Improvements of the five halves theorem of J. Boardman with respect to the decomposability degree
type: Item
instance of: scholarly article
title: Improvements of the five halves theorem of J. Boardman with respect to the decomposability degree
zbMATH Open document ID: 1305.57051
author: Patricia E. Desideri
author: Pedro L. Q. Pergher
published in: The Asian Journal of Mathematics
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ajm/1410186665
reviewed by: Parameswaran Sankaran
Mathematics Subject Classification ID: 57R85
Mathematics Subject Classification ID: 57R75
zbMATH DE Number: 6355714
zbMATH Keywords: involutions
zbMATH Keywords: five-halves theorem
zbMATH Keywords: projective space bundle
zbMATH Keywords: indecomposable manifolds
zbMATH Keywords: decomposability degree
zbMATH Keywords: Stiefel-Whitney class
zbMATH Keywords: non-dyadic partition
MaRDI profile type: Publication
DOI: 10.4310/AJM.2014.V18.N3.A3
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review text: Let \(T\) be a smooth involution on a smooth closed manifold \(M\) of dimension \(m\) and let \(F=Fix(T)\) be the set of fixed points fixed by \(T\).NEWLINENEWLINE Suppose that \(M\) is a non-bounding manifold, i.e., there does not exist a compact smooth manifold of dimension \(m+1\) such that \(\partial W=M\). Then \textit{J. M. Boardman} [Bull. Am. Math. Soc. 73, 136--138 (1967; Zbl 0153.25403)] showed that \(m\leq (5/2).n\) where \(n\) is the maximum dimension among the components of \(F\), and moreover, this estimate is sharp. There have been many improved upper bounds for \(\dim M\) (assuming \(M\) to be non-bounding) by putting restrictions on the fixed point set \(F\). Write \(F=\cup_{0\leq k\leq n}F^k\), where \(F^k\) is the union of \(k\)-dimensional components of \(F\). \textit{C. Kosniowski} and \textit{R. E. Stong} [Topology 17, 309--330 (1978; Zbl 0402.57005)] showed that if \(F=F^n\), then \(m\leq 2n\). \textit{P. L. Q. Pergher} and \textit{R. E. Stong} [Transform. Groups 6, No. 1, 79--86 (2001; Zbl 0985.57017)] showed that if \(F=F^0\cup F^n\) then \(m\leq \mathcal{M}(n)\) where \(\mathcal{M}(n)\) defined as follows: Write \(n=2^p.q, q \text{~odd},~ p\geq 0\). Then \(\mathcal{M}(n)=2n+p-q+1\) if \(p\leq q\) and \(\mathcal{M}(n)=2n+2^{p-q}\) if \(q\geq p\).NEWLINENEWLINEGiven a closed non-bounding manifold \(F=F^n\) of dimension \(n\), the decomposability degree of \(F\), denoted \(\ell(F^n)\), is the smallest number \(r\) such that there exists a non-dyadic partition \(\omega:=i_1,\dots, i_r\) of \(n\) with Stiefel-Whitney number \(\langle s_\omega(F),[F]\rangle \neq 0\). Here non-dyadic means that none of the \(i_t\) is of the form \(2^s-1\) and \(s_\omega(F)\) is the polynomial \(s_\omega(w_1,\dots, w_n)\in H^n(F;\mathbb{Z}_2)\) in the Stiefel-Whitney classes \(w_j=w_j(F)\) of \(F\) obtained from the monomial symmetric polynomial \(\sum x_1^{i_1}\dots x_r^{i_r}\) in the indeterminates \(x_1,\dots, x_n\) expressed as a polynomial \(s_\omega(\sigma_1,…
description: scientific article; zbMATH DE number 6355714