# nLab elliptic chain complex

Contents

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

## Higher genera

#### Homological algebra

homological algebra

Introduction

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 $\{E_k\}_{k \in \mathbb{Z}}$ a collection of vector bundles over $X$, a chain complex of differential operators between the spaces of sections of these bundles

$\cdots \to \Gamma(E_{k+1}) \stackrel{P_k}{\to} \Gamma(E_k) \to \cdots$

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

$\cdots \to \pi^* E_{k+1} \stackrel{\sigma(P_k)}{\to} \pi^* E_k \to \cdots$

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

For instance (Pati, def. 9.4.1).

For a single differential operator $P$ this says that $0 \to \pi^* E_1 \stackrel{\sigma(P)}{\to} \pi^* E_0 \to 0$ is exact, which means that $\sigma(P)$ is an isomorphism, hence that $P$ is an elliptic operator.

## Properties

### Atiyah-Bott lemma

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

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

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 elliptic complex. The corresponding index of an elliptic complex is the Euler characteristic

$Ind(\Omega^\bullet(X),d) = \chi(X) = \sum_{p = 0}^{dim X} (-1)^p dim H_{dR}^p(X, \mathbb{C})$

(…)

### The Dolbeault complex

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

### Spin complex

(…) index is A-hat genus (…)

• 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)