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.