Definition. Two Continuous functions $f, f': X \to Y$ are homotopoic, $f \simeq f'$, if there exists a Continuous function $F: X \times I \to Y$ ($I= [0,1]$) s.t. $F(x,0) = f(x)$, and $F(x,1) = f'(x)$.

$F$ is called a homotopy between $f$ and $f'$. It is a formalization of a "continuous deformation" from $f$ to $f'$.

Homotopoy is an Equivalence relation on the set of continuous maps from $X$ to $Y$.

See notes

Examples

every continous map $f: X \to \mathbb{R}^n$ is homotopic to a constant map, using Straight line homotopy

Continuous path $f: [0,1] \to X$, then it is homotopic to a constant path (notice this is not a Loop path, even if it may have the same image, but it definitely has a different domain ($S_1$)). However, we can also make this theory nontrivial by using Path homotopy

Can also define paths being homotopic relative to a subset A of X by requiring that the map restricted to A doesn't change during the homotopy.

Path homotopy

Define Product of paths

Fundamental group

Lemma If a continuous map $f: S^1 \to X$ extends to a cont. map $F: B^2 \to X$ (that is $F|S^1 = f$). Then $f_*: \pi_1(S^1,1) \to \pi_1(X, f(1))$ is the trivial homomorphism.

Lemma. If $f \simeq g$ (homotopic), where $f,g: S^1 \to X$. Then if $f$ extends to $B^2$ then also $g$ extends to $B^2$ (and so $f_*$ and $g_*$ are trivial).

Fundamental theorem of algebra

As can be seen in the examples at beginning of this video, homotopy has applications to Complex analysis. For instance, line integrals are often insensitive to "continuous deformations", that is to homotopies!

Also applications to Topological quantum field theory