[Math] good definition of (topological) K-Theory over arbitrary spaces

at.algebraic-topologycohomologygt.geometric-topologyhomotopy-theorykt.k-theory-and-homology

Hi

(this is my very first question here, so please don't hurt me…)

for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet been unable to find or come up with one. The definition I know goes as follows:

For X connected, compact, hausdorff one defines $V(X) = \text{ set of isoclasses of vectorbundles over} X$ which becomes a comm. monoid under direct sum and then $KO(X) = K(V(X))$ where the righthandside just means group completion of a comm monoid.

Here already the "isoclasses" of bundles bothers me, because this "set" is not really a set, is it? (?its elements being proper classes?). I guess this may be salvaged by instead looking at equi. classes of systems of transition functions taking values in $GL(\mathbb R)$, modulo some further restriction and relations !?

Anyway, this is something I might even live with, but
one goes on to show $KO(X) \cong [X, \mathbb Z \times BGL(\mathbb R)]$ for such spaces, and for a CW-complex C sets $KO(C) = [C, \mathbb Z \times BGL]$. This is the only possible defintion (up to nat. iso), when trying to end up with a cohomology-like functor; right?
Finally for a general space Z we pick (for every space simultaneously?!) a CW-substitue say C' and put $KO(Z) = [C', \mathbb Z \times BGL]$.

However, using this as defintion, there really is very little beauty left in K-Theory for me. I know that just putting $KO(X) = K(V(X))$ goes awry (bundles must be allowed to have varying dimension over different components and in turn must allow for a partition of unity and so on…)

So my question is:
Is there a way of altering the definition of V(X) sufficiently so as to give a "correct" definition of K-Theory? Or some other way of producing these groups, nicely? Nicely should in particular mean, without homotopy theory oder cell complexes, so that for instance homotopy invariance is not directly built into the definition. And if so, what about relative groups, or even higher ones?

After all for singular cohomology and bordism there also are descriptions using homotopy theory (via Eilenberg-MacLane resp. Thom-Spaces) just as above, but for an arbtrary space there are entirely different (better?) descpriptions in terms of singular chains and manifolds.

Thanks in advance

Maybe I should add that passing to spectra makes everything even worse in my opinion.

Best Answer

Some comments:

  1. You might want to look at 'Vector bundles over classifying spaces of compact Lie groups' by Jackowski and Oliver. They discuss a situation in which you can understand $K(V(X))$ quite explicitly and it is interestingly different from $[X,\mathbb{Z}\times BO]$.

  2. Picking $C'$ for all $Z$ is not a terribly big deal, because you can do it functorially: just use the geometric realisation of the singular complex of $Z$.

  3. Some things work better if you define $V(X)$ to be the set of isomorphism classes of numerable vector bundles, ie those for which there exists a trivialising cover with a subordinate partition of unity. For most spaces (compact spaces, CW complexes, metric spaces, ...) this makes no difference. In cases where it does make a difference, there is a good argument that numerability should be part of the 'right definition' of a vector bundle. I don't remember exactly how much this buys you in the foundations of $K$-theory, however. It will not affect the Jackowski-Oliver examples, as their base spaces are CW complexes.

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