Morse cohomology in a Hilbert space via the Conley index

详细信息    查看全文

• 作者：Maciej Starostka
• 关键词：Primary 53D40 ; Secondary 55N20 ; Morse homology ; Morse–Witten–Floer complex ; Conley index ; Seiberg–Witten–Floer homology
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：17
• 期：2
• 页码：425-438
• 全文大小：593 KB
• 参考文献：Abb97.Abbondandolo A.: A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces. Topol. Methods Nonlinear Anal. 9, 325-82 (1997)MATH MathSciNet
  AGP.M. Aguilar, S. Gitler and C. Prieto, Algebraic Topology from a Homotopical Viewpoint. Springer-Verlag, New York, 2002.
  C-J.Crabb M. C., Jaworowski J.: Aspects of the Borsuk-Ulam theorem. J. Fixed Point Theory Appl. 13, 459-88 (2013)MATH MathSciNet CrossRef
  Ch.K.-C. Chang, Methods in Nonlinear Analysis. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
  FOOO.K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I. AMS/IP Stud. Adv. Math. 46, American Mathematical Society, Providence, RI, International Press, Somerville, MA, 2009.
  G.G?ba K.: Fredholm σ-proper maps of Banach spaces. Fund. Math. 64, 341-73 (1969)MathSciNet
  G-G.G?ba K., Granas A.: Infinite dimensional cohomology theories. J. Math. Pures Appl. 9(52), 145-70 (1973)MathSciNet
  GIP.G?ba K., Izydorek M., Pruszko A.: The Conley index in Hilbert spaces and its applications. Studia Math. 134, 217-33 (1999)MathSciNet
  G-D.A. Granas and J. Dugundji, Fixed Point Theory. Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.
  Izy.Izydorek M.: A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems. J. Differential Equations 170, 22-0 (2001)MATH MathSciNet CrossRef
  K-Sz.Kryszewski W., Szulkin A.: An infinite-dimensional Morse theory with applications. Trans. Amer. Math. Soc. 349, 3181-234 (1997)MATH MathSciNet CrossRef
  Man.Manolescu C.: Seiberg-Witten-Floer stable homotopy type of three-manifolds with b 1 =? 0. Geom. Topol. 7, 889-32 (2003)MATH MathSciNet CrossRef
  McC.McCord C.: The connection map for attractor-repeller pairs. Trans. Amer. Math. Soc. 307, 195-03 (1988)MATH MathSciNet CrossRef
  Sal.Salamon D.: Morse theory, the Conley index and Floer homology. Bull. Lond. Math. Soc. 22, 113-40 (1990)MATH MathSciNet CrossRef
  Smol.J. Smoller, Shock Waves and Reaction-Diffusion Equations. 2nd ed., Grundlehren Math. Wiss. 258. Springer-Verlag, New York, 1994.
  Sz.Szulkin A.: Cohomology and Morse theory for strongly indefinite functionals. Math. Z. 209, 375-18 (1992)MATH MathSciNet CrossRef
  Tau.Taubes C. F.: The Seiberg-Witten equations and the Weinstein conjecture. II. More closed integral curves of the Reeb vector field. Geom. Topol. 13, 1337-417 (2009)MATH MathSciNet CrossRef
  
• 作者单位：Maciej Starostka (1) (2)
  
  1. Mathematical Institute, Polish Academy of Sciences, ul. Sniadeckich 8, 00-956, Warsaw, Poland
  2. Faculty of Technical Physics and Applied Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952, Gdansk, Poland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

The main theorem of this paper states thatMorse cohomology groups in a Hilbert space are isomorphic to the cohomological Conley index. It is also shown that calculating the cohomological Conley index does not require finite-dimensional approximations of the vector field. Further directions are discussed. Keywords Morse homology Morse–Witten–Floer complex Conley index Seiberg–Witten–Floer homology