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Results tagged with homotopy-theory
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user 14447
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
177
votes
Accepted
Do we still need model categories?
I find some of this exchange truly depressing. There is a subject of ``brave
new algebra''and there are myriads of past and present constructions and calculations that
depend on having concrete and …
- 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
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
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
25
votes
Definition of an E-infinity algebra
Drinfeld once remarked to me something to the effect
that he likes the definition of an operad because it
is so simple. One doesn't have to be a Drinfeld to
appreciate the truth of that statement. It …
- 30.4k
23
votes
Why do homotopy theorists care whether or not $BP$ is $E_\infty$?
I suppose I should try to answer since the question of whether or not $BP$ is an $E_{\infty}$ ring spectrum
was Problem 1 of "Problems in infinite loop space theory'', http://www.math.uchicago.edu/~ma …
- 30.4k
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
21
votes
Accepted
Serre fibration vs Hurewicz fibration
A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space …
- 30.4k
20
votes
Accepted
Is the $\infty$-category of spectra “convenient”?
My colleague Dylan answered first (I keep telling him not to spend too much time on this toy :) but I both agree and disagree with his "Yes of course". The same words are used with different meanings …
- 30.4k
20
votes
Accepted
A category with weak equivalences that is not a model category
A very interesting example: consider semi-simplicial sets (alias $\Delta$-sets).
These are simplicial sets without degeneracies, and there is an ``adjoin degeneracies'' functor from semi-simplicial se …
- 30.4k
19
votes
Accepted
Computing homotopies
Sometimes easy geometric pictures have awkward seeming algebraic descriptions.
On pages 6 and 7 of Concise, I gave examples where I both gave a geometric picture
and explicit formulas to make the idea …
- 30.4k
19
votes
Accepted
Homotopy extension of $E_{\infty}$-spaces
This is probably belaboring the obvious, but just take seriously the equivalence between grouplike $E_{\infty}$ spaces and connective spectra. See for example
Equivalence between $E_\infty$-spaces …
- 30.4k
19
votes
A question about the topological proofs of Bott periodicity
There is a beautiful observation of my advisor John Moore that to my mind ought to be part of the focus of any such argument: the Hopf algebra $H_{\ast}(BU;Z)$, or equivalently $H^*(BU;Z)$ is self-dua …
- 30.4k
18
votes
Accepted
How much of homotopy theory can be done using only finite topological spaces?
Vidit, thanks for the advertisement; Paul I'll answer your email shortly.
As a minor point, there is a small but subtle mistake in Clader's work
that is corrected in Matthew Thibault's 2013 Chicago t …
- 30.4k