Submission date: 8 February 2020

Abstract:

Consider an o-minimal structure on the real field. Let M be a definable C^r manifold, where r is a nonnegative integer. We first demonstrate an equivalence of the category of definable C^r vector bundles over M with the category of finitely generated projective modules over the ring C_{df}^r(M). Here, the notation C_{df}^r(M) denotes the ring of definable C^r functions on M. We also show an equivalence of the category of definable C^r bilinear spaces over M with the category of bilinear spaces over the ring C_{df}^r(M). The main theorems of this paper are homotopy theorems for definable C^r vector bundles and definable C^r bilinear spaces over M. As an application, we show that the Grothendieck rings K_0(C_{df}^r(M)), K_0(C_{df}^0(M)) and the Witt ring W(C_{df}^r(M)) are all isomorphic.

Mathematics Subject Classification: Primary 03C64, Secondary 57R22, 19A49

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Full text arXiv 2002.03081: pdf, ps.