Homotopy

Summary

A homotopy between two continuous maps $f, g \colon X \to Y$ is a continuous deformation of $f$ into $g$, parametrised by a time $t \in [0, 1]$. Formally it is a continuous map $H \colon X \times [0, 1] \to Y$ with $H(\cdot, 0) = f$ and $H(\cdot, 1) = g$. Homotopy is the basic notion of algebraic topology. Homotopic maps induce the same maps on homology and cohomology, and homotopies that fix a base point give the same maps on homotopy groups.