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On structure sets of manifold pairs

In this paper we systematically describe relations between various structure sets which arise naturally for pairs of compact topological manifolds with boundary. Our consideration is based on a deep analogy between the case of a compact manifold with boundary and the case of a closed manifold pair. This approach also gives a possibility to construct the obstruction groups for natural maps of various structure sets and to investigate their properties.


Introduction
Let be a pair of compact topological manifolds with boundary such that ∂Y ⊂ ∂X and Y ⊂ X are locally flat submanifolds with given structures of the normal bundles (cf. [8] and [9]). The pair of manifolds in (1) is a special case of a stratified manifold (cf. [9] and [11]) and a complete description of the relations between various structure sets in this situation is very helpful for investigation of the general situation of a stratified manifold. In this paper we systematically describe relations between different structure sets which arise naturally in this case. We shall work in the category of topological manifolds and assume that the dimension of all manifolds is greater than or equal to 5. We shall use surgery theory for the case of simple homotopy equivalence ( [8], [9], and [10]), and we shall tacitly assume that all obstruction groups are decorated with "s". We shall use the functoriality of the basic maps in surgery theory and the algebraic surgery theory of Ranicki (cf. [1], [3], [6], [8], and [9]).
In Section 2 we define classical structure sets for a manifold with boundary (X, ∂X) and we describe the natural relations among them. In Section 3 we define structure sets for a closed manifold pair (X n , Y n−q ) and we describe the natural relations among them. We point out a deep analogy between the case (X, ∂X) considered in Section 2 and the case of a closed manifold pair (X, Y ) considered in Section 3.
In Section 4 we present the necessary material on the realization of the structure sets introduced above, on the spectrum level ( [1], [2], [4], [5], [6], [8], and [9]). From this realization and the results of the previous sections, we derive the well-known basic diagrams of exact sequences for the structure sets (cf. [2], [4], [5], [8], [9], and [10]). In this section we also describe the analogy between (X, ∂X) and a closed manifold pair (X, Y ) on the level of the natural maps between the structure sets. This approach gives a possibility to better understand the geometric properties of natural maps in surgery theory.
In Section 5 we extend our consideration to the case of a compact manifold pair with boundary. The results of this section are known to the experts, but not all of them have been published until now. Our approach gives an opportunity to systematically describe the structure sets of such pairs and natural maps between them. The naturality of this approach also provides realization of structure sets and the natural maps between them on the spectrum level. In this way it is possible to describe all structure sets for a closed manifold pair with boundary and describe relations among them, other structure sets, and the surgery obstruction groups.
In Section 6 we point out some possible applications of our approach to the description of various obstruction groups which arise naturally when we consider the natural maps between structure sets.
2. The structure sets of (X, ∂X) Let X n be a closed n-dimensional manifold. An s-triangulation of X is a simple homotopy equivalence f : M → X, where M n is a closed n-dimensional topological manifold (cf. [8], [9], and [10]). Two s-triangulations fi : Mi → X are concordant (cf. [9]) if there exists a simple homotopy equivalence of triads where W is a compact (n + 1)-dimensional manifold with the boundary ∂W = M0 ∪ M1. The set of concordance classes is denoted by S(X).
A t-triangulation of X is a topological normal map (f, b) : M → X, where M n is a closed n-dimensional topological manifold (cf. [8] and [9]). Two t-triangulations are concordant (cf. [9]) if there exists a topological normal map of triads The set of concordance classes of t-triangulations of X is denoted by T (X).
For n ≥ 5 these sets fit into the surgery exact sequence (cf. [8], [9], and [10]) (2) · · · → Ln+1(π1(X)) → S(X) → T (X) → Ln(π1(X)) → · · · where L * (π1(X)) are the surgery obstruction groups. Thus for a closed manifold X we have two types of structures, which are denoted by S and T . We can write down the following diagram for these structures We can interpret the arrow in (3) as a relation of partial order and can say that the structure S is 'stronger' than the structure T . The diagram (3) means that, for any manifold X, a simple homotopy equivalence of n-manifolds f : M → X, which represents an element of the structure set S(X), correctly defines a structure in T (X) (cf. [10]), that is the map S(X) → T (X) in (2). The map of structures in (3) is the map of 'weakening of the structure'.
