Waldhausen's Theorem Research Articles

Comparing Heegaard splittings of non-haken 3-manifolds

Cerf theory can be used to compare two strongly irreducible Heegaard splittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collection of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfaces during which this intersection property is preserved. As sample applications we give new proofs of Waldhausen's theorem that Heegaard splittings of S 3 are standard, and of Bonahon and Otal's theorem that Heegaard splittings of lens spaces are standard. We also present a solution to the stabilization problem for irreducible non-Haken 3-manifolds: If p ⩽ q are the genera of two splittings of such a manifold, then there is a common stabilization of genus 5 p + 8 q − 9.

Boundary preserving maps of 3-manifolds

We prove an extension of Waldhausen’s theorem [5] conjectured by Hempel in [3].

Boundary-reducible $3$-manifolds and Waldhausen's theorem.

Boundary-reducible $3$-manifolds and Waldhausen's theorem.

ULC Properties in Neighbourhoods of Embedded Surfaces and Curves in E3

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

Boundary respecting maps of 3-mainfolds

This paper is about maps of compact 3-manifolds which map the boundary of the domain (possibly nonhomeomorphically) into the boundary of the range. F. Waldhausen has shown that such a map between compact, orientable, irreducible 3-manifolds with nonempty, incompressible boundary is homotopic to a homeomorphism if and only if the map induces an isomorphism at the fundamental group level. The main theorem of this paper states that the above theorem remains valid if the assumption of incompressibl e boundary is dropped. A study of disk sums of bounded 3-manifolds will be required in order to prove the above-mentioned theorem. This investigation involves theorems about disk sums of bounded 3-manifolds analogous to the classical Eneser theorem for closed 3-manifolds. The reader may wish to consult [11] for a proof of Waldhausen's theorem mentioned above and [4] and [7] for variations of Waldhausen's theorem related to the theorems proved in this paper. All spaces and maps in this paper are assumed to belong to the precise linear category, and each subspace that we shall discuss is taken to be piecewise linearly embedded. If A is a subcomplex of the simplicial complex X, we use the notation U(A, X) to denote a regular neighborhood of A in a second derived subdivision of X. If X is a manifold, we use the notation bdX and intX to denote the boundary of X and the interior of X respectively. A 3-manifold M is said to be irreducible if each 2-sphere in M is the boundary of some 3-cell in M.

Replacing certain maps of $3$-manifolds by homeomorphisms

Let f be a map of a 3-manifold M1 onto the 3-manifold M2. Call f boundary preserving if fI Bd M1 is a homeomorphism onto Bd M2 and f(Int Ml) = Int M2. Let Sf = { x I x E Ml, and f-'f (x) is nondegenerate }. In [3], McMillan announced that if M1 = S3 and Cl f(Sf) is 0-dimensional, then M2= S3. This follows by showing S3 -f-'(x) =E3 (f induces a point-like decomposition of SI). By Bing's example in [1, p. 7], M2 may be homeomorphic to M1 even though for certain points x, S3-f-'(x) #E3. In this note we wish to develop a condition on Clf(Sf) which will enable us to say M1 and M2 are homeomorphic. In Theorem 5 of [2], Hempel showed how to obtain a boundary preserving map of a cube with one knotted hole onto a cube with one handle; however, in this case, the injection of H1(Cl f(Sf); Z) into H1(M2; Z) =Z is nontrivial. This fact together with Waldhausen's Theorem 1.2 of [5] suggest the type of condition on Cl f (Sf) we are looking for. Following the definition given in [3], we say X, a subset of the interior of a 3-manifold M, is strongly n-acyclic (over Z) if for each open set UCM such that XC U there is an open set V such that XC VC U and injection of Hn(V; Z) into Hn(U; Z) is trivial.