# Finite Group Actions on Kervaire Manifolds

Diarmuid Crowley, Ian Hambleton

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

Let M4k+2 K be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1 S2k+1 of spheres. We show that a nite group of odd order acts freely on M4k+2 K if and only if it acts freely on S2k+1 S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k 6= 2j &#x100000; 1, then M4k+2 K does not admit any free TOP involutions. Free \exotic" (PL) involutions are constructed on M30 K , M62 K , and M126 K . Each smooth structure on M30 K admits a free Z=2 Z=2 action.
Original language English 88-129 42 Advances in Mathematics 283 25 Jul 2015 https://doi.org/10.1016/j.aim.2015.06.010 Published - 1 Oct 2015

### Fingerprint

Finite Group Action
Involution
Piecewise Linear
Homology
Odd
Closed
Invariant

### Keywords

• Finite group actions
• Kerviare manifold
• Piecewise linear topology
• Surgery theory
• Smoothing theory

### Cite this

Finite Group Actions on Kervaire Manifolds. / Crowley, Diarmuid; Hambleton, Ian .

In: Advances in Mathematics, Vol. 283, 01.10.2015, p. 88-129.

Research output: Contribution to journalArticle

Crowley, Diarmuid ; Hambleton, Ian . / Finite Group Actions on Kervaire Manifolds. In: Advances in Mathematics. 2015 ; Vol. 283. pp. 88-129.
title = "Finite Group Actions on Kervaire Manifolds",
abstract = "Let M4k+2 K be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1 S2k+1 of spheres. We show that a nite group of odd order acts freely on M4k+2 K if and only if it acts freely on S2k+1 S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k 6= 2j &#x100000; 1, then M4k+2 K does not admit any free TOP involutions. Free \exotic{"} (PL) involutions are constructed on M30 K , M62 K , and M126 K . Each smooth structure on M30 K admits a free Z=2 Z=2 action. ",
keywords = "Finite group actions, Kerviare manifold, Piecewise linear topology, Surgery theory, Smoothing theory",
author = "Diarmuid Crowley and Ian Hambleton",
note = "Date of Acceptance: 09/06/2015 Acknowledgements We would like to thank Bruce Williams, Jim Davis, Martin Olbermann, John Klein, Mark Behrens and Wolfgang Steimle for useful information. We would also like to thank the referee for helpful comments and suggestions.",
year = "2015",
month = "10",
day = "1",
doi = "10.1016/j.aim.2015.06.010",
language = "English",
volume = "283",
pages = "88--129",
issn = "0001-8708",

}

TY - JOUR

T1 - Finite Group Actions on Kervaire Manifolds

AU - Crowley, Diarmuid

AU - Hambleton, Ian

N1 - Date of Acceptance: 09/06/2015 Acknowledgements We would like to thank Bruce Williams, Jim Davis, Martin Olbermann, John Klein, Mark Behrens and Wolfgang Steimle for useful information. We would also like to thank the referee for helpful comments and suggestions.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - Let M4k+2 K be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1 S2k+1 of spheres. We show that a nite group of odd order acts freely on M4k+2 K if and only if it acts freely on S2k+1 S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k 6= 2j &#x100000; 1, then M4k+2 K does not admit any free TOP involutions. Free \exotic" (PL) involutions are constructed on M30 K , M62 K , and M126 K . Each smooth structure on M30 K admits a free Z=2 Z=2 action.

AB - Let M4k+2 K be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1 S2k+1 of spheres. We show that a nite group of odd order acts freely on M4k+2 K if and only if it acts freely on S2k+1 S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k 6= 2j &#x100000; 1, then M4k+2 K does not admit any free TOP involutions. Free \exotic" (PL) involutions are constructed on M30 K , M62 K , and M126 K . Each smooth structure on M30 K admits a free Z=2 Z=2 action.

KW - Finite group actions

KW - Kerviare manifold

KW - Piecewise linear topology

KW - Surgery theory

KW - Smoothing theory

U2 - 10.1016/j.aim.2015.06.010

DO - 10.1016/j.aim.2015.06.010

M3 - Article

VL - 283

SP - 88

EP - 129