Marek
Reputation
4,307
Top tag
Next privilege 5,000 Rep.
Approve tag wiki edits
 Nov 7 awarded Yearling Jun 5 awarded Nice Question Dec 16 awarded Caucus Nov 26 awarded Popular Question Nov 7 awarded Yearling Oct 28 awarded Notable Question Oct 1 awarded Popular Question Jul 2 awarded Curious Jun 6 comment Why does differentiating a polynomial reduce its degree by $1$? This is exactly Mitchell's answer above. Why post the same thing again? Jun 3 comment Decomposition of cohomology group on $S^{n}$ Can you at least please fix the typos? The question is almost unreadable as it stands. Jun 2 revised Poincare dual of unit circle Added another explanation Jun 2 revised Poincare dual of unit circle Added definition of Poincare dual Jun 2 comment Poincare dual of unit circle @PeterM: I see, I didn't get that part. I think you misunderstood the definition of Poincare dual then, let me edit it into my answer. Jun 2 answered Poincare dual of unit circle Feb 6 comment What are necessary and sufficient conditions for the product of spheres to be paralellizable? But otherwise this is a very nice and probably classical question that hopefully some local expert will answer soon. To add my gut feeling -- you can't expect parallelizability in general, tangent bundles of spheres are quite complicated and there's no a priori reason for their direct sums to be trivial besides the trivial reason of adding the normal bundle. Feb 6 comment What are necessary and sufficient conditions for the product of spheres to be paralellizable? I don't think clutching functions are helpful here, at least when used naively. That is because the contractible neighborhoods for $S^i \times S^k$ will be products of hemispheres, so there's four of them in total with mutual intersections being homotopic to either $S^{i-1}$, $S^{k-1}$ or $S^{i-1} \times S^{k-1}$. Correspondingly, there will be multiple clutching functions depending on which overlap we're talking about -- the one you mention is for passing from southern to northern hemispheres in both spheres simultaneously. Jan 16 comment existence of double covering $GL(n, {\bf C})$ has the homotopy type of $U(n)$ which has the same fundamental group as $U(1)$ (this isomorphism is induced by $\det: U(n) \to U(1)$ and backwards by the inclusion $U(1) \to U(n)$ as scalars). Jan 15 comment Dolbeault cohomology and analytic regularity Indeed, nobody can stop you from doing that. But in that case you should change the question's title and body since it has nothing to do with complexes in general and Dolbeault cohomology in particular. Jan 15 comment Dolbeault cohomology and analytic regularity I don't really understand the question, even with the edit. When you have a complex $M^{\bullet}$, all the spaces are fixed ($C^1$, say) beforehand. You can't just decide that some of them will be $C^2$ when it suits you because you need to quotient. That doesn't make any sense. Jan 14 revised How do I maximize $|t-e^z|$, for $z\in D$, the unit disk? Replaced the picture