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22
votes
10answers
6k views

What is (co)homology, and how does a beginner gain intuition about it?

This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...
11
votes
7answers
1k views

Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
24
votes
3answers
3k views

Spaces with same homotopy and homology groups that are not homotopy equivalent?

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...
9
votes
6answers
2k views

CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature. Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
18
votes
3answers
2k views

Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
11
votes
1answer
502 views

On the wikipedia entry for Borel-Moore homology

The wikipedia page on Borel-Moore homology claims to give several definitions of it, all of which are supposed to coincide for those spaces $X$ which are homotopy equivalent to a finite CW complex and ...
7
votes
1answer
427 views

Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?

Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...
4
votes
1answer
113 views

0-homologous surface bounds

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented ...
3
votes
1answer
210 views

Second betti number of compact analytic spaces

Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know: 1) if its second Betti number is non zero, 2) same question but now $V$ is a ...