HomotopyPage

Note: a homotopy is really a relation on continuous functions (see homotopic), however since any continuous map of the form $F:X\times I\rightarrow Y$ is a homotopy, and has continuous initial and final stages, it also means that, automatically, it's initial and final stages are homotopic maps. So simply by exhibiting a continuous function, $F:X\times I\rightarrow Y$, we get homotopic maps. So a homotopy warrants a definition, even if it is useless by itself.

Definition

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces and let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$. A homotopy, relative to $A$ is, in its purest form, is any continuous map:

• $F:X\times I\rightarrow Y$ (where $I:=[0,1]\subset\mathbb{R}$ - the unit interval)
  • such that $\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]$ - the homotopy is fixed on $A$.

There is some terminology used depending on whether or not $A=\emptyset$:

1. $A=\emptyset$ then we call $F$ a free homotopy or just homotopy. If however
2. $A\ne\emptyset$ then we speak of a homotopy relative to $A$ or $F\ (\text{rel}\ A)$

Stages of the homotopy

Let $t\in I$ be given, and $H:X\times I\rightarrow Y$ be a homotopy as defined above. $h_t:X\rightarrow Y$ denotes a stage of the homotopy and is defined as follows:

• $h_t:x\mapsto H(x,t)$

The family of stages, $\{h_t:X\rightarrow Y\}_{t\in I}$, are collectively called the stages of the homotopy and

1. $h_0:X\rightarrow Y$ defined by $h_0:x\mapsto H(x,0)$ is the initial stage of the homotopy.
2. $h_1:X\rightarrow Y$ defined by $h_1:x\mapsto H(x,1)$ is the final stage of the homotopy.

Note that the stages of a homotopy are continuous

Purpose

The point of a homotopy is to be a relation (in fact an equivalence relation) of (continuous) maps.

Homotopic maps

Homotopic maps/Definition