Vol. 108, No. 2, 1983

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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

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 : S2n1 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
Received: 24 July 1981
Published: 1 October 1983
Paul Sydney Selick
Department of Mathematics
University of Toronto
Toronto ON M5S 3G3