Equivariant Pontrjagin Classes and Applications to Orbit by Don Bernard Zagier (auth.)

By Don Bernard Zagier (auth.)

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The formal analogy between (7) and (22) s t ill suggests that it has a particularly nioe normal structure in sorne sense. We can hope to find some more general type of normal bundle than O(n)-bundles, for whioh (rational) Pontrjagin olasses or L-classes are still defined and such that the inclusion of the nth symmetric product of a manifold in the . ( n+1 ) st always has a normal bundle . 1n the genera11sed sense. In fact, a definition recently has been given for such a generalised bundle for homology manifolds (Martin and Maunder [2~), These objects, called "homology cobordism bundles," are (roughly) defined as projections E~B for which the inverse image of a cell in B is H-cobordant to the product of the cell with the fibre.

2a·x ( J e J -a· 2,,·x C Je J ( Hl) This sum is transformed into the one occurring in (15) by the substi-2ie tution e ,and the corollary is therefore proved. ,= We could of course write down an analogous formula for the eq uivariant L-class of the action of an element of Tn+ 1/G- on the quotient space pnt/G- by using Theorem 2 instead of Theorem 1 of §1. We will give a second application of Theorem 1 in §15. But the theorem is interesting for its own sake as well as for its applications, since it illustrates the behaviour of the equivariant L-ola5s.

E. fO •••• ,fb with each fi homogeneou~ and with fO= Z and fb = 1. fil under the intersection form, that is, olJ .. f . • [X]> l Thus J ( i,j" 0,1 , ... ,b), (3) 8 i is homogeneous of degree 2s-di , where d i " deg f i , The result of the last section was that a basi5 for H*(X(oo») is given by the elementoS where the square brackets have the meaning given in §7 and where 1] = is the element whose restriction to X{n) is (6) The map j*:H*(X(oo) -+ H*(X(n» is surjective, and a hsis for H*(X(n)) is given by the images under j* of the elements (4) for which no + ••• + ~-1 ~ n.

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