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. must haves are few because the authors take pains to set out the idea of differential varieties and the algebra required. The reader is brought to De Rham cohomology, and specific and special calculations are current as examples. subject matters lined comprise Mayer-Vietoris detailed sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This booklet should be compatible 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 strategies of differential geometry.

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

**Example text**

For all 13 column. • Thus l < 3 < n M eU. ID To go further in the same direction, one is led to introduce matrices of differential forms each coefficient of which is an element of * ft (U) . F o ran example, in ft (U) notation. t h ematrix will be denoted by (dx. ) . , . , . . , i ] 1 < l < n, 1 < ] < n with dM in conformity with the usual abuse of * M (ft (U) ) , The result i s an It-algebra: the matrix algebra M (0) n for short. k Note that i t is graded by the subspaces in ft (U), + Mn (U) , derived from that i s analogous to the multiplication of 'ordinary' matrices, except that the, product of differential forms obviously remains a n t i - commutative.

S k - 1 , d and 6 coincide as 1 < deg Y. 7) . Then because of ( i i ) and ( i i i ) for a l l i in 39 da = 2, d ( B i i A Y±) I (dSi A Tj, + ( - l ) d e g 6 i Si A dYi) . i A similar expression i s obtained for 6a, hence da = 6a on O £2 (U) . 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,

Thus l < 3 < n M eU. ID To go further in the same direction, one is led to introduce matrices of differential forms each coefficient of which is an element of * ft (U) . F o ran example, in ft (U) notation. t h ematrix will be denoted by (dx. ) . , . , . . , i ] 1 < l < n, 1 < ] < n with dM in conformity with the usual abuse of * M (ft (U) ) , The result i s an It-algebra: the matrix algebra M (0) n for short. k Note that i t is graded by the subspaces in ft (U), + Mn (U) , derived from that i s analogous to the multiplication of 'ordinary' matrices, except that the, product of differential forms obviously remains a n t i - commutative.