Surgery on a stratified space
type: Item
instance of: scholarly article
title: Surgery on a stratified space
zbMATH Open document ID: 1312.57038
published in: Mediterranean Journal of Mathematics
publication date: 2014-10-17T00:00:00Z
review text: 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 \(n \geq 5\).
reviewed by: Tibor Macko
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 19J25
Mathematics Subject Classification ID: 57N99
zbMATH DE Number: 6358085
zbMATH Keywords: homotopy classification of manifolds
zbMATH Keywords: homotopy classification of stratified spaces
zbMATH Keywords: structure sets
zbMATH Keywords: surgery obstruction groups
zbMATH Keywords: splitting obstruction groups
zbMATH Keywords: surgery exact sequence
author: Rolando Jiménez
author: Yuri V. Muranov
MaRDI profile type: Publication
full work available at URL: s00009-013-0332-7
OpenAlex ID: W2057914607
cites work: Splitting along submanifolds and \(\mathbb L\)-spectra
cites work: Surgery on a pair of transversal manifolds
cites work: Q4112651
cites work: On structure sets of manifold pairs
cites work: Splitting along a submanifold pair
cites work: Q5616706
cites work: Splitting obstruction groups and quadratic extensions of anti-structures
cites work: A spectral sequence in surgery theory and manifolds with filtrations
cites work: Q3865168
cites work: Q3924852
cites work: Q3141903
DOI: 10.1007/S00009-013-0332-7
description: scientific article; zbMATH DE number 6358085