Questions tagged [transformation-groups]

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Is the group of translations of an affine plane always commutative?

$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
Taras Banakh's user avatar
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1 vote
1 answer
114 views

About Čech cohomology in transformation groups

I'm starting a study about theory of transformation groups and equivariant cohomology, in what I read several times that Čech cohomology is the most compatible with this theory, but until now I haven'...
Ludwik's user avatar
  • 237
1 vote
0 answers
120 views

A question about fixed point set of the compact group actions

Let $G$ be an infinite compact Lie group acting on a compact space $X$. Denote $F=F(G,X)=\{x\in X$ : $gx=x$ for all $g\in G\}$. Show that if $H^*(B_{G_x};\mathbb{Q})=0$ for all $x \notin F$ and $T^1$ ...
Mehmet Onat's user avatar
  • 1,161
1 vote
0 answers
63 views

Contractible orbit space of action of compact Lie group on Euclidean space

R. Oliver proved that the following in https://www.jstor.org/stable/1970955 Theorem: Any action of a compact Lie group on a Euclidean space has contractible orbit space. My question is that this ...
Mehmet Onat's user avatar
  • 1,161
1 vote
0 answers
86 views

A question about the Conner Conjecture

In some sources, Conner conjecture is expressed as follows: Theorem [Conner Conjecture] Let $G$ be a compact Lie group, and let $X$ have the homotopy type of a finite dimensional $G$-CW complex with ...
Mehmet Onat's user avatar
  • 1,161
1 vote
1 answer
140 views

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 ...
user267839's user avatar
  • 5,938
2 votes
0 answers
64 views

Decomposition length in the stable homeomorphism conjecture

Stable homeomorphism theorem (due to Brown--Gluck, Kirby, Quinn,...) states that any orientation preserving homeomorphism $f$ of $\mathbb R^n$ is stable, that is, it can be written as a superposition $...
Dmitrii Korshunov's user avatar
3 votes
1 answer
154 views

Geometric vs cohomological dimension with families - on a proof of Lueck and Meintrup

Let $G$ be a discrete group, and let $\mathcal{F}$ be a family of subgroups of $G$ (closed under conjugation and taking subgroups). Then we may define the geometric and cohomological dimensions of $G$ ...
Mark Grant's user avatar
5 votes
0 answers
214 views

$C^1$ isometries of pseudo-Riemannian metrics are smooth?

It is well known that $C^1$ (actually even just differentiable) isometries of Riemannian manifolds are actually $C^\infty$. The proof is based on the metric structure generated by the Riemannian ...
mitsutani's user avatar
3 votes
0 answers
77 views

Can the Lie group $\text{Aff}(1)$ be extended?

Translations over $\mathbb{R}^1$ (ie. $(x\rightarrow x+b)$) form the Lie group $\mathbb{R}^{+}$. If we add the scaling operations over $\mathbb{R}^1$ , we can form the Lie group $\text{Aff}(1)$, ...
user3257842's user avatar
10 votes
4 answers
626 views

Palais's and Kobayashi's theorems on automorphism groups of geometric structures

My question concerns two results in the neighborhood of the standard theorem of Myers-Steenrod that isometry groups of Riemannian manifolds are Lie groups. Both appear in the first chapter of ...
Chris Wendl's user avatar
7 votes
0 answers
224 views

Fundamental domains for proper Lie group actions on smooth manifolds

The setting: $M$ an arbitrary smooth manifold, $G$ a Lie group acting effectively and properly on $M$ by diffeomorphisms. Motivation: when trying to figure out the homeomorphism type of the orbit ...
Russ Phelan's user avatar
11 votes
1 answer
311 views

Which surfaces have nontrivial actions of the circle (by homeomorphisms)?

