Questions tagged [equivariant-homotopy]
Equivariant homotopy theory is the study of how homotopy theory behaves when spaces are considered together with a group action on them.
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Is anything known about the equivariant homotopy theory of surfaces with the action of a finite subgroup of the mapping class group?
The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \...
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The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
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Relative $G$-equivariant homology groups
Let $X$ be a free $G$-CW-complex with $G$-equivariant cell filtration by
$n$-skeleta $X_0 \subset \dots \subset X_n \subset \dots \subset X$ (for
rigorous definition see
Chap. II, p. 98 in linked ...
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Explicit computation of the transfer in the representation ring for unitary groups
For a compact Lie group $G$ we let $R(G)$ be the ring of finite dimensional complex $G$-representations studied by Segal in http://www.numdam.org/item/PMIHES_1968__34__113_0.pdf.
This comes with extra ...
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An exact sequence involving THH
Disclaimer: I'm a relative beginner in this area. I'm trying to prove that if one has a commutative ring $R$ and a prime number $p$, then there is an exact sequence of the form
$$\DeclareMathOperator\...
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The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
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Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
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Naive Z/2-spectrum structure on E smash E?
Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it ...
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Equivariant complex $K$-theory of a real representation sphere
Consider the one-point compactification of a $U(n)$-representation $V$, denoted by $S^V$. I want to caclulate $\tilde{K}_\ast^{U(n)}(S^V)$. When $V$ is a complex $U(n)$-representation, we can use the ...
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Equivariant classifying space and manifold models
The classifying space $BS^1$ for $S^1$-bundles can be taken to be the colimit of $\mathbb{CP}^n$ which are smooth manifolds and the inclusions $\mathbb{CP}^n \hookrightarrow \mathbb{CP}^{n+1}$ are ...
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Terminology for equivariant homology
The usual $G$-equivariant homology and cohomology groups of a space $X$ with $G$-action are given by the Borel construction:
$$H_\ast^G(X)=H_\ast((X\times EG)/G),$$
$$H^\ast_G(X)=H^\ast((X\times EG)/G)...
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Equivariant spectrum with coefficients
I am curious to know whether spectra with coefficients as defined in Adams's Blue book be defined to an equivariant setting. In the non-equivariant case, for a spectrum $E$ and an abelian group $A$, ...
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Rational G-spectrum and geometric fixed points
For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...
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Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?
I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we ...
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Mostow-Palais equivariant embedding for manifolds with corners
Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into ...
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Applications of equivariant homotopy theory to representation theory
Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...
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When can the trace on cohomology be computed as the Euler characteristic of fixed points?
In this question all groups are finite, and all spaces are nice (eg, simplicial sets).
Given a $G$ space $X$, which we assume has finitely many nonzero cohomology groups, we can compute the trace of ...
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$E^G_\ast(E)$ tensored with the rationals
Lemma 17.19 of Switzer's "Algebraic topology - Homology and Homotopy" states that $E_\ast(F)\otimes\mathbb{Q}$ is isomorphic to $\pi_\ast(E)\otimes\pi_\ast(F)\otimes\mathbb{Q}$. I wanted to ...
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Is there a framed nullbordism of $T^4$ with an action of $T^4$ that extends the self-action?
Under the identification of the stable homotopy groups with the (stably) framed bordism groups, it is well known that $\eta\in\pi_1\mathbb{S}$ is represented by $S^1$ with its Lie group framing. ...
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Equivariant spectra with coefficients
In “The localization of spectra with respect to homology”, Bousfield describes localizations with respect to Moore Spectra. Given a spectrum $E$, and a group $M$, he describes the spectrum with ...
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Is this class of groups already in the literature or specified by standard conditions?
In recent work
Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators
Scott Balchin, Ethan MacBrough, and I ...
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Splitting of $BGL_1(KR)$
There are infinite loop space splittings $BGL_1(KO)\simeq BGL_1(KO)[0,2]\times Z$ and $BGL_1(KU)\simeq BGL_1(KU)[0,3]\times Z'$ where $Z$ and $Z'$ are 2 and 3 connected, respectively (i.e. they have ...
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Is the equivariant Steenrod algebra useful?
I am a newbie to the field, so please excuse any potential obvious gaps in knowledge. I have been wondering of late about the equivariant (dual) Steenrod algebra in the context of genuine $G = C_p$ ...
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What are the naive fixed points of a non-naive smash product of a spectrum with itself?
Let $X$ be an object in one of the well-known symmetric monoidal model categories of spectra. E.g., an $\mathtt S$-module in the sense of EKMM, or an orthogonal spectrum, or a $\Gamma$-space, etc.
One ...
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The slice filtration does not arise from a $t$-structure
I've heard this argument quite a bit that the slice filtration does not arise from a $t$-structure on the category of genuine $G$-spectra. Mike Hill points out in Remark 3.12 of The Equivariant Slice ...
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Model structure on dg-algebras over an "equivariant fundamental category"?
For purposes of $G$-equivariant rational homotopy theory one wants a Quillen adjunction which generalizes the classical one of Bousfield-Gugenheim from plain dg-algebras/simplicial-sets to (co-)...
