• Cieliebak, Kai ;
  • Frauenfelder, Urs
  • Received : 2009.12.11
  • Published : 2011.07.01


Given a Morse function on a manifold whose moduli spaces of gradient flow lines for each action window are compact up to breaking one gets a bidirect system of chain complexes. There are different possibilities to take limits of such a bidirect system. We discuss in this note the relation between these different limits.


Morse homology;bidirect system;direct and inverse limits


  1. K. Cieliebak and U. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math. 239 (2009), no. 2, 251-316.
  2. K. Cieliebak, U. Frauenfelder, and A. Oancea, Rabinowitz Floer homology and symplectic homology, Ann. Sci. Ec. Norm Super. (4) 43 (2010), no. 6, 957-1015.
  3. B. Eckmann and P. Hilton, Commuting limits with colimits, J. Algebra 11 (1969), 116- 144.
  4. A. Frei and J. Macdonald, Limits in categories of relations and limit-colimit commutation, J. Pure Appl. Algebra 1 (1971), no. 2, 179-197.
  5. K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I, AMS/IP Studies in Advanced Mathematics Vol. 46.1, American Mathematical Society and International Press, 2009.
  6. A. Grothendieck, Elements de geometrie algebrique. III. Etude cohomologique des faisceaux coherents. I. Inst. Hautes Etudes Sci. Publ. Math. No. 11 (1961), 167 pp, 14.05
  7. H. Hofer and D. Salamon, Floer homology and Novikov rings, The Floer memorial volume, 483-524, Progr. Math., 133, Birkhauser, Basel, 1995.
  8. H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, The Floer Memorial Volume, Birkhauser, Basel, 1995.
  9. 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.
  10. J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337-341.
  11. G. Nobeling, Uber die Derivierten des Inversen und des direkten Limes einer Modul-familie, Topology 1 (1962), 47-61.
  12. K. Ono, On the Arnold conjecture for weakly monotone symplectic manifolds, Invent. Math. 119, (1995), no. 3, 519-537.
  13. S. Piunikhin, D. Salamon, and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, Contact and symplectic geometry (Cambridge, 1994), 171-200, Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996.
  14. J. Roos, Sur les foncteurs derives de lim.Applications., C. R. Acad. Sci. Paris 252 (1961), 3702-3704.
  15. M. Schwarz, Morse Homology, Birkhauser Verlag, 1993.
  16. E. Spanier, Algebraic Topology, McGraw-Hill Book Co., New York-Toronto, Ont.- London, 1966.
  17. C. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994.

Cited by

  1. Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations vol.151, pp.03, 2011,
  2. Rabinowitz–Floer homology for superquadratic Dirac equations on compact spin manifolds vol.13, pp.1, 2013,
  3. Cuplength estimates in Morse cohomology vol.08, pp.02, 2016,
  4. Symplectic Tate homology vol.112, pp.1, 2016,
  5. Vanishing of Rabinowitz Floer homology on negative line bundles vol.285, pp.1-2, 2017,
  6. Homological approach to problems with jumping non-linearity vol.144, 2016,
  7. Bubbling phenomena in calculus of variations vol.6, pp.3, 2017,
  8. Rabinowitz Floer homology and mirror symmetry vol.11, pp.1, 2018,
  9. Symplectic homology and the Eilenberg–Steenrod axioms vol.18, pp.4, 2018,