33
votes
Accepted
Did Peter May's "The homotopical foundations of algebraic topology" ever appear?
An anonymous source told me this question is here. Dylan gave the quick answer and Tyler referred to it.
I'll use the question as an excuse to give a pontificating longer answer. When I first planned ...
- 30.4k
14
votes
What's with equivariant homotopy theory over a compact Lie group?
Regarding 2, there is no difficulty in defining $G$-spectra in the setting of $\infty$-categories. The only complications I can think of are that (1) the orbit category $\mathrm{Orb}^G$ is now an $\...
- 8,972
14
votes
Accepted
Applications of equivariant homotopy theory to representation theory
There are decades and decades of algebraic results that use techniques from equivariant homotopy theory. Some examples ...
(1) Quillen's work on ring theoretic aspects of the cohomology of finite ...
- 11.2k
13
votes
Accepted
Extending a weak version of Sullivan's generalized conjecture
The answer to the first question is negative by a result of Dror Farjoun and Zabrodsky. In Fixed points and homotopy fixed points, they prove that if a finite group $G$ is not a $p$-group, then there ...
- 148
12
votes
Accepted
Definition of $Fun^G( \mathcal C, \mathcal D)$ in the setting of quasicategories
A(n ∞-)category with $G$-action is just a functor $BG\to \mathrm{Cat}_∞$. Then, if $\mathcal{C},\mathcal{D}$ are (∞-)categories with $G$-action, we can get another (∞-)category with $G$ action $\...
- 16.5k
9
votes
Accepted
"Oriented representation" sphere
First of all, note that right before example 3.9 they prove that
$$H^G_*(S^V;\underline{\mathbb{Z}})=H_*(C^{cell}_*(S^V)^G)\,,$$
where $C^{cell}_*(S^V)$ is the cellular complex for some $G$-CW-...
- 16.5k
9
votes
Accepted
$p$-adic equivalence of spectra with $G$-action
I don't know if "$2$-stage nilpotent" is a standard term for this, but I'm sure that what the author means is this: a group $A$ that $G$ is acting on has a subgroup $B$ such that the action of $G$ ...
- 55.9k
9
votes
Accepted
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. ...
- 16.5k
9
votes
Are finite $G$-spectra idempotent complete?
I think the correct setting to look at this question is that of
W. Lück, "Transformation groups and algebraic K-theory". Lecture Notes in Mathematics, 1408. Mathematica Gottingensis. ...
- 18.2k
9
votes
Accepted
An exact sequence involving THH
Let's extract a clear question (about spectra in general) from your question, and then answer it. Let $E$ be any spectrum.
There is the degree $p$ map $p:S\to S$ from the sphere spectrum to itself. ...
- 55.9k
8
votes
Accepted
Homotopy group action and equivariant cohomology theories
From modern perspective this is much more straightforward than the "genuine" version you described above the question. Naive $G$-spaces are just functors $BG\to \cal{S}$ among infinity ...
- 4,189
8
votes
Homotopy fixed points of complex conjugation on $BU(n)$
I think the answer is yes, after Bousfield-Kan $2$-completion. For $n=1$, $BO(1) \to BU(1)^{hC_2}$ is an equivalence, since $BO(1) \simeq K(\mathbb{Z}/2, 1)$, while $BU(1) \simeq K(\mathbb{Z}(1), 2)$ ...
- 9,263
7
votes
Accepted
What are the naive fixed points of a non-naive smash product of a spectrum with itself?
In both the orthogonal world and the EKMM world one can set things up so that $G$-spectra are just spectra with an action of $G$, and the naive and genuine equivariant stable categories are the ...
- 56.9k
7
votes
What are the naive fixed points of a non-naive smash product of a spectrum with itself?
In the context of functors with smash product (FSP), or symmetric spectra, or orthogonal spectra, the spectrum with $\Sigma_2$-action $X \wedge X$ prolongs essentially uniquely to a $\Sigma_2$-...
- 9,263
7
votes
$E^G_\ast(E)$ tensored with the rationals
For any finite abelian $G$ and $H\leq G$ we have a geometric fixed-point functor $\phi^H\colon\text{Sp}_G\to\text{Sp}$ which preserves smash products and sends the equivariant sphere $S^0_G$ to $S^0$. ...
