# nLab elliptic chain complex

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

## Definitions

operator K-theory

K-homology

## Index theorems

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

The notion of elliptic chain complex is the generalization of the notion of elliptic operator from single linear maps to chain complexes of linear maps.

## Definition

For $X$ a smooth manifold and $\left\{{E}_{k}{\right\}}_{k\in ℤ}$ a collection of vector bundles over $X$, a chain complex of differential operators between the spaces of sections of these bundles

$\cdots \to \Gamma \left({E}_{k+1}\right)\stackrel{{P}_{k}}{\to }\Gamma \left({E}_{k}\right)\to \cdots$\cdots \to \Gamma(E_{k+1}) \stackrel{P_k}{\to} \Gamma(E_k) \to \cdots

is called an elliptic chain complex if the corresponding sequence of symbols

$\cdots \to {\pi }^{*}{E}_{k+1}\stackrel{\sigma \left({P}_{k}\right)}{\to }{\pi }^{*}{E}_{k}\to \cdots$\cdots \to \pi^* E_{k+1} \stackrel{\sigma(P_k)}{\to} \pi^* E_k \to \cdots

(where $\pi :{T}^{*}X\to X$ is the cotangent bundle) is an exact sequence.

For instance (Pati, def. 9.4.1).

For a single differential operator $P$ this says that $0\to {\pi }^{*}{E}_{1}\stackrel{\sigma \left(P\right)}{\to }{\pi }^{*}{E}_{0}\to 0$ is exact, which means that $\sigma \left(P\right)$ is an isomorphism, hence that $P$ is an elliptic operator.

## Properties

### Atiyah-Bott lemma

If $\left(ℰ,d\right)$ is an elliptic complex of smooth sections $ℰ={\Gamma }_{X}\left(E\right)$ of a vector bundle $E\to X$ overa compact closed manifold $X$, then the inclusion

$\left(ℰ,d\right)↪\left(\overline{ℰ},d\right)$(\mathcal{E},d) \hookrightarrow (\overline{\mathcal{E}}, d)

into the complex of distributional sections is a quasi-isomorphism, in fact a homotopy equivalence.

This is due to (Atiyah-Bott). A localized refinement (suitable for factorization algebras of local observables) appears as Gwilliam, lemma 5.2.13.

## Examples

The classical examples of elliptic complexes are discussed also in (Gilkey section 3).

### de Rham complex

Let $X$ be a compact smooth manifold. Then the de Rham complex is an ellptic complex. The corresponding index of an elliptic complex is the Euler characteristic

$\mathrm{Ind}\left({\Omega }^{•}\left(X\right),d\right)=\chi \left(X\right)=\sum _{p=0}^{\mathrm{dim}X}\left(-1{\right)}^{p}\mathrm{dim}{H}_{\mathrm{dR}}^{p}\left(X,ℂ\right)$Ind(\Omega^\bullet(X),d) = \chi(X) = \sum_{p = 0}^{dim X} (-1)^p dim H_{dR}^p(X, \mathbb{C})

(…)

### The Dolbeault complex

The index of an elliptic complex of the Dolbeault complex is the arithmetic genus?

### Spin complex

(…) index is A-hat genus (…)

## References

• V. Pati, Elliptic complexes and index theory (pdf)
• Michael Atiyah, Raoul Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374–407. MR 0212836 (35 #3701)
• Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)

Revised on February 19, 2013 15:14:58 by Urs Schreiber (80.81.16.253)