algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
The operations on an H-space (such as a topological group or a loop space) equip its homology with the structure of ring. At least for ordinary homology this is known as the Pontrjagin ring of .
Under the Hurewicz homomorphism, the commutator of the Pontrjagin product on homology is the Whitehead product on homotopy groups of a based loop space.
This is due to Samelson (1953) and for higher Whitehead brackets due to Arkowitz (1971).
In fact, in characteristic zero the Pontrjagin ring structure on connected based loop spaces is identified via the Hurewicz homomorphism with the universal enveloping algebra (see there) of the Whitehead bracket super Lie algebra of [Milnor & Moore (1965) Appendix; Whitehead (1978) Thm. X.7.10; Félix, Halperin & Thomas 2000, Thm. 16.13]. Moreover, in this case the underlying ordinary cohomology (hence the ordinary homology) vector space may be read off from any Sullivan model of (by the proposition here).
For the following examples we use these notational conventions:
denotes a field of characteristic zero,
a subscript on a generator denotes its degree,
denotes the graded -vector space spanned by the linear basis of generators listed inside the angular brackets;
denotes the underlying vector space of the free graded-commutative algebra on the set of generators listed inside the square brackets;
denotes the graded tensor algebra on a given graded vector space,
the semifree dgc-algebra on a given set of generators subject to differential relations we denote, with slight abuse of notation, by
(rational Pontrjagin algebra of loops of spheres and s)
The Sullivan model of the 2-sphere is
From this it follows (since the co-binary Sullivan differential is the dual Whitehead product) that the binary Whitehead super Lie brackets of are:
The rational Pontrjagin ring of is the universal enveloping algebra of this super Lie algebra, hence:
Therefore the underlying graded -vector space of the Pontrjagin ring of is but the product of the is deformed from to .
Similarly, the rational Pontrjagin algebra of the loop space of the 4-sphere is
whose underlying -vector space is but with the product of the deformed from to .
On the other hand, the differential of the Sullivan model of complex projective space for has vanishing co-binary part, so that
is just a plain graded-commutative algebra.
For (ie. for the classifying space ) this becomes
reflecting the fact that .
The homological version of the group completion theorem relates the Pontrjagin ring of a topological monoid to that of its group completion .
The homology Pontrjagin product of certain compact Lie groups is identified with the quantum cohomology of corresponding flag varieties (see references below).
The concept and the terminology “Pontryagin-multiplication” is due to
who name it in honor of the analogous product operation on the homology of compact Lie groups due to:
see also:
Proof that the commutator of the Pontrjagin product is the Whitehead product, under the Hurewicz homomorphism:
and in characteristic zero:
John Milnor, John Moore, Appendix (pp. 262) of: On the structure of Hopf algebras, Annals of Math. 81 (1965) 211-264 [doi:10.2307/1970615, pdf]
Yves Félix, Stephen Halperin, Jean-Claude Thomas, Thm. 16.13 in: Rational Homotopy Theory, Graduate Texts in Mathematics 205 Springer (2000) [doi:10.1007/978-1-4613-0105-9]
and slightly beyond
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]
and for higher Whitehead brackets:
and full lifting of the theorem Milnor & Moore, 1965 (Appendix), equipping the rational Pontrjagin algebra with -algebra structure and identifying it with the universal envelope of the Whitehead L-infinity algebra:
Refinement to algebra structure on the singular chain complex (Adams-Hilton model):
reviewed and further developed in:
More on the Pontrjagin rings of the classical Lie groups:
reviewed in:
