Tagged Questions
15
votes
1answer
185 views
Geometric interpretation of the map $SO(4) \to SO(3)$
Let me first explain the background of my question.
As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
4
votes
1answer
84 views
Specific homotopy between complex conjugation and the identity.
Consider the set $\mathcal{C} = C^{\infty}(\mathbb{C}^*, \mathbb{C}^*)$, where $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$.
Both $f(z) = z$ and $g(z) = \bar{z}$ can be seen as elements in ...
4
votes
1answer
199 views
Continuous maps between compact manifolds are homotopic to smooth ones
If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$.
Seems to be fairly basic, ...
4
votes
0answers
80 views
Dixmier-Douady class: computations
As far as I know, Dixmier-Douady classes represent obstrucions to spin$^c$ structures. Questions:
Could somebody prove or give a reference: manifolds of dimension lower than $5$ always have a ...
2
votes
2answers
112 views
Relation between uniform convergence of curves in a manifold and homotopy
I was working through some things in riemannian geometry and I had this doubt:
Let $M$ be a closed riemannian manifold, $H$ an embedded submanifold and $V$ be its $\varepsilon$-tubular neighborhood. ...
8
votes
1answer
460 views
Diffeomorphism group of the unit circle
I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
13
votes
3answers
429 views
How to prove a manifold is simply connected?… using geometry
I was Looking at another questions title, and given the tag of DG, I thought it would read a little more like this one. Or at least that answers to this question would be answers to that question.
...
