Isotopy

Definition

Let $X$ be a topological space and $f,g$ be homeomorphisms from $X$ to itself. An isotopy from $f$ to $g$ is a homotopy $F:X \times I \to X$ that starts at $f$, ends at $g$, and such that for any fixed $t$, the map $x \mapsto F(x,t)$ is a homeomorphism.

Loosely, it is a homotopy via homeomorphisms.

A composite of isotopies gives an isotopy, and hence being isotopic defines an equivalence relation on the group of all homeomorphisms. The quotient of the group of all homeoomorphisms by this equivalence relation gives the mapping class group of the topological space.