Description

• Title:
  Zeros of closed 1-forms, homoclinic orbits and Lusternik-Schnirelman theory
• Creator:
  Farber, Michael
• Description:
  In this paper we study topological lower bounds on the number of zeros of closed $1$-forms without Morse type assumptions. We prove that one may always find a representing closed $1$-form having at most one zero. We introduce and study a generalization ${\rm cat}(X,\xi)$ of the notion of the Lusternik-Schnirelman category, depending on a topological space $X$ and a $1$-dimensional real cohomology class $\xi\in H^1(X;\mathbb R)$. We prove that any closed $1$-form $\omega$ in class $\xi$ has at least ${\rm cat}(X,\xi)$ zeros assuming that $\omega$ admits a gradient-like vector field with no homoclinic cycles. We show that the number ${\rm cat}(X,\xi)$ can be estimated from below in terms of the cup-products and higher Massey products. < p> This paper corrects some my statements made in [< i> Lusternik–Schnirelman theory for closed $1$-forms< /i> , Comment. Math. Helv. < b> 75< /b> (2000), 156–170] and [< i> Topology of closed $1$-forms and their critical points, Topology < b> 40< /b> (2001), 235–258].< /p>
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2002.03.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 19, No 1 (March 2002); 123 - 152;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2002.008/9663
• Subject:
  Morse theory;
  Lusternik-Schnirelman theory;
  closed 1-form;
  Massey products;
  homoclinic orbits;
  37C29;
  58E05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2002.008
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other