Algebraic topology

From AoPSWiki

Algebraic topology is the study of topology using methods from abstract algebra. In general, given a topological space, we can associate various algebraic objects, such as groups and rings.

Fundamental Groups

Perhaps the simplest object of study in algebraic topology is the fundamental group. Let $X$ be a path-connected topological space, and let $x\in X$ be any point. Now consider all possible "loops" on $X$ that start and end at $x$ , i.e. all continuous functions $f:[0,1]\to X$ with $f(0)=f(1)=x$ . Call this collection $L$ . Now define an equivalence relation $\sim$ on $L$ by saying that $p\sim q$ if there is a continuous function $g:[0,1]\times[0,1]\to X$ with $g(a,0)=p(a)$ , $g(a,1)=q(a)$ , and $g(0,b)=g(1,b)=x$ . We call $g$ a homotopy. Now define $\pi_1(X)=L/\sim$ . That is, we equate any two elements of $L$ which are equivalent under $\sim$ .

Unsurprisingly, the fundamental group is a group. The identity is the equivalence class containing the map $1:[0,1]\to X$ given by $1(a)=x$ for all $a\in[0,1]$ . The inverse of a map $h$ is the map $h^{-1}$ given by $h^{-1}(a)=h(1-a)$ . We can compose maps as follows: $g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}$ One can check that this is indeed well-defined.

Note that the fundamental group is not in general abelian. For example, the fundamental group of a figure eight is the free group on two generators, which is not abelian. However, the fundamental group of a circle is ${\mathbb{Z}}$ , which is abelian.

More generally, if $X$ is an h-space, then $\pi_1(X)$ is abelian, for there is a second multiplication on $\pi_1(X)$ given by $(\alpha\beta)(t) = \alpha(t)\beta(t)$ , which is "compatible" with the concatenation in the following respect: