A celebrated result of Swan [1] states that, on a compact Hausdorff space $X$, the category of finite rank complex vector bundles is equivalent to the category of finitely generated projective $\mathcal{C}(X)$-modules, where $\mathcal{C}(X)$ is the ring of complex-valued continuous functions on $X$ (analogous statements are true over the reals and the quaternions (loc.cit.), but these will not concern us here). One can look upon this in the following ways:

  • interesting geometric objects and their morphisms-- viz. complex vector bundles of finite rank and vector bundle maps -- have a description by interesting algebraic objects and their morphisms viz. finitely generated projective modules and their morphisms;

  • just the other way round: interesting algebraic objects and their morphisms have a description by interesting geometric objects and their morphisms;

  • as a synthesis of these views one obtains that a topological invariant, the complex $K$-theory $K(X)$ of $X$, has both a geometric description by complex vector bundles and an algebraic description by projective modules.

Pondering a while upon these issues the following questions come to mind:

  • The category of finite rank vector bundles is additive, but not abelian. This may lead one to consider linear fibrations over $X$ which are locally modelled on the kernels or cokernels of vector bundle maps of finite rank vector bundles; these have fibres which are vector spaces of possibly varying finite dimension. To which algebraic objects do these correspond?
  • Are there further algebraic categories of $\mathcal{C}(X)$-modules with an interesting geometric description? In particular, do the finitely generated $\mathcal{C}(X)$-modules have an interesting geometric description?
  • Does the topological $K$-homology $K_*(X)$ of $X$ have interesting geometrical and algebraic descriptions? The $K$-theory of finiteley generated $\mathcal{C}(X)$-modules appears to fail here, since pushing forward does not seem to work.

[1] Swan, R.G. -- Vector bundles and projective modules. Trans. Amer. Math. Soc. 105, (1962), 264–277.

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    $\begingroup$ With regard to your third question, there's is a “geometric” picture of $K$-homology due to Baum and Douglas. If you don't mind a little analysis, though, the pairing between $K$-homology and $K$-theory can be beautifully understood in terms of the Atiyah–Singer index theorem and its various generalisations in index theory and noncommutative geometry. $\endgroup$ – Branimir Ćaćić Aug 9 at 23:59

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