Homeomorphic graph manifolds: A contribution to the \(\mu\) constant problem
The authors prove the following Theorem. Let \(M\) and \(N\) be two compact, connected, closed, irreducible graph 3-manifolds with infinite fundamental group. Let \(f:M\to N\) be a homology equivalence (all homologies are with integer coefficients) such that for any finite covering of \(N\) (regular or not), the induced map \({\widetilde f} :{\widetilde M}\to {\widetilde N}\) is still a homology equivalence. Then \(f\) is homotopic to a homeomorphism. As an application, they prove the following Corollary: Let \(M\) and \(N\) be as above and suppose that there exists a cobordism \(W^4\) between \(M\) and \(N\) such that (i) \(\pi_1(N)\to \pi_1(W)\) is surjective; (ii) \(W\) is obtained from \(N\) by adding handles of index \(\leq 2\); (iii) the inclusions \(M\to W\) and \(N\to W\) are homology equivalences. Then, \(M\) and \(N\) are homeomorphic. The authors say that these results were motivated by the ``\(\mu\)-constant problem in complex dimension 3'', and they give a review of this problem. They recall in particular a theorem due to Lê Dung Trang and C. P. Ramanujan, stating that if \(f_t:(U,0)\subset (C^n,0)\to (C,0)\) is a one-parameter family of holomorphic functions with isolated singularity at \(0\), for each \(t\in V(0)\subset R\), \(C^{\infty}\) in the \(t\) variable, with the same Milnor number at \(0\) (such a family is called a \(\mu\)-constant family) then, for \(n\geq 3\), \(f_0\) and \(f_t\) have the same topological type for each \(t\) in a neighborhood \(W(0)\subset V(0)\subset R\). The results of this paper imply that the Lê-Ramanujan theorem holds also in dimension 3, provided that \(\pi_1(K_t)\to \pi_1(W)\) is surjective.
scientific article; zbMATH DE number 1382622