Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Kirwan-Novikov inequalities on a manifold with boundary


Authors: Maxim Braverman and Valentin Silantyev
Journal: Trans. Amer. Math. Soc. 358 (2006), 3329-3361
MSC (2000): Primary 57R70; Secondary 58A10
DOI: https://doi.org/10.1090/S0002-9947-06-04021-9
Published electronically: February 20, 2006
MathSciNet review: 2218978
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We extend the Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we consider manifolds with boundary. Second, we allow a very degenerate structure of the critical set of the form, assuming only that the form is non-degenerated in the sense of Kirwan. In particular, we obtain a generalization of a result of Floer about the usual Morse inequalities on a manifold with boundary. We also obtain an equivariant version of our inequalities.

Our proof is based on an application of the Witten deformation technique. The main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end. This leads to a considerable simplification of the local analysis. In particular, we obtain a new analytic proof of the Morse-Bott inequalities on a closed manifold.


References [Enhancements On Off] (What's this?)

  • 1. S. Alesker and M. Braverman, Cohomology of a Hamiltonian T-space with involution, Preprint.
  • 2. M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. London, ser A 308 (1983), 523-615. MR 0702806 (85k:14006)
  • 3. -, The moment map and equivariant cohomology, Topology 23 (1984), 1-28. MR 0721448 (85e:58041)
  • 4. N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Springer-Verlag, 1992. MR 1215720 (94e:58130)
  • 5. J.-M. Bismut, The Witten complex and the degenerate Morse inequalities, J. Dif. Geom. 23 (1986), 207-240. MR 0852155 (87m:58169)
  • 6. R. Bott, Morse theory indomitable, Publ. Math. IHES 68 (1988), 99-114. MR 1001450 (90f:58027)
  • 7. M. Braverman and M. Farber, The Novikov-Bott inequalities, C.R. Acad. Sci. Paris t. 321, Série I (1995), 897-902. MR 1355849 (96i:58165)
  • 8. -, Novikov inequalities with symmetry, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 793-798. MR 1416178 (97h:57056)
  • 9. M. Braverman and M. Farber, Equivariant Novikov inequalities, $ K$-Theory 12 (1997), 293-318. MR 1485432 (98m:57043)
  • 10. -, Novikov type inequalities for differential forms with non-isolated zeros, Math. Proc. Cambridge Philos. Soc. 122 (1997), 357-375. MR 1458239 (99b:58220)
  • 11. M. Braverman, O. Milatovich, and M. Shubin, Essential selfadjointness of Schrödinger-type operators on manifolds, Russian Math. Surveys 57 (2002), 641-692. MR 1942115 (2004g:58021)
  • 12. P. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401-414. MR 0369890 (51:6119)
  • 13. C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 0511133 (80c:58009)
  • 14. H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with applications to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, 1987. MR 0883643 (88g:35003)
  • 15. A. Dold, Lectures on algebraic topology, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1972 edition. MR 1335915 (96c:55001)
  • 16. Y. Eliashberg and M. Gromov, Lagrangian intersection theory: finite-dimensional approach, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, vol. 186, Amer. Math. Soc., Providence, RI, 1998, pp. 27-118. MR 1732407 (2002a:53102)
  • 17. M. Farber and E. Shustin, Witten deformation and polynomial differential forms, Geom. Dedicata 80 (2000), no. 1-3, 125-155. MR 1762505 (2001i:58042)
  • 18. M.S. Farber, Sharpness of the Novikov inequalities, Functional Anal. Appl. 19 (1985), 49-59. MR 0783706 (86g:58029)
  • 19. A. Floer, Witten's complex and infinite dimensional Morse theory, J. Diff. Geom 30 (1989), 207-221. MR 1001276 (90d:58029)
  • 20. M. Gromov and B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. (1983), no. 58, 295-408. MR 0720933 (85g:58082)
  • 21. M. Hirsch, Differential topology, Graduate Texts in Math., vol. 33, Springer, Berlin, 1976. MR 0448362 (56:6669)
  • 22. F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 0766741 (86i:58050)
  • 23. S.P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Soviet Math. Dokl. 260 (1981), 35-35. MR 0630459 (83a:58025)
  • 24. -, The Hamiltonian formalism and a multivalued analogue of Morse theory, Russian Math. Surveys 37 (1982), 3-49, 248. MR 0676612 (84h:58032)
  • 25. A. Pazhitnov, An analytic proof of the real part of Novikov's inequalities, Soviet Math. Dokl. 293 (1987), 1305-1307. MR 0891557 (88j:57033)
  • 26. M. Reed and B. Simon, Methods of modern mathematical physics IV: Analysis of operators, Academic Press, London, 1978. MR 0493421 (58:12429c)
  • 27. M. A. Shubin, Semiclassical asymptotics on covering manifolds and Morse inequalities, Geom. Funct. Anal. 6 (1996), 370-409. MR 1384616 (97i:58171)
  • 28. -, Spectral theory of the Schrödinger operators on non-compact manifolds: qualitative results, Spectral theory and geometry (Edinburgh, 1998), Cambridge Univ. Press, Cambridge, 1999, pp. 226-283. MR 1736869 (2001d:58037)
  • 29. C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685-710. MR 1157321 (93b:58058)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R70, 58A10

Retrieve articles in all journals with MSC (2000): 57R70, 58A10


Additional Information

Maxim Braverman
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: maxim@neu.edu

Valentin Silantyev
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: v.silantyev@neu.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04021-9
Keywords: Novikov inequalities, Kirwan inequalities, Morse-Bott inequalities, Witten deformation
Received by editor(s): April 9, 2004
Published electronically: February 20, 2006
Additional Notes: This research was partially supported by NSF grant DMS-0204421
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society