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

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

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

Claim: homotopy of maps is an equivalence relation