Homotopy (object)

DRAFT PAGE:

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.

Grade: A

This page requires references, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.

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