My question is this: what is the topological analog of the Grothendieck group of coherent sheaves $G(X)$?


In Algebra/Algebraic Geometry there are two versions of the Grothendieck group of a variety $X$, one is the Grothendieck group of vector bundles, which we can denote as $K(X)$ and the other is the Grothendieck group of coherent sheaves, which we can denote by $G(X)$.

As the category of vector bundles is a subcategory of coherent sheaves there is always a group homomorphism $K(X) \to G(X)$ and if $X$ is smooth then this map is an isomorphism (basically because every coherent sheaf will be resolved by a finite complex of vector bundles as soon as $X$ is regular by Serre's Theorem).

Now let us assume that $X$ is a complex variety, so that we can compare algebraic K-theory with the topological one. That is there is a ring homomorphism $K(X) \to K^{top}(X)$, typically non-injective and non-surjective.

I am asking for groups $G^{top}(X)$ which are covariantly functorial for proper morphisms, admit natural transformations $G(X) \to G^{top}(X)$ as well as natural maps $K^{top}(X) \to G^{top}(X)$, such that the latter maps are isomorphisms in the smooth case.

Some remarks:

  1. The question makes sense for higher G-theory as well but I only bring up Grothendieck group to keep things simple and to avoid the clash of notation (e.g. $K_0(X)$ in algebra vs $K^0(X)$ in topology).

  2. A natural guess is that what I call $G^{top}(X)$ is simply what is called K-homology, could this be the case?

  3. Finally, there is a modern procedure of taking topological K-theory of a triangulated category due to Blanc (I think), and as far as I understand, $K^{top}(Perf(X)) = K^{top}(X)$, no matter whether $X$ is smooth or not, so it may be that what I want to take is $K^{top}(D^b(Coh(X))$, is this something sensible?

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    $\begingroup$ Your guess in 2) is basically correct. You can have a look at papers of Thomason for precise details. See for example projecteuclid.org/download/pdffirstpage_1/euclid.dmj/1077306718 and references there-in $\endgroup$ – user36931 Sep 29 '17 at 15:04
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    $\begingroup$ Closely related objects, providing a link between $G$-like homology classes and "true" cycles are Baum-Douglas cycles. Very briefly, such cycles on a manifold $X$ consist of a submanifold $Q\hookrightarrow X$ together with a spin$^c$ structure on $Q$; with appropriate bordism relation they indeed give $K$-homology. To obtain a "$G$-like" class out of it, take pushforward to $X$ of the tangent bundle of $Q$; under certain conditions it is represented by a finite complex of vector bundles. See the link for details and references. $\endgroup$ – მამუკა ჯიბლაძე Sep 29 '17 at 18:27
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    $\begingroup$ Topological K-theory can be defined by looking at complexes of vector bundles (or locally free sheaves) and can be endowed with covariant functoriality by using wrong way maps, as in the definition of topological index, or à la Grothendieck. $\endgroup$ – vap Sep 30 '17 at 7:14
  • $\begingroup$ user36931: I looked at the paper of Thomason, and I don't quite yet see how it's relevant. Do you mean his approach of inverting the Bott element in the Algebraic K-theory to relate to topological K-theory? $\endgroup$ – Evgeny Shinder Oct 2 '17 at 8:49
  • $\begingroup$ მამუკა ჯიბლაძე: for the Baum-Douglas cycles, is the basic assumption that $X$ is a manifold? Would then K-cohomology and K-homology canonically isomorphic (in the compact manifold case)? $\endgroup$ – Evgeny Shinder Oct 2 '17 at 8:51

I don't think K-homology will satisfy your requirements. The reason people think of G-theory as dual to K-theory is because for proper $X$, the graded space $Ext^*(F,Q)$ is finite-dimensional for $F$ a flat and $Q$ a coherent sheaf, and this defines a pairing. There is no such finite-dimensionality even for pairs of topological vector bundles over a topological manifold. In particular, there is no reason to expect there to be any sort of interesting map $KU^*(X)\to KU_{-*}(X)$ for non-smooth $X$.

Your #3 is interesting and would probably satisfy all your requirements, although I'm not sure how well functoriality of Blanc's construction works with respect to functors without any finiteness conditions (it most likely does). A simpler alternative might be as follows. Take some topological Abelian subcategory $\mathcal{C}$ of topological sheaves of modules over $C^\infty(X)$ which is stable under extensions and tensor products with bundles and which contains all coherent sheaves (extended to $C^\infty(X)$). One possibility is to model the theory of constructible sheaves and take sheaves which are repeated extensions of pushforwards of (analytic) bundles on algebraic subvarieties. Now take its topological Waldhausen K theory, and invert $u\in K_{Wald}^2(\mathbb{R}-mod)$ (which acts on the topological Waldhausen K-theory of any $\mathbb{R}$-linear category in an obvious way). If you had started with just the category of bundles on $X$, you would get $KU^*(X)$. Here you will get some new theory which will satisfy your requirements.

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