On the Surgery Theory for Filtered Manifolds

In this paper we describe some relations between various structure sets which arise naturally for a Browder-Livesay filtration of a closed topological manifold. We use the algebraic surgery theory of Ranicki for realizing the surgery groups and natural maps on the spectrum level. We obtain also new relations between Browder–Quinn surgery obstruction groups and structure sets. Finally we illustrate several examples and applications.


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set S s (X) fits into the Sullivan-Novikov-Wall surgery exact sequence (see [19], [20], and [22]) (1.1) · · · → L n+1 (π) → S s (X) → [X, G/T OP ] σ → L n (π) → · · · This sequence is the main tool for computing the structure set S s (X). The set [X, G/T OP ] coincides with the set of normal invariants T (X) which is formed by the normal bordism classes of normal maps into the manifold X. Given a normal map (also called a t-triangulation of X) (f, b) ∈ T (X), the map σ assigns a surgery obstruction σ(f ) ∈ L n (π). This is trivial if and only if there is a simple homotopy equivalence in the normal bordism class of the map (f, b). Let (X n , Y n−q , ξ) be a topological codimension q manifold pair [20, p.570], where ξ is a topological normal block bundle. We assume that n−q ≥ 5. A t-triangulation Any normal map (f, b) defines a t-triangulation of the manifold pair (X, Y, ξ), and the set T (X, Y ) of concordance classes of t-triangulations of (X, Y, ξ) coincides with T (X) [20,Proposition 7.2.3].
A simple homotopy equivalence f : M → X splits along the submanifold Y if f is homotopic to a map g : M → X which is transversal to Y with a transversal preimage N = g −1 (Y ) and whose restrictions are simple homotopy equivalences (see [19], [20, §7.2] and [22]). The splitting obstruction groups LS * (F ) are defined (see the quoted papers). If f : M → X is a simple homotopy equivalence, then an obstruction to find in its homotopy class a map transversal to Y with properties (1.3) lies in the splitting obstruction group 206 LS n−q (F ). The LS * -groups depend only on n − q (mod 4) and on the pushout square For a manifold pair (X, Y, ξ) the surgery obstruction groups LP n−q (F ) are also defined (see [19], [20], and [22]), and they depend only on n − q (mod 4) and on the square F . There are the assigning obstruction maps and the forgetful maps [20] ( . Two such maps f i : M i → X (i = 1, 2) are said to be equivalent [2] if there exists a normal bordism map with the following properties: We denote by N S s (X, Y ) (see [2]) the set of the correspondent equivalence classes. Then we have the following natural maps There are many relations between the above-defined structure sets (see [1], [2], [8], [9], [13], [17], [18], and [20, §7.2]). The surgery theory for filtered manifolds is very close to the classical surgery theory (see [3], [6], [17], and [24]). Let X be a filtration of a closed manifold X by means of closed submanifolds. We assume that dim X k = n k ≥ 5, and that every pair of manifolds of the filtration is a closed manifold pair. The filtration X in (1.8) is a stratified manifold in the sense of Browder-Quinn. Doing a topological normal map f : M → X topologically transversal to the submanifolds X i gives a t-triangulation of the filtration X (see [3], [6], [17], and [24] On the Surgery Theory for Filtered Manifolds 207 are s-triangulations. The group L BQ n k (X ) is the group of obstructions to find an s-triangulation of the filtration X in the class of the normal bordism of a ttriangulation of X . The set of concordance classes of s-triangulations of X is denoted by S s (X ). It fits into the exact sequence (see [3], [6], [17], and [24]) which generalizes the surgery exact sequence in (1.1) to the case of stratified manifolds. Some relations between various structure sets for the filtration in (1.8) were obtained in [3], [13], and [17]. Denote by F i (0 ≤ i ≤ k − 1) the square of fundamental groups in the splitting problem for the manifold pair (X i , X i+1 ) fitting into filtration (1.8). We shall call this pair of manifolds a Browder-Livesay pair if X i+1 is a one-sided submanifold of X i , and the inclusion X i+1 → X i induces an isomorphism of the fundamental groups. In this case, the group of obstructions for the splitting problem is denoted by LN * (π 1 (X i \ X i+1 ) → π 1 (X i )), and it is called the Browder-Livesay group (see [5], [7], [11], [12], [14], [16], [20], and [22]). A filtration as in (1.8), for which every pair of submanifolds (X i , X i+1 ) is a Browder-Livesay pair, is called a Browder-Livesay filtration (see [3] and [17]). The notion of Browder-Livesay filtration is very useful to investigate assembly maps, iterated Browder-Livesay invariants, surgery spectral sequences, natural maps in surgery theory, and actions of 2-groups on manifolds (see [1], [5], [7], [9], [11], [12], [14], [16], [20], and [22]).
For a Browder-Livesay filtration X (1.8) and for a simple homotopy equivalence f : M → X there is an obstruction in the group LSF n−k (X ) to find an striangulation of X in the homotopy class of the map f [3]. The groups LSF * (X ) are a natural generalization of the splitting obstruction groups LS * and relate with other surgery obstruction groups and structure sets for X by various braids of exact sequences [3]. The main connection is given by the following braid of exact sequences (1.10) where π = π 1 (X). In this paper we describe new relations between structure sets and various obstruction groups which arise for a Browder-Livesay filtration. From now on we shall consider only Browder-Livesay filtrations, but the results without difficulties can be transferred to the general case. We use the algebraic surgery theory of Ranicki for realizing the surgery groups and natural maps on the spectrum level. Finally, we discuss several examples and applications.

