Homology and Homotopy Theory Homotopy and Homology

1. TJ'he first step toward connecting these two basic concepts of topology was taken by L. E. J. Brouwer in 1912 by demonstrating that two continuous mappings of a two-dimensional sphere into itself can be continuously deformed into each other if and only if they have the same degree (that is, if they are equivalent from the point of view of homology theory). After having generalized Brouwer's result to an arbitrary number of dimensions, H. Hopf undertook a systematic study of the problem of classifying the continuous mappings of a polytope P into a polytope Q. Each mapping/induces homomorphisms of homology groups of P into the corresponding groups of Q. Two mappings / and g are said to belong to the same homology classif they induce identical homomorphisms of homology groups (for all dimensions and all coefficient domains). The mappings / and g are said to belong to the same homotopy class if they can be embedded into a common one-parameter continuous family of mappings. The homotopy class of a mapping determines its homology class, but not conversely, as shown by the example of the mappings of the sphere Ss into $2 which all belong to the same homology class although there is an infinite number of homotopy classes. The question arises: under what special conditions the homotopy classification of the mappings of P into Q coincides with their homology classification. The classical result of Hopf states that this is the case if P is a polytope of dimension n and Q the n-dimensional sphere Sn . Using cohomology groups instead of homology groups, H. Whitney gave the following elegant formulation to Hopf's theorem. Homotopy classes of mappings of an ^-dimensional polytope P into the sphere Sn are in one to one correspondence with the elements of the n-dimensional cohomology group of P with integers as coefficients.