Download e-book for kindle: Algebraic Topology via Differential Geometry by M. Karoubi, C. Leruste

By M. Karoubi, C. Leruste

ISBN-10: 0511629370

ISBN-13: 9780511629372

ISBN-10: 0521317142

ISBN-13: 9780521317146

During this quantity the authors search to demonstrate how equipment of differential geometry locate program within the examine of the topology of differential manifolds. necessities are few because the authors take pains to set out the speculation of differential kinds and the algebra required. The reader is brought to De Rham cohomology, and specific and specified calculations are current as examples. themes coated contain Mayer-Vietoris precise sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This publication may 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 thoughts of differential geometry.

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Example text

X ) \+ * IT = e U as its differential. This is where a universal abuse of notation, hallowed by tradition (its origin being the confusion between the function and the value it takes at a point), and above all extremely convenient, creeps in: one writes dx. instead of d(ir ) thus turning the correct formula U df(x) = v 3f * I -1=- (X) e. 1 i=l 3x. (cf. ,x ) e U, into 1 n df (x) = I 3 f -+- (x) dx. 1 -Definition: on U is E is not ]R , i t s t i l l is the case that : U ->- E the A differential a.

U) f (x,y) = ~Y 2 x +y and The differential form famous ('angle 1 g(x,y) = —^—j • x +y OJ = f dx + g dy c fl (U) form) under the slightly incorrect notation io = — — (-y dx + x dy) x +y 2 . 3 Theorem: {dx,dy}. by du = O. i s universally 42 Proof: Easy computation. Q This statement, which means that -— = —2- , 3y Schwarz1 Lemma, a necessary condition for to i s , because of 3x to be the differential of a O-form. However the condition i s not sufficient i t turns out to has no antecedent under d.

Dy. e ft (V) 1 with 1 0 a. e H (v) , it 1 follows immediately that (a) (x) t = V (x). (a(iKx))) v = ? 1 1 - 2, (a. ° *) the computation i s a l i t t l e more i n t r i c a t e but boils down to the same technique. F i r s t take I = (i , . . , i ) e J . To develop . 50 A ( " ( x ) ) . ) A . . n* ) (n) = ( V ( x ) . n ! ,j. ) e J # JK n J than the determinant i s none o t h e r 3*, 3x. L -(x) 3x. (x) D () (x) = J 3*. Hx) 3x 3x.

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

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