By Don Bernard Zagier (auth.)

ISBN-10: 3540060138

ISBN-13: 9783540060130

ISBN-10: 3540379886

ISBN-13: 9783540379881

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Extra resources for Equivariant Pontrjagin Classes and Applications to Orbit Spaces: Applications of the G-signature Theorem to Transformation Groups, Symmetric Products and Number Theory

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Pace i!! the average over G of the trace!! )f G. c It only remaiKII t. w e(g,X) thi~ i~ the Lefllchetz indsx formula, and oan alllo be immediately froID the equivariant Atiyah-Si~ger o~tained theorem on applying it to the de lihem cGlmplex Itf X (of. [1l] , §9, eq. linos the fixed-paint lIet of an element a oorrellponding to the partition (4) haa been found t. be - k~tk~+ ••• isomsrphic t. X e(X)k~+k2+'" and theref&re to have Euler characteristic )' n! e(X(n)) N(1T) ( )k~+k2+'" e X. k2 ! " e in agreement with equation (13) of §7.

D. • ,b-1). Let A be the graded tensor product of H**(X) with B. denote e81 (for eE H**(X») and ti to denote 1®ti (-1) in A. dn. e ]. (12) We use ~ to (i=O,. •• ,b-1); then - 42 - Since B is a paNer series ring rather than a polynomial ring and we tensored with H**(X) instead of H*(X) in forming A, it is legitimate to form power series (with rational coefficients) of elements in Ai thus gr(iX) e A is well-defined. of degree 2c of the L-class of X We write Lc(I) for the component (~ deviates from ~ ~ notation where Lc denotes a certain polynomial in Pontrjagin classes of total degree 4c), or for the corresponding element Lc(X)~1 of A.

Un > is zero. Proposition 1: Thus we have Let fO,. ,fb be a ho:nageneous basis f'illr H"'(X;Il)). Then a bSl:lis for H* (Xen);~) is. given by the set of elements (9) with nO+'''+1\ = nand ni ,,1 for all i with deg fi Odd. %[]d~O (1 txd')fd 1 - tx d~O d even + (12) dodd In particular (x= -1), the Euler characteristics of X(n) and X are related by Note: FGrmulas (12) and (13) are due to Maodonald ["2I1J. Proof of corollary: dim(\! Hr(X(n» = By the proposition, we have 'Ino, .... ~~OI nO+, .. +~:=n, nodo + ..

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Equivariant Pontrjagin Classes and Applications to Orbit Spaces: Applications of the G-signature Theorem to Transformation Groups, Symmetric Products and Number Theory by Don Bernard Zagier (auth.)


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