Complex structures on \(L(p,q) \times S^ 1\)
type: Item
instance of: scholarly article
title: Complex structures on \(L(p,q) \times S^ 1\)
zbMATH Open document ID: 0857.57008
author: Noriaki Nakagawa
published in: Hiroshima Mathematical Journal
publication date: 1997-03-11T00:00:00Z
review text: Let \(M^3\) be a compact orientable 3-manifold. Then, \(M^3 \times S^1\) has an almost complex structure, because the tangent bundle of \(M^3\) is trivial. However by [\textit{T. Matumoto}, Three and four dimensional manifolds and complex structures, Proc. Symp. `Three and four dimensional manifolds' (ed. T. Matumoto) (in Jap.), 45-74 (1987), Theorem 3.1], \(M^3 \times S^1\) cannot have any complex structure unless \(M^3\) admits a Seifert fibering structure. Moreover these complex structures are deformation equivalent except the case that \(M^3\) is homeomorphic to a lens space by [T. Matumoto, loc. cit., Theorem 3.2] and [\textit{M. Ue}, Adv. Stud. Pure Math. 20, 331-363 (1992; Zbl 0801.57007), Theorems \(C\)-\(1\) and \(C\)-\(2\)]. In this note, we determine the deformation types of all the complex structures on the product manifold \(L(p,q)\times S^1\). The purpose of this paper is to prove the following main theorem. Let \(n(N)\) denote the number of deformation types of the complex manifolds which are diffeomorphic to the manifold \(N\). Theorem. Let \(p\) and \(q\) be positive integers with \(p>1\) and \((p,q)=1\) and \(L(p,q)\) a 3-dimensional lens space. Then, \[ n\bigl( L(p,q) \times S^1\bigr) = \begin{cases} 1 \quad \text{if } & q^2 \equiv -1 \pmod p \\ 2 \quad \text{if } & q^2 \not \equiv -1 \pmod p \end{cases}. \]
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 32Q99
zbMATH DE Number: 811380
zbMATH Keywords: 3-manifold
zbMATH Keywords: almost complex structure
zbMATH Keywords: complex structure
zbMATH Keywords: lens space
zbMATH Keywords: deformation types
zbMATH Keywords: product manifold
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 811380