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Uniqueness of normal microbundle of a smooth embedding
Suppose $M$ is a topological manifold and $\iota: N\hookrightarrow M$ be a submanifold. A normal microbundle of $N$ consists of an open neighborhood $U$ of $N$ and a retraction $\pi: U \to N$ such ...
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Geometric fixed points of induction spectrum
I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4....
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A question related to injective envelope for a system of DGA's
I was trying to read Fine and Triantafillou's paper "On the equivariant formality of Kahler manifolds with finite group action".
They have defined the enlargement at $H$ of a system of DGA's ...
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Equivariant imbedding of compact manifold
Let $G$ be a compact Lie group smoothly acting on a smooth compact manifold $X$.
Is it true that there exists a smooth $G$-equivariant imbedding of $X$ into a Euclidean space acted linearly (and ...
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Double coset decomposition for compact Lie groups
The starting point of my question is the following fact: suppose $G$ is a finite group and let $H,K \leq G$ be arbitrary subgroups, then there exists an isomorphism of $G$-sets as follows
\begin{...
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Decomposition of fiber product of $G$-sets in $G$-orbits
I have posted an identical question in MSE few days ago, but maybe this site is a better adress to discuss this problem:
Let $G$ be a finite group and $K, H \leq G$ two subgroups. Then
the right ...
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Endomorphism in the rational stable $O(2)$-equivariant category of the universal space of the family of finite dihedral subgroups
Let $G$ be a compact Lie group. We can define $\mathfrak{F}G$ to be the collection of conjugacy classes of closed subgroups of $G$ whose Weyl group is finite, a bi-invariant metric on $G$ induces a ...
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Equivariant phantom maps
In the stable homotopy category a map of spectra $f\colon X \rightarrow Y$ is called phantom is the induced map between the associated homology theories $X_* \rightarrow Y_*$ is zero, it is know that ...
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Rigidity of the TMF-valued equivariant elliptic genus
Let me preface this question by saying that I wrote it at least in part to understand its statement. As such, I hope that the reader will excuse any mistakes.
$\DeclareMathOperator{\ind}{ind}\...
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Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms
In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...
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What is the pointed Borel construction of the $0$-sphere?
From what I understand, the Borel construction takes a $G$-space $X$ and produces a topological space $X\times_{G}\mathbf{E}G$―the homotopy quotient $X/\!\!/G$ of $X$ by $G$ in the $\infty$-category ...
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Projective resolution of a dual coefficient system
I was trying to read the paper "Equivariant minimal models" by G. Triantafillou(1982) and was trying to compute cohomology of a system of DGA with rational coefficient system. Given a finite ...
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Reduction to graph subgroups for Bredon homology when the $G_1\times G_2$ is $G_2$-free
I have the following problem. Let $\Gamma_{G_1\times G_2}$ be a full subcategory of the orbit category $\mathcal{O}_{G_1\times G_2}$ consisting of graph subgroups of $G_1\times G_2$. Further, let $N$ ...
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Construction of equivariant Steenrod algebra
I am reading through the calculations in Hu-Kriz "Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence" and I've got a small problem in understanding the ...
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When do non-exact functors induce morphisms on $K$-theory?
Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ ...
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Are finite $G$-spectra idempotent complete?
Question: Let $G$ be a compact Lie group (you can assume that $G$ is finite if you like). Is the category of finite $G$-spectra idempotent complete?
Here, by "finite $G$-spectra", I mean ...
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Applications of equivariant homotopy theory in chromatic homotopy theory
I usually do computations in equivariant homotopy theory, but I would like to learn chromatic homotopy theory where one may use the equivariant techniques, e.g., slice spectral sequences, etc.
For ...
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Understanding equivariance of the Tate construction $(-)^{tC_P}$
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Fun{Fun}\newcommand\Cat{\text{Cat}}\DeclareMathOperator\CoInd{CoInd}\newcommand\Spaces{\text{Spaces}}$It is stated in line 10, p76, Thomas Nikolaus, ...
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Equivariant colimit and equivariant functors
This is rather specific B.5 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on last line p147), which I am having fundamental confusion.
We have the categories $...
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Is there a 2-categorical, equivariant version of Quillen's Theorem A?
Quillen's Theorem A says that a functor $F:C \to D$ (between 1-categories) induces a homotopy equivalence of classifying spaces $BC \simeq BD$ if for every object $d$ in $D$ the fiber category $F/d$ ...
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A $d_1$-differential in the homotopy fixed points spectral sequence
I have the following problem. Let $Q$ denote $\mathbb{Z}/2$ as an abelian group and let $X$ be a $Q$-spectrum. If I want to compute the homotopy groups of $X^{hQ}$, homotopy fixed points spectrum, I ...
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How does one rigorously lift a map $Sp \rightarrow Sp$ of spectra to equivariant spectra?
This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my ...
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Computing homotopy colimit of a space with free $S^1$-action
Context. I am trying to understand the argument in B.4 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on p147).
I am still lost. But from Maxime's helpful ...
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K-theory of free $G$-sets and the classifying space, and generalization
$\newcommand\Sets{\mathrm{Sets}}\DeclareMathOperator\Nerve{Nerve}$Let $G$ be a finite group, $\mathcal{G}^0$ be the category of finite free
$G$-sets and isomorphisms between them. Then $\mathcal{G}^0$...
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