# Fundamental Group

1. Apr 21, 2010

### Mikemaths

I am doing some revision and trying to do fundamental groups and I was wondering if the fundamental group of the following space is {1} i.e. all loops based p are homotopic.

fundamental group of (X,p) = D^2\{(x,0) : 0<=x<=1} where p=(-1,0)

2. Apr 21, 2010

### rasmhop

Yes. Just consider a loop $f : [0,1] \to X$. This is homotopic to the constant map c(t)=p by the usual straight-line homotopy $(t,i) \mapsto (1-i)f(t) + ip$. You can verify yourself that this works.

The intuitive idea when dealing with fairly nice subsets of $$\mathbb{R}^2$$ is that the fundamental group is trivial if and only if the space has no holes (since then you can wrap a loop around that hole).

3. Apr 22, 2010

### Mikemaths

Ok thanks I get that as I thought.

I am trying to show that the fundamental group of a product of Topological Spaces is isomorphic to the product of fundamental groups:

pi1(X x Y , (p,q)) -> pi1(X,p) x pi1(Y,q)

I can understand this but want to construct the isomorphism and am struggling?

4. Apr 22, 2010

### rasmhop

Just construct it in the obvious way. Given two loops $f_1 : I \to X$ and $f_2 : I \to Y$ you can construct a path $I \to X \times Y$ where $t \mapsto (f_1(t),f_2(t))$. Conversely given a path $f : I \to X \times Y$ you can construct paths $I \to X$ and $I \to Y$ by projecting onto X and Y. You can show that both of these give homomorphisms of the fundamental groups and that they are inverses.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook