Topological version of K-theory of coherent sheaves

Question

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

Background:

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?

Answer 1

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.