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3
votes
1answer
218 views

Iterating Monad-Comonads structures

Let $(T, \mu , \eta )$ a monad on the category $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle: \mathscr{C}^T \to \mathscr{C}$ where ...
0
votes
0answers
118 views

automorphisms acting trivially on projective spaces

Let $K$ be a field and $\mathbb{P}=\mathbb{P}^n_K$ the projective space of dimension $n$ over $K$. Consider a linear automorphism $g$ of $\mathbb{P}$. Is it true that $g^\ast$ acts trivially on ...
6
votes
0answers
179 views

blowups and group actions

Let $X$ be a smooth projective variety over the complex numbers and assume that $X$ is equipped with the action of a finite group $G$. Denote by $Z$ the closed subscheme of fixed points of $G$ and ...
3
votes
0answers
85 views

“Spectral decomposition” action on the unitary group

Consider a matrix $U$ from the unitary group $U_N(\mathbb{C})$ and consider the map $f:U_N(\mathbb{C})\rightarrow U_N(\mathbb{C})$ where $f(U)$ is the matrix of the eigenvectors of $U$. What is ...
6
votes
1answer
314 views

Group actions with finite stabilizers and compact quotients

Let $G$ be a discrete group that acts on a contractible finite dimensional $G$-complex $X$ with the following properties: $X/G$ is compact (i.e. the action is cocompact) Each stabilizer $G_\sigma$ ...
5
votes
1answer
532 views

Characterization of amenable actions

Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...
3
votes
3answers
451 views

Group action on the real line

Hi, I was wondering about the following question: if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct a new action such ...
9
votes
3answers
584 views

The classifying space of a gauge group

Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by $$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = ...
2
votes
1answer
389 views

When does a $W^*$-algebra have a standard Borel spectrum?

EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual. This post came out a bit long, ...
4
votes
1answer
321 views

Transitive action on the sphere

Hello! From the book "Einstein manifolds" by Arthur L. Besse (at section 7.B), Lie groups $Sp(n)$, $Sp(n)\cdot U(1)$, $SU(2n)$ and $U(2n)$ constitute the complete list of Lie subgroups of $U(2n)$ ...
1
vote
1answer
237 views

Orbits of Thompson's group

Thompson's group may act by homeomorphisms on the circle. Has this action a fixed point?
0
votes
0answers
88 views

inertia stratification

Let $X$ be a nice algebraic variety (say smooth, projective) over a field of characteristic 0. Let $G$ be an abelian group acting on $X$. For each subgroup $H$ of $G$, denote by $X^H$ the closed ...
1
vote
0answers
78 views

Smoothing of a hyperquotient singularity

Let $f$ be a polynomial in $k$ complex variables, and suppose the affine variety $V$ given by $f = 0$ has an isolated singularity at the origin, but is otherwise smooth. Now assume that some cyclic ...
0
votes
1answer
106 views

The description of Hurwitz groups

Let $G$ be a Hurwitz group, i.e the automorphism group of some Hurwitz surface $C$. Then Hurwitz's automorphisms theorem shows that the quotient map of $C$ by $G$ has ramification points of indexes ...
2
votes
1answer
165 views

On the group actions on Hurwitz surfaces

Let $C$ be a Hurwitz surface, $G=\text{Aut}(C)$ and $N$ is a proper normal subgroup of $G$. Is there a simple argument (without using of classification theorems) for the fact that $N$ acts on $C$ ...
4
votes
1answer
151 views

Representations of Finite Subgroups on Homology

Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting ...
9
votes
3answers
757 views

Relation between groups and classifying spaces

Let $G$ be a nonabelian group, with classifying space $BG$. Motivation: We can compute its homology, $H_\ast(BG)=H_\ast(G)$. It would be nice to see some equivariant computations, like $H_\ast^G(BG)$ ...
2
votes
1answer
162 views

Is any $G$-set a coset geometry (in the sense of Tits-Buekenhout)?

Hi there! Let $X$ be a left $G$-set, and $\Delta=${$x_1,\ldots,x_n$} a fundamental domain of $G$ in $X$. In other words, $G$ acts on $X$ from the left, and {$Gx_1,\ldots,Gx_n$} is the orbit space ...
8
votes
2answers
338 views

Fixed point of $S^1$-action using roots of unity

Fact: For any (continuous) $S^1$-action on the closed unit disk $\mathbb{D}^n$, there is a fixed point $x_0\in\mathbb{D}^n$. I have thought of a possible argument that re-proves this, but am not sure ...
2
votes
2answers
187 views

On the definition of ‘smooth vectors’ in Rieffel's “Deformation Quantization for Actions of $ \mathbb{R}^{d} $”.

On the first page of Chapter 1 of Rieffel's Deformation Quantization for Actions of $ \mathbb{R}^{d} $, Rieffel defines a family of seminorms on the space $ A^{\infty} $ of smooth vectors of a Fréchet ...
2
votes
0answers
206 views

2 questions on Nagata's counterexample; $k[f_1,…,f_r]=k[g_1,…,g_s]$ vs. $k(f_1,…,f_r)=k(g_1,…,g_s)$

Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
6
votes
1answer
191 views

Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section. Recall that an ...
1
vote
1answer
290 views

Free and cellular G-action implies free G-complex?

