nLab Pontrjagin ring

Contents

Contents

Idea

The operations on an H-space XX (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 H *(X)H_*(X) of XX.

Properties

Relation to Whitehead product

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 ΩX\Omega X is identified via the Hurewicz homomorphism with the universal enveloping algebra (see there) of the Whitehead bracket super Lie algebra of XX [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 XX (by the proposition here).

For the following examples we use these notational conventions:

Example

(rational Pontrjagin algebra of loops of spheres and P n\mathbb{C}P^ns)
The Sullivan model of the 2-sphere is

CE(𝔩S 2)[ω 2,ω 3]/(dω 2=0 dω 3=12ω 2ω 2). CE(\mathfrak{l} S^2) \;\simeq\; \mathbb{Q}\big[ \omega_2, \omega_3 \big]\big/ \left( \begin{array}{l} \mathrm{d} \omega_2 \,=\, 0 \\ \mathrm{d} \omega_3 \,=\, \tfrac{1}{2} \omega_2 \wedge \omega_2 \end{array} \right) \,.

From this it follows (since the co-binary Sullivan differential is the dual Whitehead product) that the binary Whitehead super Lie brackets of S 2S^2 are:

[v 1,v 1]=v 2 [v 2,v 2]=0 [v 1,v 2]=0 \begin{array}{l} [v_1, v_1] = v_2 \\ [v_2, v_2] = 0 \\ [v_1, v_2] = 0 \end{array}

The rational Pontrjagin ring of ΩS 2\Omega S^2 is the universal enveloping algebra of this super Lie algebra, hence:

H (ΩS 2;𝕂)T(𝕂v 1,v 2)/(2v 1 2v 2 v 1v 2v 2v 1). H_\bullet\big( \Omega S^2 ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_1, v_2\rangle \big) \big/ \left( \begin{array}{l} 2 v_1^2 - v_2 \\ v_1 v_2 - v_2 v_1 \end{array} \right) \mathrlap{\,.}

Therefore the underlying graded 𝕂\mathbb{K}-vector space of the Pontrjagin ring of ΩS 2\Omega S^2 is 𝕂[v 1,v 2]\mathbb{K}[v_1, v_2] but the product of the v 1v_1 is deformed from v 1v 1=0v_1 \cdot v_1 = 0 to v 1v 1=12v 2v_1 \cdot v_1 = \tfrac{1}{2}v_2.

Similarly, the rational Pontrjagin algebra of the loop space of the 4-sphere is

H (ΩS 4;𝕂)T(𝕂v 3,v 6)/(2v 3 2v 6 v 3v 6v 6v 3) H_\bullet\big( \Omega S^4 ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_3, v_6\rangle \big) \big/ \left( \begin{array}{l} 2 v_3^2 - v_6 \\ v_3 v_6 - v_6 v_3 \end{array} \right)

whose underlying 𝕂\mathbb{K}-vector space is 𝕂[v 3,v 6]\mathbb{K}[v_3, v_6] but with the product of the v 3v_3 deformed from v 3v 3=0v_3 \cdot v_3 = 0 to v 3v 3=12v 6v_3 \cdot v_3 = \tfrac{1}{2}v_6.

On the other hand, the differential of the Sullivan model of complex projective space P n\mathbb{C}P^n for n2n \geq 2 has vanishing co-binary part, so that

H (ΩP n2;𝕂)T(𝕂v 2,v 2n)/(2v 1 2 v 1v 2nv 2nv 1)𝕂[v 1,v 2n] H_\bullet\big( \Omega \mathbb{C}P^{n \geq 2} ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_2, v_{2n}\rangle \big) \big/ \left( \begin{array}{l} 2 v_1^2 \\ v_1 v_{2n} - v_{2n} v_1 \end{array} \right) \;\simeq\; \mathbb{K}[v_1, v_{2n}]

is just a plain graded-commutative algebra.

For n=n = \infty (ie. for the classifying space P BU(1)\mathbb{C}P^\infty \simeq B U(1)) this becomes

H (ΩP ;𝕂)T(𝕂v 2)/(2v 1 2)𝕂[v 1], H_\bullet\big( \Omega \mathbb{C}P^{\infty} ;\, \mathbb{K} \big) \;\simeq\; T\big( \mathbb{K}\langle v_2\rangle \big) \big/ \big( 2 v_1^2 \big) \;\simeq\; \mathbb{K}[v_1] \mathrlap{\,,}

reflecting the fact that ΩP ΩBU(1)S 1\Omega \, \mathbb{C}P^\infty \,\simeq\, \Omega B U(1)\,\simeq\, S^1.

Homological group completion

The homological version of the group completion theorem relates the Pontrjagin ring of a topological monoid AA to that of its group completion ΩBA\Omega B A.

Relation to quantum cohomology

The homology Pontrjagin product of certain compact Lie groups is identified with the quantum cohomology of corresponding flag varieties (see references below).

References

General

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:

  • Hans Samelson, §3 in: Beiträge Zur Topologie der Gruppen-Mannigfaltigkeiten, Annals of Mathematics, Second Series, 42 5 (1941) 1091-1137 [jsotr:1970463]

Proof that the commutator of the Pontrjagin product is the Whitehead product, under the Hurewicz homomorphism:

and in characteristic zero:

and slightly beyond

and for higher Whitehead brackets:

and full lifting of the theorem Milnor & Moore, 1965 (Appendix), equipping the rational Pontrjagin algebra with A A_\infty -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:

In the context of string topology:

Textbook accounts:

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:

  • Jelena Grbic, Svjetlana Terzic, The integral Pontrjagin homology of the based loop space on a flag manifold, Osaka J. Math. 47 (2010) 439-460 [arXiv:math/0702113

Quantum cohomology as Pontrjagin rings

On the relation between quantum cohomology rings, hence of Gromov-Witten invariants in topological string theory, for flag manifold target spaces (such as the P 1 \mathbb{C}P^1 -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 KK 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

reviewed in

  • Jimmy Chow, Homology of based loop groups and quantum cohomology of flag varieties, talk at Western Hemisphere Virtual Symplectic Seminar (2021) [[pdf, pdf, video:YT]]

with further discussion in:

On the variant for Pontryagin products not on ordinary homology but in topological K-homology:

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 January 21, 2024 at 12:43:43. See the history of this page for a list of all contributions to it.