Algebraic topology

Algebraic topology is a branch of mathematics that uses abstract algebra to understand topological spaces.

One of the most useful tools in algebraic topology is the fundamental group, $\pi _{1}(X)$ of a topological space $X$. The definition is as follows: Let $p$ be a fixed point in $X$. Consider the space of all curves $\gamma :[0,1]\rightarrow X$ which begin and end at $p$. We consider two such curves to be (homotopically) equivalent if we can continuously deform the first curve into the second. More precisely, we consider $\gamma _{1}$ to be equivalent to $\gamma _{2}$ if there exists a mapping $H:[0,1]\times [0,1]\rightarrow X$ with $H(t,0)=\gamma _{1}(t)$ and $H(t,1)=\gamma _{2}(t)$. We can define an group structure on the resulting equivalence class of curves by declaring $\gamma _{1}\cdot \gamma _{2}$ to be the curve $\gamma _{2}$ followed by $\gamma _{1}$, and rescaled so that the resulting curve still has domain $[0,1]$. This group is called the fundamental group of $X$

Example: The most important example is the topological space $S^{1}$. It can be shown that the fundamental group of $S^{1}$ is isomorphic to the additive group of integers $\mathbf {Z}$.

In addition to aiding in the classification of topological spaces, methods in algebraic topology considerably simplify some proofs in abstract algebra. For example, the proof that there is an injection from the free group on three generators to the free group on two generators is most easily expressed in terms of the topological concept of covering spaces.