of Q, then P has the same vertices as Q.
Proof. We have P = (Q \ R) ∪ ∂R. We have to check that every point x ∈ P is of some type k. If x ∈ Q \ ∂R we are done because Q is simple. Suppose x ∈ ∂R. Let k 1 be the type of x in (M, Q). We show that x is of type k − 1 in (M, P ): in particular, P is simple.
If k = n, i.e. x ∈ Q, then x is of type (n − 1) in P because ∂R is a (n − 1)manifold. If k < n, the polyhedra ∂R and Q intersect transversely at x (in the sense of Armstrong, see [5]). See an example in Fig. 15: locally, ∂R is a horizontal K Q P Figure 14. The simple polyhedron P is obtained from Q by drilling along K.
Q R P x 6 Figure 15. At x the polyhedra Q and ∂R intersect transversely, so ∂R cuts Q into two halves. Then P is obtained by discarding the half of Q lying inside R and adding ∂R.
disc D n−1 × 0 and Q is a vertical product Y × [−1, 1]. Exercise 3.4 implies that Y ∼ = Π n−1 k−1 . The star of x in P is thus homeomorphic to

By Corollary 3.3, this is homeomoprhic to
Finally note that if K does not intersect the 1-skeleton of Q neither does R. Therefore every point in Q ∩ ∂R is of type k > 1 in Q, and hence of type k − 1 > 0 in P : no new vertices are added to Q.

Alternative definitions
Matveev's original definition of complexity c(M 3 ) for a 3-manifold M 3 was slightly different from ours. We prove here that the two definitions coincide.
A couple of natural variations might be done in our definition of complexity. The definition of "simple polyhedron" can be weakened by allowing the presence Figure 16. An almost simple spine of the boundary-connected sum (T 2 × D 1 )#(T 2 × D 1 ). It consists of two tori joined by an arc. It has no vertices.
of low-dimensional material. This choice is natural, since it allows to consider a point as a spine of D n or S n . Matveev called such polyhedra almost simple. On the other hand, the definition of "spine" can be strengthened, by allowing the removal of one ball only when strictly necessary, i.e. when the manifold is closed. We call this more restricted notion a strict spine.
We therefore get 2 × 2 = 4 possible definitions of the complexity of a manifold. Luckily, it turns out that three of them coincide. These include ours and Matveev's definition (in dimension 3). 7.1. Almost simple polyhedra. Matveev employed in dimension 3 a more relaxed notion of polyhedron, called almost simple [27]. We propose the following generalization to all dimensions. See an example in Fig 16. A vertex of P is a point whose link is homeomorphic to (S n−1 , ∂Π n−1 ). We now define c alm (M ) as the minimum number of vertices of an almost simple spine of M . Example 7.2. A point {pt} ⊂ S n is an almost simple spine of the n-sphere. Note that a point is not a vertex when n 2, and hence c alm (S n ) = 0 for all n 2. A hyperplane H ⊂ CP n is an almost simple spine of complex projective space without vertices, and hence c alm (CP n ) = 0. This spine is not simple (it has codimension 2): note that the construction of a simple spine of CP n without vertices is less immediate, see Section 3.4.
We show below that c = c alm : to prove this, we need a couple of preliminary lemmas, which show how to construct a simple spine from an almost simple one without increasing the number of vertices. This is done first by collapsing (Lemma 7.3) and then by drilling along a triangulation of the low-dimensional part K of the spine (Lemma 7.4). Lemma 7.3. Every almost simple polyhedron P ⊂ int(M n ) collapses onto Q ∪ K where Q is simple and dim K < n − 1. Every vertex of Q is also a vertex of P .
Proof. We prove this by induction on n = dim M . If n = 2 it is easy, so we turn to the case n 3. Take a triangulation of (M, P ) and collapse P as more as possible. The resulting polyhedron is some Q ∪ K ⊂ P , where Q (resp. K) is the closure of the set of all points whose link has dimension n − 2 (resp. < n − 2). Figure 17. Let Q ∪ K be a spine of M , such that Q is simple and dim K < dim Q. We obtain a simple spine of M by subdividing K into cells and then drilling them inductively. Each drilling produces a new ball in the complement. Here, K is an arc made of two 1-cells.
We now prove that Q is simple. Since P is almost simple, the link of x in (M, Q, K) is homeomorphic to (S n−1 , Q , K ) for some Q ∪ K ⊂ ∂Π n ⊂ S n−1 . The polyhedron Q ∪ K cannot be collapsed onto a proper subpolyhedron, because (Q, K) cannot. By our induction hypothesis Q is hence simple. Proposition 5.8 implies that Q ∼ = ∂Π n−2 k and hence x is of type k in Q.
Lemma 7.4. If M n has a spine Q ∪ K ⊂ M n such that Q is simple with t vertices and K has dimension < n − 1, then c(M n ) t.
