Surgery on a stratified space
Let \(Y\) be a closed \((n+q)\)-dimensional manifold with a locally flat \(n\)-dimensional submanifold \(Z \subset Y\). Let \((X,\partial X)\) be a compact \((n+1)\)-dimensional manifold with boundary \(\partial X \cong Z\). Denote by \(\mathfrak{X} = (Y \cup X;X,Y,Z)\) the stratified space obtained by gluing \(Y\) and \(X\) along \(\partial X \cong Z\). The paper under review describes the stratified structure set \(\mathcal{S} (\mathfrak{X})\) in the sense of surgery theory. The description is done by developing algebraic theory of surgery in the style of [\textit{A. Ranicki}, Exact sequences in the algebraic theory of surgery. Princeton: Princeton University Press (1981; Zbl 0471.57012)] in this setting. The structure set \(\mathcal{S} (\mathfrak{X})\) is shown to be isomorphic to the algebraic structure set of \(\mathfrak{X}\) which in turn is related to the ordinary algebraic structure set of the pair \((X, \partial X)\) and to the codimension \(q\) algebraic structure set of the pair \((Y,Z)\) in the sense of Chapter 7 of [\textit{A. Ranicki}, Exact sequences in the algebraic theory of surgery. Princeton: Princeton University Press (1981; Zbl 0471.57012)]. Both of these have a description in terms of the homology with coefficients in the \(L\)-theory spectra and ordinary \(L\)-groups via various commutative braids. The paper concludes with an example in which the stratified structure set is calculated for the case where \((Y,Z)=(S^{n+1},S^{n})\) and \(X\) is the total space of the disc bundle of the canonical line bundle over \(\mathbb R P^{n}\) and the homeomorphism \(Z \cong \partial X\) is given by the standard identification of the associated sphere bundle with \(S^{n}\), with \(n \geq 5\).
homotopy classification of stratified spaces
scientific article; zbMATH DE number 6358085
scientific article