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Results tagged with at.algebraic-topology
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user 14447
Homotopy theory, homological algebra, algebraic treatments of manifolds.
87
votes
Accepted
Proof that a local fibration is a fibration, in May
May has carelessly made a typo in transcribing a proof in one of his books to another,
and it has taken an impressive high school student in Korea to catch it. In his earlier book ``Classifying
Spac …
- 30.4k
81
votes
Accepted
Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop ...
I'm not quite certain what Peter May had in mind 40 years ago,
but probably he had in mind the fact that pushouts are a lot better
behaved in CGWH than in CGH. Specifically, CGWH is closed
under push …
- 30.4k
51
votes
Algebraic topology beyond the basics: any texts bridging the gap?
Good lord, Charles, was the reposting of this an invitation for another advertisement from me?
``More concise algebraic topology. Localization, completion, and model categories'', by Kate Ponto
and my …
47
votes
Why higher category theory?
This is addressed to Paul Siegel's answer, which I find misleading, and
not just because it is not true that maps that induce the same
map on homotopy groups are homotopic. (The famous and open Frey …
33
votes
Are spectra really the same as cohomology theories?
The answer to this question is in LMS (I.6.9 of http://www.math.uchicago.edu/~may/BOOKS/equi.pdf)
and in McClure's contribution to BMMS (VII\S1 of http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf), …
- 30.4k
33
votes
Accepted
Did Peter May's "The homotopical foundations of algebraic topology" ever appear?
An anonymous source told me this question is here. Dylan gave the quick answer and Tyler referred to it.
I'll use the question as an excuse to give a pontificating longer answer. When I first planned …
- 30.4k
30
votes
Techniques for computing cup products in singular cohomology
This is going to be a perhaps tendentious diatribe. But it is what is.
Naturality, dimensional arguments, and Poincare duality give a
reservoir of elementary examples such as spheres and projective …
- 30.4k
30
votes
Natural transformations as categorical homotopies
The homotopy analogue definition of natural transformations has been known and used regularly
since at least the late 1960's, by which time it was understood that the classifying space
functor from (s …
- 30.4k
27
votes
Accepted
Finiteness of stable homotopy groups of spheres
I agree with Ryan that Serre's proof can be viewed as perfectly conceptual, but here is a modern version. Accept from Serre that the homotopy groups of spheres are finitely generated. Let
$k\colon …
- 30.4k
26
votes
Intuition behind Thom class
You are thinking in terms of ordinary cohomology, where Mayer-Vietoris patches together the always present local orientation to produce a global one when you have it. It is more advanced, but maybe m …
- 30.4k
25
votes
origin of spectral sequences in algebraic topology
I am very surprised that nobody has yet mentioned Massey's beautiful
general theory of exact couples. This to my mind answers the
alternative version of the question, and does so without restriction …
- 30.4k
23
votes
Accepted
A toolbox for algebraic topology
The subject is really way too big (as are so many others of course). I worry a lot about students not in Cambridge or Chicago or Stanford or other places where there are people with folklore at their …
23
votes
Accepted
Modern source for spectra (including ring spectra)
[I'm a novice, and this got posted out of order: it answers Bak's question below.]
Sure, I can provide that. The cited reference was published in 1995, which
was well before details of symmetric or …
22
votes
Why study the p-completions of a space?
I'll give an answer from the point of view of an algebraic topologist,
somebody who cares about examples and computations. This all goes way back
and has nothing to do with modern generalities. Fir …
- 30.4k
22
votes
(co)homology of symmetric groups
The paradoxical answer is that it is annoying but straightforward to determine $H_k(S_n)$
for particular values of $k$ and $n$, but it is very easy to write down $H_*(\coprod BS_n)$,
encompassing all …
- 30.4k