arXiv subjects

Biplab Basak

Publications and source records attributed to Biplab Basak.

32 records · Page 2Linked to original sources

arXiv2023

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

Characterizing face-number-related invariants of a given class of simplicial complexes has been a central topic in combinatorial topology. In this regard, one of the well-known invariants is $g_2$. Let $K$ be a normal $3$-pseudomanifold such that $g_2(K) \leq g_2(lk (v)) + 9$ for some vertex $v$ in $K$. Suppose either $K$ has only one singularity or $K$ has two singularities (at least) one of which is an $\mathbb{RP}^2$-singularity. We prove that $K$ is obtained from some boundary complexes of $4$-simplices by a sequence of operations of types connected sums, bistellar $1$-moves, edge contractions, edge expansions, vertex foldings, and edge foldings. In case $K$ has one singularity, $|K|$ is a handlebody with its boundary coned off. Further, we prove that the above upper bound is sharp for such normal $3$-pseudomanifolds.

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arXiv2022

On regular genus and G-degree of PL 4-manifolds with boundary

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 spherical boundary components, the regular genus $\mathcal{G}(M)$ of $M$ is at least the weighted regular genus $\tilde{G}(M)$ of $M$ which is again at least the generalized regular genus $\bar{G}(M)$ of $M$. Another inequality states that the weighted G-degree $\tilde{D}_G (M)$ of $M$ is always greater than or equal to the G-degree $D_G (M)$ of $M$. Let $M$ be any compact connected PL $4$-manifold with $h$ number of non spherical boundary components. Then we compute the following: $$\tilde{G} (M) \geq 2 χ(M)+3m+2h-4+2 \hat{m} \mbox{ and } \tilde{D}_G (M) \geq 12(2 χ(M)+3m+2h-4+2 \hat{m}),$$ where $m$ and $\hat{m}$ are the ranks of the fundamental groups of $M$ and the corresponding singular manifold $\widehat{M}$ (obtained by coning off the boundary components of $M$) respectively. As a consequence we prove that the regular genus $\mathcal{G}(M)$ satisfies the following inequality: $$\mathcal{G} (M) \geq 2 χ(M)+3m+2h-4+2 \hat{m},$$ which improves the previous known lower bounds for the regular genus $\mathcal{G}(M)$ of $M$. Then we define two classes of gems for PL $4$-manifold $M$ with boundary: one consists of semi-simple gems and the other consists of weak semi-simple gems, and prove that the lower bounds for the weighted G-degree and weighted regular genus are attained in these two classes respectively.

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arXiv2022

A characterization of normal $3$-pseudomanifolds with $g_2\leq4$

We characterize normal $3$-pseudomanifolds with $g_2\leq4$. We know that if a $3$-pseudomanifold with $g_2\leq4$ does not have any singular vertices then it is a $3$-sphere. We first prove that a normal $3$-pseudomanifold with $g_2\leq4$ has at most two singular vertices. Then we prove that a normal $3$-pseudomanifold with $g_2 \leq 4$, which is not a $3$-sphere is obtained from some boundary of $4$-simplices by a sequence of operations connected sum, edge expansion and an edge folding. In addition, by using [17], we re-framed the characterization of normal $3$-pseudomanifolds with $g_2\leq 9$, when it has no singular vertices.

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arXiv2021

Minimal crystallizations of 3-manifolds with boundary

Let $(Γ,γ)$ 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 $\mathcal{G}(\partial M)$ be the regular genus of $\partial M$. We prove that $$\mathit k (M)\geq 3 (\mathcal{G}(M)+h-1) \geq 3 (\mathcal{G} (\partial M)+h-1).$$ These bounds for gem-complexity of $M$ are sharp for several 3-manifolds with boundary. Further, we show that if $\partial M$ is connected and $\mathit k (M)< 3 (\mathcal{G} (\partial M)+1)$ then $M$ is a handlebody. In particular, we prove that $\mathit k (M) =3 \mathcal{G} (\partial M)$ if $M$ is a handlebody and $\mathit k (M) \geq 3 (\mathcal{G} (\partial M)+1)$ if $M$ is not a handlebody. Further, we obtain several combinatorial properties for a crystallization of 3-manifolds with boundary.

