Elliptic complex

If E₀, E₁, ..., E_k are vector bundles on a smooth manifold M (usually taken to be compact), then a differential complex is a sequence

${\displaystyle \Gamma (E_{0}){\stackrel {P_{1}}{\longrightarrow }}\Gamma (E_{1}){\stackrel {P_{2}}{\longrightarrow }}\ldots {\stackrel {P_{k}}{\longrightarrow }}\Gamma (E_{k})}$

of differential operators between the sheaves of sections of the E_i such that P_i+1 ${\displaystyle \circ }$ P_i=0. A differential complex with first order operators is elliptic if the sequence of symbols

${\displaystyle 0\rightarrow \pi ^{*}E_{0}{\stackrel {\sigma (P_{1})}{\longrightarrow }}\pi ^{*}E_{1}{\stackrel {\sigma (P_{2})}{\longrightarrow }}\ldots {\stackrel {\sigma (P_{k})}{\longrightarrow }}\pi ^{*}E_{k}\rightarrow 0}$

is exact outside of the zero section. Here π is the projection of the cotangent bundle T*M to M, and π* is the pullback of a vector bundle.

• Chain complex

Atiyah, M. F.; Singer, I. M. (1968). "The Index of Elliptic Operators: I". The Annals of Mathematics. 87 (3): 484. doi:10.2307/1970715. JSTOR 1970715.