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 ...
44 votes
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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 ...
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  • 751
40 votes
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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 ...
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31 votes
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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 $F_R(M,N)$ ...
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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 ...
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  • 6,526
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.
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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, ...
27 votes
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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 ...
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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 ...
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26 votes
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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 ...
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25 votes
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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 ...
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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 $...
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24 votes
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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 ...
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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 ...
24 votes
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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 ...
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24 votes
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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. $\...
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  • 5,153
22 votes
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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 \...
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22 votes
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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(...
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22 votes
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$QS^0$ 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). ...
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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 ...
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/~...
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21 votes
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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 $\...
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21 votes
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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{...
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21 votes
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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 $\...
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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 ...
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21 votes
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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 ...
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  • 6,049
20 votes
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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 ...
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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) ...
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  • 118k
20 votes
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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 ...
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 ...
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