Homotopy

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This was to be swapped or merged with homotopyPage - don't forget, spotted more than a year later! Alec (talk) 19:31, 26 November 2017 (UTC)

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Definition

Given topological spaces $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$, and any set $A\in\mathcal{P}(X)$ a homotopy (relative to $A$) is any continuous function:

• $H:X\times I\rightarrow Y$ (where $I:=[0,1]\subset\mathbb{R}$) such that:
  • $\forall s,t\in I\ \forall a\in A[H(a,s)=H(a,t)]$

If $A=\emptyset$ then we say $H$ is a free homotopy (or just a homotopy).
If $A\neq \emptyset$ then we speak of a homotopy rel $A$ or homotopy relative to $A$.

Stages of a homotopy

For a homotopy, $H:X\times I\rightarrow Y\ (\text{rel }A)$, a stage of the homotopy $H$ is a map:

• $h_t:X\rightarrow Y$ for some $t\in I$ given by $h_t:x\mapsto H(x,t)$

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

• Note: the stages of a homotopy are continuous

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Definition

Given two topological spaces, $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ then a homotopy of maps (from $X$ to $Y$) is a continuous function: $F:X\times I\rightarrow Y$ (where $I$ denotes the unit interval, $I:=[0,1]\subset\mathbb{R}$). Note:

• The stages of the homotopy, $F$, are a family of functions, $\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\}$ such that $f_t:x\rightarrow F(x,t)$. The stages of a homotopy are continuous.
  • $f_0$ and $f_1$ are examples of stages, and are often called the initial stage of the homotopy and final stage of the homotopy respectively.

Two (continuous) functions, $g,h:X\rightarrow Y$ are said to be homotopic if there exists a homotopy such that $f_0=g$ and $f_1=h$

Claim: homotopy of maps is an equivalence relation