Elliptic complex

In

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Definition

If E₀, E₁, ..., E_k are

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

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

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.

See also

• Chain complex

References

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

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