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The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the $K$-theory of a noncommutative $C^*$-algebra that is associated to the algebra of symbols. There is a canonical map from this analytic $K$-theory group to the ordinary cohomology of the manifold, which gives a de Rham class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes’ tangent groupoid proof of the index theorem.
@article{10_4007_annals_2010_171_1647,
author = {Erik van Erp},
title = {The {Atiyah-Singer} index formula for subelliptic operators on contact manifolds. {Part} {I}},
journal = {Annals of mathematics},
pages = {1647--1681},
publisher = {mathdoc},
volume = {171},
number = {3},
year = {2010},
doi = {10.4007/annals.2010.171.1647},
mrnumber = {2680395},
zbl = {1206.19004},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1647/}
}
TY - JOUR AU - Erik van Erp TI - The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I JO - Annals of mathematics PY - 2010 SP - 1647 EP - 1681 VL - 171 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1647/ DO - 10.4007/annals.2010.171.1647 LA - en ID - 10_4007_annals_2010_171_1647 ER -
%0 Journal Article %A Erik van Erp %T The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I %J Annals of mathematics %D 2010 %P 1647-1681 %V 171 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.1647/ %R 10.4007/annals.2010.171.1647 %G en %F 10_4007_annals_2010_171_1647
Erik van Erp. The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I. Annals of mathematics, Tome 171 (2010) no. 3, pp. 1647-1681. doi: 10.4007/annals.2010.171.1647
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