Show Summary Details
More options …

IMPACT FACTOR 2017: 1.029
5-year IMPACT FACTOR: 1.147

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 1.588
Source Normalized Impact per Paper (SNIP) 2017: 0.971

Mathematical Citation Quotient (MCQ) 2017: 1.03

Online
ISSN
2169-0375
See all formats and pricing
More options …

The E-Cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on T2n

Maciej Starostka
• Corresponding author
• Ruhr-Universität Bochum, Bochum, Germany; and Gdańsk University of Technology, Gdańsk, Polen
• Email
• Other articles by this author:
/ Nils Waterstraat
• Institut für Mathematik, Naturwissenschaftliche Fakultät II, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
• Email
• Other articles by this author:
Published Online: 2019-04-16 | DOI: https://doi.org/10.1515/ans-2019-2044

Abstract

We show that the E-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals on Hilbert spaces. When applied to the setting of the Arnold conjecture, this paves the way to a short proof on tori, where it was first shown by C. Conley and E. Zehnder in 1983.

Keywords: Critical Points; Arnold Conjecture

MSC 2010: 37J10; 53D40; 58E05

References

• [1]

A. Abbondandolo, A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces, Topol. Methods Nonlinear Anal. 9 (1997), no. 2, 325–382.

• [2]

M. Chaperon, An elementary proof of the Conley–Zehnder theorem in symplectic geometry, Dynamical Systems and Bifurcations (Groningen 1984), Lecture Notes in Math. 1125, Springer, Berlin (1985), 1–8. Google Scholar

• [3]

C. C. Conley and E. Zehnder, The Birkhoff–Lewis fixed point theorem and a conjecture of V. I. Arnol’d, Invent. Math. 73 (1983), no. 1, 33–49.

• [4]

T. tom Dieck, Algebraic Topology, EMS Textbk. Math., European Mathematical Society (EMS), Zürich, 2008. Google Scholar

• [5]

Z. A. Dzedzej, K. Gȩba and W. Uss, The Conley index, cup-length and bifurcation, J. Fixed Point Theory Appl. 10 (2011), no. 2, 233–252.

• [6]

B. Fortune, A symplectic fixed point theorem for $𝐂{\mathrm{P}}^{n}$, Invent. Math. 81 (1985), no. 1, 29–46. Google Scholar

• [7]

K. Gȩba and A. Granas, Infinite-dimensional cohomology theories, J. Math. Pures Appl. (9) 52 (1973), 145–270. Google Scholar

• [8]

K. Gȩba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Studia Math. 134 (1999), no. 3, 217–233. Google Scholar

• [9]

H. Hofer, Lagrangian embeddings and critical point theory, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 6, 407–462.

• [10]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Basel, 1994. Google Scholar

• [11]

M. Izydorek, T. O. Rot, M. Starostka, M. Styborski and R. C. A. M. Vandervorst, Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces, J. Differential Equations 263 (2017), no. 11, 7162–7186.

• [12]

Y.-G. Oh, A symplectic fixed point theorem on ${T}^{2n}×𝐂{\mathrm{P}}^{k}$, Math. Z. 203 (1990), no. 4, 535–552. Google Scholar

• [13]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986. Google Scholar

• [14]

E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. Google Scholar

• [15]

M. Starostka, Morse cohomology in a Hilbert space via the Conley index, J. Fixed Point Theory Appl. 17 (2015), no. 2, 425–438.

• [16]

M. Styborski, Conley index in Hilbert spaces and the Leray–Schauder degree, Topol. Methods Nonlinear Anal. 33 (2009), no. 1, 131–148.

Revised: 2019-01-18

Accepted: 2019-03-13

Published Online: 2019-04-16

Maciej Starostka was supported by the grants Preludium9 of the National Science Centre, no. 2015/17/N/ST1/02527 as well as BEETHOVEN2 of the National Science Centre, Poland, no. 2016/23/G/ST1/ 04081. Nils Waterstraat was supported by the grant BEETHOVEN2 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081.

Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.