Triangulations of 3-dimensional pseudomanifolds with an application to state-sum invariants

Non-existence of 6-dimensional pseudomanifolds with complementarity

In a previous paper the second author showed that if $M$ is a pseudomanifold with complementarity other than the 6-vertex real projective plane and the 9-vertex complex projective plane, then $M$ must have dimension $\geq 6$, and - in case of equality - $M$ must have exactly 12 vertices. In this paper we prove that such a 6-dimensional pseudomanifold does not exist. On the way to proving our main result we also prove that all combinatorial triangulations of the 4-sphere with at most 10 vertices are combinatorial 4-spheres.

Three dimensional pseudomanifolds on eight vertices

A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normal $d$-pseudomanifolds form a broader class than triangulations of connected closed $d$-manifolds for $d \geq 3$. Here, we classify all the 8-vertex neighbourly normal 3-pseudomanifolds. This gives a classification of all the 8-vertex normal 3-pseudomanifolds. There are 73 such 3-pseudomanifolds, 38 of which triangulate the 3-sphere and other 35 are not combinatorial 3-manifolds. These 35 triangulate six distinc…

Three-dimensional normal pseudomanifolds with relatively few edges

Let $Δ$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $Δ$ is that $g_2$ of $Δ$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $Δ$ has relatively minimal $g_2$. For example, whenever the one-skeleton of $Δ$ equals the one-skeleton of the star of a vertex, then $Δ$ has relatively minimal $g_2.$ Subdividing a facet in such an example also gives a complex with relatively minimal $g_2.$ We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological t…

至多有两个奇异点的正规3-伪流形的刻画

A characterization of normal 3-pseudomanifolds with at most two singularities

本文刻画了满足$g_2$上界条件的至多含两个奇异点的正规3-伪流形，证明其可由4-单纯形边界复形经六类操作生成，并验证该上界紧致。

A general state-sum construction of 2-dimensional topological quantum field theories with defects

We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda \& Pfeiffer \cite{LP}). From the extended Pachner moves (Crane \& Yetter \cite{CY}), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter \cite{DPY}, we require that triangulations be flag-like, meaning that each simplex of the triangulation is e…

On 2-Dimensional Dijkgraaf-Witten Theory with Defects

In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face. The construction allows internal degrees of freedom in the defect curves by introducing a second gauge-group from which edges of the curve are labeled in the state-sum construction. Edges incident with the defect, but not lying in it…