Analytic and Topological Invariants

We construct skew-adjoint operators associated to nowhere zero vector elds on manifolds with vanishing Euler number. The mod 2 indices of these operators provide potentially new invariants for such manifolds. An odd index theorem for corresponding Toeplitz operators is established. This last result may be viewed as an odd dimensional analogue of the Gauss-Bonnet-Chern theorem. It is well-known that the classical Gauss-Bonnet-Chern theorem C] can be interpreted as an index theorem for de Rham-Hodge operators (cf. BGV] and LM]) and that it is nontrivial only for even dimensional manifolds. This paper arose from an attempt to search an odd dimensional analogue of this index theorem. Recall that the Gauss-Bonnet-Chern theorem is closely related to the famous Poincar e-Hopf index formula expressing the Euler number as the sum of indices of singularities of a tangent vector eld. Now for odd dimensional manifolds, or more generally for manifolds with vanishing Euler number, we take the advantage that another result of Hopf asserts that there always exist nowhere zero vector elds (see e.g. Steenrod S]). Thus let M be an odd dimensional oriented closed manifold and let V be a nowhere zero vector eld on M. Let be the one dimensional oriented vector bundle over M generated by V. Then TM== carries a canonically induced orientation. Let e(TM==) be the Euler class of TM==. For any integral element ! in H 1 (M; Q), we will take h!e(TM==); M]i as our substitute for the Euler class appeared in the Gauss-Bonnet-Chern theorem. The main result of this paper gives an analytic formula for this number as the index of certain elliptic Toeplitz operators. In fact, a general odd index theory has already been developed by Baum-Douglas BD] who pointed out that the associated index can be computed 379