nLab
Kronecker pairing

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

In ordinary homology/ordinary cohomology represented as singular homology/singular cohomology, then 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 EE, then the Kronecker pairing is a canonical pairing of the EE-generalized cohomology groups E (X)E^\bullet(X) with the EE-generalized homology groups E (X)E_\bullet(X) of suitable spaces (homotopy types/spectra) XX

, X:E 1(X)E 2(X)π 2 1(E). \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 (X)E_\bullet(X) is a projective graded module over the graded ring π (E)\pi_\bullet(E) then the adjunct

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

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

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

Definition

Let EE be a ring spectrum with product denoted μ:EEE\mu \colon E \wedge E \longrightarrow E. Let X,YX,Y be any spectra.

Definition

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

f,wE n(Y) \langle f,w\rangle \in E_n(Y)

represented by the composite

S k+nwEXYid Efid YEΣ kEYΣ kμid YΣ kEY. 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

,:E 1(X)E 2(XY)E 2 1(Y). \langle-,-\rangle \;\colon\; E^{\bullet_1}(X) \otimes E_{\bullet_2}(X \wedge Y) \longrightarrow E_{\bullet_2-\bullet_1}(Y) \,.

(and similarly for YY on the other side…)

For Y=𝕊Y = \mathbb{S} this is

,:E 1(X)E 2(X)π 2 1(E). \langle-,-\rangle \;\colon\; E^{\bullet_1}(X) \otimes E_{\bullet_2}(X) \longrightarrow \pi_{\bullet_2-\bullet_1}(E) \,.

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

Properties

Universal coefficient theorem

Proposition

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

π 0[X,EY]Hom π (E)(E (X),E (Y)) \pi_0[X, E \wedge Y] \longrightarrow Hom_{\pi_\bullet(E)}( E_\bullet(X), E_\bullet(Y) )
ff, 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 EE-module spectra

π 0[X,EY]π 0[EX,EY] EModπ Hom π (E (X),E (Y)). \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=𝕊Y = \mathbb{S} prop. gives:

Example

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

E 0(X)Hom π (E)(E (X),π (E)) E^0(X) \longrightarrow Hom_{\pi_\bullet(E)}( E_\bullet(X), \pi_\bullet(E) )
ff, f \mapsto \langle f,-\rangle

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

Pairing on Atiyah-Hirzebruch spectral sequences

Proposition

For EE a ring spectrum and XX a CW complex of finite dimension, then the Kronecker pairing ,:E (X)E (X)π (E)\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 EE-cohomology/homology

, r: r n,s n,t rπ s+t(E) \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 2\mathcal{E}_2-page this restricts to the Kronecker pairing for ordinary cohomology/ordinary homology with coefficients in π (E)\pi_\bullet(E);

  2. the differentials act as derivations

    d r(),=,d r(), \langle d_r(-),-\rangle = \langle -, d^r(-)\rangle \,,
  3. The pairing on the \mathcal{E}_\infty-page is compatible with the Kronecker pairing.

(Kochmann 96, prop. 4.2.10)

References

Presumeably named after Leopold Kronecker.

Last revised on May 26, 2017 at 12:47:36. See the history of this page for a list of all contributions to it.