Homotopy Class

Given two topological spaces  and , place an equivalence relationship on the continuous maps  using homotopies, and write  if  is homotopic to . Roughly speaking, two maps are homotopic if one can be deformed into the other. This equivalence relation is transitive because these homotopy deformations can be composed (i.e., one can follow the other).

A simple example is the case of continuous maps from one circle to another circle. Consider the number of ways an infinitely stretchable string can be tied around a tree trunk. The string forms the first circle, and the tree trunk's surface forms the second circle. For any integer , the string can be wrapped around the tree  times, for positive  clockwise, and negative  counterclockwise. Each integer  corresponds to a homotopy class of maps from  to .

After the string is wrapped around the tree  times, it could be deformed a little bit to get another continuous map, but it would still be in the same homotopy class, since it is homotopic to the original map. Conversely, any map wrapped around  times can be deformed to any other.

REFERENCES:

Aubry, M. Homotopy Theory and Models. Boston, MA: Birkhäuser, 1995.