The \(\pi_ 1\)-injectivity of self-maps of nonzero degree on 3-manifolds
type: Item
instance of: scholarly article
title: The \(\pi_ 1\)-injectivity of self-maps of nonzero degree on 3-manifolds
zbMATH Open document ID: 0793.57008
DOI: 10.1007/BF01459495
published in: Mathematische Annalen
publication date: 1994-08-25T00:00:00Z
full work available at URL: https://eudml.org/doc/165125
review text: A geometric 3-manifold \(M\) is a compact orientable 3-manifold such that either \(M\) is a Haken manifold, or a hyperbolic 3-manifold, or a Seifert manifold, or a 3-ball, or a connected sum of those 3-manifolds. A famous conjecture of Thurston is that all compact orientable 3-manifolds are geometric 3-manifolds. In this paper, we prove the following results: Theorem. Suppose \(M\) is a geometric 3-manifold which is not covered by \(S^ 3\). Then any non-zero degree proper map \(f:M \to M\) induces an injective homomorphism \(f_ *:\pi_ 1(M) \to \pi_ 1(M)\). Furthermore \(f_ *\) is an isomorphism unless \(M\) is covered by either a torus bundle over the circle, or an \(F \times S^ 1\) for some compact surface \(F\). Corollary. Suppose \(M\) is a geometric 3-manifold. Then \(M\) admits a selfmap of degree larger than 1 if and only if \(M\) is covered by either a torus bundle over the circle, or an \(F \times S^ 1\) for some compact surface \(F\), or the fundamental group of each prime summand of \(M\) is finite or cyclic. Furthermore, \(M\) admits a self-covering of degree larger than 1 if and only if \(M\) is covered by either a torus bundle over the circle, or an \(F \times S^ 1\) for some compact surface \(F\).
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 537206
zbMATH Keywords: injectivity of map induced on the fundamental group
zbMATH Keywords: geometric 3- manifold
zbMATH Keywords: Haken manifold
zbMATH Keywords: hyperbolic 3-manifold
zbMATH Keywords: Seifert manifold
zbMATH Keywords: non- zero degree proper map
zbMATH Keywords: torus bundle over the circle
zbMATH Keywords: selfmap of degree larger than 1
author: Shicheng Wang
reviewed by: Song Wang
MaRDI profile type: Publication
cites work: Deformation of maps to branched coverings in dimension two
cites work: Volume and bounded cohomology
cites work: Q5289023
cites work: Q3871581
cites work: Q3924842
cites work: π1 -Injective Mappings of Compact 3-Manifolds
cites work: Maps Inducing Isomorphisms on Fundamental Groups of Compact 3 -Manifolds
cites work: Q3733804
cites work: Degree One Maps Between Geometric 3-Manifolds
cites work: The Geometries of 3-Manifolds
cites work: The Gromov invariant of links
cites work: Q3856928
cites work: Q3308709
cites work: The existence of maps of nonzero degree between aspherical 3-manifolds
cites work: Q4192669
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description: scientific article; zbMATH DE number 537206