Surgery spectra.
First we recall some necessary results about the realization of surgery obstruction groups, structure sets, and natural maps on the spectrum level (see [1], [2], [8], [9], [10], [11], [15], [16], [17], [18], [18], [20], and [24]). We use standard notations for spectra which realize various surgery and splitting obstruction groups and 208 structure sets. Thus we have spectra L(π), LS(F ), LP (F ), LN (π → G), L BQ (F), and LSF (F), that realize the corresponding obstruction groups. For example, π i (LS(F )) = LS i (F ), and similarly for the other spectra. For any Ω-spectrum A, ΩA is the corresponding loop spectrum, and Ω −1 A represents the de-looping spectrum. Many natural maps, as the surgery transfer maps and the induced maps, are also realized on the spectra level, as well as, the corresponding relative obstruction groups. The map σ in (1.1) is realized by the cofibration of spectra with cofiber S(X) such that S n (X) = π n (S(X)), and there is an isomorphism of structure sets Surgery exact sequence (1.1) is isomorphic to the left part starting from L n (π 1 (X)) of homotopy long exact sequence of cofibration (2.1). We have similar situations for other structure sets and natural maps. For a closed manifold pair (X n , Y n−q ) the maps from (1.5), (1.6) and (1.7) are realized by the maps of spectra , For the homotopy groups of spectra that realize structure sets, we shall use the following notations and we have isomorphisms Let us consider a Browder-Livesay filtration X as in (1.8). For 0 ≤ l < j ≤ k, a restricted filtration is well-defined, and we denote it by X l j . Let X l k = X l and X 0 j = X j . The filtration X yields a filtration of manifolds with boundary which is a C-stratified manifold with boundary (see [6], [17], and [24]). Let us denote the filtration in (2.4) by X . For the spectrum L BQ (X ) (see [3] and [24]) we have isomorphisms

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The subscript * in the obstruction group L BQ * (X ) for filtration (1.8) equals the dimension n − k of the smallest manifold of the filtration.
Remark that the filtration X in (2.4) of manifolds with boundary defines a filtration of closed manifolds which we denote by ∂X . It follows from the definition that each one of the filtrations ∂X and X contains k spaces. The inductive definition of the spectra L BQ is given in [24]. The spectra L BQ (X j ) fit into the following cofibration sequence as follows immediately from the definition.
of the filtration X . Then the filtrations Y j and ∂Y j are defined as before. Let Y = Y k . Note also that the braid of exact sequences in (1.10) is also realized by the pull-back square of spectra [3] (2.9) We observe that in (2.9) the cofiber of horizontal maps is L(π 1 (X)) while the cofiber of the vertical maps is S(X ) with π i (S(X )) = S i (X ) and S n+1 (X ) = S s (X ).

