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If I remember correctly, the proof of homotopy invariance of singular homology (at least the one in Hatcher) involves cutting a (simplex)x(interval) into simplices, which perhaps can be confusing. In …

One can define the $G$-equivariant cohomology of a space $X$ as being the ordinary singular cohomology of $X \times_G EG$ --- I think this is due to Borel? (See e.g. section 2 of these notes) Alterna …

When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition". However, …

One possible answer: Stasheff proved that a (connected) space $X$ is (homotopy equivalent to) a loopspace if and only if $X$ is an algebra over the $A_\infty$ operad (or rather I should say an $A_\inf …

I found a paper which I think answers #2: Baum, Fulton, MacPherson Riemann-Roch and topological K-theory for singular varieties. They prove that the algebraic $f_!$ and the topological $f_!$ agree i …

Apparently you can compute the h^{p,q}'s of smooth things in, for example, Macaulay. Here's an example: computing the h^{p,q}'s of a quintic hypersurface in P^4.

I think Hatcher's book has a good elementary exposition of some of the differences between homology and cohomology. I believe it's in the introduction to the cohomology chapter.

I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology. Do you know any good ones?

This question is related to this one. I am wondering whether there is any sort of classification of simply connected smooth projective varieties, or any work in related directions. The reason I am i …

I recently read the original paper by Chas-Sullivan on string topology, in which they introduce some operations on homology of free loopspace LM, where M is a compact oriented manifold, giving it the …

I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard exam …

Let $f : X \to Y$ be a [fill in the blank] morphism of [fill in the blank] complex varieties. Then we have the pushforward $f_! : K(X) \to K(Y)$ which is defined by $f_!(E) = \sum_i (-1)^i [R^i f_\ast …

Are there any general methods for computing fundamental group or singular cohomology (including the ring structure, hopefully) of a projective variety (over C of course), if given the equations defini …

One application that I know of the Mumford conjecture is Teleman's proof of Givental's conjecture in this paper. Givental's conjecture states that when the quantum cohomology of a smooth projective va …

Try looking up some references on rational homotopy theory. Rational homotopy theory studies the homotopy groups tensor Q, so basically you kill all torsion information. If we focus only on homotopy g …