The action of the fundamental group on the higher homotopy groups
No. 16, 1967 (Algebraic and homotopic topology )
India ink on paper, 20x29.5 cm.
Imagine a certain spheroid in space. Imagine it tethered to some point of itself by an elastic tube. Now consider a fundamental group, a homotopy invariant of a topological space. Its elements are the classes of homotopic paths, which can be continuously deformed into each other. In this image, what hooks the two spheroids together is nothing less than a mapping function, represented intuitively. The loop represents an element of the fundamental group. Growing out of one spheroid is a long, thin tube that slides along the loop, ending up at its own beginning, leaving each spheroid to be replaced by a new spheroid. Again, the essential idea is one of mapping. Different loops will determine different mappings. Once the mapping process has ended and all transformations have been completed, the original spheroid leaves its place and wanders freely in space, while still remaining attached by a tube to its point of origin.