Structure sets of a Browder-Livesay filtration.
Here we use notations from the previous sections. For a Browder-Livesay filtra- Theorem 3.1. There exists a pullback square of spectra in which the cofibres of the vertical maps are Ω −k L(G k ), and the cofibres of the horizontal maps are S(X ). Square (3.1) defines a braid of exact sequences Proof. We have a homotopy commutative square of spectra in which the upper horizontal map is the map from (2.9), the left vertical map was constructed on the spectra level in [2], and the right vertical map is the natural forgetful map fitting in (2.7). It follows immediately from [2] and [24] that Diagram (3.3) is homotopy commutative. The fiber of the left vertical map in (3.3) is NS(X, X k ) by [2], and the fiber of the right vertical map in (3.3) is Ω −k+1 L BQ (X ) as follows from (2.7). We consider Diagram (3.3) in which the horizontal maps induce a map of fibres of the vertical maps (see [23]). Thus we obtain the pullback square in (3.1). Now the homotopy long exact sequences of the maps from (3.1) generate the braid of exact sequences in (3.2).

Proposition 3.2. Let X be a Browder-Livesay filtration as in
and (3.5) where the cofibers of all horizontal maps are naturally homotopy equivalent to Ω −k L(G k ), the fibers of vertical maps in (3.4) are naturally homotopy equivalent to , and the fibers of the vertical maps in (3.5) are naturally Each square from (3.4) and (3.5) generates a corresponding braid of exact sequences similarly to that considered in Theorem 3.1.
Note that the left vertical maps in (3.4) and (3.5) induce maps of structure sets that geometrically are given by the restrictions of a normal map to the corresponding submanifold.

On the Surgery Theory for Filtered Manifolds
Proof. We have the following homotopy commutative diagram of spectra in which all right squares are pullback. Diagram (3.6) follows if we join together pullback squares (32) from [2] for all triples of manifolds Let a Browder-Livesay filtration X be given by a triple of manifolds Z ⊂ Y ⊂ X [18]. Sometimes we shall denote this filtration by (X, Y, Z). Denote by F Z the square of fundamental groups in the splitting problem (relative to the boundary) Square (3.7) defines the braid of exact sequences Proof. Consider a homotopy commutative diagram in which the upper map is the map from (2.7) with a cofiber Ω −1 L BQ (X ). For any filtration X = (X, Y, Z) we have L BQ (X ) = LP (F Z ). Now the proof is similar to that of Theorem 3.1.

Theorem 3.4. Under the above assumptions, there exists a pullback square of spectra
in which the fibres of the vertical maps are Ω −1 LS(F Z ) and the fibres of the horizontal maps are NS(X, Z).
Square (3.9) defines the braid of exact sequences Proof. We have the natural maps of spectra (see [2], Theorem 6, and [13], Theorem 3.3): This gives a homotopy commutative square of spectra In (3.12) the fiber of the upper horizontal map is Ω −1 LS(F Z ), the fiber of the left vertical map is Ω −1 LP (F Z ) and the fiber of the right vertical map is L(π 1 (X \ Z)) by [2] and [13]. Now we can complete the proof similarly to that of Theorem 3.1.
where all rows and columns are exact sequences.
Proof. The homotopy commutative left corner square in (3.13) is obtained by natural forgetful maps. The cofibers of the maps from this square are described in Theorem 3.3 of [13] and [2]. By [1], [15], and [21] we can extend this square by a bi-infinite homotopy commutative diagram of spectra (3.13). To obtain (3.14) it is sufficient to apply π 0 to the homotopy commutative diagram of spectra in (3.13).
Note that Theorems 3.3, 3.4, and 3.5 hold, in particular, for any triple of manifolds fitting into a filtration X . Theorem 3.6. For a Browder-Livesay filtration X as in (1.8) there is a pullback square of spectra (3.15) in which the fibres of the vertical maps are Ω −k+1 LSF (X ) and the fibres of the horizontal maps are NS(X, Z).
Square (3.15) defines the braid of exact sequences Proof. Consider a homotopy commutative diagram (3.17) where the vertical maps correspond to passing from X to the subfiltration X k ⊂ X, and the horizontal maps follow from (2.7) and stratified surgery exact sequence. The fiber of the bottom composition in (3.17) is NS(X, X k ) and the same is the fiber of the upper composition. This follows from the commutative diagram in which the row and the column are cofibrations. Hence the fibers of the compositions and coincide [15]. But the first fiber is NS(X, X k ) by [2] (see also Theorem 3.1). Now the result follows by standard arguments.
Proposition 3.7. Let X be a Browder-Livesay filtration as in (1.8), and Y j a subfiltration For 1 ≤ j ≤ k there is the following homotopy commutative diagram of spectra where the rows are cofibrations and the right square is a pullback. The cofiber of both right vertical maps is Ω j−1 L(π 1 (X 0 \ X 1 )).
where all rows and columns are exact sequences.
Proof. Each of the first two columns follows from Theorem 3.6 for the filtrations X and Y = Y k . The maps of the first two columns are induced by natural forgetful maps. The middle horizontal sequence is given in [2] and the upper horizontal sequence is given in (3.3) of [13]. Then the result follows similarly as above.

