Self-covering, finiteness, and fibering over a circle (Q6567123)

A CAT manifold is called self-covering if it is CAT-isomorphic to a nontrivial covering space of itself, where CAT represents any of the following categories: smooth (DIFF), piecewise-linear (PL) and topological (TOP). It has been observed that if \(M\) is a CAT fiber bundle over \(S^1\) with monodromy \(f\), such that \(f^kg\) is CAT pseudo-isotopic to either \(gf\) or \(gf^{-1}\) for some automorphism \(g\) of the fiber \(F\), then \(M\) is self-covering. Motivated by this observation, the authors explore the conditions under which a closed self-covering manifold is a fiber bundle over \(S^1\).\N\NLet \(M\) denote a connected, closed, and self-covering CAT manifold with \(\pi_1(M)=G\times \mathbb{Z}\), where \(G\) is an abelian group. Consider \(M_k\) as the finite cyclic cover of \(M\) with \(\pi_1(M_k)=G\times (k\mathbb{Z})\). Under the assumption that \(G\) is free and \(\dim M\geq 6\), the authors prove that if there exists a homotopy equivalence \(h:M\to M_k\) for some \(k>1\) such that the induced isomorphism \(h_\sharp:\pi_1(M)\to \pi_1(M_k)\) satisfies \(h_\sharp(G\times 0)=G\times 0\), then \(M\) is a CAT fiber bundle over \(S^1\). This is also valid when \(\dim M=5\) if we further assume that \(G=0\) and CAT \(=\) TOP (Theorem B). The authors also present a closed, smooth, self-covering \(5\)-manifold \(M\) with \(\pi_1(M)=\mathbb{Z}\) which is not a smooth (piecewise linear) fiber bundle over \(S^1\) (Theorem F). In particular, they give a complete answer to the question whether a self-covering manifold with fundamental group \(\mathbb{Z}\) is a fiber bundle over \(S^1\), except for the \(4\)-dimensional smooth case. This is summarized in the table below. \begin{center} \begin{tabular}{c|c|c|c|c} & \(\dim \leq 3\) & \(\dim=4\) & \(\dim=5\) & \(\dim\geq 6\) \\\N\hline DIFF & yes & unknown yet & no & yes \\\NPL & yes & unknown yet & no & yes \\\NTOP & yes & yes & yes & yes \\\N\end{tabular} \end{center} In cases \(G\) is not free, the authors show that \(M\) may not be fiber bundles over \(S^1\) (Theorems C and D).