On closed odd-dimensional manifolds, the index of any elliptic differential operator is zero. So for interesting
examples, either "elliptic" or "differential" must be dropped. This talk will indicate what is happening in both
cases. If "elliptic" is dropped, then the examples are the Fredholm operators in the Heisenberg calculus of
contact manifolds. If "differential" is dropped, then the examples are Toeplitz operators obtained from
the splitting of the essential spectrum of the Dirac operator into its positive and negative parts. In particular, there is the Toeplitz operator index result of Louis Boutet de Monval.

Proofs of the above index formulas are by applying Bott Periodicity.

This is joint work with Erik van Erp.