Tagged Questions

2
votes
1answer
84 views

Are strongly close maps homotopic?

While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any ...
12
votes
0answers
123 views

Are locally homotopic functions homotopic?

Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
1
vote
0answers
49 views

How to prove this isotopy exist?

Let $M$ be a topological manifold and $N$ is a subset of $M$. Let $f_t$ be an isotopy from $id $ to $f$ which is a homeomorphism on $M$. Suppose $f(N)=N$ and there is an isotopy $g_t$ on $N$ such that ...
3
votes
1answer
40 views

Are these functions homotopic?

Let $\gamma$ be a smooth, simple, closed curve and let $f : \gamma \to S^1$ assign to each $x \in \gamma$ the unit normal vector there. We can find a diffeomorphism $g: \gamma \to S^1$ and define the ...
3
votes
1answer
66 views

Suspension Operation on the Pontryagin-Thom Construction

I have a feeling that this is well-known: View the Pontryagin-Thom construction as the bijective correspondence between $[M,S^r]$ and the set of (appropriate equivalence classes of) framed ...
8
votes
1answer
91 views

Are close maps homotopic?

Consider a smooth manifold $M = M^m$ and a smooth submanifold $N = N^n \subset M$. Suppose that two maps $f, g: M \to N$ are close to each other, in the sense that there exists $\epsilon > 0$ such ...
10
votes
1answer
420 views

If $f\!: X\simeq Y$, then $X\!\cup_\varphi\!\mathbb{B}^k \simeq Y\!\cup_{f\circ\varphi}\!\mathbb{B}^k$.

How can I prove that if two spaces $X$ and $Y$ are homotopy equivalent, then the corresponding spaces obtained by gluing a $k$-cell are also equivalent? In detail, if ...
4
votes
1answer
200 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, ...
8
votes
1answer
466 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 ...
1
vote
2answers
162 views

Degree of a map between product of manifolds

Let $M^m$ and $N^n$ be compact, oriented smooth manifolds without boundary. Then what is the degree of the map $$ f: M\times N \to N \times M$$ given by $f(x,y) = (y,x)$? I have the feeling it ...