## New answers tagged homotopy-theory

2
votes

### Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?

I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure of an
$E_{\infty}$ ring space or, essentially equivalently, the multiplicative ...

4
votes

Accepted

### Diagonal maps, Goodwillie calculus, and $T(n)$ local homotopy theory

This is an elaboration on my comment above. Let us consider the natural transformation induced by the diagonal
$$\Sigma^\infty\Delta\colon \Sigma^\infty X \to \Sigma^\infty X\wedge X.$$
The natural ...

3
votes

### Spaces homotopy dominated by $S^2 \times S^2\times S^2$

You're actually right. The cohomology algebra of $X=S^2\times S^2\times S^2$ with coefficients in $\mathbb{Z}$ is $H^*(X)=\mathbb{Z}[x,y,z]/(x^2,y^2,z^2)$ with $|x|=|y|=|z|=2$. The cohomology algebra ...

6
votes

Accepted

### Spaces homotopy dominated by $S^2 \times S^2\times S^2$

Put
$$ R(n)=H^*((S^2)^{\times n}) = \mathbb{Z}[x_1,\dotsc,x_n]/(x_1^2,\dotsc,x_n^2) $$
A key point is that if $u\in R(n)$ with $|u|=2$ then $u^2=0$ iff $u=0$ or $u=m\,x_i$ for some $m\in\mathbb{Z}\...

4
votes

### Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?

There are analogues of Lie's theorems in homotopy theory, primarily for rational and $p$-adic homotopy types, as well as Lie ∞-groupoids, which can be seen as smooth homotopy types.
In the rational ...

0
votes

### Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

For anyone interested, I've found a solution. It is the central construction in https://arxiv.org/abs/2203.11392.

7
votes

Accepted

### $BP$-Adams Novikov Spectral Sequence or Homotopy groups of $S/3$

For $3$-primary homotopy of $S$ there is early work by
...

4
votes

### What is $TP(\mathbb{Z}_p)$?

The calculation of $\pi_* TP(\mathbb{F}_p) = \pi_* THH(\mathbb{F}_p)^{tS^1} = \pi_* \widehat{\mathbb{H}}(S^1, THH(\mathbb{F}_p))$ (the notation has changed over the years) was first published by ...

8
votes

Accepted

### Homotopy invariance of $\ell$-adic cohomology

Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$.
Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology ...

9
votes

Accepted

### Is there a Dold-Kan theorem for circle actions?

No, they are not equivalent, even for $C = Sp$.
Indeed, the category of spectra with $S^1$-action is also the category of $\mathbb S[S^1]$-modules, and is compactly generated by a single object.
On ...

5
votes

Accepted

### Fibrant replacement of an injective model category of enriched diagrams

Section 8 of my paper All (∞,1)-toposes have strict univalent universes shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of ...

11
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\...

4
votes

### What are Koszul dualities?

I have finally found a source which puts together the pieces in a satisfactory way, at least in the stable setting, here:
Amabel, Araminta. "Poincaré/Koszul Duality for General Operads." ...

Community wiki

0
votes

Accepted

### Maps that preserve winding numbers

I found an answer to this question: the question of how a continuous function changes the winding number of a closed curve can be studied quite generally. The important concept here is the degree of ...

10
votes

### Is the decomposition of the homotopy type of a complex into a product and into a smash product unique?

Let me write $S^n/p$ for the cofibre of $p$ times the identity map on $S^n$, or in other words the mod $p$ Moore space with homology in degree $n$.
If $p$ and $q$ are coprime then $S^n/p\wedge S^m/q$ ...

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