We constructed the fundamental group by looking at the homotopy classes of loops with a fixed basepoint. The question that naturally arises is how does the fundamental group change if we move the basepoint. To begin let’s suppose we’re in a path connected space and we have two basepoints we’re concerned about, say and . Since is path connected, there exists some path from to . We can use this to convert loops into loops.

What we’ll do is take a loop, say , based at and use our path to get from to , follow the loop, then take the reverse path back to . This is illustrated in the image below (adapted from an image in Hatcher’s book).

This new path is a loop based at . Now, if two -loops, say and are homotopic, then so are the associated -loops: .

If we now construct a map by setting we see that the map will be well-defined since our construction of the -loops respects homotopy. But what happens when we concatenate two -loops?

Suppose and are loops based at . We want to see what this map does to the homotopy class of .

So our map is in fact a homomorphism. Notice that implies , which means that our map is injective.

To show the map is surjective, let be an -loop, then is an -loop. Applying we have , and so our map is surjective. Together with injectivity above we have shown that in a path connected space the fundamental group of and are isomorphic. We then simply write since the choice of basepoint is basically irrelevant.

If our space is not path connected, but and are in the same path component, the above gives us that .

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