## Homotopy groups of spheres

No. 253, 1971

(Algebraic topology and homotopy)

Oil on art board, 50x70 cm.

Beneath an eerie light, within this strange cosmic space, rays of an ancient sun illuminate a fantastic castle, poised on the precipice of a rock-hewn cliff Above hovers a great sphere, collapsing in as space undulates and folds in around it. A mapping function is at work, and under its efforts a large sphere crumples inward into a fairly complicated structure, enveloping a smaller inner sphere with several layers. The visible structure represents an intuitive projection. Shown too are types of singularities, degeneracies, and folds that can arise from just such a mapping.

The mathematical idea is one of computing the homotopy groups of spheres, formed by mapping one sphere onto another, and in this case a large sphere onto a smaller one. These homotopy groups are important invariants of topological spaces that aid us in distinguishing one space from another.