Homotopy

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 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