Topologically rigid non-Haken 3-manifolds. (English) Zbl 0806.57006

Hass, J.; Menasco, W.

J. Aust. Math. Soc., Ser. A 55, No. 1, 60-71 (1993).

A closed irreducible 3-manifold \(M\) is called topologically rigid if any homotopy-equivalent irreducible 3-manifold is homeomorphic to \(M\). It is known that all Haken 3-manifolds and all Seifert 3-manifolds with infinite fundamental group are topologically rigid. The authors construct infinitely many non-Haken, non-Seifert 3-manifolds which are topologically rigid. All of them can be obtained by Dehn surgeries of \(S^ 3\) along an explicitly presented alternating 3-component link \(L\). The proof that the manifolds are non-Haken is based on an earlier paper of the second author [Topology 23, 37-44 (1984; Zbl 0525.57003)]; the proof that they are topologically rigid follows from results of the first author and P. Scott [ibid. 31, 493-517 (1992; Zbl 0771.57007)].

Reviewer: S.V.Matveev (Chelyabinsk)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

incompressible surface; non-Seifert 3-manifolds; non-Haken 3-manifolds; closed irreducible 3-manifold; topologically rigid; homotopy-equivalent; Dehn surgeries

Citations:

Zbl 0525.57003; Zbl 0771.57007