Novikov–Shubin invariant

In mathematics, a Novikov–Shubin invariant, introduced by Sergei Novikov and Mikhail Shubin (1986), is an invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on square-integrable differential forms on its universal cover.

The Novikov–Shubin invariant gives a measure of the density of eigenvalues around zero. It can be computed from a triangulation of the manifold, and it is a homotopy invariant. In particular, it does not depend on the chosen Riemannian metric on the manifold.

References

• Cheeger, Jeff; Gromov, Mikhail (1985), "On the characteristic numbers of complete manifolds of bounded curvature and finite volume", in Chavel, Isaac; Farkas, Hershel M. (eds.), Differential geometry and complex analysis, Berlin, New York: Springer-Verlag, pp. 115–154, ISBN 978-3-540-13543-2, MR 0780040
• Efremov, A. V. (1991), "Cell decompositions and the Novikov-Shubin invariants", Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 46 (3): 189–190, doi:10.1070/RM1991v046n03ABEH002800, ISSN 0042-1316, MR 1134099
• Farber, Michael S. (1996), "Homological algebra of Novikov–Shubin invariants and Morse inequalities", Geometric and Functional Analysis, 6 (4): 628–665, CiteSeerX 10.1.1.252.2307, doi:10.1007/BF02247115, MR 1406667

• Gromov, Mikhail; Shubin, Mikhail A. (1991), "von Neumann spectra near zero", Geometric and Functional Analysis, 1 (4): 375–404, doi:10.1007/BF01895640, ISSN 1016-443X, MR 1132295
• Lück, Wolfgang (2002). L²-invariants: theory and applications to geometry and K-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Vol. 44. Berlin: Springer-Verlag. ISBN 978-3-540-43566-2.
• Novikov, Sergei P.; Shubin, Mikhail A. (1986), "Morse inequalities and von Neumann II₁-factors", Doklady Akademii Nauk SSSR, 289 (2): 289–292, ISSN 0002-3264, MR 0856461