Topology of cosymplectic manifolds
type: Item
instance of: scholarly article
title: Topology of cosymplectic manifolds
zbMATH Open document ID: 0845.53025
author: Domingo Chinea
author: Manuel de León
author: Juan Carlos Marrero
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1996-09-11T00:00:00Z
review text: A \(2n\)-dimensional compact Kähler manifold \(M^{2n}\) has the following five topological properties: 1) The Betti numbers \(b_{2i}\), \(1\leq i\leq n\) are non-zero. 2) The Betti numbers \(b_{2i- 1}\) are even. 3) The dimension of the space of effective harmonic \(p\)-forms is \(b_p- b_{p- 2}\), for \(p\leq n+ 1\), and consequently \(b_{p- 2}\leq b_p\). 4) \(M^{2n}\) has the strong Lefschetz property, viz. the mapping \(L^{n- p}\) of harmonic \(p\)-forms to harmonic \((2n- p)\)-forms induced by \(L\alpha= \alpha\wedge \Omega\) is an isomorphism, \(\Omega\) being the fundamental 2-form of the Kähler structure. 5) The minimal model of \(M^{2n}\) is formal (so in particular all Massey products of \(M^{2n}\) vanish). The purpose of this paper is to obtain similar results for a compact cosymplectic manifold \(M^{2n+ 1}\). In the reviewer's paper with \textit{S. I. Goldberg} [J. Differ. Geom. 1, 347-354 (1967; Zbl 0163.43902)] it was shown that: 1) The Betti numbers \(b_i\) are non-zero for all \(i\). 2) The dimension of the space of effective harmonic \(p\)-forms is \(b_p- b_{p- 2}\), for \(p\leq n+ 1\), and consequently \(b_{p- 2}\leq b_p\) (an error in the proof is corrected by a stronger result in the present paper). The main results of the present paper are: 1) On a compact cosymplectic manifold \(M^{2n+ 1}\), \(b_0\leq b_1\leq\cdots\leq b_n= b_{n+ 1}\) and \(b_{n+ 1}\geq b_{n+ 2}\geq\cdots \geq b_{2n+ 1}\). 2) On a compact cosymplectic manifold \(M^{2n+ 1}\) the differences \(b_{2p+ 1}- b_{2p}\) with \(0\leq p\leq n\) are even and so in particular the first Betti number of \(M^{2n+ 1}\) is odd. 3) A strong Lefschetz property. 4) A compact cosymplectic manifold \(M^{2n+ 1}\) is formal. The notion of cosymplectic used in this paper is the stronger one, viz. a normal almost contact metric manifold for which both the fundamental 1-form and fundamental 2-form are closed. The other notion of a cosymplectic manifold in the literature is a manifold \(M^{2n+ 1}\) admitting…
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 811435
zbMATH Keywords: formality
zbMATH Keywords: compact cosymplectic manifold
zbMATH Keywords: Betti numbers
zbMATH Keywords: strong Lefschetz property
reviewed by: David E. Blair
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 811435