0
votes
1answer
52 views

Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?

Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$? For reasonable spaces $X$, e.g. CW-complexes, one has ...
2
votes
0answers
76 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
0
votes
0answers
34 views

existance of loop with finitely many point of intersection

for every loop on compact orientable surface exists freely homotopic loop with finitely many points of intersection. I see that it have to be true, but I can't prove it. I know Thom's theorem, Sard's ...
2
votes
1answer
83 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
1answer
96 views

$[M,\mathbb CP^\infty]=[M,\mathbb CP^2]$ where $M$ is a smooth closed orientable 3-manifold!

Prove the above result, where $[X,Y]$ means the set of all homotopy classes of maps from X to Y, two topological spaces. I have answered it below.
14
votes
1answer
190 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
1answer
163 views

homotopy type of the manifold minus the boundary

Let $X$ be a topological manifold with boundary.What is the idea behind the fact that emoving the boundary doesn't change the homotopy type of the manifold; i.e.,that is the manifold $X$ has the same ...
2
votes
2answers
217 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 ...