Contents

Idea

Where a homology theory is a covariant functor and a cohomology theory is a contravariant functor on some category of spaces, a bivariant cohomology theory is a bifunctor, hence a functor of two variables, contravariant in the first, and covariant in the second.

Examples:

Axiomatization in homotopy theory

Here are some notes on a proposal for how to usefully formalize bivariant cohomology theory in stable homotopy theory. (This is in generalization of the structure of KK-theory, while the original axioms of (Fulton-MacPherson 81) are a little different1. Aspects of the following appear in (Nuiten 13, Schreiber 14). See also at dependent linear type theory the section on secondary integral transforms).

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Let $E$ be an E-∞ ring, write $GL_1(E)$ for its ∞-group of units. With $\mathbf{H}$ the ambient (∞,1)-topos, write $\mathbf{H}_{/\mathbf{B}GL_1(E)}$ for the slice (∞,1)-topos over the delooping of this abelian ∞-group. This is the (∞,1)-category of spaces equipped with (∞,1)-line bundles over $E$. Consider an (∞,1)-functor

$\Gamma^\ast \;\colon \; \mathbf{H}_{/\mathbf{B}GL_1(E)} \to E Mod$

to the (∞,1)-category of (∞,1)-modules over $E$, which form $E$-modules of co-sections of $E$-(∞,1)-module bundles (generalized Thom spectra).

This is well understood for $\mathbf{H} =$ ∞Grpd in which case $\Gamma \simeq \underset{\to}{\lim} \circ i$ is the (∞,1)-functor homotopy colimits in $E Mod$ under the canonical embedding $\mathbf{B} GL_1(E) \simeq E Line \hookrightarrow E Mod$. But one can consider similar constructions $\Gamma$ for more general ambient (∞,1)-toposes $\mathbf{H}$.

Definition

For $\chi_i \colon X_i \to \mathbf{B}GL_1(E)$ two objects of $\mathbf{H}_{/\mathbf{B}GL_1(E)}$, the $(\chi_1,\chi_2)$-twisted bivariant $E$-cohomology on $(X_1,X_2)$ is

$E^{\bullet + \chi_2 - \chi_1}(X_1,X_2) \;\coloneqq\; Hom_{E Mod}\left(\Gamma^\ast_{X_1}\left(\chi_1\right), \Gamma^\ast_{X_2}\left(\chi_2\right)\right) \in E Mod \,.$
Example

By the general discussion at twisted cohomology, following (ABG, def. 5.1) we have

• for $X_2 = \ast$ the point, the above bivariant cohomology is the $\chi_1$-twisted $E$-cohomology of $X_1$;

$E^{\bullet + \chi_1}(X_1, \ast) \simeq E^{\bullet + \chi_1}(X_1) \,.$
• for $X_1 = \ast$ the point, the above bivariant cohomology is the $\chi_2$-twisted $E$-homology of $X_2$;

$E^{\bullet + \chi_2}(\ast, X_2) \simeq E_{\bullet + \chi_2}(X_2) \,.$
Example

KK-theory is a model for bivariant twisted topological K-theory over differentiable stacks (hence 1-truncated suitably representable objects in $\mathbf{H} =$ Smooth∞Grpd, see Tu-Xu-LG 03). According to (Joachim-Stolz 09, around p. 4) the category $KK$ first of all is naturally an enriched category $\mathbb{KK}$ over the category $\mathcal{S}$ of symmetric spectra and as such comes with a symmetric monoidal enriched functor

$\mathbb{KK} \to KU Mod \,.$

This sends an object to its operator K-theory spectrum, hence to the $E$-dual of the $E$-module of co-sections.

Remark

Generally, one may want to consider in def. the dualized co-section functor

$\Gamma = [\Gamma^\ast(-), E] \;\colon\; \left(\mathbf{H}_{/\mathbf{B}GL_1(E)}\right)^{op} \to E Mod \,.$
Example

A correspondence in $\mathbf{H}_{/\mathbf{B}GL_1(E)}$

$\array{ && Q \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} \\ X_1 && \swArrow_{\xi} && X_2 \\ & {}_{\mathllap{\chi_1}}\searrow && \swarrow_{\mathrlap{\chi_2}} \\ && \mathbf{B}GL_1(E) }$

is a morphism of “twisted $E$-motives” in that it is a correspondence in $\mathbf{H}$ between the spaces $X_1$ and $X_2$ equipped with an $(i_1^\ast \chi_1, i_2^\ast \chi_2)$-twisted bivariant $E$-cohomology cocycle $\xi$ on the correspondence space $Q$. Under the co-sections / Thom spectrum functor this is sent to a correspondence

$\Gamma_{X_1}(\chi_1) \stackrel{\xi}{\rightarrow} \Gamma_Q(i_2^\ast \chi_2) \stackrel{i_2^\ast}{\leftarrow} \Gamma_{X_2}(\chi_2)$

in $E Mod$. If the wrong-way map of this is orientable in $E$-cohomology then we may form its dual morphism/Umkehr map to obtain the corresponding “index

$\Gamma_{X_1}(\chi_1) \stackrel{(i_2)_! \xi}{\to} \Gamma_{X_2}(\chi_2)$

in $E Mod$. Identifying correspondences that yield the same “index” this way yields a presentation of bivariant cohomology by motive-like structures. This is how (equivariant) bivariant K-theory is presented, at least over manifolds, see at KK-theory – References – In terms of correspondences.

A general introduction to bivariant cohomology theories is in

A general construction of bivariant theories on smooth manifolds from cohomology theories by geometric cycles, generalizing the construction of K-homology by Baum-Douglas geometric cycles, is in

• Martin Jakob, Bivariant theories for smooth manifolds, Applied Categorical Structures 10 no. 3 (2002)

A similar construction for PL manifolds is in

• S. Buoncristiano, C. P. Rourke and B. J. Sanderson, A geometric approach to homology theory, Cambridge Univ. Press, Cambridge, Mass. (1976)

A study of bivariant theories in the context of motivic stable homotopy theory, and more generally in the broader framework of Grothendieck six functors formalism is in

• F. Déglise, Bivariant theories in motivic stable homotopy, (arXiv:1705.01528)

References related to the discussion in Axiomatization in homotopy theory above include the following

1. Thanks to Thomas Nikolaus for patiently emphasizing this.