Proof. Take a triangulation T of (M, Q, K). Let σ 1 , . . . , σ k be the simplexes of K that are not contained in Q, and not contained in any higher-dimensional simplex of K. We have Q ∪ K = Q ∪ σ 1 ∪ . . . ∪ σ k .
Each σ i is a cell (i.e. a disc) of dimension < n − 1. We want to drill inductively along each σ i . Since each σ i is a cell, each drilling produces a new open ball in the complement, so the final simple polyhedron is a spine. In order to not create new vertices, at each step we put the cells in general position. See Fig. 17.
More precisely, we construct for each i = 0, . . . , k a simple polyhedron Q i with t vertices and some cells σ i 1 , . . . , σ i k−i of dimension < n − 1 intersecting themselves and Q i only at their boundaries, such that Q i ∪ σ i 1 ∪ . . . ∪ σ i k−i is a spine of M n . For i = 0, take Q 0 = Q and σ 0 j = σ j . Let now Q i and σ i 1 , . . . , σ i k−i be defined for some i < k. Since dim σ i j < n − 1, we can perturb the cells so that they do not intersect the 1-skeleton of Q. To do this, we use a collar of P (see Section 4.2): we lift the cells to M Qi , perturb them slightly, and project them back. The perturbed polyhedron is still a spine.
Let Q i+1 be obtained from Q i by drilling along σ i k−i . The polyhedron Q i+1 is simple with t vertices by Lemma 6.1. To drill we use a triangulation T of the whole data (M, Q i , σ i 1 , . . . , σ i k−i ): this ensures that σ i+1 j = σ i j \ Q i+1 is still a cell. Finally, Q k is a simple spine of M with t vertices.   . We now show the converse. Let P be a minimal almost simple spine, i.e. with c alm (M ) vertices. We now construct a strict one without increasing the number of vertices.
If P is not strict, the complement M \ P has some redundant balls: for each such B, there must be a (n − 1)-component of P which is adjacent on B to one side and to another component of M \ P on the other (because M is connected). By removing a small open (n − 1)-ball from this (n − 1)-component we get an almost simple spine with one ball less in its complement. After finitely many such removals we get an almost simple strict spine Q ⊂ P with the same vertices as P . Therefore c alm str (M ) c alm (M ). Finally, note that we have c str (M ) > c(M ) in some cases. That is, a manifold M may not have a simple strict spine with c(M ) vertices: for instance, S 2 and D 2 do not have a simple strict spine at all, and hence c str (S 2 ) = +∞; we also have c str (S 3 ) = 1 (the abalone is a strict spine with one vertex) and c str (S n ) = 0 for every n 4 (a generalized Bing's house without vertices in dimension n 4 is constructed in [25]). A reasonable complexity should be zero on spheres, so we do not investigate c str here.

Drilling spheres, handles, surgery, and connected sums
We can finally employ the techniques introduced in the previous sections to study how complexity changes under the most common topological operations. 8.1. Drilling. We consider first the effect of drilling along curves. Let M be a compact manifold and γ ⊂ M a properly embedded 1-manifold. The drilled manifold M γ is M γ = M \ R(γ) for some regular neighborhood R(γ). We have the following.
Theorem 8.1. Let M n be a compact manifold of dimension n 4, and γ ⊂ M n a properly embedded 1-manifold. We have c(M n γ ) c(M n ). Proof. Let P be a minimal spine of M . If γ lies in int(M ), we can isotope it inside P and disjoint from the 1-skeleton (because dim P 3). By drilling P along γ we thus get a spine Q for M γ . Lemma 6.1 implies that Q has no more vertices than P , and hence c(M γ ) c(M ).
If γ intersects ∂M , we can isotope it so that γ = γ ∪ λ where γ lies in P as before and λ consists of arcs connecting P to ∂M , and each arc is a fiber of a collar of P , see Fig. 18.
A spine P of M \ R(γ ) is constructed from P by drilling along γ . The polyhedron P intersects λ transversely in some points. An almost simple spine of M γ is � �' Figure 18. Put γ as γ ∪ λ with γ ⊂ P and λ vertical. Then drill along γ and make a hole around λ. γ Figure 19. By making a hole on a (n − 1)-component C of P we get a spine of M γ with γ intersecting P transversely in one point. After collapsing, we kill all the vertices adjacent to C. There is a simple closed curve γ ⊂ M n such that c(M n γ ) < c(M n ). Proof. Let P be a minimal spine of M . Since c(M ) > 0, it has at least one vertex v. Let C be a (n − 1)-component of P incident to v. By removing a small open (n − 1)-ball from C and then collapsing as more as possible we get an almost simple polyhedron Q ⊂ P ⊂ M with strictly less vertices than P , since v has been "killed" during the collapse.
