Invariant incompressible surfaces in reducible 3-manifolds

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$ . As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz, and R. Ures: a reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-\text{id}$ , or the mapping torus of a hyperbolic automorphism.

TOTALLY GEODESIC SURFACES AND QUADRATIC FORMS

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

LIFTED HEEGAARD SURFACES AND VIRTUALLY HAKEN MANIFOLDS

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [A. Casson and C. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283] and extends [J. Masters, W. Menasco and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004) 133–150].

2-STRING FREE TANGLES AND INCOMPRESSIBLE SURFACES

Suppose k is a connected sum of two knots, one of which admits a 2-string essential free tangle decomposition, then the exterior of k contains an incompressible surface of genus n for each positive integer n.

KNOTS WITH INFINITELY MANY INCOMPRESSIBLE SEIFERT SURFACES

We show that a knot in S3 with an infinite number of incompressible Seifert surfaces contains a closed non-peripheral incompressible surface in its complement.