Lectures on surgical methods in rigidity
type: Item
instance of: scholarly article
title: Lectures on surgical methods in rigidity
zbMATH Open document ID: 0883.57035
published in: Lectures on Mathematics and Physics. Mathematics. Tata Institute of Fundamental Research
publication date: 1997-08-04T00:00:00Z
review text: The Borel Conjecture asserts that for two closed aspherical manifolds \(M\) and \(N\) with isomorphic fundamental groups, there exists a homeomorphism \(f:M\to N\) inducing the given isomorphism of the fundamental groups. The Generalized Borel Conjecture asserts that \(|{\mathcal S} (M,\partial) |=1\) for a compact aspherical manifold \(M\) with boundary \(\partial M\). Here, \({\mathcal S} (M,\partial)\) is the structure set of the equivalence classes of pairs \((N,f)\), where \(N\) is a compact manifold with boundary \(\partial N\), \(f:(N,\partial N)\to (M, \partial M)\) is a homotopy equivalence which restricts to a homeomorphism \(f|_{\partial N}: \partial N\to \partial M\), and two pairs \((N,f)\) and \((N',f')\) are equivalent if there exists a homeomorphism \(h:(N, \partial N) \to(N', \partial N')\) such that \(f'\circ h\) is homotopic to \(f \text{ rel } \partial N\). The book under review originated from two courses given by the author at the Tata Institute of Fundamental Research. The book gives an exposition of the Farrell-Jones topological rigidity theorem (originally proven in [Proc. Symp. Pure Math. 54, Part 3, 229-274 (1993; Zbl 0796.53043)] that reads as follows. For a closed non-positively curved \(m\)-dimensional Riemannian manifold \(M\), \(|{\mathcal S} (M\times D^n, \partial) |=1\) when \(m+n \geq 5\) (Theorem 14.1). The result gives a partial verification of the Borel Conjecture and the Generalized Borel Conjecture. The proof uses a surgical method for analyzing \({\mathcal S} (M \times D^n, \partial)\). As ingredients, the method involves a result on an assembly map \([M \times D^n, \partial;G/ \text{Top}] \to L^2_{m+n} (\pi_1M)\) (Theorem 7.1), a result on vanishing of the Whitehead group \(\text{Wh} (\pi_1M)\) (Theorem 14.2), and a surgery result describing a condition under which a map \(f:N \to M\) is homotopic to a homeomorphism (Proposition 14.6). The exposition of the rigidity theorem is given in Lectures 1-14 of the book. Background m…
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57Q25
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 57R80
Mathematics Subject Classification ID: 57R60
zbMATH DE Number: 1043672
zbMATH Keywords: topological rigidity
zbMATH Keywords: non-positively curved Riemannian manifold
zbMATH Keywords: aspherical manifold
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: Wall groups
zbMATH Keywords: Whitehead group
zbMATH Keywords: surgery
author: F. Thomas Farrell
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1043672