Let (X n , ∂X n ) be a compact manifold with boundary. In this case there are the following five structures on (X, ∂X): T T , T S, T D, SS and SD.
The two first structures on (X, ∂X) in (4) correspond to T -structures on X whose restrictions to the boundary give the structures T and S. The third structure corresponds to T -structures on X whose restriction to ∂X is a homeomorphism and the equivalence relation is considered rel the boundary. The two second structures on (X, ∂X) in (4) are considered similarly. For a compact manifold with boundary (X n , ∂X), we recall the definition of the structure sets which correspond to the structures from (4) (cf. [4], [9], and [10]).
(T T ) This is the relative case in the sense of [9] and [10]. Let with ∂V = ∂M0 ∪ ∂M1. The set of concordance classes is denoted by T (X, ∂X).
(T S) This is the mixed structure on a manifold with boundary in the sense of [4] and [10]. Let f : (M, ∂M ) → (X, ∂X) be a t-triangulation of (X, ∂X) such that the restriction f | ∂M : ∂M → ∂X is an s-triangulation. Two such maps fi : (Mi, ∂Mi) → (X, ∂X), i = 0, 1, are concordant if there exists a t-triangulation (5) of the 4 − ad such that V = F −1 (∂X × I) and the restriction F |V is an s-triangulation The set of equivalence classes of such maps is denoted by T S(X, ∂X) (cf. [4]).
(T D) This is the rel ∂ -case in the sense of [9] and [10] (cf. also [4] and [5]). A t ∂triangulation of (X, ∂X) is a t-triangulation whose restriction ∂f to the boundary is a homeomorphism ∂M → ∂X. Two The set of concordance classes is denoted by T ∂ (X, ∂X).
(SS) This is the relative case of s-triangulations in the sense of [9] and [10]. Let The set of concordance classes is denoted by S(X, ∂X).
(SD) This is the case of s ∂ -triangulation in the sense of [4], [5], and [9]. An s ∂triangulation of (X, ∂X) is an s-triangulation of the pair (X, ∂X) The set of concordance classes is denoted by S ∂ (X, ∂X).
For a compact manifold with boundary X ←֓ ∂X we can write down the following diagram of structures SS −→ T S −→ T T in which, similarly to the diagram (3), the arrows provide natural maps of 'weakening of the structure' (cf. [4], [9], and [10]), and we can interpret every arrow (or their composition) as a relation of a partial order. For any structure in (9), the right symbol which corresponds to the structure on the boundary is 'stronger than or equal' to the left symbol. Every arrow in (9) gives a natural map of 'weakening of the structure' on a manifold with boundary (X, ∂X). From this it follows that any sequence of arrows from (9) also gives a map of 'weakening of the structure'. Hence, for any (X, ∂X), the diagram (9) provides the following commutative diagram of structure sets (compare with the diagram on page 116 of [10]) The diagram (10) gives an opportunity to construct several commutative diagrams which have the form of the square or of the triangle, for example we have S(X, ∂X) −→ T (X, ∂X).
Excluding the trivial cases when the restriction of the structure to the boundary is a homeomorphism, we obtain the following relations between structures on (X, ∂X) and structures on ∂X (cf. [4], [5], [9], and [10] ) (12) SS → S, T S → S, T T → T .
In (12) the left pairs of symbols correspond to the structures on (X, ∂X) and the right symbols correspond to the structures on ∂X which is the restriction of the structure on (X, ∂X) to the boundary. The relations in (12) and the diagram (9) provide the following diagram of structures (cf. [4], [9], and [10]) 3. The structure sets of a closed manifold pair (X, Y ) Let (X n , Y n−q ) be a closed manifold pair without boundary. In this section we systematically describe various structures on this pair. It follows from our description that, for a pair of closed manifolds, structures and relations between them are similar to structures and relations for a manifold with boundary. In fact it is clear from our consideration, that in a number of cases the submanifold 'plays the role of a boundary' [2]. For a pair (X n ←֓ Y n−q ), the list of structures almost coincides with the corresponding list of structures for (X, ∂X). Below we explain the difference. First we remark that all structures from (4) are realized for manifold pairs (X, Y ). Now we define these structures (cf. [2], [9], and [10]).
