47
votes

### What is modern algebraic topology(homotopy theory) about?

While I think that Andre is right in saying that homotopy theory (or algebraic topology) is ready to study everything that fits into the framework of abstract homotopy theory, some things have still ...

Community wiki

44
votes

Accepted

### Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

Yes, such M exists. The boundary connected sum of 28 copies of the Milnor plumbing has boundary diffeomorphic to $S^7$ so it can be closed off with $D^8$ and you can let $M$ be the connected sum of ...

40
votes

Accepted

### Morava K-theories for dummies?

This is a result in algebraic topology, where we study the structure of topological spaces $X$. One early way to do this is to calculate a thing called $H_*(X)$, the ordinary homology of $X$. Later ...

31
votes

Accepted

### Endomorphism ring spectrum of the Eilenberg-MacLane spectrum

No, $A$ is not an $H\Bbb Z$-algebra.
Suppose $R$ is an $H\Bbb Z$-algebra. Then the category of left $R$-modules is $H\Bbb Z$-linear: for any $R$-modules $M$ and $N$, the function spectrum $F_R(M,N)$ ...

28
votes

### Stable homotopy theory and physics

There's an interesting application of stable homotopy theory to condensed-matter physics, and it makes heavy use of integral and torsion information, contradicting your 4th assumption.
Within the ...

28
votes

### Why not a Stacks project for Homotopy Theory?

Easy. Who's gonna write it?
JDJ (Johan de Jong) has written almost the entire stacks project himself.

28
votes

### References and resources for (learning) chromatic homotopy theory and related areas

Preliminaries (i.e. Advanced Algebraic Topology)
General References
Advanced Algebraic Topology, Alexander Kupers;
More Concise Algebraic Topology, J. Peter May;
Introduction to Homotopy Theory, ...

Community wiki

27
votes

Accepted

### A refinement of Serre's finiteness theorem on unstable homotopy groups of spheres

$P(\eta) = [i_n,i_n] \circ \eta$, where $[i_n,i_n]: S^{2n-1} \rightarrow S^n$ is the Whitehead product of the identity map with itself. So you are asking if this composite is null.
I don't know if ...

27
votes

### Latest results in chromatic homotopy theory

I want to mention five directions where in the last years significant progress has been made in chromatic homotopy theory. This is of course not exclusive!
Unstable chromatic homotopy theory
Among the ...

26
votes

Accepted

### What happened to the last work Gaunce Lewis was doing when he died?

Lewis wrote, but never published, a very influential paper setting foundations for the multiplicative theory for Mackey functors. The paper is called "The Theory of Green functors" and Doug ...

25
votes

Accepted

### Idempotent ring spectrum

There are actually quite a number of other examples. In particular, this is necessary and sufficient for $R$ to be a so-called smashing localization of the sphere $\Bbb S$, and there are several ...

24
votes

### What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

Write $S = \bigsqcup_n BS_n$ for the symmetric monoidal category of finite sets and bijections under disjoint union, and write $\mathbb{S}$ for the sphere spectrum, thought of as a symmetric monoidal $...

24
votes

Accepted

### Nilpotence of the stable Hopf map via framed cobordism

Answer Summary
Let $\eta$ be the framed 1-manifold which is the Lie group framing on the circle and let $\nu$ be the Lie group framing on $S^3 = Spin(3)$. I am probably going to conflate these framed ...

24
votes

### What is modern algebraic topology(homotopy theory) about?

Abstract homotopy theory allows one to use the tools of homotopy theory (e.g. inverting weak equivalences, computing homotopy colimits, doing Bousfield localization, taking fibrant and cofibrant ...

Community wiki

24
votes

Accepted

### Who computed the third stable homotopy group?

The error is that Rokhlin claimed that $\pi_6(S^3)=\mathbb{Z}/6$, but Hilton, in his review, points out that the paper instead shows that $\pi_6(S^3)/\pi_5(S^2) = \mathbb{Z}/6$. The error lies in a ...

24
votes

Accepted

### Has anyone seen a nice map of multiplicative cohomology theories?

I'm not sure I understand what "the" map is here, but I'll attempt to answer
the questions that were asked in the body of the question. Sorry if I'm just
saying things that you already know.
$\...

