Finite Group Actions on Kervaire Manifolds

Diarmuid Crowley; Ian  Hambleton

doi:10.1016/j.aim.2015.06.010

Finite Group Actions on Kervaire Manifolds

Diarmuid Crowley, Ian Hambleton

Mathematical Science

McMaster University

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

12 Downloads (Pure)

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 􀀀 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
Pages (from-to)	88-129
Number of pages	42
Journal	Advances in Mathematics
Volume	283
Early online date	25 Jul 2015
DOIs	https://doi.org/10.1016/j.aim.2015.06.010
Publication status	Published - 1 Oct 2015

Bibliographical 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.

Keywords

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

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Actions_on_Kervaire_Manifolds
NOTICE: this is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, VOL 283 (2015) DOI: 10.1016/j.aim.2015.06.010 This manuscript is distributed in accordance with the Creative Commons Attribution Non Commercial-No Derivs (CC BY-NC-ND 4.0) license, which permits others to distribute this work non-commercially provided the original work is properly cited and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc-nd/4.0/
Accepted author manuscript, 492 KB

Cite this

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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 􀀀 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. ",

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author = "Diarmuid Crowley and Ian Hambleton",

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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.

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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 􀀀 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 􀀀 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

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VL - 283

SP - 88

EP - 129

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Finite Group Actions on Kervaire Manifolds

Abstract

Bibliographical note

Keywords

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