Existence and properties of incompressible surfaces in 3-dimensional manifolds are surveyed. Some conjectures of Waldhausen and Thurston concerning such surfaces are stated. An outline is given of the proof that such surfaces can be pulled back by non-zero degree maps between 3-manifolds. The effect of surgery on immersed, incompressible surfaces and on hierarchies is discussed. A characterisation is given of the immersed, incompressible surfaces previously studied by Hass and Scott, which arise naturally with cubings of non-positive curvature.

A closed irreducible 3-manifold M is topologically rigid if any homotopy equivalent irreducible 3-manifold is homeomorphic to M. A construction is given which produces infinitely many non-Haken topologically rigid 3-manifolds

There are 11 closed 4-manifolds which admit geometries of compact type (S4, CP2 or S2 × S2) and two other closely related bundle spaces (S2 × S2 and the total space of the nontrivial RP2-bundle over S2). We show that the homotopy type of such a manifold is determined up to an ambiguity of order at most 4 by its quadratic 2-type, and this in turn is (in most cases) determined by the Euler characteristic, fundamental group and Stiefel-Whitney classes. In (at least) seven of the 13 cases, a PL 4-manifold with the same invariants as a geometric manifold or bundle space must be homeomorphic to it.

The results here concern bijective continous functions from one connected separable n-manifold M to another N. If M has the property that every such function is necessarily a homeomorphism, then M is said to be strongly reversible. Strongly reversible manifolds having only compact boundary components are completely charaterized.

Let X be a closed, oriented, smooth 4-manifold with a finite fundamental group and with a non-vanishing Seiberg-Witten invariant. Let G be a finite group. If G acts smoothly and freely on X, then the quotient X/G cannot be decomposed as X1#X2 with (Xi) > 0, i = 1, 2. In addition let X be symplectic and c1(X)2 > 0 and b+2(X) > 3. If σ is a free anti-symplectic involution on X then the Seiberg-Witten invariants on X/σ vanish for all spinc structures on X/σ, and if η is a free symplectic involution on X then the quotients X/σ and X/η are not diffeomorphic to each other.

A 2-handle addition on the boundary of a hyperbolic 3-manifold M is called degenerating if the resulting manifold is not hyperbolic. There are examples that some manifolds admit infinitely many degenerating handle additions. But most of them are not ‘basic’. (See Section 1 for definitions). Our first main theorem shows that there are only finitely many basic degenerating handle additions. We also study the case that one of the handle additions produces a reducible manifold, and another produces a ∂-reducible manifold, showing that in this case either the two attaching curves are disjoint, or they can be isotoped into a once-punctured torus. A byproduct is a combinatorial proof of a similar known result about degenerating hyperbolic structures by Dehn filling.

In this paper we present a short definition of the Witten invariants of 3-manifolds. We also give simple proofs of invariance of those obtained for r = 3 and r = 4. Our definition is extracted from the 1993 paper of Lickorish and the Prasolov-Sossinsky book, where it is dispersed over 20 pages. We show by several examples that it is indeed convenient for calculations.

We show that diagrammatically reducible two-complexes are characterized by the property: every finity subconmplex of the universal cover collapses to a one-complex. We use this to show that a compact orientable three-manifold with nonempty boundary is Haken if and only if it has a diagrammatically reducible spine. We also formulate an nanlogue of diagrammatic reducibility for higher dimensional complexes. Like Haken three-manifolds, we observe that if n ≥ 4 and M is compact connected n-dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover of M is homeomorphic to Euclidean n-space.

Let N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

We provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.

Examples of hyperbolic knots in S3 are given such that their complements contain quasi-Fuchsian non-Fuchsian surfaces. In particular, this proves that there are hyperbolic knots that are not toroidally alternating.

Let N be a closed s-Hopfian n-manifold with residually finite, torsion free π1 (N) and finite H1(N). Suppose that either πk(N) is finitely generated for all k ≥ 2, or πk(N) ≅ 0 for 1 < k < n – 1, or n ≤ 4. We show that if N fails to be a co-dimension 2 fibrator, then N cyclically covers itself, up to homotopy type.

We study weak and strong peripheral 1-acyclicity, a homology version of D. R. McMillan, Jr.'s weak cellularity criterion and cellularity criterion, for embeddings of compacta in 3-manifolds. In contrast with the two cellularity criteria we prove that the two peripheral acyclicities are equivalent and moreover, for compacta of dimension at most 1, independent of the embedding. We also give some results concerning regular neighborhoods of compact polyhedra in 3-manifolds.

In this paper we prove a realizability theorem for Quinn’s mapping cylinder obstructions for stratified spaces. We prove a continuously controlled version of the s-cobordism theorem which we further use to prove the relation between the torsion of an h-cobordism and the mapping cylinder obstructions. This states that the image of the torsion of an h-cobordism is the mapping cylinder obstruction of the lower stratum of one end of the h-cobordism in the top filtration. These results are further used to prove a theorem about the realizability of end obstructions.

If P is a closed 3-manifold the covering space associated to a finitely presentable subgroup ν of infinite index in π1(P) is finitely dominated if and only if P is aspherical or . There is a corresponding result in dimension 4, under further hypotheses on π and ν. In particular, if M is a closed 4-manifold, ν is an ascendant, FP3, finitely-ended subgroup of infinite index in π1(M), π is virtually torsion free and the associated covering space is finitely dominated then either M is aspherical or or S3. In the aspherical case such an ascendant subgroup is usually Z, a surface group or a PD3-group.

We prove that the Witten–Reshetikhin–Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unknot.

A stereological formula for the Euler number involving projections of the set in thin parallel slabs is considered. Sufficient conditions for the validity of this formula are derived.