# Mathematics Prelims

## March 15, 2009

### Homotopy Classes

Filed under: Topology — cjohnson @ 1:11 pm

If we have two continuous maps, $f : X \to Y$ and $g : X \to Y$, we would like to view $f$ and $g$ as essentially being the same map if they are homotopic to one another; we want to consider $f$ and $g$ as being equivalent with respect to homotopy.  To do this we need to show that homotopy is an equivalence relation on the set of continuous $X \to Y$ maps.  This is a pretty simple thing to verify.

Suppose $f, g, h : X \to Y$ are all homotopic to one another with $\{ \alpha_t : X \to Y \}$ being the homotopy from $f$ to $g$, and $\{ \beta_t : X \to Y \}$ the homotopy from $g$ to $h$.  To show that $f \simeq f$ we can simply take the identity homotopy: $\{ i_t : X \to Y \}$ where $i_t \equiv f$.  For reflexivity we’ll construct a new homotopy $\{ \overline{\alpha_t} : X \to Y \}$ where $\overline{\alpha_t}(s) = \alpha_{1 - t}(s)$.  This simply reverses the direction of the homotopy: instead of going from $f$ to $g$, we start at $g$ and go to $f$.  For transitivity we construct the homotopy $\{ \gamma_t : X \to Y \}$ with $\gamma_t = \beta_t \circ \alpha_t$.  Then $\{ \gamma_t \}$ takes us to $h$ by first taking us $g$.

This shows that homotopy is an equivalence relation.  We will refer to equivalence classes here as being homotopy classes.

Just as general homotopy is an equivalence relation, so too is path homotopy.  Since path homotopy has the additional requirement of being anchored down at the path’s endpoints, though, path homotopy is an equivalence relation on all paths with two fixed endpoints.

If a function is in the same homotopy class as a constant function, we say that the function is null-homotopic.  This means that we can squeeze the function’s graph down to a single point with a homotopy.  Notice that the only way a path can be null-homotopic (when we say homotopic and are talking about paths, we mean path homotopic) is if its initial and terminal points are the same.  Such a path is called a loop and the homotopy classes of loops is what we will actually place a group structure on to construct the fundamental group.