Unstable homotopy theory
This is part of an algebraic topology problem
list, maintained by Mark Hovey.
- 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
- 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.
- 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.
Department of Mathematics