By S. Yamamuro

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Pi ) of degree one has i. 197]. /% ~ by X . i 57 o = 1 P1 24 = - 5760 (-4P2 + 7p ) A (Z~ 1 967,680 3 = Remarks: (i) Since (~ (16P3 - 44P2Pl + 31p3) is a sum of c o h o m o l o g y classes of e v e n A dimension, w h e n i is odd C~[X] = 0 . Hence Spin(X) = O when is odd. (ii) As p o i n t e d out above, Spin(X) is an invariant of the c o n f o r m a l s t r u c t u r e on the r i e m a n n i a n m a n i f o l d X ~ thus A the formula Spin(X) = C0[X] may be r e g a r d e d as r e l a t i n g this structure to the P o n t r y a g i n classes of the fact that Spin(X) t r i v i a l result that is an integer, Cb[x] X .

S 1 Here v ~v 1 refers to the covariant derivative in the direction associated to the riemannian structure on X , and the indicated product is Clifford multiplication. In local terms, one has V( j~ ajej) = iJ [ ei bei(aJeJ) = It follows that V /k n X . /k~ , and eiaj~ei e J " and ~ - spinors depends on the Locally, a change in orientation replaces by an isomorphic space & to i'~, J is a first order partial differential operator i. The distinction between ~ orientation of ~a I ~J e i ~ J + '/kn ; however, -- /k n is isomorphic to /k + n h~ is isomorphic 52 and that its symbol is g i v e n ~(x, Since ~2 morphism A of the = _ <~, ~ and h e n c e simple inner orthonormal 9) basis a, b e C n order k one on shows Rn Since for Cn by ~ + O is an i s o - operator.

12]). - 3p4) when written to write For them computations , to the H o d g e example, we see that operator, with the the index theorem, the Hirzebruch yields fact that when when n = 4k it is signature theorem. Theorem: dimension Let 4k X . in . ) these D + = Sign X applied (381P4 quickly it is m o r e factored - 13PlP2 be compact smooth orientable Then S i g n X = L[X] manifold of 43 where L = ~- Lj classes of Example: X is the universal associated When k = 1 polynomial to the p o w e r the signature 1 X = CP 2 is the c o m p l e x m u s t b e of the form class Cp1c dual to .