By I. Madsen, B. Oliver

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Additional resources for Algebraic Topology Aarhus 1982. Proc. conf. Aarhus, 1982

Example text

5, (6) and (8)). It is a good exercise to show that the morphism of vector bundles determined by ϕ according to n. 9 coincides with f. In conclusion, if a fiber-wise linear map f : Eξ → Eπ over f is smooth as a map between manifolds, then it is a vector bundle morphism (even in the algebraic sense). 21 and n. 25 allow us to define a faithful functor VB g → VB that associates, with each π, the vector bundle determined by Γ(π) and, with each morphism, the corresponding morphism obtained by means 25 Note that the identification homomorphisms C∞ (M ) → C∞ (Eπ ), P ∨ → C∞ (Eπ ) induce a C∞ (M )–algebra homomorphism ι : S (P ∨ ) → C∞ (Eπ ), where ι : S (P ∨ ) denotes the symmetric algebra of P ∨ (Sr will denote an r-th symmetric power).

The graded module with graded components Λs (P ) will be denoted by Λ• (P ). A (ordinary) differential form on A will be a differential form on A with values in A itself. A differential form on a smooth manifold M will be a differential form on the algebra C∞ (M ). The C∞ (M )–module Λ• (C∞ (M )) of all differential forms on M will be denoted simply by Λ• (M ) (and its graded components by Λs (M )). 2 Cotangent Bundle Let M be a manifold and A = C∞ (M ). Arguing as in n. 40], one deduces that Λ1 (M ) is projective, finitely generated, and determines an equidimensional pseudobundle πΛ1 (M ) : Λ1 (M ) → M which is, therefore, a vector bundle.

X ◦ πN Therefore, X ∈ Im ιD(M )N if and only if X vanishes on the image of the homomorphism ∗ πN : C∞ (N ) → C∞ (M × N ) that defines C∞ (M × N ) as a C∞ (N )–algebra. Hence X ∈ Im ιD(M )N if and only if X is a C∞ (N )–derivation of C∞ (M × N ) into itself: Im ιD(M )N = DC∞ (N ) (C∞ (M × N )) . October 8, 2008 14:20 World Scientific Book - 9in x 6in 50 Fat Manifolds and Linear Connections The module DC∞ (N ) (C∞ (M × N )) of all derivations of the C∞ (N )– algebra C∞ (M × N ) will be also denoted by DN (M × N ) .

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