On the Nielsen numbers of slide homeomorphisms on 3-manifolds.
Every closed oriented \(3\)-manifold can be written uniquely as a connected sum of prime manifolds. Following the work of \textit{D. McCullough} [Geometric and algebraic topology, Banach Cent. Publ. 18, 61--76 (1986; Zbl 0637.57009)], every self-homeomorphism on a reducible \(3\)-manifold \(M\) is a composition of homeomorphisms of four types, namely, homeomorphisms preserving summands, those interchanging summands, spins of \(S^1 \times S^2\), and slide homeomorphisms. Write \(M=W \cup (\bigcup {M_i}')\) where \(W\) is a punctured \(3\)-cell, either \(M_i'=M_i- \text{int}(D_i)\) where \(M_i\) is irreducible and \(D_i\) is a closed \(3\)-disk, or \(M_i'\) is homeomorphic to \(S^2 \times [0,1]\). Let \(S\) be an essential \(2\)-sphere in \(M\), which is isotopic to a boundary component of \(W\) and \(\alpha :[0,1] \to M\) be a path such that \(\alpha ((0,\varepsilon))\cup \alpha ((1-\varepsilon,1))\subset W-\partial W\) and \(\alpha ([0,1])\cap S=\{\alpha (0), \alpha (1)\}\) for some \(\varepsilon >0\). Take two regular neighborhoods \(N_1, N_2\) of \(S \cup \alpha ([0,1])\) in \(M\) such that \(N_1 \subset \text{int}(N_2)\). Then \(\text{int}(N_2-N_1)\) has two components, one of which, denoted by \(T(S,\alpha)\), is homeomorphic to \(T^2\times (0,1)\) and the other to \(S^2 \times (0,1)\). Let \(c:T(S,\alpha) \to T^2\times (0,1)\) be the homeomorphism. By a slide homeomorphism, we mean a map \(s:M\to M\) such that \(s(x)=c^{-1}(\theta +2\pi t, \varphi, t)\) if \(x=c^{-1}(\theta +2\pi t, \varphi, t)\in T(S,\alpha)\) and \(s(x)=x\) otherwise. The purpose of the paper under review is to study the fixed point set of composites of slide homeomorphisms, from the point of view of Nielsen fixed point theory. First, the author shows that the Nielsen number of a slide homeomorphism is always zero while that of a composite of two slide homeomorphisms need not be even though such composites must have zero Lefschetz number. Furthermore, he shows, by means of an example, that the Nielsen number of a composite of two slide homeomorphisms can be made arbitrarily large while the Lefschetz number of such homeomorphism is necessarily zero.
scientific article; zbMATH DE number 2037251