By M. Karoubi, C. Leruste

During this quantity the authors search to demonstrate how tools of differential geometry locate program within the research of the topology of differential manifolds. must haves are few because the authors take pains to set out the idea of differential types and the algebra required. The reader is brought to De Rham cohomology, and specific and unique calculations are current as examples. subject matters coated comprise Mayer-Vietoris precise sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This booklet may 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 advent to the concepts of differential geometry.

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**Additional resources for Algebraic Topology via Differential Geometry**

**Example text**

By induction hypothesis. II Existence Condition If a = I a dx (v) d e f i n e s e Q (U) , d where a efJ°(U), set n da= (cf. Y , d a A d x = L -jk. I I I e J n Y da. ^ . I ,. l < i , < . . < i,

8 x with x 1 k ad) a(k) I It is denoted by S (E) . ,k}. 3 Theorem: Let F be a vector space and k-linear map. 4 Definition: (i) (ii) (iii) For any K-vector space E, set S°(E) = K SX(E) = E S(E) = ® S k (E). 5 Theorem: The multiplication in T(E) induces on S(E) the structure of a commutative graded algebra, known as the symmetric algebra of If a commutative algebra C and a linear map E. 6 Theorem: Let E,F be two vector s p a c e s . There i s a canonical isomorphism of graded algebras S(E © F) = S(E) 8 s(F) .

4. I t in other Q a l g e b r a of a d i r e c t t h a t we need an a l g e b r a s t r u c t u r e on the t e n s o r product 22 of two a l g e b r a s . 10 Theorem: If In t h e n o n - g r a d e d c a s e , A and B a r e two a l g e b r a s , b, b Proof: Start (where a , a 1 £ A, from t h e composite Id A X T x I d B A i s an a l g e b r a w i t h e B) . PA»PB AxB xA xB where A» B (a 53 b) ( a ' K b ' ) = a a ' & b b 1 a m u l t i p l i c a t i o n d e f i n e d by 1 t h i s i s done n a t u r a l l y : x (resp.