- 56.9k
7
votes
Rational G-spectrum and geometric fixed points
This is implied by Theorem 3.10 in Wimmer's "A model for genuine equivariant commutative ring spectra away from the group order". Taking $R = \Bbb Q$ and $\mathcal F$ to be the full family ...
- 52.6k
6
votes
K-theory of free $G$-sets and the classifying space, and generalization
The general point is just that if $\mathcal{U}$ is equivalent to the free symmetric monoidal category $F\mathcal{C}$ generated by $\mathcal{C}$ then $K(\mathcal{U})\simeq\Sigma^\infty_+\mathcal{C}$. ...
- 56.9k
6
votes
Accepted
$RO(Q)$-graded homotopy fixed point spectral sequence
For a based $G$-space or $G$-spectrum $X$, the homotopy fixed point object $X^{hG}$ is by definition $F_G(EG_+,X)$. Suppose we write $EG$ as the colimit of a sequence of $G$-subspaces $A_k$. This ...
- 56.9k
6
votes
Accepted
When do non-exact functors induce morphisms on $K$-theory?
As suggested by Dustin Clausen in his answer, polynomial functors induce maps on $K$-theory. In the setting of stable $\infty$-categories, you proved this in your joint work with Barwick, Glasman, and ...
6
votes
How does one rigorously lift a map $Sp \rightarrow Sp$ of spectra to equivariant spectra?
Let $C$ be a complete $\infty$-category.
Let $U:Fun(BC_n,C)\to C$ denote the forgetful functor, $\mathrm{CoInd}$ its right adjoint, and $(-)^{triv}$ the functor given by precomposition along $BC_n\to *...
- 15.9k
6
votes
The adjoint representation of a Lie group
For Question 2, there are only two kinds of such examples: the case where $G$ is commutative and even-dimensional, and the case where all simple factors of $G$ come in isomorphic pairs. (In the ...
- 12.9k
6
votes
Accepted
$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group.
Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$.
Since $\B Ω G≃G$, this ∞-category is ...
- 37.8k
5
votes
Is the Milnor construction contractible
Oliver Straser is correct in that Milnor himself in 1956 [1] only showed his model for $EG$ is weakly contractible (that is, all its homotopy groups vanish) with his coarse topology on the join. (For ...
- 687
5
votes
Homotopy group action and equivariant cohomology theories
Much has already been said in the other answers and comments, but let me summarize a few points.
One way to obtain from a category a 'homotopy theory' (aka an $\infty$-category) is to specify a notion ...
- 12.1k
5
votes
Are finite $G$-spectra idempotent complete?
To complement Oscar's more systematic answer, let me expand my comment about the case $G = \mathbf{Z}/p\mathbf{Z}$ for a prime number $p$, where the answer is no when $\tilde{K}_0(\mathbf{Z}[G]) \neq ...
5
votes
Equivariant complex $K$-theory of a real representation sphere
Here is one possible approach, which is specific to the adjoint representation. Let $X$ denote $U(n)$, regarded as a $U(n)$-space by the rule $g.x=gxg^{-1}$. Put $X_k=\{x\in X:\text{rank}(x-1)\leq k\...
- 56.9k
5
votes
Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
$\newcommand{\Z}{\mathbf{Z}}\newcommand{\Map}{\mathrm{Map}}$Let $\sigma$ denote the sign representation of $\Z/2$, and let $S^{d\sigma}$ denote the one-point compactification of $\sigma^{\oplus d}$. ...
- 5,760
5
votes
Serre spectral sequence of Borel construction
It's terrible notation to use $p$ for both the prime in question and for the index in the spectral sequence, so I'll write the spectral sequence as
$$E_2^{nm} = H^n(G;H^m(X)).$$
Since $X$ is a mod-$p$ ...
- 44.8k
4
votes
Accepted
Fibre preserving maps of Borel constructions
The answer to the question in the edit is no. Take $G=\mathbb Z$, $X$ to be a point and $Y=\mathbb R$ with the action $n\cdot x = x+n$. Then there are no equivariant maps from $X$ to $Y$ but there are ...
- 959
4
votes
Accepted
Is a $G$-cell complex always a $G$-CW complex?
As Najib says in the comments to the question, the proof of the classical statement can be easily-ish adapted to the equivariant case. Let's see the details
Lemma Let $X$ be a $G$-CW-complex and let $...
- 16.5k
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