Topological rigidity and H1–negative involutions on tori

Connolly, Frank; Davis, James F; Khan, Qayum

doi:10.2140/gt.2014.18.1719

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Topological rigidity and $H_1$–negative involutions on tori

Frank Connolly, James F Davis and Qayum Khan

Geometry & Topology 18 (2014) 1719–1768

DOI: 10.2140/gt.2014.18.1719

Abstract

We show, for $n \equiv 0, 1 (mod 4)$ or $n = 2, 3$ , there is precisely one equivariant homeomorphism class of $C_{2}$ –manifolds $(N^{n}, C_{2})$ for which $N^{n}$ is homotopy equivalent to the $n$ –torus and $C_{2} = {1, σ}$ acts so that $σ_{*} (x) = - x$ for all $x \in H_{1} (N)$ . If $n \equiv 2, 3 (mod 4)$ and $n > 3$ , we show there are infinitely many such $C_{2}$ –manifolds. Each is smoothable with exactly $2^{n}$ fixed points.

The key technical point is that we compute, for all $n \geq 4$ , the equivariant structure set $S_{TOP} (ℝ^{n}, Γ_{n})$ for the corresponding crystallographic group $Γ_{n}$ 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

Volume 18, issue 3 (2014)