A locally regular heptagon–dodecahedron embedded in 3-space

We construct a polyhedral 2-manifold of genus 2 embedded in Euclidean 3-space, and hence oriented, built up of 12 planar (but nonconvex) heptagons, three meeting at each vertex, i.e., locally regular or equivelar. The polyhedron is face-minimal among all equivelar polyhedra of genus g≥2.It has a threefold symmetry axis and exists in two chiral versions.     

Spatial polyhedra without diagonals

We provide new polyhedra without diagonals, and we discuss their embeddings in euclidean 3-space with maximal symmetries starting with a complete classification of their combinatorial properties: orientable neighborly 2-pseudomanifolds with 9 vertices or Mendelsohn triple systemsS ₂(2, 3, 9). This article was motivated by the longstanding and still open question: find a triangulated 2-manifold which can not be embedded in 3-space. Furthermore, we applied tested and improved algorithms for realizing oriented matroids or finding final polynomials.     

A Note on Surfaces Bounding Hyperbolic 3-Manifolds

We consider the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold (as some or as the only boundary component). We discuss some explicit examples of hyperbolic surfaces, in particular the surface associated to the small stellated dodecahedron (one of the four Kepler-Poinsot polyhedra) which is the boundary of a hyperbolic icosahedral 3-manifold.     

A Note on Surfaces Bounding Hyperbolic 3-Manifolds

We consider the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold (as some or as the only boundary component). We discuss some explicit examples of hyperbolic surfaces, in particular the surface associated to the small stellated dodecahedron (one of the four Kepler-Poinsot polyhedra) which is the boundary of a hyperbolic icosahedral 3-manifold.     

Representation of torus homeotopies by simple polyhedra with boundary

In 1991, Turaev and Viro constructed a quantum topological linear representation of mapping class groups of closed surfaces. To the mappings of a surface into itself, they assigned simple polyhedra whose boundaries consisted of two simple graphs cutting the surface into cells. The computational complexity of the Turaev-Viro representations strongly depends on the choice of suitable sets of simple polyhedra. In this paper, simple polyhedra for the torus are constructed. One of the reasons why they are convenient is that they all are obtained by gluing along boundary of copies of the same simple polyhedron. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 533–539, October, 1999.     

The multidirectional Neumann problem in ℝ4

The Neumann problem as formulated in Lipschitz domains with L^p boundary data is solved for harmonic functions in any compact polyhedral domain of ℝ⁴ that has a connected 3-manifold boundary. Energy estimates on the boundary are derived from new polyhedral Rellich formulas together with a Whitney type decomposition of the polyhedron into similar Lipschitz domains. The classical layer potentials are thereby shown to be semi-Fredholm. To settle the onto question a method of continuity is devised that uses the classical 3-manifold theory of E. E. Moise in order to untwist the polyhedral boundary into a Lipschitz boundary. It is shown that this untwisting can be extended to include the interior of the domain in local neighborhoods of the boundary. In this way the flattening arguments of B. E. J. Dahlberg and C. E. Kenig for the H¹_at Neumann problem can be extended to polyhedral domains in ℝ⁴. A compact polyhedral domain in ℝ⁶ of M. L. Curtis and E. C. Zeeman, based on a construction of M. H. A. Newman, shows that the untwisting and flattening techniques used here are unavailable in general for higher dimensional boundary value problems in polyhedra.     

A Böchner technique for harmonic mappings from a 3-manifold to a surface

Let be a harmonic mapping from a Riemannian 3-manifold to a Riemannian 2-manifold. A smooth function on M is associated to , derived from the eigenvalues of the first fundamental form, the vanishing of which is equivalent to being a harmonic morphism. The Laplacian of this function is computed and a maximum principle applied to derive criteria when a harmonic map must be a harmonic morphism.     

Dihedral Angles of Convex Polyhedra

Convex polyhedra in H ³ are not determined by (their combinatorics and) their edge lengths. Convex space-like polyhedra in the de Sitter space S ³ ₁ are determined neither by their dihedral angles nor by their edge lengths. The same holds of convex polyhedra in S ³ . Received November 16, 1998, and in revised form March 8, 1999.     

PROPER AFFINE DEFORMATIONS OF THE ONE-HOLED TORUS

A Margulis spacetime is a complete at Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface ∑ homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when ∑ is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of ∑. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces.     

Seiberg—Witten—Floer Homology and Gluing Formulae

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spin^C structure.Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.     

