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Surgery on a pair of transversal manifolds. (English) Zbl 1257.57034
The main focus of classical surgery theory is the classification, up to homeomorphism, of closed manifolds homotopy equivalent to a closed manifold. There are variants of the classical theory for manifolds with boundary, for pairs of manifolds or, even more general, for manifold \(n\)-ads. In the paper under review, the authors study a certain case of stratified spaces. More precisely, the authors study the case when the space is the union of two manifolds over a common locally flat submanifold. It is assumed that the two manifolds are of the same dimension and all three manifolds that appear are connected. The authors point out that their methods, being functorial enough, work without these assumptions. The approach uses the classical approach. First, the correct definition of normal maps (called \(t\)-triangulation) in this setting is given. Essentially, these are maps between the ambient spaces such that the pre-images of the pieces are the corresponding pieces and the restriction to the two manifold pairs that appear are \(t\)-triangulations of pairs [M. Cencelj, Yu. V. Muranov and D. Repovš, Homology Homotopy Appl. 11, No. 2, 195–222 (2009; Zbl 1185.57030); A. Ranicki, Exact sequences in the algebraic theory of surgery. Mathematical Notes, 26. Princeton, New Jersey: Princeton University Press; University of Tokyo Press. (1981; Zbl 0471.57012)]. The equivalence relation between such maps is concordance. It is shown that the right spectrum whose homotopy groups, in the right dimension, are the equivalence classes of the \(t\)-structures, is the pullback of the \(\mathbf{L}_{\bullet}\)-homology spectra of the two manifold pieces over the \(\mathbf{L}_{\bullet}\)-homology spectrum of the intersection with a dimension shift, depending on the codimension of the intersection in each space. A similar construction works for the obstruction groups. The spectrum whose homotopy groups are the obstruction groups is the pullback of the spectra of the \(\mathbb{L}P\)-spectra of the corresponding pairs (and their normal data) over the \(\mathbb{L}\)-spectrum of the intersection. Alternatively, the authors characterize the obstruction spectrum as the pullback of the \(\mathbb{L}P\)-spectrum of one of the piece, the \(\mathbb{L}\)-spectrum of the other piece with a dimension shift, over the \(\mathbb{L}\)-spectrum of the inclusion of the complement of the intersection in the last piece to the whole submanifold, again with the same dimension shift. The dimension shift is positive and to the order of the codimension of the intersection. All the above constructions ensure that the objects defined for the stratified space fit into braids of exact sequences that connect them with the classical objects. Furthermore, the homotopy groups of the spectra that are defined also fit into the corresponding exact sequence that generalizes the classical one.
As an application the authors give the example of the space \(\mathcal{X} = (\mathbb{R}P^n, \mathbb{R}P^n; \mathbb{R}P^{n-1})\), \(n \geq 6\). In this case, the components for building up the objects of \(\mathcal{X}\) are known. Using the fact that \(\mathbb{R}P^n \setminus \mathbb{R}P^{n-1} \cong D^n\), direct calculation for the structure sets shows that: \[ \mathcal{S}^{\text{TOP}}(\mathcal{X}) \cong \mathcal{S}^{\text{TOP}}(\mathbb{R}P^n, \mathbb{R}P^{n-1}) \cong \mathcal{S}^{\text{TOP}}(\mathbb{R}P^{n-1}). \]

MSC:
57R67 Surgery obstructions, Wall groups
19J25 Surgery obstructions (\(K\)-theoretic aspects)
57N99 Topological manifolds
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