The component C is adjacent to one or two distinct components of M \ P . Each such component is an n-ball. If it is adjacent to two distinct balls, these glue to form a single ball in M \ Q, and hence Q is a spine of M : a contradiction, since it has less vertices than P . Therefore C is adjacent to a single ball, and Q is a spine of M γ where γ ⊂ M is a closed curve intersecting P transversely in one point of σ. See an example in Fig. 19.
In other words, every closed manifold of positive complexity is obtained by filling a manifold of strictly smaller complexity. Remark 8.3. In contrast with the previous result, Theorem 8.2 also holds in dimension 3. For instance, a lens space (whic may have arbitrarily high complexity) is obtained by filling a solid torus, which has complexity zero. The Matveev-Fomenko-Weeks smallest closed hyperbolic 3-manifold has complexity 9 [27,24] and can be obtained by filling the figure-eight knot sibling, which has complexity 2 [9]. Note that the hyperbolic volume (and hence Gromov norm) satisfies the opposite inequality Vol(M 3 γ ) > Vol(M 3 ) for any γ. 8.2. Handles. We have the following. We are left with the case i < n − 1. Let P be a minimal spine of M , that is a spine with c(M ) vertices. By using a collar for P (see Section 4.2), we attach the core disc D i of the handle directly to P and get a spine P ∪ D i of N . Then c(N ) c(M ) by Lemma 7.4.

8.3.
Drilling along spheres. In Section 8.1 we showed the effect of drilling along a curve. Drilling along higher-dimensional spheres gives the opposite inequality. As above we set M S = M \ R(S). In the following, the sphere S ⊂ M is not necessarily locally flat. Proof. The manifold M is obtained from M S by adding a (n − k)-handle and a n-handle: the result then follows from Theorem 8.4. To prove the first assertion, represent R(S) as a block bundle [35] over S (block bundles play the rôle of normal bundles in the PL category). A n-block over a k-simplex σ of S is a (n − k)-handle H. The complement R(S) \ H collapses onto the disc S \ σ, and hence to a point: it is therefore a disc, i.e. a n-handle.

Surgery.
Let M n be a manifold of dimension n 2. A surgery along a simple closed curve γ ⊂ M n whose regular neighborhood is homeomorphic to D n−1 × S 1 consists of substituting this regular neighborhood with S n−2 × D 2 via some gluing map on the boundaries, both homeomorphic to S n−2 × S 1 . When M is simply connected, a surgery is just a connected sum with S 2 × S n−2 . In dimension n = 4 there is no simply connected manifold M of positive complexity, see Theorem 12.1. We do not know if there is one such manifold in dimension 4. If so, the following holds. Proof. Making a connected sum corresponds to removing two n-handles from the (disconnected) manifold M 1 M 2 , adding one 1-handle and one n-handle. Noone of these operations can increase the complexity when n 3. Similarly, a ∂-connected sum is the addition of one 1-handle.
Complexity is actually additive on connected sums in dimension n = 3, see [27]. Actually, we do not know any example of non-additivity in higher dimension. If there were a closed simply connected 4-manifold with c(M ) > 0, then Corollary 8.8 would yield a non-additive connected sum.

Coverings and products
We study how complexity changes under finite coverings and products. 9.1. Coverings. We have the following. In contrast with Gromov norm, we often get a strict inequality c( M ) < dc(M ). For instace, lens spaces may have arbitrarily high complexity while the complexity of their universal covering S 3 is zero. The following consequence is worth mentioning.

Normal hypersurfaces
When n = 3, Matveev proved [27] that complexity is non-increasing when cutting a 3-manifold along an incompressible surface. This was done by putting the surface in normal position with respect to the handle decomposition induced by a minimal spine, and by showing that a simple spine can be "cut" along a normal surface.
To extend this result, we define here normal discs in simplexes ∆ of any dimension. These can be used to define normal hypersurfaces with respect to triangulations or simple spines in any dimension: we do this in the simple case where every simplex contains at most one normal disc. In the dual setting of simple spines, this means that the normal surface is actually a subpolyhedron of the spine.
10.1. Normal discs. We extend the usual definition of normal discs in a tetrahedron, used in 3-dimensional topology, to all dimensions. Let ∆ = ∆ n+1 be the (n + 1)-simplex.
Definition 10.1. A normal disc in ∆ is a subpolyhedron dual to some partition P = {V 1 , V 2 } of the vertices of ∆ into two non-empty subsets, see Definition 5.1. By Proposition 5.2 a normal disc is homeomorphic to Π n n ∼ = D n and is hence indeed a disc. It is a subcomplex of the subdivided ∆ . The type of a normal disc is the unordered pair (#V 1 , #V 2 ).

10.2.
