Related papers: Trisections of 5-manifolds
We prove the long-standing Montesinos conjecture that any closed oriented PL 4-manifold M is a simple covering of S^4 branched over a locally flat surface (cf [J M Montesinos, 4-manifolds, 3-fold covering spaces and ribbons, Trans. Amer.…
The triangulation complexity of a compact 3-manifold is the minimal number of tetrahedra in any triangulation of the 3-manifold. We compute the triangulation complexity of all elliptic 3-manifolds and all sol 3-manifolds, to within a…
This paper provides a topological method to construct all simply-connected, spin, smooth $6$-manifolds with torsion-free homology using simply-connected, smooth $4$-manifolds as building blocks. We explicitly determine the invariants that…
We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex…
In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3--manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first…
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed…
The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), ..., O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex…
Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such…
A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over…
We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space and in a given linear order in the Euclidean 3-space. Similarly, we estimate the number…
The problem of finding necessary and sufficient conditions to decompose a complete tripartite graph $K_{r,s,t}$ into 5-cycles was first considered by E.S. Mahmoodian and Maryam Mirzakhani (1995). They stated some necessary conditions and…
In [4]: `The Riley slice of Schottky space', (Proc. London Math. Soc. 69 (1994), 72-90), Keen and Series analysed the theory of pleating coordinates in the context of the Riley slice of Schottky space R, the deformation space of a genus two…
We characterise which simplicial surfaces can be folded onto a triangle. We define a notion of folding that incorporates the non-intersection-properties of real materials. All of the surfaces foldable onto a triangle admit a…
Given an $S\cong \mathbb{R}P^2$ smoothly embedded in a 4-manifold $X^4$ with Euler number 2 or -2, the Price twist is a surgery operation on $\nu(S)$ yielding (up to) three different 4-manifolds: $X^4,\tau_S(X^4),\Sigma_S(X^4)$. This is of…
We offer a new proof that two closed oriented 4-manifolds are cobordant if their signatures agree, in the spirit of Lickorish's proof that all closed oriented 3-manifolds bound 4-manifolds. Where Lickorish uses Heegaard splittings we use…
We consider topologically twisted N=4 supersymmetric Yang-Mills theory on a four-manifold of the form V = W \times R_+ or V = W \times I, where W is a Riemannian three-manifold. Different kinds of boundary conditions apply at infinity or at…
It is conjectured that every cusped hyperbolic 3-manifold admits a geometric triangulation, i.e. it is decomposed into positive volume ideal hyperbolic tetrahedra. Here, we show that sufficiently highly twisted knots admit a geometric…
Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit…
In this article, we study the geometry of plane curves obtained by three sections and another section given as their sum on certain rational elliptic surfaces. We make use of Mumford representations of semi-reduced divisors in order to…
The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in two earlier papers (math.SG/0101206 and math.SG/0105202). This four-dimensional theory also…