Homotopy Class (original) (raw)
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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.
See also
Homotopy, Homotopy Group, Homotopy Theory, Topological Space
This entry contributed by Todd Rowland
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References
Aubry, M. Homotopy Theory and Models. Boston, MA: Birkhäuser, 1995.
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Cite this as:
Rowland, Todd. "Homotopy Class." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HomotopyClass.html