Homotopy asks if one map can melt into another
Two continuous maps between spaces are homotopic if one can be smoothly morphed into the other, and the morph itself is called a homotopy. The word joins Greek homós, same, and tópos, place, and the idea underpins homotopy groups, key invariants of algebraic topology.
Formally, a homotopy from f to g is a single continuous map defined on the domain crossed with the interval from 0 to 1. Read the second input as time: at 0 you have f, at 1 you have g, and in between the function slides continuously, like a slider control. A classic animation deforms a torus embedded as a doughnut into one shaped like a coffee mug. Being homotopic is an equivalence relation, and it survives composition: compose homotopic maps with homotopic maps and the results are still homotopic.
Simple examples abound. The functions sending x to (x, x³) and to (x, eˣ) are joined by blending the second coordinates in proportions that shift with time. In a convex region, two paths with the same endpoints can always be connected by a straight-line homotopy. On the unit disk, the identity map can be shrunk to the map sending every point to the centre by scaling each point by one minus t.
Two spaces are homotopy equivalent, or share a homotopy type, if maps back and forth compose to something homotopic to the identity on each side. Intuitively, one can be bent, shrunk or stretched into the other, and a space equivalent to a single point is called contractible. Every homeomorphism is a homotopy equivalence, but not the reverse: a solid disk is equivalent to a point yet cannot be matched with it one to one, and the Möbius strip and a plain untwisted band both shrink to a circle without being homeomorphic. All of n-dimensional real space is likewise equivalent to a point.
Homotopies can misbehave on some spaces, so algebraic topologists usually work with compactly generated spaces, CW complexes or spectra.
Source: Homotopy
