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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.

MSC:
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
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References:
[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
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