Differential homological algebra
Joseph A. Neisendorfer · Cambridge University Press eBooks · 2010
Differential homological algebra is used in algebraic topology to study the homology of fibrations. The Serre spectral sequence determines the homology of the total space given the homologies of the base and fibre. In general, given the homologies of two out of the three, base, fibre, and total space, various techniques have been developed to determine the homology of the third. Two cases of historical interest are the following: (1) Given the homology of a space, determine the homology of the loop space. (2) Given the homology of a topological group, determine the homology of the classifying space. In a real sense, these problems are dual to one another and have specific solutions. The problem of determining the homology of the loop space is solved with the help of the cobar construction introduced by Frank Adams. This differential algebra, when applied to the chains on a space, is homologically equivalent to the chains on the loop space together with the multiplication induced by loop multiplication. The problem of determining the homology of the classifying space is solved with the help of the bar construction introduced by Eilenberg and MacLane and followed up in a geometric form by John Milnor. This differential coalgebra, when applied to the chains on a topological group, is homologically equivalent to the chains on the classifying space together with the coalgebra structure given by the diagonal.