Intrinsic transversality structures
type: Item
instance of: scholarly article
title: Intrinsic transversality structures
zbMATH Open document ID: 0661.57005
author: Norman Levitt
author: Andrew Ranicki
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: This paper tackles the fundamental problem of the existence and classification of topological manifold structures on Poincaré duality spaces \(X^ n\), with Spivak normal fibration \(\nu\), through various notions of transversality. Classically, this is a two step problem. The first step is to check if \(\nu\) has a TOP reduction. This can be accomplished by viewing \(\nu\) as a high, but finite dimensional spherical fibration and requiring that all maps f: \(M^ n\to T(\nu)\) can be made Poincaré-transverse in a mutually consistent way, or, in the terminology of the authors, \(X^ n\) has an extrinsic transversality structure. It is shown by \textit{N. Levitt} and \textit{J. Morgan} [Bull. Am. Math. Soc. 78, 1064-1068 (1972; Zbl 0267.55021)] that \(X^ n\) has a TOP- reducible Spivak normal fibration iff it has an extrinsic transversality structure. The second step is to see whether such a reduction exists such that the resulting surgery obstruction in \(L_ n(\pi_ 1X)\) vanishes. The authors show that this can be accomplished by considering the notion of an intrinsic transversality structure. Very roughly speaking, \(X^ n\) has an intrinsic transversality structure if, for any PL-stratified space Q, all maps f: \(X^ n\to Q\) can be made Poincaré-transverse to the stratification of the range Q. After making this notion precise the authors prove: Theorem. Intrinsic transversality structures on \(X^ n\) are in one-to-one correspondence with topological manifold structures on \(X^ n\).
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 57N75
Mathematics Subject Classification ID: 57Q65
Mathematics Subject Classification ID: 57N80
zbMATH DE Number: 4080422
zbMATH Keywords: topological manifold structures on Poincaré duality spaces
zbMATH Keywords: Spivak normal fibration
zbMATH Keywords: transversality
zbMATH Keywords: Poincaré-transverse
zbMATH Keywords: TOP-reducible Spivak normal fibration
zbMATH Keywords: extrinsic transversality structure
zbMATH Keywords: surgery obstruction
zbMATH Keywords: intrinsic transversality structure
zbMATH Keywords: PL-stratified space
reviewed by: R. J. Stern
MaRDI profile type: Publication
full work available at URL: pjm.1987.129.85
OpenAlex ID: W2066793041
DOI: 10.2140/PJM.1987.129.85
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description: scientific article; zbMATH DE number 4080422