(T T ) For a closed manifold pair (X, Y ) this structure is given by the structure set T (X, Y ) = T (X) (cf. [9]) and it consists of concordance classes of t-triangulations of the manifold X.
(T S) This structure is given by the structure set N S(X, Y ) introduced in [2]. Its restriction to X gives the T structure on X and its restriction to Y gives the S-structure [2]. This structure is similar to the mixed structure on the manifold with boundary (cf. [2], [4], and [10]). Let with the following properties: The set of equivalence classes of such maps is denoted by N S(X, ∂X) (cf. [2]).
(SS) This structure is given by the structure set S(X, Y, ξ) where ξ is a topological normal block bundle of Y in X (cf. [9]). The restriction of this structure to X is an S-structure, and the restriction to Y also is an S-structure. We give now an explicit definition following [9].
An s-triangulation of a manifold pair S(X, Y, ξ) is a t-triangulation which is transversal to Y with N = f −1 (Y ) and for which the maps are s-triangulations (cf. [2], [5], and [9]). Two such s-triangulations The set of concordance classes is denoted by S(X, Y, ξ).
(SD) In this case we shall consider S-structures on X whose restriction to a tubular neighborhood of Y provide D-structures. The structure set in this case is given by the rel ∂ s-triangulations of the manifold with boundary (X \ Y, ∂(X \ Y )) that is [4], [5], [9], and [10]).
The relations (9) take place for the structures introduced above in this section (cf. [2], [4], [5], [9], and [10]). Thus, for a closed manifold pair (X, Y ), the diagram (9) provides the following commutative diagram of structure sets which is similar to diagram (10). In a similar way (cf. [2], [4], [5], and [9]), using the restriction of a structure on the manifold X to the submanifold Y , we obtain the relations between structures on (X, Y ) and structures on Y given in (12). Now in (12) the left pairs of symbols for any arrow correspond to structures on (X, Y ) and the right symbols correspond to the structures on Y . The results of [2], [9], and [10] provide the relations (13) between structures for a closed manifold pair (X, Y ). Thus, for any closed manifold pair (X, Y ), the diagram (13) provides the following commutative diagram of structure sets that is similar to (14) However, there is a difference between structures for the pairs (X, ∂X) and (X, Y ). For a manifold pair (X, Y ) we can consider also the S-structure on X as ST -structure on the pair (X, Y ). There is no analogue structure for the case of manifolds with boundary. In this case the restriction of the structure to the submanifold gives a 'weaker structure' than the structure on the ambient manifold. This structure fits into the following commutative diagram of structures (cf. [2] and [9]) which has no analog for the structures on manifolds with boundary. For any closed manifold pair (X, Y ) the diagram (17) provides the following commutative diagram of structure sets The structure ST fits also into the following diagram of structures in which the vertical arrows correspond to the restriction of the corresponding structure to the submanifold. For any closed manifold pair (X, Y ), the diagram (19) gives the following commutative diagram of structure sets

The spectrum level. The diagrams of structure sets
In accordance with the algebraic surgery theory of Ranicki (cf. [1], [4], [6], and [8]) the structure sets introduced above, various surgery obstruction groups, and natural maps are realized on the spectrum level. We shall use the following notations.
The groups L * (π1(X)) are the surgery obstruction groups of Wall [10]. For a closed manifold X n with n ≥ 5, we have For the structure sets of a compact manifold (X n , ∂X n ) with boundary ∂X there exists the following homotopy commutative diagram of spectra (cf. [4], [5], [8], and Note that the commutative square * in (26) is isomorphic to the commutative square which follows from the diagram (10), and the commutative square * * is isomorphic to the commutative square which follows from (14).
