The absence of efficient dual pairs of spanning trees in planar graphs

A spanning tree T in a finite planar connected graph G determines a dual spanning tree T* in the dual graph G such that T and T* do not intersect. We show that it is not always possible to find T in G, such that the diameters of T and T* are both within a uniform multiplicative constant (independent of G) of the diameters of their ambient graphs.Comment: 7 pages, 3 figure

3-manifolds with(out) metrics of nonpositive curvature

In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.Comment: 16 page

A Note on Real Tunneling Geometries

In the Hartle-Hawking ``no boundary'' approach to quantum cosmology, a real tunneling geometry is a configuration that represents a transition from a compact Riemannian spacetime to a Lorentzian universe. I complete an earlier proof that in three spacetime dimensions, such a transition is ``probable,'' in the sense that the required Riemannian geometry yields a genuine maximum of the semiclassical wave function.Comment: 5 page

Homologous non-isotopic symplectic tori in a K3-surface

For each member of an infinite family of homology classes in the K3-surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic submanifolds in many other symplectic 4-manifolds including the elliptic surfaces E(n) for n>2.Comment: 15 pages, 9 figures; v2: extended the main theorem, gave a second construction of symplectic tori, added a figure, added/updated references, minor changes in figure

Spherical structures on torus knots and links

The present paper considers two infinite families of cone-manifolds endowed with spherical metric. The singular strata is either the torus knot ${\rm t}(2n+1, 2)$ or the torus link ${\rm t}(2n, 2)$. Domains of existence for a spherical metric are found in terms of cone angles and volume formul{\ae} are presented.Comment: 17 pages, 5 figures; typo

Reconstructing the global topology of the universe from the cosmic microwave background

If the universe is multiply-connected and sufficiently small, then the last scattering surface wraps around the universe and intersects itself. Each circle of intersection appears as two distinct circles on the microwave sky. The present article shows how to use the matched circles to explicitly reconstruct the global topology of space.Comment: 6 pages, 2 figures, IOP format. To be published in the proceedings of the Cleveland Cosmology and Topology Workshop 17-19 Oct 1997. Submitted to Class. Quant. Gra

The Generalized Ricci Flow for 3D Manifolds with One Killing Vector

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riemannian metric with one Killing vector. In particular, we show that the RG flow with De Turck terms can be reduced to two equations: the continual Toda flow solved by Bakas, plus its linearizaton. We find exact solutions which flow to homogeneous but not always isotropic geometries

Comments on Closed Bianchi Models

We show several kinematical properties that are intrinsic to the Bianchi models with compact spatial sections. Especially, with spacelike hypersurfaces being closed, (A) no anisotropic expansion is allowed for Bianchi type V and VII(A\not=0), and (B) type IV and VI(A\not=0,1) does not exist. In order to show them, we put into geometric terms what is meant by spatial homogeneity and employ a mathematical result on 3-manifolds. We make clear the relation between the Bianchi type symmetry of space-time and spatial compactness, some part of which seem to be unnoticed in the literature. Especially, it is shown under what conditions class B Bianchi models do not possess compact spatial sections. Finally we briefly describe how this study is useful in investigating global dynamics in (3+1)-dimensional gravity.Comment: 14 pages with one table, KUCP-5

Circles in the Sky: Finding Topology with the Microwave Background Radiation

If the universe is finite and smaller than the distance to the surface of last scatter, then the signature of the topology of the universe is writ large on the microwave background sky. We show that the microwave background will be identified at the intersections of the surface of last scattering as seen by different ``copies'' of the observer. Since the surface of last scattering is a two-sphere, these intersections will be circles, regardless of the background geometry or topology. We therefore propose a statistic that is sensitive to all small, locally homogeneous topologies. Here, small means that the distance to the surface of last scatter is smaller than the ``topology scale'' of the universe.Comment: 14 pages, 10 figures, IOP format. This paper is a direct descendant of gr-qc/9602039. To appear in a special proceedings issue of Class. Quant. Grav. covering the Cleveland Topology & Cosmology Worksho

Complexity of links in 3-manifolds

We introduce a natural-valued complexity c(X) for pairs X=(M,L), where M is a closed orientable 3-manifold and L is a link contained in M. The definition employs simple spines, but for well-behaved X's we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0,2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0,2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.Comment: 24 pages, 6 figure