7
votes
0answers
131 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
3
votes
0answers
35 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
1
vote
1answer
23 views

Being smooth homotopic relation: proof

Suppose we have an open set $U$ in the plane and two $\cal C^\infty$ paths $\gamma,\eta:[a,b]\to U$ with the same endpoints (i.e., $P:=\gamma(a)=\eta(a)$ and $Q:=\gamma(b)=\eta(b)$). We say that ...
2
votes
1answer
65 views

Are diffeomorphic sets smoothly deformable into each other?

Given connected, bounded and open sets $U, V\subset \mathbb{R}^n$ and an orientation preserving diffeomorphism $F:U\to V$, is there always an isotopy $H:[0,1]\times \mathbb{R}^n\to\mathbb{R}^n$, s.t. ...
2
votes
0answers
67 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
1
vote
0answers
43 views

homeomorphism still isotopic to the identity after the deletion of points.

I have a surface $M$ (without boundary) and a homeomorphism $f:M \rightarrow M$, which is isotopic to the identity on $M$. If I delete two points $x$ and $f(x)$ from the surface, I get a ...
2
votes
1answer
82 views

negative Euler characteristic $\Rightarrow$ homotopy unique up to homotopy

In a paper by John Franks I stumbled upon the following: Let $M$ be a surface and $f:M \rightarrow M$ be a homeomorphism, which is homotopic to the identity on $M$. That means, that there is ...
1
vote
0answers
64 views

finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
5
votes
2answers
149 views

Framed Cobordism Classes of links in $\mathbb R^3$

We know that every link in $S^3$ is framed cobordant to the unknot with some framing. The idea is to study smooth homotopy classes of maps from $S^3$ to $S^2$. Actually in the title I have given ...
0
votes
1answer
120 views

Find a regular homotopy

Firstly, we define a regular homotopy between regular closed curves as a continuous map $F:$I x I$\rightarrow \mathbb{R}^n$ satisfying the following conditions: (i) for each fixed u $\in$ I, the map ...
15
votes
1answer
281 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
105 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 ...
6
votes
1answer
416 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
94 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 ...
3
votes
2answers
158 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
803 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
505 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. ...