The mixed structure sets relate to other structure sets introduced above by the following braids of exact sequences [4] (31) which are realized on the spectrum level. Note that the commutative square * in (31) is isomorphic to the left commutative square in the diagram (10), the commutative square * * in (32) is isomorphic to the right commutative square in the diagram (14), the commutative triangle ⋆ in (32) follows from the right square of (10), and the commutative triangle * in the diagram (33) is isomorphic to the commutative triangle

t t t t t t t t T S(X, ∂X)
which follows from the diagram (14). Now let (X n , Y n−q , ξ) be a closed manifold pair (cf. [9, §7.2]). An s-triangulation f : M → X splits along Y if it is homotopic to a map which is an s-triangulation of the manifold pair (X, Y, ξ) (the definition was given in Section 3). The splitting obstruction group LSn−q(F ) is defined in [9, §7.2]. Let ∂U be the boundary of a tubular neighborhood U of Y in X. The groups LSn−q(F ) depend only on n − q mod 4 and a pushout square A of fundamental groups with orientations. If f : M → X is a t-triangulation then by [9, §7.2] a group LPn−q(F ) of obstructions is defined. This group provides obstructions to the existence of an s-triangulation of (X, Y, ξ) in the class of the normal bordism of the map f . The groups LPn−q(F ) depend only on n − q mod 4 and the square F as well. There exists a homotopy commutative diagram of spectra (cf. [2] and [9]) where the spectrum LP(F ) is the spectrum for obstruction groups LPi(F ), and all rows and columns in (36) are cofibrations. Denote πi(S(X, Y, ξ)) = Si(X, Y, ξ), and we have an isomorphism [9] (37) Sn+1(X, Y, ξ) ∼ = S(X, Y, ξ). Note that the commutative square * in (38) is isomorphic to the commutative square which follows from the diagram (15). The commutative square * * in (38) is isomorphic to the commutative square which follows from the diagram (19). Thus the diagram (38) is similar to the diagram (26) which was constructed for a manifold with boundary. For a closed manifold pair (X n , Y n−q ), we define a spectrum NS(X, Y ) as the homotopy fiber of the map L(π1(X \ Y )) −→ S(X, Y, ξ) which is the composition of maps from the extended diagram (36) [2]. It follows from (36) (cf. [2]) that this spectrum fits into the following cofibrations Note that the cofibrations in (41) for (X, Y ) are similar to cofibrations in (30) for (X, ∂X).
Denote πi(NS(X, Y )) = N Si(X, Y ), and we have an isomorphism N S(X, Y ) ∼ = N Sn(X, Y ). The structure sets N S(X, Y ) relate to the other structure sets for a manifold pair (X, Y ) by the following braids of exact sequences [2] (42) : The diagrams (42) -(44) are realized on the spectrum level and follow from the diagrams (36) (cf. [1], [2], and [7]). The diagram (42) is similar to the diagram (31), the diagram (43) is similar to the diagram (32), and the diagram (44) is similar to the diagram (33).
For the case of a manifold pair (X, Y, ξ) we can consider the following pair of topological spaces (X, X \ Y ), where as usual X \ Y denotes the closure of X \ U where U is a tubular neighborhood of Y in X. Then the commutative diagram of exact sequences from page 560 of [9] gives the diagram The structure sets for a manifold pair (X, Y ) also fit into the following commutative diagrams of exact sequences [9, §7.2]: 7 7 n n n n n n n n n n n n ' ' y y y y y y y y y y y y which are realized on the spectrum level. The commutative triangle * in (46) corresponds to the following diagram of structures The commutative triangle * in (47) corresponds to the following diagram of structures For the case (X, ∂X), there are no diagrams that are similar to the diagrams (45)-(48). In this case we have a simple homotopy equivalence (X \ ∂X) → X, and the theory is degenerated.

The structure sets of a manifold pair with boundary
In this section we describe various structure sets which arise naturally for a compact manifold pair with boundary in (1). We shall use notations that are similar to notations of previous sections. Thus, for example, denotes a structure on Q in (1) which is a T T -structure on (X, ∂X) whose restriction to (Y ←֓ ∂Y ) (the bottom pair of symbols) gives an SD-structure, and whose restriction to (∂X ←֓ ∂Y ) (right vertical pair of symbols) gives a T D-structure. We shall denote by F the pushout square for the manifold pair (X, Y ) as in (35). The manifold pair with boundary Q in (1) defines a pair of closed manifolds ∂Y ⊂ ∂X with a pushout square A of fundamental groups for the splitting problem. A natural inclusion δ : ∂X → X induces a map ∆ : F ∂ → F of squares of fundamental groups. The relative splitting obstruction groups LS * (∆) and the relative surgery obstruction groups LP * (∆) are defined in [3], [5], [9], and [10]. First we describe structures for which there is a similarity between the pairs (X, ∂X) and (X, Y ). In this case for a structure » AB CE we have that B and C are stronger than or equal to A, and E is stronger than or equal to B and C. In such a way it is sufficiently easy to describe all possible natural structures on a manifold pair with boundary in (1) (cf. [5] and [9, pp. 585-587]). In this case we obtain the following list of possible structures The » SD SD --structure on a pair in (1) corresponds to rel ∂ -case structure set defined in [9] (cf. also [5]). In this case, consider an s-triangulation which is transversal to (Y, ∂Y ) with a transversal preimage (N, ∂N ) such that ∂f : ∂M → ∂X is a homeomorfism and the maps are s-triangulations (cf. [2], [5], and [9]). The set of concordance classes rel boundary of maps in (59) gives the structure set in (58).
