The classification of 3-manifolds with 4-plane, quasi-finite immersed surfaces

In this paper, we prove that homotopy equivalence implies homeomorphism equivalence for a pair of closed, orientable, irreducible 3-manifolds, one of which contains an immersed incompressible surface with some combinatorial constraints on its self-intersections. In particular, we generalize the result of Paterson (preprint), so that we require only the 4-plane property and the condition of quasi-finiteness to hold for the immersed surface. We prove this result by first homotoping the surface into a canonical form for its homotopy class.