On self-mapping degrees of <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>-geometry manifolds

In this paper we determined all of the possible self-mapping degrees of the manifolds with S3-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self-mapping degrees of all closed orientable 3-manifold in Thurston＇s picture.     

A volume-preserving counterexample to the Seifert conjecture

We prove that every 3-manifold possesses aC ¹, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving,C ^∞ flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the first result. The proof uses a Dehn-twisted Wilson-type plug which also preserves volume. The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.     

Non-Zero Degree Maps Between <Emphasis Type="Italic">2n</Emphasis>-Manifolds

Abstract Thom–Pontrjagin constructions are used to give a computable necessary and sufficient condition for a homomorphism ϕ : H ⁿ (L;Z) → H ⁿ (M;Z) to be realized by a map f : M → L of degree k for closed (n − 1)-connected 2n-manifolds M and L, n > 1. A corollary is that each (n − 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms; in particular, there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M. Both authors are supported by MSTC, NSFC. The comments of F. Ding, J. Z. Pan, Y. Su and the referee enhance the quality of the paper     

Critical Heegaard surfaces

In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.     

Regular homotopy classes of immersions of 3-manifolds into 5-space

 We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and of the 3-torus T ³ with the following properties: (1) is regularly homotopic to F ₈ for some immersion , and (2) the immersion h as above cannot be chosen from a regular homotopy class containing an embedding. Received: 29 March 2001     

Cell Decompositions of the Projective Plane with Petrie Polygons of Constant Length

We study dual pairs of combinatorial face-to-face cell decompositions of the real projective plane P ² such that their canonically induced cell decompositions on the 2-sphere S ² form dual pairs of combinatorical types of convex polyhedra, and such that these dual pairs share two natural properties with those induced by dual pairs of Platonic solids: (1) Every Petrie polygon is a finite simple closed polygon of length 2(n-1) for some fixed n. (2) Every pair of Petrie polygons has precisely two common edges. Such pairs of face-to-face cell decompositions of the projective plane are in one-to-one correspondence with n-element pseudoline arrangements. We study in particular those dual pairs of cell decompositions in which has only 3-valent vertices, i.e., via the above one-to-one correspondence: p ₃-maximal pseudoline arrangements. A p ₃ -maximal pseudoline arrangement with n elements in turn determines a neighborly 2-manifold with Euler characteristic χ = n(7-n)/6, and vice versa, this neighborly 2-manifold uniquely determines its generating p ₃ -maximal pseudoline arrangement. We provide new inductive constructions for finding infinite example classes of p ₃-maximal pseudoline arrangements from small existing ones, we describe an algorithm for generating them, we provide a complete list of existence up to n = 40, and we discuss their properties. Received July 18, 1996, and in revised form October 28, 1996.     

Fundamental groups of manifolds with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-category 2

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁,W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹ → M ⁿ . We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.      

Q-curvature flow on 4-manifolds

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.     

Extremal Metrics for Quadratic Functional of Scalar Curvature on Closed 3-Manifolds

In this paper, we first show the global existence of the three-dimensionalCalabi flow on any closed 3-manifold with an arbitrary background metric g ₀. Second, we show the asymptotic convergence of a subsequence ofsolutions of the Calabi flow on a closed 3-manifold with Yamabe constant Q < 0 or Q = 0 and Q > 0, up to conformal transformations. With itsapplication, we prove the existence of extremal metrics for quadraticfunctional of scalar curvature on a closed 3-manifold which is served asan extension of the Yamabe problem on closed manifolds. Moreover, theexistence of extremal metrics on complete noncompact 3-manifolds willdiscuss elsewhere.     

The computational complexity of basic decision problems in 3-dimensional topology

We study the computational complexity of basic decision problems of 3-dimensional topology, such as to determine whether a triangulated 3-manifold is irreducible, prime, ∂-irreducible, or homeomorphic to a given 3-manifold M. For example, we prove that the problem to recognize whether a triangulated 3-manifold is homeomorphic to a 3-sphere, or to a 2-sphere bundle over a circle, or to a real projective 3-space, or to a handlebody of genus g, is decidable in nondeterministic polynomial time (NP) of size of the triangulation. We also show that the problem to determine whether a triangulated orientable 3-manifold is irreducible (or prime) is in PSPACE and whether it is ∂-irreducible is in coNP. The proofs improve and extend arguments of prior author’s article on the recognition problem for the 3-sphere.