bivariant cohomology theory




Special and general types

Special notions


Extra structure





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.


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).


Let EE be an E-∞ ring, write GL 1(E)GL_1(E) for its ∞-group of units. With H\mathbf{H} the ambient (∞,1)-topos, write H /BGL 1(E)\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 EE. Consider an (∞,1)-functor

Γ *:H /BGL 1(E)EMod \Gamma^\ast \;\colon \; \mathbf{H}_{/\mathbf{B}GL_1(E)} \to E Mod

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

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


For χ i:X iBGL 1(E)\chi_i \colon X_i \to \mathbf{B}GL_1(E) two objects of H /BGL 1(E)\mathbf{H}_{/\mathbf{B}GL_1(E)}, the (χ 1,χ 2)(\chi_1,\chi_2)-twisted bivariant EE-cohomology on (X 1,X 2)(X_1,X_2) is

E +χ 2χ 1(X 1,X 2)Hom EMod(Γ X 1 *(χ 1),Γ X 2 *(χ 2))EMod. 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 \,.

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

  • for X 2=*X_2 = \ast the point, the above bivariant cohomology is the χ 1\chi_1-twisted EE-cohomology of X 1X_1;

    E +χ 1(X 1,*)E +χ 1(X 1). E^{\bullet + \chi_1}(X_1, \ast) \simeq E^{\bullet + \chi_1}(X_1) \,.
  • for X 1=*X_1 = \ast the point, the above bivariant cohomology is the χ 2\chi_2-twisted EE-homology of X 2X_2;

    E +χ 2(*,X 2)E +χ 2(X 2). E^{\bullet + \chi_2}(\ast, X_2) \simeq E_{\bullet + \chi_2}(X_2) \,.

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

𝕂𝕂KUMod. \mathbb{KK} \to KU Mod \,.

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


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

Γ=[Γ *(),E]:(H /BGL 1(E)) opEMod. \Gamma = [\Gamma^\ast(-), E] \;\colon\; \left(\mathbf{H}_{/\mathbf{B}GL_1(E)}\right)^{op} \to E Mod \,.

A correspondence in H /BGL 1(E)\mathbf{H}_{/\mathbf{B}GL_1(E)}

Q i 1 i 2 X 1 ξ X 2 χ 1 χ 2 BGL 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 EE-motives” in that it is a correspondence in H\mathbf{H} between the spaces X 1X_1 and X 2X_2 equipped with an (i 1 *χ 1,i 2 *χ 2)(i_1^\ast \chi_1, i_2^\ast \chi_2)-twisted bivariant EE-cohomology cocycle ξ\xi on the correspondence space QQ. Under the co-sections / Thom spectrum functor this is sent to a correspondence

Γ X 1(χ 1)ξΓ Q(i 2 *χ 2)i 2 *Γ X 2(χ 2) \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 EModE Mod. If the wrong-way map of this is orientable in EE-cohomology then we may form its dual morphism/Umkehr map to obtain the corresponding “index

Γ X 1(χ 1)(i 2) !ξΓ X 2(χ 2) \Gamma_{X_1}(\chi_1) \stackrel{(i_2)_! \xi}{\to} \Gamma_{X_2}(\chi_2)

in EModE 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

A similar construction for PL manifolds is in

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

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

  1. Thanks to Thomas Nikolaus for patiently emphasizing this.