nLab Kronecker pairing

Idea

In ordinary homology/ordinary cohomology represented as singular homology/singular cohomology, Kronecker pairing refers to the defining pairing of a chain with a cochain.

More generally, in generalized (Eilenberg-Steenrod) cohomology/generalized homology represented by a ring spectrum $E$, then the Kronecker pairing is a canonical pairing of the $E$-generalized cohomology groups $E^\bullet(X)$ with the $E$-generalized homology groups $E_\bullet(X)$ of suitable spaces (homotopy types/spectra) $X$

$$\langle-,-\rangle_X \;\colon\; E^{\bullet_1}(X) \otimes E_{\bullet_2}(X) \longrightarrow \pi_{\bullet_2-\bullet_1}(E) \,.$$

The combination of the Kronecker pairing with a diagonal map yields the cap product pairing in generalized (co-)homology.

If $E_\bullet(X)$ is a projective graded module over the graded ring $\pi_\bullet(E)$ then the adjunct

$$E^0(X) \longrightarrow Hom_{\pi_\bullet(E)}(E_\bullet(X), \pi_\bullet(E))$$

of the Kronecker pairing is an isomorphism and hence exhibits $E$-generalized cohomology as the $\pi_\bullet(E)$-linear dual of the $E$-generalized homology of $X$; an instance of a universal coefficient theorem for generalized (co-)homology (prop. below).

On CW-complexes $X$ of finite dimension, the Kronecker pairing induces a pairing of the corresponding Atiyah-Hirzebruch spectral sequences (prop. below).

Definition

Let $E$ be a ring spectrum with product denoted $\mu \colon E \wedge E \longrightarrow E$. Let $X,Y$ be any spectra.

Definition

Given $[f] \in E^k(X)$ with representative $f \colon X \longrightarrow \Sigma^k E$ and given $[w] \in E_{n+k}(X \wedge Y)$ with representative $w \colon S^{n+k} \longrightarrow E \wedge X \wedge Y$, then their Kronecker pairing is the element

$$\langle f,w\rangle \in E_n(Y)$$

represented by the composite

$$S^{k+n} \stackrel{w}{\longrightarrow} E\wedge X \wedge Y \stackrel{id_E \wedge f \wedge id_Y}{\longrightarrow} E \wedge \Sigma^k E \wedge Y \stackrel{\Sigma^k \mu \wedge id_Y}{\longrightarrow} \Sigma^k E \wedge Y \,.$$

This yields a homomorphism of graded abelian groups

$$\langle-,-\rangle \;\colon\; E^{\bullet_1}(X) \otimes E_{\bullet_2}(X \wedge Y) \longrightarrow E_{\bullet_2-\bullet_1}(Y) \,.$$

(and similarly for $Y$ on the other side…)

For $Y = \mathbb{S}$ this is

$$\langle-,-\rangle \;\colon\; E^{\bullet_1}(X) \otimes E_{\bullet_2}(X) \longrightarrow \pi_{\bullet_2-\bullet_1}(E) \,.$$

(e.g. Kochman 96, (4.2.1), Schwede 12, construction 6.13)

Properties

Universal coefficient theorem

Proposition

If $E_\bullet(X)$ is a projective graded module over the graded ring $\pi_\bullet(E)$ then the adjunct

$$\pi_0[X, E \wedge Y] \longrightarrow Hom_{\pi_\bullet(E)}( E_\bullet(X), E_\bullet(Y) )$$

$$f \mapsto \langle f,-\rangle$$

of the Kronecker pairing, def. , is an isomorphism.

(e.g. Schwede 12, chapter II, prop.6.20)

Proof idea

By the formula for adjuncts, the morphism factors through the free-forgetful adjunction for $E$-module spectra

$$\pi_0[X, E \wedge Y] \stackrel{\simeq}{\longrightarrow} \pi_0[E\wedge X, E \wedge Y]_{E Mod} \stackrel{\pi_\bullet}{\longrightarrow} Hom_{\pi_\bullet}( E_\bullet(X), E_\bullet(Y) ) \,.$$

Hence one is reduced to showing that under the given conditions the second morphis is an iso. (…)

This may be regarded as a universal coefficient theorem (Adams 74, part III, around prop. 13.5).

For $Y = \mathbb{S}$ prop. gives:

Example

If $E_\bullet(X)$ is a projective graded module over the graded ring $\pi_\bullet(E)$ then the adjunct

$$E^0(X) \longrightarrow Hom_{\pi_\bullet(E)}( E_\bullet(X), \pi_\bullet(E) )$$

$$f \mapsto \langle f,-\rangle$$

of the Kronecker pairing, def. , is an isomorphism.

Pairing on Atiyah-Hirzebruch spectral sequences

Proposition

For $E$ a ring spectrum and $X$ a CW complex of finite dimension, then the Kronecker pairing $\langle -,-\rangle \colon E^\bullet(X)\otimes E_\bullet(X)\to \pi_\bullet(E)$ of def. passes to a page-wise pairing of the corresponding Atiyah-Hirzebruch spectral sequences for $E$-cohomology/homology

$$\langle-,-\rangle_r \;\colon\; \mathcal{E}_r^{n,-s} \otimes \mathcal{E}^r_{n,t} \longrightarrow \pi_{s+t}(E)$$

such that

1. on the $\mathcal{E}_2$-page this restricts to the Kronecker pairing for ordinary cohomology/ordinary homology with coefficients in $\pi_\bullet(E)$;

2. the differentials act as derivations

   $$\langle d_r(-),-\rangle = \langle -, d^r(-)\rangle \,,$$

3. The pairing on the $\mathcal{E}_\infty$-page is compatible with the Kronecker pairing.

(Kochman 96, prop. 4.2.10)

References

Presumeably named after Leopold Kronecker.

• Frank Adams, Part III, section 13 of Stable homotopy and generalised homology, 1974

• Stanley Kochman, (4.2.1) and prop. 4.2.10 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Stefan Schwede, chapter II, section 6 (Construction 6.13) of Symmetric spectra, 2012 (pdf)