Anick's fibration and the odd primary homotopy exponent of spheres

TL;DR

This paper constructs Anick's fibration for primes p>=3 using new methods and leverages it to determine the p-primary homotopy exponent of odd-dimensional spheres, simplifying previous approaches.

Contribution

It introduces a new construction of Anick's fibration for p>=3 and applies it to establish the p-primary homotopy exponent of spheres, extending prior results.

Findings

Constructed Anick's fibration for p>=3 using novel methods

Derived the p-primary homotopy exponent of spheres from the fibration

Simplified previous proofs of the exponent result

Abstract

For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick further developed this for p>=5 by constructing a homotopy fibration S^2n-1 --> T^2n+1(p^r) --> Loop S^2n+1 whose connecting map is degree p^r on the bottom cell. A much simpler construction of such a fibration for p>=3 was given by Gray and the author using new methods. In this paper the new methods are used to start over, first constructing Anick's fibration for p>=3, and then using it to obtain the exponent result for spheres.