Morse-Bott homology
HTML articles powered by AMS MathViewer
- by Augustin Banyaga and David E. Hurtubise PDF
- Trans. Amer. Math. Soc. 362 (2010), 3997-4043 Request permission
Abstract:
We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the homology of the chain complex is independent of the Morse-Bott-Smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a Floer-type continuation theorem.References
- Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
- Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
- Michèle Audin, Torus actions on symplectic manifolds, Second revised edition, Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 2004. MR 2091310, DOI 10.1007/978-3-0348-7960-6
- D. M. Austin and P. J. Braam, Morse-Bott theory and equivariant cohomology, The Floer memorial volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 123–183. MR 1362827
- David M. Austin and Peter J. Braam, Equivariant Floer theory and gluing Donaldson polynomials, Topology 35 (1996), no. 1, 167–200. MR 1367280, DOI 10.1016/0040-9383(95)00004-6
- Augustin Banyaga and David E. Hurtubise, The Morse-Bott inequalities via a dynamical systems approach, Ergodic Theory Dynam. Systems 29 (2009), no. 6, 1693–1703. MR 2563088, DOI 10.1017/S0143385708000928
- Augustin Banyaga and David Hurtubise, Lectures on Morse homology, Kluwer Texts in the Mathematical Sciences, vol. 29, Kluwer Academic Publishers Group, Dordrecht, 2004. MR 2145196, DOI 10.1007/978-1-4020-2696-6
- Augustin Banyaga and David E. Hurtubise, A proof of the Morse-Bott lemma, Expo. Math. 22 (2004), no. 4, 365–373. MR 2075744, DOI 10.1016/S0723-0869(04)80014-8
- Jean-François Barraud and Octav Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. (2) 166 (2007), no. 3, 657–722. MR 2373371, DOI 10.4007/annals.2007.166.657
- Stefan Bauer, Electronic posting, Topology Listserv (2005).
- Raoul Bott, Nondegenerate critical manifolds, Ann. of Math. (2) 60 (1954), 248–261. MR 64399, DOI 10.2307/1969631
- Raoul Bott and Hans Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029. MR 105694, DOI 10.2307/2372843
- F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
- Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1993. MR 1224675, DOI 10.1007/978-1-4757-6848-0
- J. L. Bryant, Triangulation and general position of PL diagrams, Topology Appl. 34 (1990), no. 3, 211–233. MR 1042280, DOI 10.1016/0166-8641(90)90039-5
- S. K. Donaldson, Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics, vol. 147, Cambridge University Press, Cambridge, 2002. With the assistance of M. Furuta and D. Kotschick. MR 1883043, DOI 10.1017/CBO9780511543098
- Michael Farber, Topology of closed one-forms, Mathematical Surveys and Monographs, vol. 108, American Mathematical Society, Providence, RI, 2004. MR 2034601, DOI 10.1090/surv/108
- A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212 (1993), no. 1, 13–38. MR 1200162, DOI 10.1007/BF02571639
- Andreas Floer, An instanton-invariant for $3$-manifolds, Comm. Math. Phys. 118 (1988), no. 2, 215–240. MR 956166, DOI 10.1007/BF01218578
- John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
- John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
- Urs Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not. 42 (2004), 2179–2269. MR 2076142, DOI 10.1155/S1073792804133941
- Kenji Fukaya, Floer homology of connected sum of homology $3$-spheres, Topology 35 (1996), no. 1, 89–136. MR 1367277, DOI 10.1016/0040-9383(95)00009-7
- Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
- F. Reese Harvey and H. Blaine Lawson Jr., Finite volume flows and Morse theory, Ann. of Math. (2) 153 (2001), no. 1, 1–25. MR 1826410, DOI 10.2307/2661371
- H. Hofer, K. Wysocki, and E. Zehnder, A general Fredholm theory II: Implicit function theorems, Geom. Funct. Anal. To appear.
- H. Hofer, K. Wysocki, and E. Zehnder, A general Fredholm theory. I. A splicing-based differential geometry, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 841–876. MR 2341834, DOI 10.4171/JEMS/99
- David E. Hurtubise, The flow category of the action functional on $\scr LG_{N,N+K}(\textbf {C})$, Illinois J. Math. 44 (2000), no. 1, 33–50. MR 1731380
- —, Multicomplexes and spectral sequences, J. Algebra Appl. To appear.
- Peter J. Kahn, Pseudohomology and homology, arXiv:math/0111223v1 (2001).
- R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
- Janko Latschev, Gradient flows of Morse-Bott functions, Math. Ann. 318 (2000), no. 4, 731–759. MR 1802508, DOI 10.1007/s002080000138
- Gang Liu and Gang Tian, On the equivalence of multiplicative structures in Floer homology and quantum homology, Acta Math. Sin. (Engl. Ser.) 15 (1999), no. 1, 53–80. MR 1701133, DOI 10.1007/s10114-999-0060-x
- Juan Margalef Roig and Enrique Outerelo Domínguez, Differential topology, North-Holland Mathematics Studies, vol. 173, North-Holland Publishing Co., Amsterdam, 1992. With a preface by Peter W. Michor. MR 1173211
- William S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. MR 1095046, DOI 10.1007/978-1-4939-9063-4
- Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2004. MR 2045629, DOI 10.1090/coll/052
- Jean-Pierre Meyer, Acyclic models for multicomplexes, Duke Math. J. 45 (1978), no. 1, 67–85. MR 486489
- Lars Tyge Nielsen, Transversality and the inverse image of a submanifold with corners, Math. Scand. 49 (1981), no. 2, 211–221 (1982). MR 661891, DOI 10.7146/math.scand.a-11931
- Yongbin Ruan and Gang Tian, Bott-type symplectic Floer cohomology and its multiplication structures, Math. Res. Lett. 2 (1995), no. 2, 203–219. MR 1324703, DOI 10.4310/MRL.1995.v2.n2.a9
- P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
- Matthias Schwarz, Morse homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel, 1993. MR 1239174, DOI 10.1007/978-3-0348-8577-5
- Matthias Schwarz, Equivalences for Morse homology, Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999) Contemp. Math., vol. 246, Amer. Math. Soc., Providence, RI, 1999, pp. 197–216. MR 1732382, DOI 10.1090/conm/246/03785
- Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297, DOI 10.1007/978-1-4757-1799-0
- Joa Weber, The Morse-Witten complex via dynamical systems, Expo. Math. 24 (2006), no. 2, 127–159. MR 2243274, DOI 10.1016/j.exmath.2005.09.001
Additional Information
- Augustin Banyaga
- Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, University Park, Pennsylvania 16802
- MR Author ID: 30715
- Email: banyaga@math.psu.edu
- David E. Hurtubise
- Affiliation: Department of Mathematics and Statistics, The Pennsylvania State University, Altoona, Altoona, Pennsylvania 16601-3760
- Email: Hurtubise@psu.edu
- Received by editor(s): October 11, 2007
- Published electronically: March 23, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 3997-4043
- MSC (2010): Primary 57R70; Secondary 58E05, 57R58, 37D15
- DOI: https://doi.org/10.1090/S0002-9947-10-05073-7
- MathSciNet review: 2608393