Elliptic complex

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Definition

Symbol-free definition

A differential complex is said to be elliptic if its sequence of symbols is exact.

Definition with symbols

Let $M$ be a differential manifold and $E_i$ be smooth vector bundles over $M$. Let $P_i:\mathrm{\Gamma }(E_i)\to \mathrm{\Gamma }(E_{i+1})$ form a differential complex (viz $P_i\circ P_{i-1}=0$. Then this differential complex is said to be elliptic if the following sequence of symbols is exact:

$\dots {\pi }^{*}(E_{i-1}){\to }^{\sigma (P_{i-1})}{\pi }^{*}(E_i){\to }^{\sigma (P_i)}{\pi }^{*}{E}_{i+1}\dots$