Unstable homotopy theory

This is part of an algebraic topology problem list, maintained by Mark Hovey.

  1. The Johnson question. This says that if X is a space, and x is in BP_n (X), then x is not v_n torsion. My guess is that one should consider this question as a test case for whether there can be a really powerful analog of the Lannes theory of unstable algebras over the Steenrod algebra using BP instead. Such a theory has been constructed by Boardman, Johnson, and Wilson, but so far it has been unable to resolve this question.

  2. Determine the v_1 -exponents for the spheres. Recall that Cohen, Moore, and Neisendorfer showed that the p-torsion in the homotopy of S^2n+1 is all killed by p^n, but not p^n-1, for odd p. Determine the analogous v_1-exponent. I am not quite clear on the statement of this problem; I think we want to look at maps from the Moore space into X, so that the Adams map actually acts. There is probably a conjecture out there about what the exponent should be, but again I do not know it. Hopefully somebody will let me know and I can clean up this problem.

  3. Suppose X is a simply connected finite complex. Do the Steenrod reduced powers P^t act trivially on the mod p cohomology of the loop space of X when p is sufficiently large? This question is apparently due to Wilkerson, and is taken from Chuck McGibbon's problem list on phantom maps, in Stable and unstable homotopy, Fields Institute Communications 19 (a book published by the AMS), where there are other questions as well.

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Mark Hovey
Department of Mathematics
Wesleyan University