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

A homotopy between two topological spaces, [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath], is a continuous function:

• [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I[/ilmath] denotes the unit interval, [ilmath][0,1]\subset\mathbb{R} [/ilmath])

A homotopy is relative to [ilmath]A\in\mathcal{P}(X)[/ilmath] if [ilmath]F(a,t)[/ilmath] is independent of [ilmath]t[/ilmath] for all [ilmath]a\in A[/ilmath]

Terminology

The family of functions [ilmath]\{f_t:X\rightarrow Y\ \vert\ \forall t\in[0,1],\ f_t:x\mapsto F(x,t)\} [/ilmath] are called the stages of the homotopy. So we might say:

• Let [ilmath]f_t[/ilmath] be a stage of the homotopy [ilmath]F[/ilmath] or something similar

OLD ATTEMPT AT PAGE

(Scrapped because I didn't like the layout)

Definition

A homotopy from the topological spaces [ilmath](X,\mathcal{ J })[/ilmath] to [ilmath](Y,\mathcal{ K })[/ilmath] is a continuous function:

• [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I[/ilmath] denotes the unit interval, [ilmath][0,1]\subseteq\mathbb{R} [/ilmath])

For each [ilmath]t\in I[/ilmath] we have a function:

• [ilmath]F_t:X\rightarrow Y[/ilmath] defined by [ilmath]F_t:x\mapsto F(x,t)[/ilmath] - these functions, the [ilmath]F_t[/ilmath] are called the stages of the homotopy.

Applications

Homotopic maps

---

Conditions

Relative homotopic maps

---

Conditions

Path homotopy

---

Conditions