All Questions
3,701 questions
176
votes
7
answers
19k
views
Proofs of Bott periodicity
K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of ...
- 6,308
147
votes
10
answers
16k
views
What non-categorical applications are there of homotopical algebra?
(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)
More ...
125
votes
4
answers
8k
views
What do the stable homotopy groups of spheres say about the combinatorics of finite sets?
The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
$\mathbb{Z}\...
- 23k
119
votes
6
answers
10k
views
What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
- 5,759
106
votes
4
answers
13k
views
What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...
- 42.4k
101
votes
6
answers
15k
views
Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets ...
- 25.6k
88
votes
4
answers
8k
views
Is there an accepted definition of $(\infty,\infty)$ category?
For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
- 54.6k
84
votes
4
answers
22k
views
Do we still need model categories?
One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
- 25.6k
83
votes
0
answers
3k
views
Which finite abelian groups aren't homotopy groups of spheres?
Someone asked me if all finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know ...
- 22.3k
80
votes
10
answers
11k
views
What are the uses of the homotopy groups of spheres?
Pete Clark threw down the challenge in his comment to my answer on Why the heck are the homotopy groups of the sphere so damn complicated?:
Have the homotopy groups of spheres ever been applied to ...
- 26.8k
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 2,392
70
votes
6
answers
8k
views
third stable homotopy group of spheres via geometry?
It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...
- 20.9k
66
votes
4
answers
6k
views
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
- 1,187
63
votes
5
answers
18k
views
What is modern algebraic topology(homotopy theory) about?
At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
62
votes
9
answers
9k
views
There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?
Quadratic forms play a huge role in math. This leads one to wonder: Is there a theory of cubic forms, quartic forms, quintic forms and so on? I have failed to discover any. Is there any such theory? ...
- 705
61
votes
7
answers
5k
views
What are surprising examples of Model Categories?
Background
Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three ...
- 13k
55
votes
3
answers
5k
views
What are the higher homotopy groups of Spec Z ?
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
- 8,004
53
votes
8
answers
9k
views
Analogue to covering space for higher homotopy groups?
The connection between the fundamental group and covering spaces is quite fundamental. Is there any analogue for higher homotopy groups? It doesn't make sense to me that one could make a branched ...
- 13.6k
52
votes
2
answers
4k
views
What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
My question is as in the title:
Does anyone have an example (supposing one exists) of an
$\infty$-topos which is known not to be equivalent to sheaves on a
Grothendieck site?
An $\infty$-topos is as ...
- 27.2k
51
votes
3
answers
3k
views
Is each squared finite group trivial?
A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
Problem: Is each squared finite group ...
- 41.9k
51
votes
5
answers
5k
views
What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...
- 40.2k
51
votes
3
answers
12k
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 ...
- 10.1k
50
votes
6
answers
7k
views
Total spaces of $TS^2$ and $S^2 \times R^2$ not homeomorphic
Hello,
I'm looking for an invariant to distinguish the homeomorphism types of homotopy equivalent spaces. Specifically, how does one show that the total spaces of the tangent bundle to $S^2$ and the ...
- 931
49
votes
2
answers
6k
views
Why should I care about topological modular forms?
There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...
- 491
48
votes
6
answers
4k
views
Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)?
In his 1967 paper A convenient category of topological spaces,
Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces
as a good replacement of the category Top topological ...
- 43.2k
48
votes
2
answers
8k
views
Why is Voevodsky's motivic homotopy theory 'the right' approach?
Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...
- 1,098
46
votes
6
answers
7k
views
Why does one think to Steenrod squares and powers?
I'm studying Steenrod operations from Hatcher's book. Like homology, one can use them only knowing the axioms, without caring for the actual construction. But while there are plenty of intuitive ...
- 14.7k
46
votes
5
answers
3k
views
‘Naturally occurring’ $K(\pi, n)$ spaces, for $n \geq 2$
[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg–MacLane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (Or if not a manifold, say some ...
- 2,734
46
votes
8
answers
11k
views
Non-examples of model structures, that fail for subtle/surprising reasons?
An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...
- 21.3k
46
votes
2
answers
4k
views
What are the potential applications of perfectoid spaces to homotopy theory?
This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ...
46
votes
0
answers
6k
views
Cochains on Eilenberg-MacLane Spaces
Let $p$ be a prime number, let $k$ be a commutative ring in which $p=0$, and let
$X = K( {\mathbb Z}/p {\mathbb Z}, n)$ be an Eilenberg-MacLane space.
Let $F$ be the free $E_{\infty}$-algebra over $k$ ...
- 17.8k
45
votes
1
answer
9k
views
Homotopy Type Theory: What is it?
My question is, broadly, what is the main project of Homotopy Type Theory (HoTT). I asked a professor who is likely to be correct and he say the following:
There are three directions:
...
- 551
44
votes
9
answers
3k
views
Homotopy as a general organizing principle
One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories ...
44
votes
5
answers
6k
views
What is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
- 63.9k
44
votes
4
answers
5k
views
Integral cohomology (stable) operations
There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...
- 3,726
43
votes
4
answers
6k
views
Why the Dold-Thom theorem?
Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$
It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...
- 17.5k
42
votes
12
answers
7k
views
Why is the definition of the higher homotopy groups the "right one"?
If someone asked me the question for the fundamental group, I would answer as follows:
The connection to classification of covering spaces.
The fundamental group of many spaces is an object of ...
42
votes
2
answers
5k
views
Homotopy groups of $S^2$
in the paper
Foundations of the theory of bounded cohomology,
by N.V. Ivanov, the author considers the complex of bounded singular cochains on a simply connected CW-complex $X$, and constructs a ...
- 3,749
42
votes
5
answers
4k
views
What are the main structure theorems on finitely generated commutative monoids?
I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's ...
- 22.3k
41
votes
4
answers
8k
views
Definition of an E-infinity algebra
Can anyone give me a plain-and-simple
definition of an E-infinity algebra without using
the words "operad," "ring spectrum," or
"stable homotopy"?
Sorry, but I honestly couldn't find it using
all on-...
- 3,267
41
votes
1
answer
3k
views
Are there any "homotopical spaces"?
This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense.
[Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ...
- 23.3k
41
votes
4
answers
2k
views
What is the probability two random maps on n symbols commute?
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
- 38.6k
41
votes
1
answer
10k
views
Why not a Roadmap for Homotopy Theory and Spectra?
MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up ...
- 1,162
41
votes
0
answers
1k
views
Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?
Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...
- 63.9k
41
votes
0
answers
1k
views
Homotopy type of TOP(4)/PL(4)
It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(...
- 6,210
40
votes
16
answers
11k
views
"Homotopy-first" courses in algebraic topology
A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...
40
votes
3
answers
7k
views
Timeline of "foundational" advances in homotopy theory?
As an interested outsider, I have been intrigued by the number of times that homotopy theory seems to have revamped its foundations over the past fifty years or so. Sometimes there seems to have been ...
40
votes
1
answer
2k
views
What can topological modular forms do for number theory?
Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E_\infty$ ring spectra ...
- 3,799
39
votes
3
answers
6k
views
Why do finite homotopy groups imply finite homology groups?
Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
- 2,782
39
votes
4
answers
6k
views
What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?
In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...
- 8,512