# Questions tagged [equivariant-maps]

Questions about or involving equivariant maps, the natural maps between $G$-sets.

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### Example of CW-complex with $G$-action, which is not $G$-CW-complex

Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, ...
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### isomorphic groups that are G-equivariant [closed]

Start with two finite groups $A$ and $B$ and a group isomorphism $f$ between them. Let a finite group $G$ act on both $A$ and $B$. By definition $f$ is $G$-equivariant if $g(f(a))=f(g(a))$. Do I ...
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### Confusion about induced bundles, equivariant Vector bundles and representations

I am writing this question to try to unconfuse myself in two different directions. Throughout we work over $\mathbb{C}$ and the choice of topology should not matter, but say we are using the etale ...
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### About equivariant vector valued forms on principal bundle

Let $\pi: M \to E$ be a $G$-equivariant vector bundle and let us adopt the notation $$C^{\infty}(M,E)^{G} = \{\tilde{s} \in C^{\infty}(M,E) | \ \forall g \in G \ \tilde{s} \cdot g = \tilde{s}\}$$ ...
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### Existence of a transitive smooth G-action of $G\times G$

This question arose from an attempt of using the Equivariant Rank Theorem to prove question 7.1 of Lee's Introduction to Smooth Manifolds. The Equivariant Rank Theorem is stated: Let M and N be ...
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### Strange notation of fibre product in an article of Cattaneo, Felder and Tomassini.

I'm doing my bachelor thesis on explaining an article of Cattaneo, Felder and Tomassini, and here is the link to the article: https://arxiv.org/pdf/math/0012228.pdf. I'm currently working on the page ...
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### Equivariant rank theorem.

I was reading this post that explains why $$\phi :O_{n}\times H\rightarrow GL_{n}(\mathbb{R})$$ $$\phi(B,A)=BA$$ is a diffeomorphism. Here $O_{n}$ is the orthogonal group and $H$ is the group of all ...
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### Pushforward of equivariant sheaf

I work over an algebraically closed field of characteristic zero. Let $G$ be an algebraic group, $X,Y$ varieties with $G$-actions, and $\phi:X\to Y$ a $G$-equivariant morphism. Let $\mathcal{F}$ be ...
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### Why is the Tate map lax monoidal (just for abelian groups?)

Let $A$ be an abelian group with $G$ action, then I call the tate map $$(-)^{tG}: Ab^G \rightarrow Ab$$ $$A \mapsto \operatorname{coker}(Nm:A_G \rightarrow A^G)$$ where $Nm:x \mapsto \sum_g gx$, is ...
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### Understanding implications of the equivariant Darboux-Weinstein theorem

I am trying to understand the implications of the equivariant Darboux-Weinstein theorem, stated here: The book that states this (Hamiltonian Group Actions and Equivariant Cohomology) gives an example ...
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### Complex group representations as an enriched category?

In my lecture notes it says: ‘Complex representations of a given group G, together with intertwiners, form a category enriched over the complex numbers.’ Is it true that the category is enriched ...
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### long exact sequence of representations

Let $G$ be a group and $V_1,V_2$ be $G$-representations. Are the ${\rm Ext}^i(V_1,V_2)$ $G$-representations as well? Once established this, suppose I have a short exact sequence 0\to V_1\to V_2\...
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### What does it mean for right equivariant maps to be left translations?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
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In Sets and Groups by Green a question 5 from the chapter 3 reads: Write $\theta_{a,b}$ for the map of the preceding exercise [which is $\theta(x)=ax+b=\theta_{a,b}$]. Prove $\theta_{a,b}\theta_{c,d}=... 1 vote 1 answer 267 views ### If pushforward by equivariant map of structure sheaf is structure sheaf and the space of sections isomorphic, are they isomorphic as G-modules? Apologies for what may very well be a trivial question from a non AG person. Suppose I have a morphism of varieties$f: X\rightarrow Y$, with$Y$affine, which is equivariant with respect to the ... • 13 4 votes 1 answer 2k views ### Does equivariance of the MLE require the function be invertible? My statistics text states this theorem as if it works for any function$g$: Let$\tau = g(\theta)$be a function of$\theta$. Let$\hat{\theta}_n$be the MLE (Maximum Likelihood Estimator) of$\...
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I'm reading a paper where they have an appendix about homotopy fixed point sets of a $G$-space, and at some point they claim that if $f:X\to Y$ is a $G$-map that is an ordinary (non-equivariant) ...