Fixed point thorems from a de Rham perspective

Atiyah and Bott ([AB1],[AB2]) generalized this theorem to complexes of elliptic operators; we briefly recall (under mild restrictions) their theorem. Let E0, E1, · · · , EN be a sequence of smooth hermitian vector bundles over M , equipped with a sequence of first order differential operators Di : Γ(Ei) → Γ(Ei+1). This sequence, denoted Γ(E), is called an elliptic complex if for all i, Di+1Di = 0, and D∗ iDi +Di−1D ∗ i−1 is elliptic. Here we set Di = 0 for i 6∈ [0, N − 1]. Set H(Γ(E)) = KerDp/ImDp−1. Given a smooth map f and smooth bundle homomorphisms φp : (f ∗E)p → Ep, we may define endomorphisms Tp : Γ(Ep) → Γ(Ep) by Tps = φpf ∗s.