HomotopyPage

Note: a homotopy is really a relation on continuous functions (see homotopic), however since any continuous map of the form [ilmath]F:X\times I\rightarrow Y[/ilmath] 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, [ilmath]F:X\times I\rightarrow Y[/ilmath], we get homotopic maps. So a homotopy warrants a definition, even if it is useless by itself.

Definition

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

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

There is some terminology used depending on whether or not [ilmath]A=\emptyset[/ilmath]:

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

Stages of the homotopy

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

• [ilmath]h_t:x\mapsto H(x,t)[/ilmath]

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

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