Fundamental group

Proposition

Given paths:

The following hold:

1. relative to
2. relative to
3. relative to

Proof

(1) Let interpolate between paths and be the path variable.
Consider the following image:

^
|---0---|-1-|-2-|
|
t
|
|-0-|-1-|---2---|
*------s-------->

If we linearly scaled the bottom intervals to the top ones,
the interval lengths would be (as a function of ): , , .
This motivates the homotopy:

(2) Same strategy, with the following image

---0---
-0-|-c-

Interval lengths: and

(3) Now this one is a bit different, but also easier. We will just make paths that go up to and back.

Theorem

Let be a space and a point. Let be the set of homotopy classes of loops in based at .
Then using

gives a group structure

Proof

Directly from previous proposition.

Definition

is the fundamental group of based at .
Fundamental group isomorphisms