2008 | OriginalPaper | Buchkapitel
Lefschetz Distribution of Lie Foliations
verfasst von : Jesús A. Álvarez López, Yuri A. Kordyukov
Erschienen in: C*-algebras and Elliptic Theory II
Verlag: Birkhäuser Basel
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Let
$$ \mathcal{F} $$
be a Lie foliation on a closed manifold
M
with structural Lie group
G
. Its transverse Lie structure can be considered as a transverse action Φ of
G
on (
M
,
$$ \mathcal{F} $$
);
i.e.
, an “action” which is defined up to leafwise homotopies. This Φ induces an action Φ* of
G
on the reduced leafwise cohomology
$$ \bar H\left( \mathcal{F} \right) $$
. By using leafwise Hodge theory, the supertrace of Φ* can be defined as a distribution
L
dis
(
$$ \mathcal{F} $$
) on
G
called the Lefschetz distribution of
$$ \mathcal{F} $$
. A distributional version of the Gauss-Bonett theorem is proved, which describes
L
dis
(
$$ \mathcal{F} $$
) around the identity element. On any small enough open subset of
G
,
L
dis
(
$$ \mathcal{F} $$
) is described by a distributional version of the Lefschetz trace formula.