Homology. The homology group H n (P n ; Z 2Z ) of a simple polyhedron P n can be naturally interpreted as the set of all closed n-submanifolds of P . We prove this fact first locally and then globally. Represent Π n ⊂ ∆ dually to ∆ = ∆ n+1 . Every normal disc D is contained in Π n . Its boundary ∂D = D ∩ ∂Π n is a (n − 1)-sphere. Consider the maps Proof. Since dim ∂Π n = n − 1, a cycle α ∈ H n−1 (∂Π n ; Z/ 2Z ) is represented by a unique subpolyhedron Y ⊂ ∂Π n ⊂ S n . We must prove that Y = ∂D for a normal disc D, induced by some partition of V into two sets. The polyhedron Y is the closure of the union of some (n−1)-components of ∂Π n . Since H n−1 (S n ; Z/ 2Z ) ∼ = H 1 (S n ; Z 2Z ) = {0} for n 2, the connected components of S n \ Y may in fact be partitioned into two sets in a unique way, such that every (n − 1)-component of Y is adjacent to components belonging to distinct sets. The vertices of ∆ are thus partitioned in two sets, and Y is the normal disc dual to the partition, as required.
Corollary 10.5. Let P n be a simple polyhedron. Closed n-submanifolds of P are in natural bijection with H n (P n , Z/ 2Z ).
Proof. Take a triangulation of P . Every cycle is represented by a subcomplex S ⊂ P . It intersects the link of every point x in P into a cycle: by Proposition 10.4 this is a sphere and the star of S in x is a normal disc. Therefore S is submanifold. Different submanifolds yield different subcomplexes and hence different cycles.
10.3. Cutting along normal surfaces. Let S ⊂ M be a closed submanifold in M of codimension 1. As above, we set M S = M \ R(S). A key property of 3-dimensional complexity, proved by Matveev in [27], is that c(M S ) c(M ) whenever S is an incompressible surface in a 3-manifold M . We prove here a kind of generalization of this fact.
The powerful notion of incompressible surface unfortunately does not extend easily to higher dimensions. On the other hand, in dimension 3, every homology class in H 2 (M, Z) is represented by an incompressible surface. We propose the following generalization of Matveev's result. Let P be obtained from Q by drilling along S, see Section 6. The polyhedron P is a simple spine of M S : however, S intersects the 1-skeleton of Q, so we cannot use Lemma 6.1 to conclude that c(M S ) c(M ).
The manifold S is the union of the colsure of some (n − 1)-components of Q. Let R be a small regular neighborhood of S. The proof of Lemma 6.1 shows that the Figure 21. When the normal disc D is of type (n, 1), there is an adjacent 1-component not lying in D. Therefore the drilling deletes a vertex and produces a new one (left). If D is of some other type, it contains all the adjacent 1-components: therefore a vertex is deleted and noone is created (right). Here, n = 2.
new vertices of P lie in the transverse intersection of the hypersurface ∂R with the 1-skeleton of Q. There is precisely one such intersection for every pair (v, e) such that v is a vertex of Q contained in S and e is an oriented edge (i.e. 1-component) exiting from v and not contained in S. Here ∂R intersects e transversely near v.
Let (v, e) be one such pair. The star of v in (M, Q) is homeomorphic to (∆ n , Π n−1 ) and it intersects S into a normal disc D ⊂ Π n−1 , determined by some partition P = {V 1 , V 2 } of the vertices of ∆ n . The edge e is dual to a facet of ∆ n . Since e ⊂ D, all the n vertices lying in this facet belong to the same set of the partition. Therefore the normal disc is of type (n, 1) as in Fig 21-left. As we see in Fig. 21-left, one new vertex is created but v is destroyed: the total number of vertices in the spine does not increase.

Nerve
We study here the nerve N of a pair (M, P ) consisting of a closed manifold M and a simple spine P ⊂ int(M ), see Section 2.5. We prove that the nerve map ϕ : M → |N | induces an isomorphism on fundamental groups when π 1 (M ) has no torsion. Therefore N carries many topological informations on M .
We have dim |N | dim M , and dim |N | < n precisely if P has no vertices. Actually, the n-dimensional part of |N | has a kind of singular triangulation (see Remark 5.6), with one singular n-simplex "dual" to each vertex of P .
The facts listed in Subsection 11.1 will be used to prove most of the results stated in Sections 12, 13, and 14. The informations collected in Sections 11.2 and 11.3 are only needed to prove Theorem 13.2. 11.1. Basic properties. We will need the following lemma.  Figure 22. The pre-nerve of a spine P of the torus T with two vertices is homeomorphic to the cone over P , and is hence simply connected: the vertex of the cone is the 2-component f = T \ P , which is incident to all the other components.
have the following commutative diagram (with appropriate basepoints): By Proposition 4.4, every component of M P is either a disc or a product which maps to a collar of a component N of ∂M . Therefore (f * • i * )(π 1 ( C)) is either trivial or contained in i * (π 1 (N )). Since f : C → C is a covering, the subgroup f * (π 1 ( C)) has finite index in π 1 (C): therefore i * (π 1 (C)) is a finite extension of (i * • f * )(π 1 (C)) = (f * • i * )(π 1 ( C)).
