nLab Pontrjagin ring

Contents

Context

Algebraic topology

Idea
Properties

Relation to Whitehead product
Homological group completion
Relation to quantum cohomology

Related concepts
References

General
Quantum cohomology as Pontrjagin rings

Idea

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

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

For the following examples we use these notational conventions:

$\mathbb{K}$ denotes a field of characteristic zero,
a subscript on a generator denotes its degree,
$\mathbb{K}\langle\cdots \rangle$ denotes the graded $\mathbb{K}$ -vector space spanned by the linear basis of generators listed inside the angular brackets;
$\mathbb{K}[\cdots]$ denotes the underlying vector space of the free graded-commutative algebra on the set of generators listed inside the square brackets;
$T(-)$ denotes the graded tensor algebra on a given graded vector space,
the semifree dgc-algebra on a given set $\{\cdots\}$ of generators subject to differential relations we denote, with slight abuse of notation, by

$\mathbb{K}[\cdots]\big/ \left( \begin{array}{c} \mathrm{d}(-) = \ldots \\ \vdots \end{array} \right)$

Example

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

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 $S^2$ are:

\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 $\Omega S^2$ is the universal enveloping algebra of this super Lie algebra, hence:

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

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

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

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

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

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 $\Omega \, \mathbb{C}P^\infty \,\simeq\, \Omega B U(1)\,\simeq\, S^1$ .

Homological group completion

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

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).

Related concepts

homology of loop spaces

References

General

The concept and the terminology “Pontryagin-multiplication” 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]

who name it in honor of the analogous product operation on the homology of compact Lie groups due to:

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

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 $\mathbb{C}P^1$ -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 $K$ is essentially the quantum cohomology ring of the flag variety of its complexification by its Borel subgroup is attributed (“Peterson isomorphism”) to

Dale H. Peterson (notes by Arun Ram), Quantum Cohomology of $G/H$ , MIT (1997) $[$ web, pdf, pdf $]$