Questions tagged [transformation-groups]
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48
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Is the product of exponentiated elliptic curve basis elements invariant under FFT of scalars?
I am working with an elliptic curve defined over a finite field $\mathbb{F}_p$ and have a basis set of points ${g_0, g_1, \ldots, g_n}$. When I perform the FFT on these points, I obtain a new basis ${...
3
votes
1
answer
357
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A question about spectral sequences
In the following proof (from The pontrjagin numbers of an orbit map and generalized G-signature theorem by Hsu-Tung Ku & Mei-Chin Ku https://link.springer.com/chapter/10.1007/BFb0085610), it is ...
- 1,301
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75
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A question about the localization theorem of Borel-Hsiang and spectral sequence
Suppose that $T$ is a torus acting on a topological space $X
$. Let $T\longrightarrow E_{T}\longrightarrow B_{T}$ be the universal $T$-bundle. Let $X\longrightarrow X_{T}\longrightarrow B_{T}$ be the ...
- 1,301
10
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1
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344
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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, ...
- 41.1k
1
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1
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132
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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'...
- 237
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127
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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 $T^1$ ...
- 1,301
1
vote
0
answers
70
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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 ...
- 1,301
1
vote
0
answers
92
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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 ...
- 1,301
1
vote
1
answer
144
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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 ...
- 5,780
2
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0
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65
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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 $...
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3
votes
1
answer
158
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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 $G$ ...
- 35.5k
5
votes
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216
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$C^1$ 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 ...
- 51
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78
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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 $\text{Aff}(1)$, ...
- 151
10
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4
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674
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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 ...
- 476
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247
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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 ...
- 183