Using the same line of argument as in Section 4 (cf. also [2] and [5]), we can define » T D SD --structures on a pair in (1), which corresponds to the structure set N S ∂ (X, Y ) (N S-structure set relative boundary, as usual we suppose that the restriction to the boundary is a homeomorphism). with The set of equivalence classes of such maps is denoted by N S ∂ (X, Y ) (cf. [2]).
which is the rel ∂ -version of the diagram (38). Two left squares in (61) correspond to the following commutative diagram of structures Proof. We can use the diagram (60) similarly to the diagram (36) to define a spectrum NS ∂ (X, Y ) fitting into the cofibrations which are similar to the cofibrations in (41). From here the result follows using standard argument (cf. [2], [5], [9], and [10]).
Wall pointed out the existence of mixed structures (cf. pages 115 and 116 in [10]) of n-ads. Now, similarly to the definition of the T S-structure on (X, ∂X) in Section 2, we can define a T S-structure on (X ; ∂1X) in (68). This is the T -structure on X whose restriction to (∂1X, ∂ ∅ X) is the S-structure.
Definition 5.6. Let f : M → X be a t-triangulation (68) such that the restriction is an s-triangulation. The set of equivalence classes of such maps is denoted by T S(X ; ∂1X).
Example 5.15. The third structure in (54) is given by the classes of concordance of t-triangulations of (X, ∂X) whose restrictions to the boundary provides classes of concordance of s-triangulations ∂X which are split along ∂Y . Denote the corresponding structure set by T S(X; ∂X, ∂Y ). This structure set is realized by the spectrum TS(X; ∂X, ∂Y ) with πi(TS(X; ∂X, ∂Y )) = T Si(X; ∂X, ∂Y ), T Sn(X; ∂X, ∂Y ) = T S(X; ∂X, ∂Y ) (cf. [5] and [9]).
Consider the commutative diagram of structures on Q in (1) (78) here all the maps in the upper square and the bottom horizontal map are the maps of 'weakening of the structure', and the vertical maps in the bottom square are given by the restriction of the structure to the submanifold. The diagram (78) induces the following commutative diagram of structure sets (cf. [5], [8], [9], and where the first row is the surgery exact sequence of the triple in (74) for the case rel ∂U with L rel ∂U * = L * (π1(∂X \ ∂Y ) → π1(X \ Y )) and L rel * = L * (π1(∂Y ) → π1(Y )). The second row of (79) follows from the diagram 3.34 of [5]. The bottom row in (79) is the surgery exact sequence of the manifold pair (Y, ∂Y ).
The diagram (79) is realized on the spectrum level and we have the following isomorphisms which follows from (9) the obstruction groups have not been introduced so far. Now it is an easy exercise to introduce the surgery spectra for the corresponding obstruction groups and commutative diagrams and braids of exact sequences, similarly to above.
For the case of a manifold pair (X, Y ) we can write down the maps of 'weakening of the structure' and corresponding cofibrations similarly to Example 6.2. The diagram (15) on the spectrum level provides the maps of spectra for structure sets, and as cofibres of the maps we shall obtain spectra for corresponding obstruction groups. In this case we can also write additional cofibrations of spectra for structure sets (with cofibres which are spectra for corresponding obstruction groups) which provides commutative diagram of structures (17).
For a manifold pair with boundary we have a great number of maps of 'weakening of the structure', but now it is sufficiently easy to construct spectra for corresponding obstruction groups and describe theirs properties. Many of such examples are given in Section 5. where the groups LP Si(∆) = πi(LPS(∆)) are the corresponding obstruction groups.