Loop Structures on the Homotopy Type of S~3 Revisited

摘要

In an attempt to understand Lie groups from a homotopy theory point of view, Rector suggested studying Lie groups through their classifying spaces. Using S~3 as a test case, he proved in his pioneering paper that there are uncountably many homotopically distinct deloopings of S~3. These deloopings form the so-called genus of the classifying space BS~3. To be more precise, for a nilpotent finite type space X, the genus of X is defined to be the set of homotopy types of nilpotent finite type spaces Y such that the p-completions of X and Y are homotopy equivalent for each prime p and also their rationalizations are homotopy equivalent. When considering genus, one often ignores the difference between a homotopy type and a space with that homotopy type.