algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $E$, 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) $X$
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
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).
Let $E$ be a ring spectrum with product denoted $\mu \colon E \wedge E \longrightarrow E$. Let $X,Y$ be any spectra.
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
represented by the composite
This yields a homomorphism of graded abelian groups
(and similarly for $Y$ on the other side…)
For $Y = \mathbb{S}$ this is
(e.g. Kochman 96, (4.2.1), Schwede 12, construction 6.13)
If $E_\bullet(X)$ is a projective graded module over the graded ring $\pi_\bullet(E)$ then the adjunct
of the Kronecker pairing, def. , is an isomorphism.
(e.g. Schwede 12, chapter II, prop.6.20)
By the formula for adjuncts, the morphism factors through the free-forgetful adjunction for $E$-module spectra
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:
If $E_\bullet(X)$ is a projective graded module over the graded ring $\pi_\bullet(E)$ then the adjunct
of the Kronecker pairing, def. , is an isomorphism.
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
such that
on the $\mathcal{E}_2$-page this restricts to the Kronecker pairing for ordinary cohomology/ordinary homology with coefficients in $\pi_\bullet(E)$;
the differentials act as derivations
The pairing on the $\mathcal{E}_\infty$-page is compatible with the Kronecker pairing.
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 of Symmetric spectra, 2012 (pdf)