Self-maps of loop spaces. I

Campbell, H. E. A.; Peterson, F. P.; Selick, P. S.

doi:10.1090/S0002-9947-1986-0814910-1

Self-maps of loop spaces. I
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by H. E. A. Campbell, F. P. Peterson and P. S. Selick

Trans. Amer. Math. Soc. 293 (1986), 1-39

DOI: https://doi.org/10.1090/S0002-9947-1986-0814910-1

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Abstract:

In this paper we study self-maps of ${\Omega ^k}{S^{m + 1}}$ and show that, except for the cases $m = 1, 3, 7$, or $p$ and $m$, if $f$ induces an isomorphism on ${H_{m + 1 - k}}({\Omega ^k}{S^{m + 1}}; Z/pZ)$ with $k < m$, then ${f_{(p)}}$ is a homotopy equivalence.

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