Recall that a CW-complex $X$ with an action of a group $G$ which permutes the cells (i.e., for any $g \in G$ and any cell $\sigma \subseteq X$, $g\sigma$ is a cell) is called a $G$-complex. If the ...
7
votes
1answer
322 views

Free group actions on varieties and algebras of coinvariants

Suppose $k$ is an algebraically closed field of characteristic zero and $A$ is a finitely generated commutative associative reduced $k$-algebra. Suppose the group $\mathbb{Z}_2$ acts on $A$ in such a ...
8
votes
0answers
343 views

“Homogeneity” of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]

My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
6
votes
1answer
187 views

volume of exceptional group orbits

Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by ...
3
votes
5answers
506 views

Examples of manifolds with effective circle actions?

I would like to know examples of smooth compact connected manifolds, on which there exists an effective smooth circle action preserving a positive smooth volume, besides the simple example: $[0,1]^d ...
10
votes
2answers
755 views

Symmetric group action on squarefree polynomials

The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest. Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...
1
vote
1answer
136 views

Find an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ which is compatible with the fraction linear transform of $SL(2,\mathbb{R})$

There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by: $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw]. $$ Let ...
11
votes
3answers
685 views

Actions on Sⁿ with quotient Sⁿ

What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$? Comments. I am mostly interested in (maybe trivial) properties of such ...
15
votes
0answers
399 views

Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer. I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order. Or equivalently, ...
3
votes
1answer
315 views

Can a non-trivial action of a connected group on a reduced scheme be trivial on a dense open?

It is well-known that if a reduced algebraic group $G$ acts on a separated reduced scheme $X$, and $G$ acts trivially on a dense open subscheme $U\subseteq X$, then the action is trivial. If $X$ is ...
3
votes
2answers
304 views

Invariant for group actions

Hello everybody! Define the action of $SL_4({\mathbb{Z}})$ on alternating 2-forms or simply skew-symmetric matrices of degree 4 according to the following: For $B \in SL_4({\mathbb{Z}})$ and an ...
1
vote
2answers
256 views

Group action on spin^c 4-manifold.

[edit] I'll try to be more precise. In paper N.Nakamura, "Bauer–Furuta invariants under $Z_2$-actions" there is an assumption that $Z_2$ action "lifts to spin^c structure". What i think it means: ...
0
votes
1answer
200 views

eigen-bundles of a trivial vector bundle

Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on ...
4
votes
2answers
266 views

Is the Turing equivalence relation the orbit equiv. relation of the action of a countable group?

The Turing equivalence relation on $\cal P(\mathbb{N})$ is defined by $A\equiv_T B$ iff $A\leq_T B$ and $B\leq_T A$. This is a countable Borel equivalence relation on the polish space $\cal ...
3
votes
1answer
483 views

Abstract definition of properly discontinuous action

A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$. Is there a more abstract ...
1
vote
0answers
87 views

invariant lines avoiding fixed subvarieties

Could anybody help me with the following question ? Assume we are given: (1) a finite order (linear) automorphism $g$ of the complex projective space $\mathbb{P}^r$, (2) a closed algebraic ...
4
votes
1answer
275 views

Faithful transitive actions by large groups on small sets

How large is the largest transitive subgroup of $S_n$ other than itself and $A_n$? In particular, does its size grow at least exponentially in $n$?
1
vote
1answer
296 views

Isomorphism between the set of classes of Principal Homogeneous spaces and non-Abelian H^1(G,A) cohomology

Let A be a G-group, i.e. a set on which G acts on, has a group structure and satisfies $^s(xy)=^s x ^s y$ for all $x,y \in A \ , s \in G$. A homogeneous principal space P is a non-empty G-set on which ...
3
votes
0answers
121 views

Minimality of time-t minimal flows

This question is mainly motivated by the question Transitivity of a flow and its time-1 map Let $M$ be a closed smooth manifold and $\Phi\colon\mathbb{R}\times M\to M$ be a smooth minimal flow, i.e. ...
7
votes
1answer
391 views

Free $\mathbb{Z}_2$-actions match at some point

I have in front of me a proof of this lemma: If $f$ and $g$ are free $\mathbb{Z}_2$-actions on $S^1$, then $f(x)=g(x)$ for some $x \in S^1$. A $\mathbb{Z}_2$-action on the unit circle $S^1$ is a ...
3
votes
1answer
316 views

Action on a compact group

If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?
1
vote
1answer
244 views

a question on continuity of $G$-module for a profinite group $G$

I have seen the following statment somewhere, for example in Appendix B2 on Silverman's book "The Arithmetic of Elliptic Curves" : Let $M$ be an abelian group with discrete topology and $G$ be a ...
6
votes
2answers
284 views

$S^1$-action in three dimensions

Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action. What does this imply for $M$? What are examples except for (products of) spheres?
1
vote
1answer
226 views

Does the semi-stable set determine the linearization of a GIT quotient?

Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group. Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the ...
1
vote
1answer
257 views

is a closed subscheme of the projective line closed under the action of Gal(Qbar/Q)

Let $S$ be a non-empty closed subscheme of $P^1_K$, where $K$ is a number field. Assume that the cardinality of $S$ is finite. Is $S$ closed under the action of the absolute Galois group of the ...
4
votes
4answers
950 views

What about the empty torsor?

Let $G$ be a group. A $G$-torsor is a set $X$ together with an action of $G$ such that for all $x,y \in X$ there is exactly one $g \in G$ such that $gx=y$. This looks like a group which has forgotten ...
3
votes
0answers
500 views

Untwisting the Cohomology with Twisted Coefficients

This question is set on a finite $2$-group $G$ and a subgroup $H$ of index $2$ (but perhaps the question could be answered for arbitrary orders/indexes). It was asked here on MO whether ...
2
votes
1answer
387 views

A question regarding Lie group actions

Can you give me an example of a Lie group acting on a compact metric connected space transitively so that it has a closed finite index subgroup which does not act transitively?