Let $S$ be a $2$-dimensional manifold such that the circle $S^1$ acts nontrivially on $S$ by homeomorphisms. What can we conclude about $S$? If $S$ is a compact oriented surface on which $S^1$ acts ...
Laura's user avatar
  • 111
3 votes
0 answers
173 views

Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...
Robert Wegner's user avatar
2 votes
0 answers
31 views

On the number of connected functional digraphs recoverable from the preimage set size structure

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, $P_j=\left[f^{-j}(...
bmf's user avatar
  • 23
1 vote
0 answers
56 views

Are there multiple conjugacy classes of order 2 elements in the smooth automorphism group of $\mathbb{R}$?

Consider the group $\text{Aut}\mathbb{R}$ of smooth invertible maps from $\mathbb{R}$ to $\mathbb{R}$. If $f\in\text{Aut}\mathbb{R}$ has order 2 ($f$ is an involution), is $f$ conjugate to $g(x)=-x$? ...
Anon E. Mous's user avatar
0 votes
1 answer
84 views

Primage structures: induced domain partitioning by itterated inverse (reference request)

I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$. For example, the j-th such preimage list ...
bmf's user avatar
  • 23
5 votes
1 answer
198 views

The Hausdorff dimension of the union of singular orbits and exceptional orbits

Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
user avatar
8 votes
1 answer
355 views

Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones

Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...
user avatar
16 votes
1 answer
619 views

Relationship between Smith's special homology groups and equivariant homology theory

EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is ...
Andy Putman's user avatar
  • 43.3k
9 votes
0 answers
219 views

Fixed-points of a topological circle action

Suppose the circle group $G = S^1$ acts on $X$. If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. ...
Just Me's user avatar
  • 343
6 votes
0 answers
226 views

Recovering SU(2)-space when the orbit space is a 3 sphere with 3 singular orbits

Background: Consider $SU(2)$ action on the 6-dim flag manifold $M=SU(3)/T^2$ via left multiplication. We view $SU(2)$ as a subgroup of $SU(3)$ corresponding to $2\times 2$-block. The action is just ...
Yuhang Liu's user avatar
24 votes
1 answer
523 views

Homotopy equivalence of diffeomorphism groups

Let $M$ be a closed connected smooth manifold and let ${\rm Diff}^r(M)$ be the group of $C^r$-diffeomorphisms equipped with the compact-open $C^r$-topology. I am looking for a reference to the fact ...
Jarek Kędra's user avatar
  • 1,772
9 votes
1 answer
803 views

6-manifolds admitting SO(3) action with 2 orbit types

Let $M^6$ be a 6-dimensional smooth manifold, on which the group $G=SO(3)$ acts smoothly with 2 orbit types $SO(3)/SO(2)$ and $SO(3)$, such that the orbit space $X=M/SO(3)$ is a 3-ball $B^3$, whose ...
Yuhang Liu's user avatar
15 votes
0 answers
465 views

Diffeomorphisms of $\mathbf R^n$

Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies. Question: Is there an example to ...
Jarek Kędra's user avatar
  • 1,772
16 votes
3 answers
1k views

SO(3) action on (simply connected) 6 manifold with discrete fixed point

If a 6-dimensional orientable smooth manifold $M$ admits a smooth effective $SO(3)$ action with discrete fixed point set, what can we say about the topology of $M$? What if we assume that in addition ...
Yuhang Liu's user avatar
7 votes
0 answers
227 views

Higher homotopy of diffeomorphism groups from singularities

In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \operatorname{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth ...
Nati's user avatar
  • 1,971
1 vote
1 answer
130 views

Does the orbit map induce an isomorphism (or a monomorphism) in Alexander -Spanier cohomology?

Let $G$ be a compact and totally disconnected group acting on a paracompact space $X$. Does the orbit map $X \rightarrow X/G$ induce an isomorphism (or a monomorphism) in Alexander-Spanier ...
Mehmet Onat's user avatar
  • 1,161
4 votes
0 answers
418 views

Localization in equivariant cohomology theory for groups other than ($p$-)tori

Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups: Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite ...
user3158840's user avatar
2 votes
0 answers
89 views

Irreducible real curves on ${\mathbb C}P^1$ invariant under the finite group action

