A Comprehensive Introduction to Differential Geometry, Vol. - download pdf or read online

By Michael Spivak

ISBN-10: 0914098721

ISBN-13: 9780914098720

E-book by means of Michael Spivak, Spivak, Michael

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Extra info for A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition

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We can then find another such family (Vi)iEI such that Vi ~ For each i E I, we have a section s, of A over Ui , such that s, coincides with s over U, n Z . For each x EX, let Yx be an open neighborhood of x in X such that the set T x of i, for which Y x n Vi is non-empty, is a finite set. By shrinking Yx if necessary, we may assume that x E U, for i E T x . For each i,j E T x , the sections s, and Sj agree over some neighborhood N i j of X. Let W x be the intersection of Yx with all N i j (for i,j E T x ) ; then over W x , we have a well-defined section u x of A, which at every y is equal to Si for every i such that y E Wx n Ui.

11 that there exists a smooth function gi on Ui, with support in Vi, such that 9i(X) > 0 for all x E Vi . Setting 9i(X) = 0 for x ~ Ui, we obtain a smooth function gi on M, with support in V i, which is positive on Vi. 4. DE RHAM COHOMOLOGY We summarize the results we have obtained for manifolds modeled on ILH spaces. 15. Theorem. Let M be a paracompact manifold, modeled on an ILH space E satisfying (1-25). Then the sheaves A~ are soft, and we have canonical isomorphisms If M is a convex open set of E, these groups are 0 for p > O.

Since A is injective, A is a direct summand of the sheaf I. But a direct summand of a flabby sheaf is also flabby. This proves (1). We will now show that given an exact sequence 0 -+ F -+ G -+ H -+ 0 in AB(X) with F soft, the map I'(X, G) -+ I'(X, H) is surjective. Let s E I'(X, H) . As X is paracompact, there exists a locally finite open covering (Ui)iEI of X and a section t i of Gover U, such that t; projects to s. Next we have an open covering (~) such that ~ ~ Ui. Set Zi = ~. Consider the set of pairs (J, tJ) consisting of a subset J of I and a section tJ of G over the closed subset ZJ := UiEJZi which projects to s.

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A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition by Michael Spivak

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