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Revision as of 15:21, 23 August 2006

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

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Homology and Cohomology

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