10
votes
3answers
407 views

Why care about group actions?

Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when ...
1
vote
1answer
196 views

Smooth diffeomorphisms preserving common symmetries

Let $A$ and $B$ be two $C^\infty$ orientation-preserving diffeomorphic, connected, bounded, open subsets of ${\mathbb R}^n$, with finitely many ends and $G$ the group of isometries of ${\mathbb R}^n$ ...
1
vote
0answers
39 views

Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
0
votes
0answers
109 views

Questions (doubts) on: Group Action on Manifolds

There are 2 questions that are bugging me in differential topology and I'd be glad if the same could be cleared up: Let $X = x\frac{\partial}{\partial y}$ be a vector field on $M = R^2$, where $R$ ...