Finite Group Actions on Kervaire Manifolds

Diarmuid Crowley, Ian Hambleton

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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 languageEnglish
Pages (from-to)88-129
Number of pages42
JournalAdvances in Mathematics
Volume283
Early online date25 Jul 2015
DOIs
Publication statusPublished - 1 Oct 2015

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Keywords

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

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