Let $G$ be a finite subgroup of a Möbius group with a standart action on a real algebraic variety ${\mathbb C}{\mathbf P^1}.$ How one can describe $G$-invariant irreducible real algebraic curves? ...
probably's user avatar
  • 403
7 votes
3 answers
312 views

Cyclic groups acting on balls, and interior fixed points

Let a finite cyclic group $G = \mathbb Z/n$ act continuously on an open $d$-ball $B^d$. Suppose further that this action extends to the closed ball $\overline{B^d}$. Is there necessarily a fixed point ...
Jens Reinhold's user avatar
0 votes
1 answer
120 views

free action on product of two spaces [closed]

Let $G$ be a compact Lie group acting freely on $X\times Y$ , product of two Hausdorff spaces. Is is true that $G$ must act freely on one of the factor spaces ($X$ or $Y$). For example the group $\...
user168639's user avatar
9 votes
1 answer
907 views

Rational homology and finite group actions

I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...
Lisa's user avatar
  • 93
1 vote
0 answers
180 views

Gysin sequence for $\mathbb S^3$ bundle

Let the group $G=\mathbb S^3$ act semi freely on a paracompact space $X$. Then exercise 12 of G.E. Breadon's book , Introduction to compact transformation groups pg 169 asks to derive the following ...
user168639's user avatar
1 vote
0 answers
134 views

free action on mod p cohomology sphere

It is well known that the group $G=\mathbb Z_2\oplus\mathbb Z_2$ cannot act freely on mod 2 cohomology n-sphere. Is it also true that this group $G$ cannot act freely on any mod p cohomology n-...
user168639's user avatar
13 votes
3 answers
1k views

Why do we need a $G$-universe?

Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$. Question: Why do we need a $G$-universe? A $G$-universe is defined to be a countably ...
H. Shindoh's user avatar
12 votes
1 answer
722 views

Which topological spaces are coset spaces of locally compact groups?

What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ? Such a space $X=G/H$ necessarily ...
Hannes Thiel's user avatar
  • 3,305
42 votes
5 answers
7k views

What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, ...
David Corwin's user avatar
  • 15.1k
4 votes
1 answer
365 views

Collapsing of Riemannian manifolds with a group action

Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold ...
Acky's user avatar
  • 643
1 vote
1 answer
116 views

Free involutions and equivariant maps

The following paper of Conner and Floyd does not include proofs of many theorems/ results. 'Fixed point free involution and equivariant maps , Bull. Amer. Math. Society vol 66, no. 6( 1960)'. I would ...
Jasp's user avatar
  • 11
2 votes
0 answers
336 views

Do non-ordinary Bredon cohomology theories extend?

As shown by Lewis, May, and McClure (MR0598689), the ordinary equivariant Bredon cohomology theory $H^*_G(-; M)$ extends to an $RO(G)$-graded cohomology theory precisely when the coefficient system $M$...
Bill Kronholm's user avatar
0 votes
1 answer
312 views

Intersections of conjugates of the icosahedral group in SO(3)

(Related question) Let $I$ be the group of orientation preserving symmetries of a regular icosahedron. This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A_5$. It is ...
Mark Grant's user avatar
5 votes
3 answers
2k views

Dimensions of orbit spaces

Let $G$ be a compact Lie group acting effectively on a compact, Hausdorff topological space $X$. I am looking for results of the type If $X$ is a ... and the action is ... then $\dim(X/G)\leq \dim(X)...
Mark Grant's user avatar
10 votes
2 answers
1k views

Does the Borel functor take equivariant fibrations to fibrations?

Let $p\colon X\to B$ be a fibration. Let $G$ be a topological group acting continuously on $X$ and $B$, and assume that the map $p$ is $G$-equivariant. We can apply the Borel functor $EG\times_G-$ ...
Mark Grant's user avatar
7 votes
1 answer
935 views

Is there a smooth free circle action on the Klein bottle?

Can the circle group $S^1$ act smoothly and freely on the Klein bottle? I'm sure there is some obvious reason why the answer is no, which eludes me right now. We can view $K$ as the quotient of $S^1\...
Mark Grant's user avatar