Homotopy for Maps vs Paths

Homotopy (of maps)

A homotopy is a family of maps $f_t: X\to Y$ , $t\in I$ , such that the associated map $F:X\times I\to Y$ given by $F(x,t)=f_t(x)$ is continuous. Two maps $f_0, f_1:X\to Y$ are called homotopic, denoted $f_0\simeq f_1$ , if there exists a homotopy $f_t$ connecting them.

Homotopy of paths

A homotopy of paths in a space $X$ is a family $f_t: I\to X$ , $0\leq t\leq 1$ , such that

(i) The endpoints $f_t(0)=x_0$ and $f_t(1)=x_1$ are independent of $t$ .
(ii) The associated map $F:I\times I\to X$ defined by $F(s,t)=f_t(s)$ is continuous.

When two paths $f_0$ and $f_1$ are connected in this way by a homotopy $f_t$ , they are said to be homotopic. The notation for this is $f_0\simeq f_1$ .

The above two definitions are related, since a path is a special kind of map $f: I\to X$ .

Author: mathtuition88

Math and Education Blog View all posts by mathtuition88

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