Homotopy Class

Given two topological spaces M and N, place an equivalence relationship on the continuous maps f:M->N using homotopies, and write f_1∼f_2 if f_1 is homotopic to f_2. 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 n, the string can be wrapped around the tree n times, for positive n clockwise, and negative n counterclockwise. Each integer n corresponds to a homotopy class of maps from S^1 to S^1.

After the string is wrapped around the tree n 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 n times can be deformed to any other.

See also

This entry contributed by Todd Rowland

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Rowland, Todd. "Homotopy Class." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HomotopyClass.html

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