The method of killing spaces in homotopic topology
No. 44, 1968
(Homotopic topology)
India ink and color pencil on paper, 31.5x43.5 cm.
A great stone ledge projects out into a tumultuous sea of vapor and flame, oozing and whirling in places, though calmer in others. On the ledge, endless rows of nameless souls hang their heads in prayer, searching for meaning and asking for redemption. Strangely, underlying this image is a method for computing topological invariants, or homotopy groups. The many successive levels of the long ledge, filled with unending series of human beings, represent homotopy groups, each one of which corresponds to the next killing space. Such killing spaces are a sequence of special spaces designed to study a certain object, say a polyhedron. If necessary, we can first try to compute the homology groups, then the homotopy groups, of the polyhedron. The first nontrivial homotopy group turns out to be isomorphic to a homology group. Subsequently, we can construct a new polyhedron whose homotopy groups coincide with those of the original space. The idea is that the first nontrivial homotopy group of the original polyhedron is killed. In the resulting space, we compute subsequent homology groups, following a certain algorithm and find the second homotopy group of the original polyhedron. The process continues on infinitely.
