$X$ Topological space, $x_0 \in X$

Definition of the fundamental group with base point at $x_0$:

$\pi_1(X,x_0) = \{[f] | f: I \to X \text{ a path} | f(0) = f(1) = x_0\}$ (Homotopy classes of Closed paths starting/ending at $x_0$).

Here we are using Path homotopy as we are using paths.

See also this video, and these notes

Product

The product for making it into a Group is the Product of paths lifted to their homotopy classes (as explained in Path homotopy). It is always defined for the above set because they all start and end in the same point.

$[f]*[g] = [f*g]$

(where $*$ for paths is the Product of paths)

The unit element is the constant path $c_{x_0}$.

$[f]^{-1} = [f^{-1}]$

Example For R^n, it is the Trivial group using Straight line homotopy

pi_1 of circle $\cong \mathbb{Z}$

Fundamental group from Triangulation

The group of equivalence classes of edge loops under 'elementary deformations' for a Triangulation of space X is a finite combinatorial object, easier to compute, and it is isomorphic to the fundamental group of the space X.

See here

How to map paths from foundamental group with one base point to another with another base point, given a path connecting them. Shows that the groups are isomorphic.

Therefore, if $X$ is Path connected, then all fundamental groups are isomorphic. However, in general, there is not a canonical/natural isomorphism, it depends on the path. See this other vid

Simply connected

Functor

Def:

Let $h: X \to Y$ be a Continuous function with $h(x_0) = y_0$. Define

$h_*: \pi(X,x_0) \to \pi_1(Y,y_0)$

$h_* ([f]) = [h \circ f]$

This is a Group homomorphism induced by $h$.

Functorial properties of the map

$X \to_f Y \to_g Z$

$f(x_0) = y_0$, $g(y_0) = z_0$

=> $(g \circ f)_* = g_* \circ f_*$

identity map $(id_X)_* = id_{\pi_1(X,x_0)}$

See also this vid.

Theorem. If $f: X \to Y$ is a Homeomorphism, $f(x_0) = y_0$, then $f_*: \pi_1(X,x_0) \to \pi_1 (Y,y_0)$ is n Isomorphism.

Remark. I $X$ and $Y$ are path-connected and $\pi_1(X,x_0)$ is not isomorphic to $\pi_1(Y,y_0)$ then $X$ and $Y$ are not homeomorphic.

Fundamental group $(\pi(X,x_0),h_*)$ is a Functor from the Category of topological spaces with base points and continuous base-point-preserving maps to the category of groups and group homomorphisms.

Examples of fundamental groups

Fundamental group of a product of spaces is isomorphic to the product of the fundamental groups

Van Kampen's theorem