algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
The homology of (iterated) based loop spaces (ordinary homology or generalized homology) carries special structure, reflecting the ∞-group-structure of based loop spaces.
In particular, under mild technical conditions (recalled e.g. in Halperin 92) the Pontrjagin ring-structure induced by concatenation of loops enhances the homology coalgebra induced by the diagonal maps to that of a Hopf algebra.
Lev Pontrjagin, Homologies in compact Lie groups, Rec. Math. [Mat. Sbornik] N.S., 1939 Volume 6(48), Number 3, Pages 389–422 (mathnet:5835)
Eldon Dyer, Richard Lashof, Homology of Iterated Loop Spaces, American Journal of Mathematics Vol. 84, No. 1 (Jan., 1962), pp. 35-88 (jstor:2372804)
Samuel Eilenberg, John Moore, Homology and fibrations, Comment. Math. Helv., 40 (1966), pp. 199-236 (pdf)
William Browder, Homology Rings of Groups, American Journal of Mathematics, Vol. 90, No. 1 (Jan., 1968) (jstor:2373440)
Stephen Halperin, Universal enveloping algebras and loop space homology, Journal of Pure and Applied Algebra Volume 83, Issue 3, 11 December 1992, Pages 237-282 (doi:10.1016/0022-4049(92)90046-I)
Jonathan A. Scott, Algebraic Structure in the Loop Space Homology Bockstein Spectral Sequence, Transactions of the American Mathematical Society Vol. 354, No. 8 (Aug., 2002), pp. 3075-3084 (arXiv:stable/3073034)
See also
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