The \(\pi_ 1\)-injectivity of self-maps of nonzero degree on 3-manifolds
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 \(F\).
injectivity of map induced on the fundamental group
scientific article; zbMATH DE number 537206
scientific article