ATRACTOR

A path connected surface is characterized by the existence of a path connecting any two points of the surface. We can prove that the fundamental group of path connected surfaces doesn't change with the choice of base point, up to isomorphism.

The idea for the proof is simple. Consider any two points \(P\) and \(Q\) of the path connected surface, and a path between \(P\) and \(Q\) - which existence follows from the surface being path connected. For each closed path \(f\) with base point \(P\) we define a new closed path: \[f\rightarrow g^{-1}.f.g\]

This new path starts in \(Q\), goes to \(P\) by \(g^{-1}\), continues through \(f\) returning to \(P\), and goes back to \(Q\) by \(g\); that is, it is a closed path with base point \(Q\).

Given an homotopy class of paths \(b\) between \(P\) and \(Q\), we define the function \(\varphi_{b}\), from the fundamental group with base point \(P\) to the fundamental group with base point \(Q\), given by \(\varphi_{b}(a)=b^{-1}.a.b\), where \(a\) is a homotopy class of closed paths with base point \(P\) ( and \(\varphi_{b}(a)\) is a homotopy class of closed paths with base point \(Q\). This function is an isomorphism (equivalence of groups), and, therefore, the set of equivalence classes - the fundamental groups - with base point \(P\) and with base point \(Q\) are equivalent.

Note that the isomorphism depends on the choice of the homotopy class \(b\), and so it is not necessarily unique.