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 (see Milnor-Moore 65, p. 262, 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.
The Pontrjagin product is due to:
Discussion of homology of (iterated) loop spaces:
William Browder, Homology and Homotopy of H-Spaces, Proceedings of the National Academy of Sciences of the United States of America, Vol. 46, No. 4 (Apr. 15, 1960), pp. 543-545 (3 pages) (jstor:70867)
Eldon Dyer, Richard Lashof, Homology of Iterated Loop Spaces, American Journal of Mathematics Vol. 84, No. 1 (Jan., 1962), pp. 35-88 (jstor:2372804)
John Milnor, John Moore, p. 262 & Appendix of: On the structure of Hopf algebras, Annals of Math. 81 (1965), 211-264 (doi:10.2307/1970615, pdf)
Samuel Eilenberg, John Moore, Homology and fibrations, Comment. Math. Helv., 40 (1966), pp. 199-236 (pdf, doi:10.1007/BF02564371)
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 (jstor:3073034)
Victor Buchstaber, Jelena Grbić, Hopf algebras and homology of loop suspension spaces (pdf, pdf) in: V. Buchstaber et al. (eds.), Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014, American Mathematical Society Translations - Series 2
Advances in the Mathematical Sciences, 2014 (ISBN:978-1-4704-1871-7)
See also:
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