zbMATH — the first resource for mathematics

A non-linear version of Swan’s theorem. (English) Zbl 0760.58001
Swan’s theorem [R. G. Swan, Trans. Am. Math. Soc. 105, 264-277 (1962; Zbl 0109.416)]asserts that there is an equivalence between the category of smooth vector bundles over a compact connected smooth finite dimensional manifold \(K\) with the bundle morphism and the category of finitely generated projective modules over the algebra \(\mathcal A\) (\(=C^ \infty(K)\)) of smooth functions on \(K\) with morphisms \(\mathcal A\)-linear maps.
The author defines \(\mathcal A\)-maps between finitely generated projective \(\mathcal A\)-modules to be smooth maps whose derivatives are \(\mathcal A\)-linear and shows the equivalence between the category of smooth maps over \(K\) with fibre preserving smooth maps and the category of finitely generated projective \(\mathcal A\)-modules with \(\mathcal A\)-maps. The author then defines the notion of \(\mathcal A\)-manifolds which represent the nonlinear correspondents of the finitely generated projective \(\mathcal A\)-modules and establishes the equivalence between the category of deformations over \(K\) and a full subcategory of \(\mathcal A\)-manifolds with \(\mathcal A\)-maps.

58A05 Differentiable manifolds, foundations
57R19 Algebraic topology on manifolds and differential topology
57R22 Topology of vector bundles and fiber bundles
58D29 Moduli problems for topological structures
Full Text: DOI EuDML
[1] [H] Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc.7, 65–222 (1982) · Zbl 0499.58003
[2] [S] Swan, R.G.: Vector bundles and projective modules. Trans. Am. Math. Soc.105, 264–277 (1962) · Zbl 0109.41601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.