elliptic chain complex

noncommutative topology, noncommutative geometry

noncommutative stable homotopy theory

**genus, orientation in generalized cohomology**

(also nonabelian homological algebra)

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.

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.

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.

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

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

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

(…)

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

(…) index is A-hat genus (…)

- V. Pati,
*Elliptic complexes and index theory*(pdf)

- Peter Gilkey,
*The Atiyah-Singer Index Theorem – Chapter 5*(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)