# Algebraic Topology via Differential Geometry by M. Karoubi, C. Leruste By M. Karoubi, C. Leruste

During this quantity the authors search to demonstrate how equipment of differential geometry locate software within the research of the topology of differential manifolds. necessities are few because the authors take pains to set out the idea of differential kinds and the algebra required. The reader is brought to De Rham cohomology, and particular and certain calculations are current as examples. subject matters coated comprise Mayer-Vietoris special sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This e-book can be appropriate for graduate scholars taking classes in algebraic topology and in differential topology. Mathematicians learning relativity and mathematical physics will locate this a useful creation to the ideas of differential geometry.

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Algebraic Topology via Differential Geometry

During this quantity the authors search to demonstrate how equipment of differential geometry locate software within the examine of the topology of differential manifolds. necessities are few because the authors take pains to set out the idea of differential kinds and the algebra required. The reader is brought to De Rham cohomology, and particular and targeted calculations are current as examples.

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S 1 p £ 2, [A(Ke ) K---B A(KE ) ] a up w i t h t h e p r o d u c t s for {e. H. . S £'. } A(Ke. ) = A°(K£. ) and 1 of degree 0, In t h e s e k the singleton t h i s b e i n g an e l e m e n t w i t h r e p e t i t i o n s A P (K£. ) = {0} dim Ak(E) < i algebras. 12 as g r a d e d n = dim E; and a b a s i s whose elements are the products If Proof: k vector space; E. n If £. 13 Theorem: Let {E D under the k a S... H e with a. = . . = a. = 1, a. = 0 25 Finally, as with tensor algebras, a linear map induces a homomorphism between e x t e r i o r algebras: Let any integer E,F be two vector spaces, k > 1, A k (f) f : E -> F a linear map.

And D = V, both a t H * Remark: Writing 0 * then * * l e v e l and a t U l e v e l c r e a t e s no r i s k of 53 confusion: the context makes i t clear which i s meant. 1 ensures t h a t *(Bk(V)) c Bk(U) for a l l k e W. k k < } > (Z (V)) c z (U) and The r e s t i s Standard s t u f f from elementary l i n e a r algebra. D 5 HOMOTOPY We wish to study the situation created when a differential form i s allowed to vary differentiably depending on a parameter. 9. 1 Definition: form depending on a parameter i s a C -map : U x P •+• A k ( E * ) a where A k-differential U is as before an open subset of a finite-dimensional vector space E.

3x. ) a € \$2 (U) ; (cf. 8). are zero. What remains can be rewritten using = 0 d(d(f dCda) T , ^ . 2) f, dx dx ) ; (cf. a e 0 (U) . £- * v Now dx n = ( - 1 ) P dxT A dfi (iv) (1) Take with A n da A g = ( da dx X Jd a i A The second double sum i s none o t h e r £ A J . 7). + f d(dg) = dfA dg b e c a u s e of (iii) and ( 1 ) . Then dg) ) = d ( d f A dg) = d(df) A dg - df A d(dg) = 0 and d(da) = O. (3) Write any 1 < deg y. Suppose k > 2 and ( i v ) e s t a b l i s h e d up t o d e g r e e k r a £ J2 (U) a s a = £ B.