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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Topological rigidity and $H_1$–negative involutions on tori

Frank Connolly, James F Davis and Qayum Khan

Geometry & Topology 18 (2014) 1719–1768
Abstract

We show, for n 0,1(mod4) or n = 2,3, there is precisely one equivariant homeomorphism class of C2–manifolds (Nn,C2) for which Nn is homotopy equivalent to the n–torus and C2 = {1,σ} acts so that σ(x) = x for all x H1(N). If n 2,3(mod4) and n > 3, we show there are infinitely many such C2–manifolds. Each is smoothable with exactly 2n fixed points.

The key technical point is that we compute, for all n 4, the equivariant structure set STOP(n,Γn) for the corresponding crystallographic group Γn in terms of the Cappell UNil–groups arising from its infinite dihedral subgroups.

Keywords
equivariant rigidity, torus, surgery
Mathematical Subject Classification 2010
Primary: 57S17
Secondary: 57R67
References
Publication
Received: 20 February 2012
Revised: 15 November 2013
Accepted: 15 December 2013
Published: 7 July 2014
Proposed: Steve Ferry
Seconded: Peter Teichner, Ronald Stern
Authors
Frank Connolly
Department of Mathematics
University of Notre Dame
255 Hurley
Notre Dame, IN 46556
USA
James F Davis
Department of Mathematics
Indiana University
Rawles Hall
831 E. 3 St.
Bloomington, IN 47405
USA
Qayum Khan
Department of Mathematics
Saint Louis University
220 North Grand Blvd
St Louis, MO 63103
USA