K-theory of non-compact spaces

Question

This is a question on nomenclature of $K$-theory in the topological category.

The $K$-theory of a compact space $X$ is defined as the Grothendieck group of the vectorbundles on $X$. The Atiyah-Jänich Theorem states that this is the same thing as the homotopy classes of maps $X\rightarrow \Phi(\mathbb{H})$, where $\Phi(\mathbb{H})$ is the space of Fredholm operators on some separable infinite-dimensional Hilbert space.

Now, for a non-compact space $Y$ the $K$ theory $K(Y)$ is not defined as the Grothendieck group of vector bundles on $Y$. One can do a couple of things:

1. Define it as the $K$-theory of its one point compactification $K(Y):=K(Y_+)$. One needs to assume $Y$ is locally compact for this to make sense.
2. Another option is to assume that $Y$ is nice, for example an infinite $CW$ complex such as $CP^\infty$, and to define the $K$-theory of $Y$ as the limit of its finite subcomplexes. Note that $CP^\infty$ is not locally compact.
3. Yet another option is to define the $K$-theory of $Y$ as maps $Y\rightarrow \Phi(\mathbb{H})$. I believe this is called representable $K$-theory.

I have a couple of questions.

Why is 1. a good definition? I like my cohomology theories to be functorial under maps. Theory 1. clearly is not. I believe it is functorial with respect to proper maps. This reminds me of compactly supported cohomology. But why is this then not called compactly supported $K$-theory?

And:

How are 2. and 3. related?

More specifically:

What exactly does 3. describe? Are these virtual vector bundles that admit numerable trivializations?

finally:

Is there a reference where all these definitions are discussed?

Answer 1

2 and 3 are not equivalent, because of a phenomenon known as "phantom maps".You can have a map of a CW complex $X$ to $Y$ which is non-trivial in homotopy, but homotopy trivial when restricted to every finite subcomplex of $X$.

That this actually occurs for K-cohomology is shown in an old paper of Anderson and Hodgkin, "the K-theory of Eilenberg Maclane Complexes" http://www.sciencedirect.com/science/article/pii/0040938368900098 See corollary 1 of that paper, which gives examples for Eilenberg-Maclane complexes.

Also, in answer to your question about an interpretation of 3 as virtual vector bundles, I don't think you will get any joy there. See the paper by Jackowski and Oliver quoted by Neil Strickland in his answer to this earlier MO question Is there a good definition of (topological) K-theory over arbitrary spaces

Answer 2

I think that definition 1 has a different nature with respect to the other two, because it defines K-theory with compact support. Even in ordinary cohomology, one thing is the singular cohomology of a space, another thing is the compactly-supported one. For example, $H^{1}_{cpt}(\mathbb{R}) = \mathbb{Z}$, but $H^{1}(\mathbb{R}) = 0$.

If I am not wrong, K-theory is defined as the Grothendieck group of vector bundles on compact Hausdorff spaces, in particular on finite CW-complexes. Then, as every cohomology theory defined on such a category, it can be extended to spaces having the same homotopy type of a compact Hausdorff space, in particular to spaces having the same homotopy type of a finite CW-complex. For this extension you need to impose homotopy invariance, that does not hold for compactly supported cohomology ($\mathbb{R}$ has the same homotopy type of a point, hence of a finite CW-complex, but the compactly supported cohomology is different).

You can define the extension as Atiyah has shown at the beginning of the paper Atiyah-Hirzebruch, "Analytic cycles on complex manifolds". Given a pair $(X, A)$, you define an element of $K^{\bullet}(X, A)$ as a functor $\xi$ that, given a pair $(Y, B)$ of compact Haudorff spaces (or a finite CW-pair) and a map $f: (Y, B) \rightarrow (X, A)$, assigns to $f$ an element of $K^{\bullet}(Y, B)$, that you call $f^{*}\xi$ (imagining the pull-back of the class $\xi$, but here it is a definition), in such a way that:

• given a morphism $g: (Y, B) \rightarrow (Y', B')$ between $f: (Y, B) \rightarrow (X, A)$ and $f': (Y', B') \rightarrow (X, A)$ (i.e., $f = f' \circ g$), we have that $f^{*}(\xi) = g^{*}(f')^{*}(\xi)$;
• $f^{*}\xi$ only depends on the homotopy class of $f$.

In the case of an infinite CW-complex, such a definition coincides with definition 2 in your question, but I think it is not a cohomology theory on such a category (maybe the long exact sequence fails). I suppose it is also equivalent to 3, but I am not sure.

Answer 3

What exactly does 3. describe? Are these virtual vector bundles that admit numerable trivializations?

Virtual vector bundles, when defined as formal differences (i.e., elements in the homotopy group completion) of vector bundles, do not form a sheaf when considered on noncompact manifolds. The reason for this is that a compatible system of such differences considered on some open cover might use differences of higher and higher dimensional vector bundles as one approaches the infinity, whereas any virtual vector bundle is a difference of two bundles of fixed dimension.

Put differently, the homotopy group completion functor, being a left adjoint, does not preserve the sheaf condition, which is defined using a homotopy limit.

However, the problem can be easily rectified by replacing the above presheaf with its sheafification. With such an extended notion of virtual vector bundles one then proves that their isomorphism classes are represented by the connective K-theory spectrum.

(The above is literally true for smooth manifolds with the standard Grothendieck topology; for topological spaces one must take the numerable Grothendieck topology, which produces numerable bundles.)