相关论文: Lower bounds for regular genus and gem-complexity …

Within crystallization theory, two interesting PL invariants for $d$-manifolds have been introduced and studied, namely {\it gem-complexity} and {\it regular genus}. In the present paper we prove that, for any closed connected PL…

In this article, we introduce two new PL-invariants: weighted regular genus and weighted G-degree for manifolds with boundary. We first prove two inequalities involving some PL-invariants which state that for any PL-manifold $M$ with non…

The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called {\it semi-simple} and {\it weak semi-simple crystallizations},…

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular…

Let $(\Gamma,\gamma)$ be a crystallization of connected compact 3-manifold $M$ with $h$ boundary components. Let $\mathcal{G}(M)$ and $\mathit k (M)$ be the regular genus and gem-complexity of $M$ respectively, and let $\mathcal{G}(\partial…

Gems are a particular type of edge-colored graphs, dual to colored triangulations, which represent compact PL-manifolds of arbitrary dimension, both in the closed and boundary case. In the present paper, gem theory is used to approach…

In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms $f:M \to M$ for a certain class of PL-manifolds $M$. These yield upper bounds for gem-complexity and regular genus of a large class of…

(d+1)-colored graphs, i.e. edge-colored graphs that are (d+1)-regular, have already been proved to be a useful representation tool for compact PL d-manifolds, thus extending the theory (known as crystallization theory) originally developed…

In 2018, M. Chu and S. Tillmann gave a lower bound for the trisection genus of a closed 4-manifold in terms of the Euler characteristic of $M$ and the rank of its fundamental group. We show that given a group $G$, there exist a 4-manifold…

Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present paper, we prove that any…

Given a closed four-manifold $X$ with an indefinite intersection form, we consider smoothly embedded surfaces in $X \setminus $int$(B^4)$, with boundary a knot $K \subset S^3$. We give several methods to bound the genus of such surfaces in…

Let $X$ be a closed indefinite $4$-manifold with $b_+(X) = 3 \; ({\rm mod} \; 4)$ and with non-vanishing mod $2$ Seiberg--Witten invariants. We prove a new lower bound on the genus of a properly embedded surface in $X \setminus B^4$…

Let N be a closed irreducible 3-manifold and assume N is not a graph manifold. We improve for all but finitely many S^1-bundles M over N the adjunction inequality for the minimal complexity of embedded surfaces. This allows us to completely…

This article focuses on a class of properly edge-colored graphs, which arise from topological combinatorics, and investigates their embeddings onto surfaces. Specifically, these graphs are known as the dual graphs of balanced normal…

In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold $X$ with second positive Betti number $b_2^+(X)=1$. We study the…

In this paper, we prove the existence of $H^2$-regular coordinates on Riemannian $3$-manifolds with boundary, assuming only $L^2$-bounds on the Ricci curvature, $L^4$-bounds on the second fundamental form of the boundary, and a positive…

The idea of studying trisections of closed smooth $4$-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and…

We provide new branched covering representations for bounded and/or non-compact 4-manifolds, which extend the known ones for closed 4-manifolds. Assuming $M$ to be a connected oriented PL 4-manifold, our main results are the following: (1)…

We define the notion of $(p_0,p_1,\dots,p_d)$-type semi-equivelar gems for closed connected PL $d$-manifolds, related to the regular embedding of gems $\Gamma$ representing $M$ on a surface $S$ such that the face-cycles at all the vertices…

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case…