48 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 ...
30 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

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 ...
25 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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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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24 votes
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Why do we study complex orientable cohomology theories

There is a sort of a priori reason why one would consider the cohomology theory $MU$, without first knowing of its connection to manifold geometry, to formal groups, … . Since complex-oriented ...
21 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
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Are complex-oriented ring spectra determined by their formal group law?

The following is a communal answer from the algebraic topology Discord [1], primarily put forward by Irakli Patchkoria (correcting previous half-answers by Tyler Lawson and me). Kiran suggested it be ...
18 votes
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Why does elliptic cohomology fail to be unique up to contractible choice?

So the issue is with this: all of the groups occuring in the Goerss--Hopkins obstruction theory vanish In "generic terms", for the obstruction theory that you're running in either the $K(1)...
17 votes

What is modern algebraic topology(homotopy theory) about?

In response to Ryan Budney's comment, let me try to say something about topological data analysis, and other recent applications of algebraic topology outside of traditional mathematics. Applied ...
17 votes
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Uniqueness of Complex Orientation of Morava K-theory

First, I claim that if we ignore the ring structure and complex orientation, then there is a unique spectrum (up to homotopy equivalence) that deserves to be called $K(n)$. I do not know whether ...
16 votes

Why the sphere spectrum is more correct than $\mathbb{Z}$?

For this to work, it is best to identify connective spectrum with spaces equipped with a group-like $E_\infty$-algebra structure (these are equivalent). From this point of view: $\mathbb{Z}$ is the ...
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15 votes

What is so 'coloured' on Chromatic Homotopy Theory

This term is surely due to Doug Ravenel. In the mid 1970's, he and collaborators Steve Wilson and Haynes Miller constructed and exploited a "chromatic spectral sequence" for computing Ext ...
13 votes

Can the Bousfield class of projective space be computed directly?

Here is an easy argument which sometimes works. I have updated and extended it to incorporate comments from Dylan Wilson and Maxime Ramzi. For any finite spectrum $X$, we have (co)unit maps $S\...
12 votes
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Are the AHSS and Adams spectral sequence the same when computing connective Morava K-theory of a space?

Tyler's comment answers my question. A bit more detail: the Postnikov tower of $k(n)$ is an Adams resolution, because the `bottom class' map $k(n) \rightarrow H\mathbb F_p$ is onto in mod $p$ ...
12 votes

Morava $K$-theory of $K( \mathbb{Z}/p^2)$

$L_{K(1)}K({\Bbb Z}/p^n)$ is always trivial. This is Proposition 2.14 in the recent preprint by Bhatt, Clausen and yourself. Alternatively, this also appears in recent work by Land, Meier and Tamme.
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12 votes
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Completed and uncompleted operations for Morava $E$-theory

No, this map is not injective. To see this, put $W=\mathbb{W}(\mathbb{F}_{p^n})$, which is a free module of finite rank over $\mathbb{Z}_p$. It is standard that $\mathbb{Z}_p\otimes\mathbb{Z}_p$ ...
11 votes

What is higher equivariant homotopy?

I would also like to know the answer to your question. Since no one has given an answer yet, I'll speculate recklessly and irresponsibly on how this might work (by riffing off of the final paragraph ...
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11 votes
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Localization at the Johnson-Wilson spectrum and rationalization

First, recall that any rational spectrum is a wedge of suspended copies of $H\mathbb{Q}$. It follows that $H\mathbb{Q}$ is a retract of $E(n)\mathbb{Q}$ and so is $E(n)$-local. From this we see that ...
11 votes
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Latest results in chromatic homotopy theory

In 2017, the long-standing problem of whether the Brown–Peterson spectrum $\mathrm{BP}$ admits the structure of an $E_\infty$-ring was shown in the negative. The question dates back to 1975, ...
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11 votes
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Does the spectrum of Morava E-theory depend only on height?

Here's an argument that Eric Peterson and I came up with showing that the homotopy type of Morava $E$-theory only depends on the choice of perfect char $p$ field $k$ and the height $n<\infty$. $\...
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10 votes

What is modern algebraic topology(homotopy theory) about?

I'm going to give an algebraist's perspective. First let's discuss homological algebra (which has roots in topology). There's a quote (attributed, I think, to Connes) that a great mystery of ...
10 votes
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Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum

See Lemma 2.3 of the following paper, and the surrounding discussion: ...
10 votes
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For which $n$ does there exist a closed manifold of (chromatic) type $n$?

After discussing this with Tim we came up with the following answer: The first steifel whiteny class $\omega_1$ of $M$ can be written as the following composition: $$M \to BO(n) \to BO \to BAut(\...
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10 votes
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On the sparsity of the descent spectral sequence computing homotopy groups of the K(n)-local sphere

This sparsity holds even without the assumptions that $p>2$ and $2(p-1)>n^2$. (Those are used to get a horizontal vanishing line). As in the comments, it comes down to the fact that you can use $...
10 votes

Why the sphere spectrum is more correct than $\mathbb{Z}$?

An elementary answer to the first part of your question: Finite sets are more fundamental than their cardinalities. Consider the category of finite sets and bijective functions. Its geometric ...
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9 votes
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Map between homology of spectra

For any $m$, there is a Kunneth spectral sequence $$ Tor_{BP_*} (K(m)_*, BP_*(X)) \Rightarrow K(m)_* X. $$ For your $X$, $BP_*(X)$ is acted on nilpotently by $v_m$ for $0 \leq m \leq n$, and so this ...
9 votes
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Applications of equivariant homotopy theory in chromatic homotopy theory

The canonical answer to this question is of course the celebrated solution by Hill, Hopkins and Ravenel to the Kervaire invariant one problem Hill, Michael A., Michael J. Hopkins, and Douglas C. ...
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9 votes
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Are Landweber exact spectra determined by their coefficient ring?

Put $R_*=\mathbb{Z}_{(p)}[x,y,y^{-1}]$ with $|x|=|y|=2$. Define $f,g\colon BP_*\to R_*$ by \begin{align*} f(v_i) &= \begin{cases} y^{p-1} & \text{ if } i = 1 \\ 0 &...
8 votes

Why do we study complex orientable cohomology theories

A surface-level motivation for complex oriented cohomology theories is that they are precisely those that admit a theory of generalized chern classes. More surprisingly, the universal complex ...
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