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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Kervaire invariants and selfcoincidences

Ulrich Koschorke and Duane Randall

Geometry & Topology 17 (2013) 621–638
Abstract

Minimum numbers decide, eg, whether a given map f : Sm SnG from a sphere into a spherical space form can be deformed to a map f such that f(x)f(x) for all x Sm. In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In the stable dimension range these numbers coincide. But already in the first nonstable range (when m = 2n 2) the Kervaire invariant appears as a decisive additional obstruction which detects interesting geometric coincidence phenomena. Similar results (involving, eg, Hopf invariants taken mod 4) are obtained in the next seven dimension ranges (when 1 < m 2n + 3 8). The selfcoincidence context yields also a precise geometric criterion for the open question whether the Kervaire invariant vanishes on the 126–stem or not.

Keywords
Kervaire invariant, coincidence, Nielsen number
Mathematical Subject Classification 2010
Primary: 55M20, 55P40, 55Q15, 55Q25, 57R99
Secondary: 55Q45
References
Publication
Received: 9 August 2011
Revised: 9 August 2012
Accepted: 26 September 2012
Published: 9 April 2013
Proposed: Shigeyuki Morita
Seconded: Ralph Cohen, Steve Ferry
Authors
Ulrich Koschorke
FB 6 - Mathematik V
Universität Siegen
Emmy Noether Campus
Walter-Flex-Str. 3
D-57068 Siegen
Germany
http://www.math.uni-siegen.de/topologie/
Duane Randall
Loyola University New Orleans
Mathematical Sciences
6363 St. Charles Ave.
Campus Box 35
New Orleans, LA 70118
USA