Following Gromov [17], a set X ⊂ M n in a manifold M n is amenable if for every path-connected component X of X the image of the map i * : π 1 (X ) → π 1 (M ) is an amenable group (see [17] for a definition). Simple examples of amenable groups are finite and abelian groups. Subgroups and finite extensions of amenable groups are amenable. We have the following.
Proposition 11.2. Let P be a spine of a compact M . Let ϕ : M → |N | be a nerve map of (M, P ).
(1) If M is closed or with amenable boundary, the fiber ϕ −1 (x) of every point x ∈ |N | is amenable. (2) If M is closed and π 1 (M ) is torsion-free, the map ϕ * : is an isomorphism.
Proof. The fiber ϕ −1 (x) of a point is contained in some component of (M, P ), since each fiber of the pre-nerve map does, see Section 2.5. Since subgroups and finite extensions of amenable groups are amenable, point (1) follows from Lemma 11.1. We turn to (2). The map ϕ : M → |N | is surjective and the fiber ϕ −1 (x) over a point of |N | is path-connected: these two facts imply that ϕ * is surjective. On the other hand, Lemma 11.1 implies that i * (ϕ −1 (x)) < π 1 (M ) is a finite group. If π 1 (M ) has no torsion, this finite group is trivial: this easily implies that ϕ * is injective.
Remark 11.3. We note that Proposition 11.2-(2) does not hold in general for the pre-nerve map ϕ 0 : M → |N 0 |: we really need Stein factorization here. See Exercise 2.3 and Fig. 22.
Remark 11.4. Some hypothesis on π 1 (M ) is indeed necessary in Proposition 11.2- (2). For instance, a hyperplane H ⊂ RP n is a spine of RP n , but the nerve of (RP n , H) is a segment: hence ϕ * : Z/ 2Z → {e} is not an isomorphism.
Remark 11.5. Let P be a spine of a closed manifold M dual to some triangulation T as in Section 5.2 (with the discrete partition v ∼ v ⇔ v = v ) . The pre-nerve map ϕ : T → N 0 is an isomorphism. Therefore N ∼ = N 0 ∼ = T and the nerve map ϕ : M → |N | is a homeomorphism.
Concerning the dimension of the nerve, we have the following. Proof. Set n = dim M . We have dim N = dim N 0 . A k-simplex in N 0 is determined by some components C 0 < . . . < C k . In particular, we have 0 dim C 0 < . . . < dim C k n. This implies that dim N 0 n. If P has no vertices, then dim C 0 1 and hence dim N 0 < n. If P has a vertex v, there is a chain {v} = C 0 < . . . < C n and hence dim N = n.
11.2. Fundamental class. Let P ⊂ int(M ) be any simple polyhedron in a compact manifold (possibly with boundary). Let ϕ : M → |N | be a nerve map of (M, P ). We show that N looks like a pseudomanifold. In particular, we can define the notion of fundamental class. Let us start with the following.
Proposition 11.7. Every (n − 1)-simplex in N is adjacent to either zero or two n-simplexes.
Proof. Let σ be a (n−1)-simplex in N . The counterimage ϕ −1 (σ * ) of its barycenter σ * is either a point or a connected 1-manifold in int(M ). If it is a point or a segment then σ is adjacent to two n-simplexes. If it is a circle then σ is not adjacent to any n-simplex.
The nerve N looks like a pseudomanifold. Note however that the adjacencies of the n-simplexes along the (n − 1)-simplexes do not necessarily form a connected graph: for instance, |N | might consist of a bouquet of two n-manifolds.
A fundamental class for N is an element of H n (|N |, Z) which may be written (in simplicial homology) as a sum of all n-simplexes of N , each with an appropriate sign ±1. We indicate such a class as [N ]. A fundamental class induces an orientation on each n-simplex of N . Proof. Let T be the sufficiently subdivided triangulation of (M, P ) defining the nerve map ϕ : T → N . Since ϕ is surjetive with connected fibers, every n-simplex σ in N is the image of precisely one n-simplex η of T . The class [M, ∂M ] is fundamental class, i.e. it is represented by a sum of all the simplexes of M with appropriate signs, and hence (using simplicial homology) also ϕ * ([M, ∂M ]) is a sum of all the simplexes of N with appropriate signs. 11.3. Singular simplexes. Let P ⊂ int(M ) be any simple polyhedron in a manifold and ϕ : M → |N | a nerve map. We have just seen that N looks like a pseudomanifold. We now prove that the n-dimensional part of N is the twice barycentric subdivision of a singular triangulation (see Remark 5.6), with one singular simplex corresponding "dually" to each vertex of P .
Let We want a singular simplex in |N |. We therefore try to lift β v to N along the projection g : N → N 0 . The following result says that there is a natural way do to this.
