Now I learn the Atiyah-Singer index theorem. When I read an article on wikipedia of the index theorem, I saw an fact "Index of elliptic differential operators vanish in odd dimensional manifolds". I tried to prove it but failed.
Please tell me why index of elliptic differential operators vanish in odd dimensional manifolds and how to prove it.
This is proved in Spin Geometry by Lawson and Michelsohn, Theorem $13.12$ of chapter III.
The idea is that the diffeomorphism $c : TX \to TX$ given by $c(v) = -v$ acts on two of the three terms in (their formulation of) the formula for the topological index. One of the terms doesn't change, and the other is negated, so $\operatorname{ind}(P) = -\operatorname{ind}(P)$.