Manifolds compatible with the point with respect to involutions
type: Item
instance of: scholarly article
title: Manifolds compatible with the point with respect to involutions
author: Jessica C. R. R. Costa
author: Pedro L. Q. Pergher
author: Renato M. Moraes
published in: Journal of Fixed Point Theory and Applications
publication date: 2022-12-19T00:00:00Z
review text: The authors introduce the following notion of Property CP (\textit{compatible with the point}): A compact smooth manifold \(F\) is said to have Property CP (``compatible with a point'') if there is a (smooth) involution \((M,T)\) (that is, \(T:M\to M\) is a smooth involution on a compact of a smooth compact manifold \(M\),) such that the fixed point set of \(T\) is a disjoint union \(F\sqcup \{p\}\) where \(p\) is a point. It was shown by Conner and Floyd in 1964 that the only spheres which have the property CP are \(\mathbb S^n, n=1,2,4,8\), and \(\dim M=2\dim F=2n.\) Extending this result, Pergher classified products of spheres which have property CP. The real projective space \(\mathbb RP^n\) has the property CP, as can be seen by considering the involution \(T: \mathbb RP^{n+1}\to \mathbb RP^{n+1}\) induced by the reflection about the hyperplane \(\mathbb R^{n+1}\subset \mathbb R^{n+1}\). The same argument also shows that \(\mathbb CP^n\) and \(\mathbb HP^n\) have the property CP. The property CP is not an invariant of the unoriented cobordism. It is also not preserved by taking products. The authors consider class of Dold manifolds \(P(m,n)\), which is the quotient of \(\mathbb S^m\times \mathbb CP^n\) by the involution \((v,[z_0,\ldots,z_n])\mapsto (-v,[\bar z_0,\ldots, \bar z_n])\). Their main results, proved in Theorems 3.1-3.4, are the following: Let \(m,n\ge 1\). Then \(P(m,n)\) does not have property CP if any of the following hold: (1) \(m\) is odd, (2) \(m\equiv 0 \pmod 4, n,\) odd, (3) \(m=2, n,\) even, and, (4) \(6<m\le n\), and \(n\) odd. A crucial ingredient needed in the proofs are the Conner-Floyd theory, [\textit{P. E. Conner} and \textit{E. E. Floyd}, Differentiable periodic maps. Springer-Verlag, Berlin (1964; Zbl 0125.40103)] especifically, relation between the \(\mathbb Z_2\)-equivariant cobordism of an involution \((M,T)\) and its fixed point data, and the exactness of the Conner-Floyd exact sequence \(0\to \mathcal N_m^{\mathbb Z_2}(*)\to \…
Mathematics Subject Classification ID: 57R85
Mathematics Subject Classification ID: 57R75
zbMATH DE Number: 7634038
zbMATH Keywords: involution
zbMATH Keywords: fixed data
zbMATH Keywords: property \textit{CP}
zbMATH Keywords: equivariant cobordism class
zbMATH Keywords: characteristic number
zbMATH Keywords: real projective bundle
zbMATH Keywords: Hopf line bundle
zbMATH Keywords: Dold manifold
zbMATH Keywords: Conner-Floyd exact sequence
zbMATH Keywords: Wu formula
zbMATH Keywords: Conner formula
zbMATH Keywords: Steenrod operation
reviewed by: Parameswaran Sankaran
MaRDI profile type: Publication
full work available at URL: s11784-022-01014-y
OpenAlex ID: W4311318725
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DOI: 10.1007/S11784-022-01014-Y
description: scientific article; zbMATH DE number 7634038