Proposition 11.9. There is a natural simplicial map α v : ∆ → N such that Proof. We can formalize this as follows. The restriction of ϕ 0 to the (closed) star of v in T splits naturally into two simplicial maps The map ϕ v sends every vertex to the local component to which it belongs. The map ϕ v has connected fibers. Therefore there is a unique simplicial map α v : ∆ → N which makes the following diagram commute.
The map α v is actually well-defined only up to a permutation of the vertices of ∆, induced from the chosen identification of the open star of v with int(∆, Π n−1 ).
The following propositions show that the singular simplexes α v at the varying of v among the vertices of P form a singular triangulation of the n-dimensional part of |N | whose double barycentric subdivision yields the original triangulation of N .  Proof. We have a commutative diagram of topological maps ) lies in the interior of the star of v in T . Therefore ϕ −1 (α v (x)) is also contained in st(v, T ), and is connected since ϕ has connected fibers. Therefore ) is connected. The map α v is finite-toone by construction, thus this set is also finite. Therefore it consists of only one point {x}.
Proposition 11.12. Every n-simplex of N is contained in α v (∆) for some v.
Proof. Let σ be a n-simplex of N . Since the nerve map ϕ : T → N has connected fibers, the preimage of σ consists of a single n-simplex ϕ −1 (σ). We show that ϕ −1 (σ) lies in the star st(v, T ) of some vertex v: this implies that σ is contained in α v (∆).
Letσ be the n-simplex of T which contains ϕ −1 (σ). Consider the maps The image ϕ 0 (σ) = g(ϕ(σ)) is a n-simplex because g preserves the dimension of simplexes. That is, the vertices ofσ lie in distinct components C 0 < . . . < C n of (M, P ). This implies that dim C i = i, and thus C 0 = {v} is a vertex. Thereforeσ lies in st(v, T ). The edge e induces a simplicial isomorphism ψ : f 1 → f 2 defined by taking track along e of the incident components. We have the following.
Suppose M is oriented. By Proposition 11.8 the orientation on M induces a fundamental class [N ] in N and hence an orientation on its n-simplexes. A singular simplex α v can thus be orientation-preserving or reversing, and we set σ v = 0 or 1 correspondingly.
We would like to say that (−1) σv α v is a cycle representing [N ]. However, this argument is wrong: a singular triangulation, without any additional data, does not yield a cycle, because boundary maps do not necessarily cancel algebraically (for that reason, the slightly richer notion of semisimplicial complex [11] or ∆-complex [19,Page 103] is usually employed in the literature). We solve this problem by averaging each α v over its (n + 1)! different representations. That is, we definẽ Proof. First, note that vα v is indeed a cycle: the terms in the boundary cancel in pairs thanks to Proposition 11.13. The twice subdivision of vα v yields the fundamental class [N ], each simplex being counted (n + 1)! times with coefficient 1/(n + 1)!.

Homotopy type
We study here the relations between the complexity and some homotopy invariants of a manifold. First, note that the complexity is not a homotopy invariant, since it distinguishes homotopically equivalent lens spaces: we have c(L 7,1 ) = 4 and c(L 7,2 ) = 2, see [27].  The smallest known fundamental group of a manifold with positive complexity is Z/ 4Z : the lens space L 4,1 has c(L 4,1 ) = 1. We do not know if there are manifolds of positive complexity with fundamental group Z/ 2Z or Z/ 3Z .
On the other hand, every finitely presented group is the fundamental group of a closed 4-manifold of complexity zero: see Corollary 15.2.
12.2. Essential manifolds. Following Gromov [18] a non-simply connected closed manifold M is essential if one of the following equivalent conditions holds: (1) the image f * ([M ]) ∈ H n (K(π, 1)) of its fundamental class in the corresponding Eilenberg-MacLane space is non-trivial; (2) there is no map f : M → X onto a lower-dimensional polyhedron X which induces an isomorphism on fundamental groups. The equivalence between this definitions was proved by Babenko [6]. A group is virtually torsion-free if it contains a finite-index torsion-free subgroup. Proof. There is a finite covering M → M such that π 1 ( M ) is infinite and torsionfree. We have c( M ) = 0 by Corollary 9.2. Let P be a spine of M without vertices. By Proposition 11.6, the nerve |N | of ( M , P ) has dimension smaller than dim M . On the other hand, the nerve map ϕ : M → |N | yields an isomorphism on fundamental groups by Proposition 11.2. Therefore M is not essential. This implies that M is not essential.
A manifold is aspherical if π i (M ) is trivial for all i > 1. Equivalently, the universal cover M is contractible. Closed aspherical manifolds and real projective spaces are essential. The following fact is well-known. Proposition 12.3. An aspherical manifold has torsion-free fundamental group.
