noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
and
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
is called an elliptic chain complex if the corresponding sequence of symbols
(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
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
(…)
The index of an elliptic complex of the Dolbeault complex is the arithmetic genus
(…) index is A-hat genus (…)