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**5**

votes

**1**answer

95 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 ...

**5**

votes

**1**answer

104 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 ...

**16**

votes

**1**answer

314 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 ...

**9**

votes

**0**answers

135 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 $X$. ...

**5**

votes

**0**answers

169 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 ...

**24**

votes

**1**answer

333 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 ...

**10**

votes

**1**answer

472 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 ...

**13**

votes

**0**answers

311 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 ...

**15**

votes

**3**answers

804 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 ...

**7**

votes

**0**answers

207 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 ...

**1**

vote

**1**answer

102 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 ...

**4**

votes

**0**answers

284 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 ...

**2**

votes

**0**answers

80 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?
...

**7**

votes

**3**answers

200 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 ...

**0**

votes

**1**answer

101 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 $\...

**8**

votes

**1**answer

537 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 ...

**1**

vote

**0**answers

122 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 ...

**1**

vote

**0**answers

121 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-...

**13**

votes

**3**answers

839 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 ...

**11**

votes

**1**answer

462 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 ...

**40**

votes

**5**answers

5k 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, ...

**4**

votes

**1**answer

304 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 ...

**1**

vote

**1**answer

99 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 ...

**2**

votes

**0**answers

322 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 $M$...

**0**

votes

**1**answer

276 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 ...

**3**

votes

**2**answers

975 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)...

**9**

votes

**2**answers

702 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 $EG\times_G-$ ...

**6**

votes

**1**answer

588 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\...