Homotopy of Maps & Paths

Continuous deformation of maps, path homotopy rel endpoints, and the equivalence relation.

The idea

A homotopy is a continuous deformation of one map into another. It gives precise form to questions that homeomorphism is too rigid to ask — whether a loop can be slid off a hole, whether a map can be shrunk to a constant — and unlike homeomorphism it relates maps, not spaces.

To deform $f$ into $g$ is to give a family of maps between them, one for each instant, packaged as a single map with the time as an extra input. Two continuous maps $f, g \colon X \to Y$ are homotopic when there is a continuous $H \colon X \times [0,1] \to Y, \qquad H(x, 0) = f(x), \quad H(x, 1) = g(x).$ Fixing $t$ gives a snapshot $f_{t} = H(\,\cdot\,, t)$, a map from $X$ to $Y$; letting $t$ run from $0$ to $1$ slides $f$ through the intermediate maps $f_{t}$ into $g$, as the figure shows inside the target $Y$. The definition requires $H$ to be continuous in $x$ and $t$ together, so the snapshots change without jumping.

For paths — maps out of $[0,1]$ — we add one condition. Two paths sharing a start $p$ and an end $q$ are path homotopic when a homotopy between them holds both ends fixed: $H(0,t) = p$ and $H(1,t) = q$ for every $t$. Without this condition the notion says nothing about paths: in a connected space the ends could be dragged together, making any two paths homotopic.

Ways to work on it

• Walkthrough. The definition of homotopy, the straight-line homotopy in convex sets, and the equivalence relation.
• Practice. Decide whether a given deformation is a valid (path) homotopy.
• Hardest. Show a reparameterized path is path-homotopic to the original, endpoints held fixed.

Not sure where to start? Take the ten-question placement test.