Fundamental Group

Definition

If the starting and ending point of a path are the same, then this path is called a loop The common starting and ending point $x_0$ is referred to as the basepoint

The set of all homotopy classes $[f]$ of loops $f:I\to X$ at the basepoint $x_0$ is denoted ${\pi }_1(X,x_0)$

Proposition 1.3 ( Algebraic Topology, p.35)

${\pi }_1(X,x_0)$ is a group with respect to the product $[f][g]=[f\cdot g]$

This group is called the fundamental group of $X$ at the basepoint $x_0$

Further we can show that the map ${\beta }_h:{\pi }_1(X,x_1)\to {\pi }_1(X,x_0)$ is an isomorphism

Thus if $X$ is path-connected, the group ${\pi }_1(X,x_0)$ is, up to isomorphism, independent of the choice of basepoint $x_0$. In this case the notation ${\pi }_1(X,x_0)$ is often abbreviated to ${\pi }_1(X)$ or ${\pi }_{1}X$.

geometric examples

an infinite cyclic group (free group on one generator)

AB are not linked nonabelian free group on two generators

AB are linked free abelian group on two generators

Important

One can often show that two spaces are not homeomorphic by showing that their fundamental groups are not isomorphic,

Check this nice visualization video on YouTube

Circle

We show that ${\pi }_1(S_1)\approx \mathbb{Z}$ ($\approx$ meaning isomorphism here) Claim

Theorem 1.7 Algebraic Topology, p.38

${\pi }_1(S^1)$ is an infinite cyclic group generated by the homotopy class of the loop $\omega (s)=(\cos 2\pi s,\sin 2\pi s)$ based at $(1,0)$

Covering Space

Definition

Given a space $X$, a covering space of $X$ consists of a space $\tilde{X}$ and a map $p:\tilde{X}\to X$ satisfying the following condition:

For each point $x\in X$ there is an open neighborhood $U$ of $x$ in $X$ such that $p^{-1}(U)$ is a union of disjoint open sets each of which is mapped homeomorphically onto $U$ by $p$. Such a $U$ is called evenly covered

Example

Think $X$ as $S^1$ and $\tilde{X}$ as $\mathbb{R}$, then we can construct $p$ (this spiral shape is called a helix) we define $p:\mathbb{R}\to S^1$ with $p(s)=(\cos (2\pi s),\sin (2\pi s))$ each section on $S_1$ is mapped from multiple (infinitely many) sections on $\mathbb{R}$ Each map from a section on $R$ to section on $S^1$ is a homeomorphism

Define ${\tilde{\omega }}_n:I\to \mathbb{R}$ as $p({\tilde{\omega }}_n(t)={\omega }_n(t)$ , then it is a lift of loop ${\omega }_n$ a) For each path $f:I\to X$ starting at a point $x_0\in X$ and each ${\tilde{x}}_0\in p^{-1}(x_0)$, there is a unique lift $\tilde{f}:I\to \tilde{X}$ starting at ${\tilde{x}}_0$. b) For each homotopy $f_t:I\to X$ of paths starting at $x_0$ and each ${\tilde{x}}_0\in p^{-1}(x_0)$, there is a unique lifted homotopy ${\tilde{f}}_t:I\to \tilde{X}$ of paths starting at ${\tilde{x}}_0$.

Link to original

We now prove the theorem