Fundamental Group of (X,p): D^2\{(x,0) : 0<=x<=1}

[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)

 

[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).

 

[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?

 

[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.