Homotopy

A homotopy is a continuous deformation between two continuous functions from one topological space to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.

Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.

Definition

If $f,g:X\to Y$ are continuous functions between topological spaces $X$ and $Y$, a homotopy $H$ from $f$ to $g$ is a continuous map

$$H:X\times [0,1]\to Y$$

such that for all $x\in X$:

• $H(x,0)=f(x)$,

• $H(x,1)=g(x)$.

Two continuous functions $f$ and $g$ are called homotopic if there exists a homotopy between them, denoted $f\simeq g$.

Homotopy equivalence

Two topological spaces $X$ and $Y$ are homotopy equivalent if there exist continuous maps

$$f:X\to Y\text{and}g:Y\to X$$

such that

• $g\circ f\simeq {\mathrm{id}}_X$,

• $f\circ g\simeq {\mathrm{id}}_Y$.

In this case, $X$ and $Y$ have the same "essential shape" from the perspective of homotopy. This concept is distinct from homeomorphism, which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid sphere is homotopy equivalent to a single point, but they are not homeomorphic.