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arXiv2020

Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary

Let $M$ be a connected compact PL 4-manifold with boundary. In this article, we have given several lower bounds for regular genus and gem-complexity of the manifold $M$. In particular, we have proved that if $M$ is a connected compact $4$-manifold with $h$ boundary components then its gem-complexity $\mathit{k}(M)$ satisfies the following inequalities: $$\mathit{k}(M)\geq 3χ(M)+7m+7h-10 \mbox{ and }\mathit{k}(M)\geq \mathit{k}(\partial M)+3χ(M)+4m+6h-9,$$ and its regular genus $\mathcal{G}(M)$ satisfies the following inequalities: $$\mathcal{G}(M)\geq 2χ(M)+3m+2h-4\mbox{ and }\mathcal{G}(M)\geq \mathcal{G}(\partial M)+2χ(M)+2m+2h-4,$$ where $m$ is the rank of the fundamental group of the manifold $M$. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL $4$-manifold with boundary. Further, the sharpness of these bounds has also been shown for a large class of PL $4$-manifolds with boundary.

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arXiv2020

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 type of $3$-dimensional $Δ$ with relatively minimal $g_2$ whenever $Δ$ has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body. Complete combinatorial descriptions of $Δ$ with $g_2(Δ) \le 2$ are due to Kalai [12] $(g_2=0)$, Nevo and Novinsky [13] $(g_2=1)$ and Zheng [21] $(g_2=2).$ In all three cases $Δ$ is the boundary of a simplicial polytope. Zheng observed that for all $d \ge 0$ there are triangulations of $S^d \ast \mathbb{RP}^2$ with $g_2=3.$ She asked if this is the only nonspherical topology possible for $g_2(Δ)=3.$ As another application of relatively minimal $g_2$ we give an affirmative answer when $Δ$ is $3$-dimensional.

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arXiv2019

Regular genus and gem-complexity of some mapping tori

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 PL-manifolds. The bound for the regular genus is sharp for the mapping torus of some (PL) homeomorphisms $f:M \to M$, where $M$ is $\mathbb{RP}^2$, $\mathbb{RP}^2\#\mathbb{RP}^2$, $\mathbb{S}^1\times \mathbb{S}^1$, $\mathbb{RP}^3$, $\mathbb{S}^{2} \times \mathbb{S}^1$, $\mathbb{S}^{\hspace{.2mm}2} \mbox{$\times \hspace{-2.6mm}_{-}$} \, \mathbb{S}^{\hspace{.1mm}1}$ or $\mathbb{S}^d$. In particular, for $M=\mathbb{S}^{d-1} \times \mathbb{S}^1$ or $\mathbb{S}^{\hspace{.2mm}d-1} \mbox{$\times\hspace{-2.6mm}_{-}$} \, \mathbb{S}^{\hspace{.1mm}1}$, our construction gives a crystallization of a mapping torus of a (PL) homeomorphism $f:M \to M$ with regular genus $d^2-d$. As a consequence, we prove the existence of an orientable mapping torus of a (PL) homeomorphism $f:(\mathbb{S}^{2} \times \mathbb{S}^1)\to (\mathbb{S}^{2} \times \mathbb{S}^1)$ with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product $\mathbb{RP}^3 \times \mathbb{S}^1$ among closed connected prime orientable PL $4$-manifolds.

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arXiv2018

Genus-minimal crystallizations of PL 4-manifolds

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 genus of every orientable surface equals its genus, and the regular genus of every 3-manifold equals its Heegaard genus. For every closed connected PL $4$-manifold $M$, it is known that its regular genus $\mathcal G(M)$ is at least $2 χ(M) + 5m -4$, where $m$ is the rank of the fundamental group of $M$. In this article, we introduce the concept of "weak semi-simple crystallization" for every closed connected PL $4$-manifold $M$, and prove that $\mathcal G(M)= 2 χ(M) + 5m -4$ if and only if $M$ admits a weak semi-simple crystallization. We then show that the PL invariant regular genus is additive under the connected sum within the class of all PL 4-manifolds admitting a weak semi-simple crystallization. Also, we note that this property is related to the 4-dimensional Smooth Poincaré Conjecture.

