S. D. V. Stephens

notes / Math 55b / chapter 4: The Fundamental Group and Covering Spaces / From my notebook

N19 Homotopy

Homotopy is the notion of continuous deformation, parametrized by (the “time” variable in the homotopy).

Definition N19.1. Let be two continuous maps. A homotopy between and is a continuous map such that and for all . If this exists, then we say and are homotopic and write . If is homotopic to a constant map, we say it is nullhomotopic. (All this: a family of continuous maps that vary over time.)

The square X x I with a vertical segment above a point x, mapped by H into Y; the bottom edge maps to the curve f, the top edge to the curve g, and the segment to a path from f(x) through H(x,t) to g(x).

Paths in are continuous maps , so in particular we can compare two paths .