Abstract
For a Riemannian foliation \(\mathcal F\) on a closed manifold M, we define L 2-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of \(\mathcal F\) to the universal covering of M is used in the definitions. These invariants are natural extensions of the L 2-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant by foliated homotopy equivalences, and they are computed for several examples.
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References
Álvarez López J.A. (1989). Duality in the spectral sequence of Riemannian foliations. Am. J. Math. 111: 905–926
Álvarez López J.A. (1989). A finiteness theorem for the spectral sequence of a Riemannian foliation. Illinois J. Math. 33: 79–92
Álvarez López J.A. (1992). The basic component of the mean curvature of Riemannian foliations. Ann. Global Anal. Geom. 10: 179–194
Álvarez López J.A. and Kordyukov Y.A. (2000). Adiabatic limits and spectral sequences. Geom. Funct. Anal. 10: 977–1027
Álvarez López J.A. and Kordyukov Y.A. (2001). Long time behavior of leafwise heat flow for Riemannian foliations. Comp. Math. 125: 129–153
Álvarez López, J.A., Masa, X.M.: Morphisms of pseudogroups, foliation maps, and spectral sequence of Riemannian foliations, Preprint
Atiyah M.F. (1976). Elliptic operators, discrete groups and von Neumann algebras. Astérisque 32(33): 43–72
Borel A. (1963). Compact Clifford-Klein forms of symmetric spaces. Topology 2: 111–122
Bott R. and Tu L. (1982). Differential Forms in Algebraic Topology. Springer-Verlag, New York
Cheeger J. and Gromov M. (1985). Bounds on the von Neumann dimension of L 2-cohomology and the Gauss–Bonnet theorem for open manifolds. J. Differential Geom. 21: 1–34
Cheeger J., Gromov M. and Taylor M. (1982). Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17: 15–54
Chernoff P.R. (1973). Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12: 401–414
Dodziuk J. (1982). L 2 harmonic forms on complete manifolds. In: Yau, S.-T. (eds) Seminar on Differential Geometry. Annals of Math. Studies No. 102, pp 291–302. Princeton University Press, Princeton, NJ
Gromov M. and Shubin M.A. (1991). von Neumann spectra near zero. Geom. Funct. Anal. 1: 375–404
Lück W. (2002). L 2-Invariants: Theory and Applications to Geometry and K-theory. Springer-Verlag, Berlin
Lott J. (1992). Heat kernels on covering spaces and topological invariants. J. Differential Geom. 35: 471–510
Masa X.M. (1992). Duality and minimality in Riemannian foliations. Comment. Math. Helv. 67: 17–27
Molino P. (1988). Riemannian Foliations, Progr. in Math. 73. Birkhäuser, Basel
Novikov S.P. and Shubin M.A. (1986). Morse inequalities and von Neumann II 1-factors. Dokl. Akad. Nauk SSSR 289: 289–292
Roe J. (1987). Finite propagation speed and Connes’ foliation algebra. Math. Proc. Cambridge Philos. Soc. 102: 459–466
Roe J. (1988). Elliptic Operators, Topology and Asymptotic Methods. Wiley, New York
Roe J. (1988). An index theorem on open manifolds I. J. Differential Geom. 27: 87–113
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Sanguiao, L. L 2-invariants of Riemannian foliations. Ann Glob Anal Geom 33, 271–292 (2008). https://doi.org/10.1007/s10455-007-9085-5
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DOI: https://doi.org/10.1007/s10455-007-9085-5