18
votes
5answers
2k views

A toolbox for algebraic topology

This question has a very general part and a rather concrete part. General: When one wants to prove something in algebraic topology (actually in all parts of mathematics) one obviously needs some ...
16
votes
3answers
973 views

What do whitehead towers have to do with physics?

First let me say something that I don't completely understand, since I do not know enough physics. If I say anything wrong, someone please tell me: For the spinning particle, there is a sigma-model, ...
14
votes
6answers
2k views

Why chain homotopy when there is no topology in the background?

Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow ...
6
votes
1answer
559 views

Analogue to Serre spectral sequence for cofiber sequences and homotopy

(This is a follow-up question to this one). As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well ...
8
votes
2answers
1k views

Why does homotopy behave well with respect to fibrations and homology with respect to cofibrations?

(I apologize that this is a vague question). I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups ...
25
votes
10answers
2k views

Are infinite dimensional constructions needed to prove finite dimensional results?

Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me ...
12
votes
4answers
2k views

Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...
3
votes
1answer
366 views

Morava's “Motives and cell bundles”?

Hello, do you know more about, or some exposition of Morava's talk?
4
votes
4answers
1k views

Geometric interpretation of the fundamental groupoid

Motivation The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy ...
16
votes
3answers
809 views

Natural setting for characteristic classes?

In my mind, algebraic topology is comprised of two components: Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks". ...
5
votes
2answers
513 views

visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...
19
votes
4answers
2k views

Visualizing how Cech cohomology detects holes

I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space. How can we directly visualize how and in what sense the Cech cohomology of a cover does this? In ...