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

By M. Karoubi, C. Leruste

During this quantity the authors search to demonstrate how tools of differential geometry locate software within the learn of the topology of differential manifolds. necessities are few because the authors take pains to set out the speculation of differential varieties and the algebra required. The reader is brought to De Rham cohomology, and particular and specific calculations are current as examples. themes coated comprise Mayer-Vietoris precise sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This publication 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 creation to the ideas of differential geometry.

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More g e n e r a l l y , i f U has m connected components, 0 m H (U) =]R . 4 Theorem: <>| : U -+ V If U i s an open s u b s e t of n R , V an open s u b s e t of p ]R and a C -map, then t h e DGA morphism <()* : fl*(V) -»• H*(U) i n d u c e s a l i n e a r map of degree * * If : H (V) -»• H (U) . T i s an open s u b s e t of ($ ° ty) = \j> If n = p o 3R and \j> : T •> U a C -map, then : H (V) •+ H (T) . 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 n £. ik , 1 e j l < . < i I k E. A . . A e. \ 1 iff i = j r onto of k distinct * (e. l ) = ^"p J q £ (j, , •- . ) i s a permutation of ( i , .

I a dy £ fir(V) , 6 = I g T dy e flS(V) I £J J £J p p Because o f ( i ) Now l e t a = where a , Q e Q (V) . I J **(ct) = T <(>*(a dy ) X I I £Jr P I, I , 4>*(a )<|>*(dy ) X I I £ JJr r P as we have just seen. Similarly 4>(B) = y * ( g T ) * ( d y ) J J J £ JS P Using t h e same remarks a g a i n t o g e t h e r w i t h ( i v ) and ( i ) , i t follows that (a) A $ ( g ) = ][ I I ( a ) ((> ( 6 ) <(> ( d y T ) A ((> J J J J (dy J ) 48 4> ( d Y l I J I J dy = ( a A A dy ) g) A (iii) Recall that, strictly speaking, dy.

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