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Vector bundles and regulous maps. (English) Zbl 1288.14042
The authors investigate the relationships between pre-algebraic and algebraic \(\mathbb F\)-vector bundles on a real algebraic set \(X\), where \(\mathbb F\) stands for \(\mathbb R\), \(\mathbb C\) or \(\mathbb H\) (the quaternions). It should be mentioned that pre-algebraic vector bundles and algebraic vector bundles have been called algebraic vector bundles and strongly algebraic vector bundles, respectively, in the literature predating the publication of [J. Bochnak et al., Real algebraic geometry. Transl. from the French. Rev. and updated ed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 36. Berlin: Springer. (1998; Zbl 0912.14023)]. In fact, pre-algebraic vector bundles have been viewed as pathological objects and almost completely neglected, with a single exception: [M. Coste and M. M. Diop, Boll. Unione Mat. Ital., VII. Ser., A 6, No. 2, 249–254 (1992; Zbl 0802.14022)], which contains a proof that any pre-algebraic vector bundle on a nonsingular variety is a Nash vector bundle.
The article under review, which is very interesting and admirably written, contains several results. Firstly it is proved that given a pre-algebraic \(\mathbb F\)-vector bundle \(\xi\) on an affine real algebraic set \(X\) there exists a nonsingular real algebraic set \(X'\) and a sequence \(\pi:X'\to X\) of finitely many blowing ups such that the pullback bundle \(\pi^*\xi\) on \(X'\) is algebraic. The proof involves, of course, Hironaka’s desingularization theorem, and a clever use of the so called regulous (a mixed of regular plus continuous) functions.
Employing this fundamental result the authors prove that the Stiefel-Whitney classes of any pre-algebraic \(\mathbb R\)-vector bundle are algebraic. Moreover, they derive that the Chern classes of any pre-algebraic \(\mathbb C\)-vector bundle and the Pontryagin classes of any pre-algebraic \(\mathbb R\)-vector bundle are blow-\(\mathbb C\)-algebraic. Given a compact non singular affine real algebraic set \(X\), a cohomological class \(u\) in \(H^{2k}(X;\mathbb Z)\) is said to be blow-\(\mathbb C\)-algebraic if there exists a finite sequence of blowing ups \(\pi:X'\to X\) such that the cohomological class \(\pi^*(u)\) is \(\mathbb C\)-algebraic.
The article contains also many enlightening and well chosen examples.

MSC:
14P05 Real algebraic sets
14P25 Topology of real algebraic varieties
14P99 Real algebraic and real-analytic geometry
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