Free involutions on \(S ^{1} \times S ^{n }\)
type: Item
instance of: scholarly article
title: Free involutions on \(S ^{1} \times S ^{n }\)
zbMATH Open document ID: 1229.57031
author: Bjørn Jahren
author: Slawomir Kwasik
published in: Mathematische Annalen
publication date: 2011-10-25T00:00:00Z
review text: The authors study the classification problem of free involutions on \(S^1\times S^n\) up to conjugation for \(n\geq 3\). It is known that the quotients of the four simplest standard involutions are \(S^1\times S^n\), \(S^1\tilde{\times} S^n\), \(S^1\times \mathbb R\text{P}^n\) or the connected sum \(\mathbb R\text{P}^{n+1}\#\mathbb R\text{P}^{n+1}\), where \(S^1\tilde{\times} S^n\) denotes the non-trivial \(S^n\)-bundle over \(S^1\). First, the authors show that any free involution of \(S^1\times S^n\) is homotopically conjugate to one of the four standard involutions. Next, they prove that two free involutions on this space are topologically conjugate if and only if the two quotients are homeomorphic. In particular, they show that every involution is topologically conjugate to the standard involution if its quotient is of type \(S^1\times S^n\) or of type \(S^1\tilde{\times} S^n\). Moreover, they prove that there is a one-one correspondence between the set of conjugacy classes of free involutions and \[ \begin{cases} (\mathbb Z /2)^{\lfloor \frac{n-1}{2}\rfloor +\lfloor \frac{n-2}{4}\rfloor} &\text{ if }n\not\equiv 3 \mod 4\\ \mathbb Z_{\geq 0}\times (\mathbb Z/2)^{\frac{3n-2}{4}} &\text{ if }n\equiv 3 \mod 4 \end{cases} \] if its quotient is of type \(S^1\times \mathbb R\text{P}^n\). Their proof is based on surgery theory and some computations of homotopy groups by using the Fedeler-Schults spectral sequence.
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57R67
Mathematics Subject Classification ID: 55P10
zbMATH DE Number: 5963442
zbMATH Keywords: free involution
zbMATH Keywords: quotient manifold
zbMATH Keywords: conjugacy class
zbMATH Keywords: surgery
MaRDI profile type: Publication
full work available at URL: s00208-010-0599-y
OpenAlex ID: W1970230221
cites work: Classification of free involutions on surfaces
cites work: Manifolds homotopy equivalent to \(P^{n} \# P^{n}\)
cites work: Fixed point free involutions on homotopy spheres
cites work: A splitting theorem for manifolds
cites work: On the calculation of UNil
cites work: The surgery obstruction groups of the infinite dihedral group
cites work: Manifolds with π 1 = G x α T
cites work: Q4040346
cites work: On Siebenmann periodicity
cites work: Three-dimensional surgery theory, UNil-groups and the Borel conjecture.
cites work: Manifolds homotopy equivalent to $RP^4 \# RP^4$
cites work: On s-cobordisms of metacyclic prism manifolds
cites work: A survey of 4-manifolds through the eyes of surgery
cites work: Q5616706
cites work: Mappings of manifolds and the notion of degree
cites work: Q4003859
cites work: A composition formula for manifold structures
cites work: Wall's surgery obstruction groups for GxZ
cites work: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88)
cites work: Q5607104
cites work: Involutions on S 1 × S 2 and other 3-Manifolds
cites work: Q5624419
cites work: The homotopy of spaces of maps between real projective spaces
DOI: 10.1007/S00208-010-0599-Y
description: scientific article; zbMATH DE number 5963442