Involutions whose top dimensional component of the fixed point set is indecomposable
type: Item
instance of: scholarly article
title: Involutions whose top dimensional component of the fixed point set is indecomposable
zbMATH Open document ID: 1206.57040
author: Pedro L. Q. Pergher
published in: Geometriae Dedicata
publication date: 2010-06-16T00:00:00Z
review text: In 1967, \textit{J. Boardman} proved the famous Five Halves Theorem [Bull. Am. Math. Soc. 73, 136--138 (1967; Zbl 0153.25403)] which states: if \(M^m\) is a smooth closed \(n\)-dimensional manifold, and if \(T:M^m \to M^m\) is a smooth involution on \(M^m\) for which the fixed point set \(F = \displaystyle {\bigcup_{j =0}^{n}} F^j \) does not bound, then \(m \leq\frac{5}{2}n\) (here, \(F^j\) denotes the union of those components of \(F\) having dimension \(j\)). Further, in this generality, this estimative is best possible. In recent years, the author of the present paper obtained several results in the following relevant direction: can we obtain improvements for the Boardman bounds if we impose additional conditions on \(F\)? In the present paper, the author deals with this question when the top dimensional component of \(F\), \(F^n\), is indecomposable; we recall that a closed manifold is indecomposable if its unoriented cobordism class cannot be expressed as a sum of products of lower dimensional cobordism classes. In this case, the author proves that the Boardman bound \(m \leq\frac{5}{2}n\) can be improved to \(m \leq 2n+1\). To do that, the author discovered a very special characteristic class associated to the total space of projective bundles. The author also constructs, for each \( n \geq 2\) not of the form \(2^t - 1\), a special involution \((M^{2n+1},T)\) so that the dimension of the top dimensional component of the fixed point set is \(n\) and with this top dimensional component being indecomposable, thus showing that the bound in question is best possible.
reviewed by: Alice K. M. Libardi
Mathematics Subject Classification ID: 57R85
Mathematics Subject Classification ID: 57R75
zbMATH DE Number: 5721540
zbMATH Keywords: involution
zbMATH Keywords: projective space bundle
zbMATH Keywords: indecomposable manifold
zbMATH Keywords: splitting principle
zbMATH Keywords: Stiefel-Whitney class
zbMATH Keywords: characteristic number
MaRDI profile type: Publication
full work available at URL: s10711-009-9419-5
OpenAlex ID: W2077062407
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DOI: 10.1007/S10711-009-9419-5
description: scientific article; zbMATH DE number 5721540