Kervaire invariants and selfcoincidences

TL;DR

This paper explores how Kervaire invariants influence minimum numbers and Nielsen numbers in sphere maps, revealing new geometric coincidence phenomena and criteria related to the Kervaire invariant's behavior.

Contribution

It introduces the role of Kervaire invariants as obstructions in nonstable dimensions and compares them with Nielsen numbers, providing new geometric insights.

Findings

Kervaire invariant acts as an obstruction in nonstable dimensions

Nielsen numbers coincide with minimum numbers in stable ranges

Provides criteria for the vanishing of the Kervaire invariant on the 126-stem

Abstract

Minimum numbers decide e.g. whether a given map f: S^m --> S^n/G from a sphere into a spherical space form can be deformed to a map f' such that f(x) not equal f'(x) for all x in S^m. 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 e.g. Hopf invariants, taken mod 4) are obtained in the next seven dimension ranges (when 1<m-2n+3<9). The selfcoincidence context yields also a precise geometric criterion for the open question whether the Kervaire invariant vanishes on the 126-stem or not.