Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

Marek Izydorek; Thomas O. Rot; Maciej Starostka; Marcin Styborski; Robert C.A.M. Vandervorst

doi:10.1016/j.jde.2017.08.007

Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

Marek Izydorek
, Thomas O. Rot^*
, Maciej Starostka
, Marcin Styborski
, Robert C.A.M. Vandervorst

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

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Abstract

In this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse–Conley–Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.

Original language	English
Pages (from-to)	7162-7186
Number of pages	25
Journal	Journal of Differential Equations
Volume	263
Issue number	11
Early online date	23 Aug 2017
DOIs	https://doi.org/10.1016/j.jde.2017.08.007
Publication status	Published - 5 Dec 2017

Funding

M. Izydorek, T.O. Rot and M. Starostka were partially supported by DAAD and MNISW PPP-PL Grant no. 57217076 . M. Starostka was also supported by National Science Centre grant UMO-2015/17/N/ST1/02527 .

Funders	Funder number
Deutscher Akademischer Austauschdienst
Narodowe Centrum Nauki	UMO-2015/17/N/ST1/02527
Ministerstwo Nauki i Szkolnictwa Wyższego	57217076

Keywords

(Local) Morse homology
Conley index
LS-flows
Morse–Conley–Floer homology

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10.1016/j.jde.2017.08.007

Homotopy invariance of the Conley index and local Morse homology in Hilbert spacesFinal published version, 419 KBLicence: Article 25fa Dutch Copyright Act

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title = "Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces",

abstract = "In this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse–Conley–Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.",

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T1 - Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

AU - Izydorek, Marek

AU - Rot, Thomas O.

AU - Starostka, Maciej

AU - Styborski, Marcin

AU - Vandervorst, Robert C.A.M.

PY - 2017/12/5

Y1 - 2017/12/5

N2 - In this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse–Conley–Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.

AB - In this paper we introduce a new compactness condition — Property-(C) — for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse–Conley–Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.

KW - (Local) Morse homology

KW - Conley index

KW - LS-flows

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JO - Journal of Differential Equations

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Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

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