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Register a new userA characterization of tightly triangulated 3-manifolds
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For a field $mathbb{F}$, the notion of $mathbb{F}$-tightness of simplicial complexes was introduced by Kuhnel. Kuhnel and Lutz conjectured that any $mathbb{F}$-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only $mathbb{F}$-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is $mathbb{F}$-tight if and only if it is $mathbb{F}$-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is $mathbb{F}$-tight if and only if it is $mathbb{F}$-orientable, neighbourly and stacked. In consequence, the Kuhnel-Lutz conjecture is valid in dimension $leq 3$.
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It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the K{u}hnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $beta_1$, then $(n-4)(617n- 3861) leq 15444beta_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.
This paper gives a combinatorial description of spin and spin^c-structures on triangulated PL-manifolds of arbitrary dimension. These formulations of spin and spin^c-structures are established primarily for the purpose of aiding in computations. The novelty of the approach is we rely heavily on the naturality of binary symmetric groups to avoid lengthy explicit constructions of smoothings of PL-manifolds.
Let $Delta$ be a triangulated homology ball whose boundary complex is $partialDelta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $Delta$, $mathbb F[Delta]$, is isomorphic to the Stanley--Reisner module of the pair $(Delta, partialDelta)$, $mathbb F[Delta,partial Delta]$. This result implies that an Artinian reduction of $mathbb F[Delta,partial Delta]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $mathbb F[Delta]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.
We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite number of times.
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.
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