homology of loop spaces in nLab

Contents

Contents

Idea

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.

Examples

• weight systems are cohomology of loop space of configuration space

• Pontrjagin ring

References

The “Pontryagin-multiplication” on loop/path spaces is due to

• Raoul Bott, Hans Samelson, On the Pontryagin product in spaces of paths, Commentarii Mathematici Helvetici 27 (1953) 320–337 [doi:10.1007/BF02564566]

named in honor of the analogous construction over compact Lie groups in:

• Lev Pontrjagin, Homologies in compact Lie groups, Rec. Math. [Mat. Sbornik] N.S., 1939 Volume 6(48), Number 3, Pages 389–422 (mathnet:5835)

Further discussion of homology of (iterated) loop spaces:

• 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 83 3 (1992) 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 354 8 (2002) 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:

• Wikipedia, Pontryagin product