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Homotopy theory, homological algebra, algebraic treatments of manifolds.

21 votes

Rational homotopy groups of $S^2\vee S^2$

Piotr, I think the question you really want to ask is: Given a map $f:S^n \to S^2 \vee S^2$, how do I prove that it is homotopically nontrivial? This is the so-called homotopy period problem, and it …
Fedya's user avatar
  • 934
14 votes
1 answer
1k views

Whitehead products in homotopy groups of spheres

Here is what I know about Whitehead products in homotopy groups of spheres: $[\mathrm{id}_{S^{2n}},\mathrm{id}_{S^{2n}}]$ has Hopf invariant (EDIT: $\pm$) two. No element that survives into the stab …
3 votes

A variant of $\ell^2$-cochains

I'm not sure if this should be a comment rather than an answer, since I'm not attempting to answer the hard questions (2) and (3). But it's too long to be a comment. That said: First of all, I just …
Fedya's user avatar
  • 934
7 votes
Accepted

A homeomorphism with a prescribed action on the fundamental group - decidable or not?

This is undecidable. For a construction, see the first page of Nabutovsky and Weinberger, "Algorithmic aspects of homeomorphism problems", Rothenberg Festschrift 1998.
Fedya's user avatar
  • 934