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, <math>\pi_{1}(X)</math> of a topological space <math>X</math>. The definition is as follows: Let <math>p</math> be a fixed point in <math>X</math>. Consider the space of all curves <math>\gamma:[0,1]\rightarrow X</math> which begin and end at <math>p</math>. We consider two such curves to be (homotopically) equivalent if we can continuously deform the first curve into the second. More precisely, we consider <math>\gamma_1</math> to be equivalent to <math>\gamma_2</math> if there exists a map <math>H:[0,1]\times[0,1]\rightarrow X</math> with <math>H(t,0) = \gamma_1(t)</math> and <math>H(t,1) = \gamma_2(t)</math>. We can define an group structure on the resulting equivalence class of curves by declaring <math>\gamma_1\cdot\gamma_2</math> to be the curve <math>\gamma_2</math> followed by <math>\gamma_1</math>, and rescaled so that the resulting curve still has domain <math>[0,1]</math>. This group is called the fundamental group of <math>X</math>

Example: The most important example is the topological space <math>S^1</math>. It can be shown that the fundamental group of <math>S^1</math> is the ring of integers <math>\mathbf{Z}</math>.