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 $\mathbf {Z}$.