Examples and Applications.
In this section we illustrate several examples and applications of the obtained results. We maintain all notations from the previous sections.
and for m even we have an exact sequence where ξ is a topological normal block bundle of X k in X.
Proof. For the pair (X, X k , ξ) the commutative diagram from [2, Theorem 3] has the following form

Mathematics and Statistics 1(4): 204-219, 2013
since the codimension k of X k in X is greater than 2. The map is trivial for m odd and has image Z/2 for m even. In fact, the map is trivial for m odd and has image Z/2 for m even (see, for example, Theorem 3.4 from [11] (see also [12]). Now the statement is proved.
Proof. Using a commutative diagram similar to (4.3), it follows that the map is trivial since the Assembly map [X, G/T OP ] → L 4l+1 (π 1 (X)) is trivial (see [12] and Theorem 3.4 from [11]). The same arguments used before allow us to complete the proof.
Now we compute various structure sets for a Browder-Livesay filtration X given by a natural filtration of real projective space RP n with n − k ≥ 5. We recall here well-known results about some structure sets for the filtration X in (4.6) from [4], [14], and [22]: 1 Z 2 ⊕ Z, since these structure sets fit into the following surgery exact sequences (4.9) [20] we obtain the exact sequence (4.10) · · · → S m (RP n \ RP n−1 ) → S m (RP n , RP n−1 , ξ) → S m−1 (RP n−1 ) → . . .
Proof. We use the upper horizontal exact sequence in (3.19). An induction argument on the number of manifolds in the filtration and the isomorphism S * (RP i \ RP i−1 ) = 0 in (4.11) provide the result.
Proof. If X = RP n and Y = RP n−1 , then X \ Y = D n and π 1 (D n ) = 1.
In case i) for n = 4k + 1, let us consider the part of commutative diagram (23) from [2]: (4.13) where LN i (1 → Z/2 + ) = LS i (F + ). Here F ± is a square of fundamental groups in the splitting problem for the pair (X, Y ) = (RP n , RP n−1 ), where the orientation is + for n odd and − otherwise. In (4.13) we use isomorphisms L 1 (1) = L 1 (Z/2 + ) = 0 and (4.9). Now the statement i) for n = 4k + 1 follows. The case n = 4k + 3 can be obtained using similar arguments. So case i) is completely proved.

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In case ii) the diagram, which is similar to that in (4.13), has the form Here we use (4.8) and (4.9) and the computation of Wall groups in [23]. Now statement ii) follows. In case iii) the diagram, which is similar to that in (4.13), has the form Here we use (4.8) and (4.9) and the computation of Wall groups in [23]. Now statement iii) follows by a diagram chasing. Thus the theorem is completely proved.