Proof. Suppose M is aspherical and π 1 (M ) contains a non-trivial finite cyclic group C p . Let M → M be the covering determined by C p . The manifold M is also aspherical. Therefore H i ( M , Z) ∼ = H i (C p , Z). However, the group H i (C p , Z) is nontrivial for infinitely many values of i: a contradiction because M is a manifold. In particular, the n-torus S 1 × · · · × S 1 has positive complexity. Products of S 1 and/or closed surfaces of positive genus also have positive complexity.
Remark 12.5. Some hypothesis on π 1 (M ) is necessary in Theorem 12.2, since RP n is essential and c(RP n ) = 0 for n 2. Proof. Since c(M ) = 0, M has a spine P without vertices. Let N be the nerve of (M, P ). We have dim N < dim M by Proposition 11.6. Proposition 11.2 implies that M fibers on a low-dimensional polyhedron with amenable fibers. Therefore M can be covered by at most n amenable (not necessarily connected) open sets. If M is closed, Gromov's Vanishing Theorem [17] implies that ||M || = 0. If M has boundary, every boundary component lies in one fiber and the fibration extends naturally to the double DM . Therefore ||DM || = 0. Since ∂M is amenable, we have ||DM || = 2||M || by [21], so ||M || = 0.
The hypothesis on ∂M is indeed necessary: for instance, a 3-dimensional handlebody of genus 2 has complexity zero and positive norm (because its boundary has positive Gromov norm). Now π 1 (M ) is torsion-free. Let P be a minimal spine of M , i.e. a spine with c(M ) vertices. Let N be the nerve of (M, P ). A nerve map ϕ : M → |N | induces an isomorphism ϕ * : π 1 (M ) → π 1 (|N |) by Proposition 11.2. Gromov's Mapping Theorem [17] says that an isomorphism on fundamental groups yields an isomorphism and isometry on bounded cohomology, and hence ||ϕ * (α)|| = ||α|| for every cycle α ∈ H * (M ). In particular, Proposition 11.14 shows a cycle that represents ϕ * ([M ]) whose norm equals the number c(M ) of vertices of P . Therefore ||M || c(M ). Theorem 13.2 is actually the only tool we have to prove the following.  Proof. Each boundary component N n−1 ⊂ ∂M n corresponds to a cusp of M and is a flat (n − 1)-manifold. By Bieberbach Theorem π 1 (N ) has a finite-index subgroup isomorphic to Z n−1 and is hence amenable. We have ||M || > 0 because M admits a complete hyperbolic metric [17] and therefore c(M ) > 0 by Theorem 13.1.
This result is sharp since the Gieseking 3-manifold (non-orientable with a Klein bottle cusp) has complexity 1, see [9].
Remark 14.2. We do not know if Corollary holds for hyperbolic manifolds with geodesic boundary; it does in dimensions 2 and 3 (because every compact irreducible, ∂-irreducible, and anannular 3-manifold has positive complexity [27]).
Theorem 13.2 in turn implies the following finiteness result. Proof. In dimension n = 3, there are finitely many irreducible manifolds of any given complexity [27], so we are done. In dimension n = 3 there are finitely many hyperbolic manifolds (up to homeomorphism) of bounded volume. The volume of a hyperbolic manifold is proportional to its Gromov norm. Since π 1 (M ) is torsionfree, Theorem 13.2 gives ||M || c(M ) and we are done.
Note that the same assertion for Gromov norm (or equivalently hyperbolic volume) is not true for n = 3.
Remark 14.4. We do not know if Corollary 14.3 holds for hyperbolic manifolds with cusps and/or geodesic boundary: it does in dimensions 2 and 3 (because there are finitely many compact irreducible, ∂-irreducible, and anannular 3-manifolds having a fixed complexity [27]).
14.2. Manifolds of non-positive curvature. A closed riemannian manifold with non-positive sectional curvature is aspherical because of Cartan-Hadamard's Theorem. Corollary 12.4 then implies the following. Remark 14.6. In contrast with the hyperbolic case, Corollary 14.5 does not hold for every compact manifold M whose interior admits a complete metric of non-positive (or even negative) curvature and finite volume (see Corollary 14.2). For instance, a product of a closed surface and a bounded surface, both with χ < 0, admits such a metric and has complexity zero by Theorem 9.3. 14.3. Geometric invariants. Complexity is related to other interesting geometric invariants. Let (M, g) be a riemannian manifold. The volume entropy of (M, g) is the limit λ(M, G) = lim R→∞ log Vol (B(p, R) R where Vol(B(p, R)) is the volume of the ball of radius R around a point p in the universal cover M , taken with respect to the lifted metricg. Such a quantity does not depend on p, see [22]. The systole L(M, g) is the length of the shortest closed geodesic which is not homotopically trivial.
The volume entropy λ(M ) and systolic constant σ(M ) of M are defined respectively as the infimum of the volume entropies and systoles among all metrics g on M of volume 1. The spherical volume T (M ) is defined by Besson, Courtois, and Gallot in [7]. Proof. The manifold M is not essential by Theorem 12.2. Babenko showed in [6] that a non-essential manifold M has λ(M ) = σ(M ) = 0. Moreover λ(M ) = 0 implies T (M ) = 0, see [7].
