Algebraic topology

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.

Higher Homotopy Groups

(I know next to nothing about these. Please fill in if you know about them.)

Homology and Cohomology

(This is for when I'm feeling braver. Or, better yet, when someone else is feeling braver.)

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Algebraic topology

Fundamental Groups

Higher Homotopy Groups

Homology and Cohomology