Further early discussion:
Raoul Bott, The space of loops on a Lie group, Michigan Math. J. 5 1 (1958) 35-61 [doi:10.1307/mmj/1028998010]
(for loop spaces of Lie groups)
William Browder, Homology and Homotopy of H-Spaces, Proceedings of the National Academy of Sciences of the United States of America 46 4 (1960) 543-545 [jstor:70867]
William Browder, p. 36 of: Torsion in H-Spaces, Annals of Mathematics, Second Series 74 1 (1961) 24-51 [jstor:1970305]
William Browder, Homology Rings of Groups, American Journal of Mathematics 90 1 (1968) [jstor:2373440]
On the effect on Pontrjagin rings of group completion of topological monoids:
Michael Barratt, Stewart Priddy, On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici, 47 1 (1972) 1–14 [doi:10.1007/BF02566785, eudml:139496]
Dusa McDuff, Graeme Segal: Homology fibrations and the “group-completion” theorem, Inventiones mathematicae 31 (1976) 279-284 [doi:10.1007/BF01403148]
Pontrjagin rings in the context of string topology:
Moira Chas, Dennis Sullivan, §3 in: String Topology [arXiv:math/9911159]
Ralph L. Cohen, John D. S. Jones, Jun Yan, pp. 15 of: The loop homology algebra of spheres and projective spaces, in Categorical Decomposition Techniques in Algebraic Topology, Progress in Mathematics 215, Birkhäuser (2003) [arXiv:math/0210353, doi:10.1007/978-3-0348-7863-0_5]
Gwénaël Massuyeau, Vladimir Turaev, Brackets in the Pontryagin algebras of manifolds, Mém. Soc. Math. France 154 (2017) [arXiv:1308.5131]
Textbook accounts:
George W. Whitehead, p. 98 in: Elements of Homotopy Theory, Springer (1978) [doi:10.1007/978-1-4612-6318-0]
Allen Hatcher, §3.C, pp. 287 in: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
Lecture notes
See also
Relating the Pontrjagin algebra on loop groups of compact Lie groups to their Langlands dual groups:
Computation of the Pontryagin products for (loop spaces of) flag manifolds:
On the relation between quantum cohomology rings, hence of Gromov-Witten invariants in topological string theory, for flag manifold target spaces (such as the -sigma model) and Pontrjagin rings (homology-Hopf algebras of based loop spaces):
That the Pontryagin ring-structure on the ordinary homology of the based loop space of a simply-connected compact Lie group is essentially the quantum cohomology ring of the flag variety of its complexification by its Borel subgroup is attributed (“Peterson isomorphism”) to
see also
and proven in
Thomas Lam, Mark Shimozono, §6.2 in: Quantum cohomology of and homology of affine Grassmannian, Acta Mathematica 204 (2010) 49–90 arXiv:0705.1386, doi:10.1007/s11511-010-0045-8
Chi Hong Chow, Peterson-Lam-Shimozono’s theorem is an affine analogue of quantum Chevalley formula arXiv:2110.09985
Chi Hong Chow, On D. Peterson’s presentation of quantum cohomology of arXiv:2210.17382
reviewed in
with further discussion in:
On the variant for Pontryagin products not on ordinary homology but in topological K-homology:
Thomas Lam, Changzheng Li, Leonardo C. Mihalcea, Mark Shimozono, A conjectural Peterson isomorphism in K-theory, Journal of Algebra 513 (2018) 326-343 doi:10.1016/j.jalgebra.2018.07.029, arXiv:1705.03435
Takeshi Ikeda, Shinsuke Iwao, Toshiaki Maeno, Peterson Isomorphism in K-theory and Relativistic Toda Lattice, International Mathematics Research Notices 2020 19 (2020) 6421–6462 arXiv:1703.08664, doi:10.1093/imrn/rny051
Syu Kato, Loop structure on equivariant K-theory of semi-infinite flag manifolds arXiv:1805.01718
Syu Kato, On quantum -groups of partial flag manifolds arXiv:1906.09343
Syu Kato, Darboux coordinates on the BFM spaces arXiv:2008.01310
Syu Kato, Quantum -groups on flag manifolds, talk at IMPANGA 20 (2021) pdf
On the example of the CP^1 sigma-model: LLMS18, §4.1, Kato21 p. 17, Chow22 Exp. 1.4.
See also:
Relation to chiral rings of D=3 N=4 super Yang-Mills theory:
Last revised on July 8, 2024 at 18:23:10. See the history of this page for a list of all contributions to it.