The hypothesis on π 1 (M ) is necessary for the vanishing of σ, since c(RP n ) = 0 and σ(RP n ) > 0. We do not know if it is necessary for the vanishing of T and λ. It is not necessary for the vanishing of || · || because of Theorem 13.1.
14.4. Thin riemannian manifolds. The cut locus of a riemannian manifold is sometimes a simple spine. Alexander and Bishop [1,2] proved that the cut locus of a thin riemannian manifold is a simple spine without vertices: a thin riemannian manifold has therefore complexity zero. This happens for instance if we assign a product metric to M × [0, 1] which is very small on [0, 1].
Following [1], a riemannian manifold M is thin if the radii of all metric balls in M are small relative to K M and κ ∂M , where K M is the sectional curvature of the interior and κ ∂M is the normal curvature of the boundary ∂M . To get a scale-free measure of the width of a manifold, Alexander and Bishop used the curvature-normalized inradius where J is the supremum over all points in M of their distances to ∂M . Theorem 14.8 (Alexander and Bishop [1]). There are universal constants a 2 < a 3 < . . . such that if a bounded riemannian manifold M n has (curvature-normalized) inradius less than a n , then c(M n ) = 0.
Proof. As shown in [1], there exists a sequence of universal constants a 2 < a 3 < . . . such that if the curvature-normalized inradius of M is less than a h and h n + 1, then the cut locus of M is simple and every point is of type n + 1 − h. Therefore it has no vertices for h < n + 1. The cut locus is a spine, hence c(M ) = 0 in that case.

Four-manifolds
We describe here some families of 4-manifolds having complexity zero. The various results proved in this paper show that the set of all 4-manifolds of complexity zero contains all products N × N with non-empty boundary or N ∈ {S 2 , S 3 } and is closed under connected sums, finite coverings, addition of handles of index = 3, and surgery along simple closed curves.
We show that Fintushel-Stern's infinitely many exotic K3's and Gompf's symplectic manifolds with arbitrary π 1 are constructed by adding handles of index = 3 to a product: they thus have complexity zero. The double of a 2-handlebody has also complexity zero, because it is obtained by surgerying along a link in a connected sum of some copies of S 3 × S 1 .

M S E'(n)
E'(n) 1 3 � Figure 24. A manifold constructed by taking a product M 3 × S 1 with ∂M 3 consisting of some tori and capping the boundary components with some copies of E (n) has complexity zero. Among these manifolds we find infinitely many exotic K3 surfaces (Theorem 15.6) and symplectic manifolds with arbitrary fundamental group (Theorem 15.7).
15.1. Arbitrary fundamental group. Closed 4-manifolds with complexity zero may have arbitrary (finitely presented) fundamental groups.

15.2.
Capping with elliptic surfaces. Consider the elliptic surface E(n), described as a fiber-connected sum of n copies of E(1) as in [16]. The elliptic surface has an elliptic fibration E(n) → S 2 , whose regular fiber is a torus. The regular neighborhood R(T ) of a regular fiber T is homeomorphic to a product T × D 2 . Define E (n) = E(n) \ int(R(T )).
This is a simply connected manifold with a 3-torus as its boundary. It has a decomposition with only 0-and 2-handles [16,Figg 8.9 and 8.10]. This manifold is sometimes used to cap some boundary component of a 4-manifold, see Fig 24. We have the following. Every such manifold is necessarily simply connected. However, it is still unknown whether every simply connected 4-manifold has a handle decomposition without 3handles (or even without 1-and 3-handles).
For every knot K ⊂ S 3 , Fintushel and Stern constructed an exotic K3 surface X K whose Seiberg-Witten invariant is roughly the Alexander polynomial of the knot [12]. Among them there are infinitely many distinct exotic K3 surfaces.
Theorem 15.6. The manifold X K has complexity zero for every knot K.
Proof. The manifold X K is constructed by capping (S 3 \ R(K)) × S 1 with one copy of E (2), see [12]. The result follows from Lemma 15.4.
We do not know if X K admits a handle decomposition without 3-handles. 15.4. Symplectic manifolds. Gompf constructed in [15] a family of closed symplectic 4-manifolds with arbitrary fundamental group. It turns out that these manifolds have complexity zero. We can therefore strengthen Corollary 15.2.
Theorem 15.7. Every finitely presented group is the fundamental group of a closed symplectic 4-manifold of complexity zero. The manifold can be chosed to be spin or nonspin.
Proof. Gompf's construction starts with a particular 3-manifold M 3 bounded by some tori, and hence caps the boundary components of M 3 × S 1 with copies of E (n), see [15]. The result follows from Lemma 15.4.