Transverse noncommutative geometry of foliations
Abstract
We define an L^{2}signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate BealsGreiner calculus, satisfying the usual axioms, gives rise to a semifinite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the ConnesChern character. The ConnesMoscovici hypoelliptic signature operator yields an example of such a triple and gives the differential definition of our "L^{2}signature". For Galois coverings of bounded geometry foliations, we also define an AtiyahConnes semifinite spectral triple which generalizes to Riemannian bifoliations the Atiyah approach to the L^{2}index theorem. The compatibility of the two spectral triples with respect to Morita equivalence is proven, and by using an Atiyahtype theorem proven in [7], we deduce some integrality results for Riemannian foliations with torsionfree monodromy groupoids.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 December 2018
 DOI:
 10.1016/j.geomphys.2018.08.011
 arXiv:
 arXiv:1804.06837
 Bibcode:
 2018JGP...134..161B
 Keywords:

 19L47;
 19M05;
 19K56;
 Mathematics  Geometric Topology;
 58H05;
 46L80;
 51P05
 EPrint:
 doi:10.1016/j.geomphys.2018.08.011