A reformulation of the Arf invariant one mod p problem and applications to atomic spaces

Paul Sydney Selick

Vol. 108 (1983), No. 2, 431–450

DOI: 10.2140/pjm.1983.108.431

Abstract

A (mod p) atomic space is one whose lowest nonvanishing (mod p) homology group has dimension 1 and which has the property that all self-maps which induce isomorphisms on this lowest nonvanishing group are homotopy equivalences. An atomic space cannot be decomposed, up to homotopy, into a produce of other spaces and thus is, in some sense, an atom. In this paper we show that if p is an odd prime and n > 1 then Ω³S²ⁿ⁺¹ and the homotopy-theoretic fibre of the double suspension Σ² : S²ⁿ⁻¹ → Ω²S²ⁿ⁺¹ are (mod p) atomic. Some indecomposability results are also obtained for the homotopy-theoretic fibre of the degree p map of ΩS²ⁿ⁺¹.

Mathematical Subject Classification 2000

Primary: 55P99

Secondary: 55Q40

Milestones

Received: 24 July 1981

Published: 1 October 1983