22
votes

Accepted

### Cohomology of the image of J spectrum

This was answered by Don Davis, 1975 Bol. Soc. Mat. Mex. In modern notation, the answer is
(at p=2) $H^* j = (A \oplus \Sigma^7 A)/I$, where $I$ is the ideal generated by $Sq^1 \iota_0$, $Sq^2 \...

22
votes

Accepted

### Is there a homomorphism between $\pi_8(S^5)$ and $\pi_8(SO(6))$?

There is a fibration $p : SO(m+1) \to S^m$ with fibre $SO(m)$ which induces a long exact sequence in homotopy
$$\dots \to \pi_n(SO(m)) \to \pi_n(SO(m+1)) \xrightarrow{p_*} \pi_n(S^m) \to \pi_{n-1}(SO(...

22
votes

Accepted

### $QS^0$ isn't a product of Eilenberg-Mac Lane

If $X$ is a product of Eilenberg-MacLane spaces then the map
$$ \eta^*\colon \pi_2(X) = [S^2,X]\to[S^3,X]=\pi_3(X) $$ is easily seen to be zero (where $\eta\colon S^3\to S^2$ is the Hopf map). ...

22
votes

### Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

The question used the phrase "still needed." This is a very loaded term, and the answer that you give will depend very strongly on how you interpret it.
If, as Dylan does, we interpret this as asking ...

Community wiki

21
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/~...

21
votes

Accepted

### What's the stabilization of the $\infty$-category of $\infty$-categories?

In a project in progress with Matan Prasma and Joost Nuiten concerning the abstract cotangent complex formalism we compute the stabilization of the $\infty$-category $\infty\mathrm{Cat}_{/C}$ of $\...

21
votes

Accepted

### What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?

I don't think this "finite limits and finite colimits coincide" business can be taken very far. If you take any small category $S_0$ you can add an initial and a terminal object to form $S = \mathrm{...

21
votes

Accepted

### Is a spectrum with trivial homology groups trivial?

If $K(n)$ is the $n$-th Morava K-theory for $n>0$, then $K(n)\otimes H\mathbb{Z}=0$ because, via the 2 complex orientations of $K(n)\otimes H\mathbb{Z}$, there are two formal groups over the ring $\...

21
votes

### The $K$-theory homology of the Eilenberg-MacLane spectrum

We have $KU_*(H\mathbb Z) = \pi_*(KU\wedge H\mathbb Z)$; this ring is concentrated in even dimensions and carries an isomorphism between the additive and multiplicative formal group law, hence is ...

21
votes

Accepted

### Is $[X, \_]$ a homology theory?

This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a ...

20
votes

Accepted

### Motivation and potential applications of spectral algebraic geometry

This is not really an answer to your question, just an attempt to address your question from the comments.
There are various flavours of homotopical or higher algebraic geometry that are commonly ...

20
votes

### Why not a Stacks project for Homotopy Theory?

According to the about page of Kerodon:
Kerodon is an online textbook on categorical homotopy theory and related mathematics. It currently consists of a single chapter, but should grow (slowly) ...

20
votes

Accepted

### References and resources for (learning) chromatic homotopy theory and related areas

I am not sure whether it is in the spirit of the original question, but let me add a wordy version of Theo's extensive and excellent bibliography -- a bit more of a road map. Let me divide to this ...

Community wiki

20
votes

### The $K$-theory homology of the Eilenberg-MacLane spectrum

Since for any two spectra $E,F$ we have
$$
E_n(F)=[\mathbb{S}^n,E\wedge F]\cong [\mathbb{S}^n,F\wedge E] = F_n(E),
$$
you may as well ask about $H_n(KU;\mathbb{Z})$, the integral homology of the ...

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

stable-homotopy × 486at.algebraic-topology × 310

homotopy-theory × 213

reference-request × 41

ring-spectra × 35

stable-homotopy-category × 35

motivic-homotopy × 33

chromatic-homotopy × 31

ag.algebraic-geometry × 28

kt.k-theory-and-homology × 28

ct.category-theory × 27

infinity-categories × 26

model-categories × 24

higher-category-theory × 22

cobordism × 22

cohomology × 21

spectral-sequences × 21

equivariant-homotopy × 20

triangulated-categories × 19

homological-algebra × 18

algebraic-k-theory × 14

higher-algebra × 12

motives × 10

bousfield-localization × 9

homology × 8