**121**

votes

**7**answers

10k 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 ...

**108**

votes

**10**answers

7k 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 ...

**85**

votes

**4**answers

3k 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:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
...

**75**

votes

**6**answers

4k 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 ...

**52**

votes

**9**answers

5k 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 ...

**51**

votes

**6**answers

5k 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 ...

**47**

votes

**6**answers

3k 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 ...

**47**

votes

**2**answers

2k 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**

votes

**4**answers

7k 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 ...

**41**

votes

**6**answers

3k 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 ...

**41**

votes

**3**answers

2k 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 ...

**39**

votes

**1**answer

1k 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 ...

**37**

votes

**6**answers

4k 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 ...

**37**

votes

**9**answers

2k 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 ...

**37**

votes

**1**answer

4k 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 $k$ ...

**37**

votes

**0**answers

934 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 ...

**36**

votes

**6**answers

2k 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 ...

**36**

votes

**2**answers

986 views

### Is there an analog of Sperner's lemma for the Hopf invariant?

Recall that Sperner's lemma is essentially a combinatorial version of the topological statement "A map from $S^n$ to $S^n$ with degree one cannot be nullhomotopic."
My question is, does there exist ...

**35**

votes

**3**answers

5k 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 ...

**34**

votes

**3**answers

3k 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. ...

**34**

votes

**2**answers

3k 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 ...

**34**

votes

**1**answer

2k 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 ...

**33**

votes

**2**answers

2k 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 ...

**33**

votes

**1**answer

942 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 ...

**33**

votes

**0**answers

1k views

### Finite complexes whose homotopy groups are not “finitely generated”

I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$.
It seems likely that ...

**32**

votes

**3**answers

5k views

### What is the “intuition” behind “brave new algebra”?

Y.I. Manin mentions in a recent interview
the need for a “codification of efficient new intuitive tools, such as … the “brave new algebra” of homotopy theorists”. This makes me puzzle, because I ...

**32**

votes

**4**answers

2k views

### Why is it so hard to compute $\pi_n(S^n)$?

Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how ...

**32**

votes

**3**answers

2k views

### Are there pairs of highly connected finite CW-complexes with the same homotopy groups?

Fix an integer n. Can you find two finite CW-complexes X and Y which
* are both n connected,
* are not homotopy equivalent, yet
* $\pi_q X \approx \pi_q Y$ for all $q$.
In Are there two ...

**32**

votes

**0**answers

925 views

### Functor that maps to both $KO^n$ and $KO^{-n}$

(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ...

**31**

votes

**6**answers

3k 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 ...

**31**

votes

**5**answers

2k views

### `Naturally occuring' $K(\pi, n)$ 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 space ...

**31**

votes

**2**answers

1k views

### Maps which induce the same homomorphism on homotopy and homology groups are homotopic

I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...

**31**

votes

**5**answers

3k 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 ...

**31**

votes

**4**answers

2k 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 ...

**30**

votes

**5**answers

3k 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 ...

**30**

votes

**4**answers

4k 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 ...

**29**

votes

**1**answer

582 views

### Classifiying sphere eversions

For a year I have been giving lectures on a (probalby) new way to present an explicit sphere eversion. These lectures include a review of many other explicit eversions that have been described, as ...

**29**

votes

**4**answers

2k 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 ...

**29**

votes

**2**answers

1k views

### What do loop groups and von Neumann algebras have to do with elliptic cohomology?

Recall that complex $K$-theory is a cohomology theory on topological spaces, which can be described in several equivalent ways:
Given a finite complex $X$, $K^0(X)$ is the Grothendieck group of ...

**29**

votes

**3**answers

2k views

### For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms

This is a spinoff of Can anyone give me a good example of two interestingly different ordinary cohomology theories? . By an ordinary homology theory, I mean a functor on topological spaces which ...

**28**

votes

**3**answers

2k views

### Topological Langlands?

In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...

**28**

votes

**2**answers

1k views

### Is there a “simplification” functor in algebraic topology?

Recall that a space (=CW complex) is called simple if it is connected, the fundamental group is abelian, and the fundamental group acts trivially on all higher homotopy groups. Call Simp(X) a ...

**27**

votes

**6**answers

3k views

### How to think about model categories?

I've read about model categories from an Appendix to one of Lurie's papers.
What are the examples of model categories? What should be my intuition about them?
E.g. I understand the typical examples ...

**27**

votes

**5**answers

2k views

### How can I visualize the nontrivial element of $\pi_4(S^3)$ and $\pi_5(S^3)$ ?

I've read in the textbooks that the non-trivial generator $\eta_n$ of $\pi_{n+1}(S^n)$ is the suspension of the Hopf map $S^3\to S^2$, and the generator $\chi$ of $\pi_5(S^3)$ is given by $\eta_3 ...

**27**

votes

**4**answers

2k 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 ...

**26**

votes

**15**answers

7k 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 ...

**26**

votes

**3**answers

2k views

### Are the higher homotopy groups of the Hawaiian earring trivial?

The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not ...

**26**

votes

**3**answers

2k views

### In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.
He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie ...

**26**

votes

**3**answers

2k views

### How should a homotopy theorist think about sheaf cohomology?

As a student of homotopy theory or algebraic topology, I have a certain outlook as to how one ought to think of a cohomology theory. There are axioms that help us with rudimentary computations, there ...

**26**

votes

**2**answers

1k views

### What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...