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arXiv2017

3-regular colored graphs and classification of surfaces

Motivated by the theory of crystallizations, we consider an equivalence relation on the class of $3$-regular colored graphs and prove that up to this equivalence (a) there exists a unique contracted 3-regular colored graph if the number of vertices is $4m$ and (b) there are exactly two such graphs if the number of vertices is $4m+2$ for each $m\geq 1$. Using this, we present a simple proof of the classification of closed surfaces.

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arXiv2016

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

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 $4$-manifold $M$, its gem-complexity $\mathit{k}(M)$ and its regular genus $ \mathcal G(M)$ satisfy: $$\mathit{k}(M) \ \geq \ 3 χ(M) + 10m -6 \ \ \ \text{and} \ \ \ \mathcal G(M) \ \geq \ 2 χ(M) + 5m -4,$$ where $rk(π_1(M))=m.$ These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of {\it semi-simple crystallizations} is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of "standard type", involved in existing crystallization catalogues, and their connected sums.

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arXiv2015

An algorithmic approach to construct crystallizations of $3$-manifolds from presentations of fundamental groups

We have defined weight of the pair $(\langle S \mid R \rangle, R)$ for a given presentation $\langle S \mid R \rangle$ of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of $(\langle S \mid R \rangle, R)$ is $n$ then our algorithm constructs all the $n$-vertex crystallizations which yield $(\langle S \mid R \rangle, R)$. As an application, we have constructed some new crystallizations of 3-manifolds. We have generalized our algorithm for presentations with three generators and certain class of relations. For $m\geq 3$ and $m \geq n \geq k \geq 2$, our generalized algorithm gives a $2(2m+2n+2k-6+δ_n^2 + δ_k^2)$-vertex crystallization of the closed connected orientable $3$-manifold $M\langle m,n,k \rangle$ having fundamental group $\langle x_1,x_2,x_3 \mid x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$. These crystallizations are minimal and unique with respect to the given presentations. If `$n=2$' or `$k\geq 3$ and $m \geq 4$' then our crystallization of $M\langle m,n,k \rangle$ is vertex-minimal for all the known cases.

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arXiv2015

Simple crystallizations of 4-manifolds

Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of "standard" type, that is, all connected sums of $\mathbb{CP}^2$, $S^2 \times S^2$, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of $\mathbb{CP}^2$ is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of $\mathbb{CP}^2$ by the minimum of four edge contractions.

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arXiv2014

Equilibrium and equivariant triangulations of some small covers with minimum number of vertices

Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal $\mathbb{Z}_2^2$-equivariant triangulations of $2$-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{RP}^3 \# \mathbb{RP}^3$, $S^1 \times \mathbb{RP}^2$ and a nontrivial $S^1$ bundle over $\mathbb{RP}^2$. We construct some nice equilibrium triangulations of the real projective space $\mathbb{RP}^n$ with $2^n +n+1$ vertices. The main tool is the theory of small covers.

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arXiv2013

Minimal crystallizations of 3-manifolds

We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold $M$ is at least the weight of $π(M, \ast)$. This lower bound is sharp for the 3-manifolds $\mathbb{R P}^3$, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$, $S^2 \times S^1$, $S^2 \mbox{$\times \hspace{-2.8mm}_{-}$} S^1$ and $S^3/Q_8$, where $Q_8$ is the quaternion group. Moreover, there is a unique such facet minimal pseudotriangulation in each of these seven cases. We have also constructed contracted pseudotriangulations of $L(kq-1,q)$ with $4(q+k-1)$ facets for $q \geq 3$, $k \geq 2$ and $L(kq+1,q)$ with $4(q+k)$ facets for $q\geq 4$, $k\geq 1$. By a recent result of Swartz, our pseudotriangulations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even. In 1979, Gagliardi found presentations of the fundamental group of a manifold $M$ in terms of a contracted pseudotriangulation of $M$. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold $M$, we construct a contracted pseudotriangulation of $M$. So, our construction of a contracted pseudotriangulation of a 3-manifold $M$ is based on a presentation of the fundamental group of $M$ and it is computer-free.

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