| Russian Mathematical Surveys |
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Brief communications A lower bound for triangulation complexity for compact 3-manifolds with boundary D. D. Nigomedyanovab, E. A. Fominykhabc a Saint Petersburg State University b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow c St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
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     The notion of complexity is an important organising principle in the study of 3-manifolds. Let $M$ be a connected compact 3-manifold with boundary. An ideal triangulation of $M$ is a realization of the interior of $M$ as a gluing of a finite number of copies of the standard tetrahedron $\Delta$ with all its vertices removed, induced by a simplicial face-pairing of the corresponding copies of $\Delta$. The triangulation complexity $c_{\Delta}(M)$ of $M$ is the minimum number of tetrahedra in any ideal triangulation of $M$. Upper complexity bounds arise usually from explicit constructions of triangulations, while finding lower bounds is a hard problem in general. Here we mention the lower bounds on the triangulation complexity of manifolds with boundary presented in [1]–[3]. These lower bounds allow one to determine the complexity of infinite families of 3-manifolds. Our contribution to this line of work is a new lower complexity bound obtained via $\mathbb{Z}_2$-homology. More precisely, we prove the following result. Theorem 1. Let $M$ be a connected compact $3$-manifold with boundary. Then $c_{\Delta}(M) \geqslant \beta_1(M,\mathbb{Z}_2)$. To prove Theorem 1 we use an equivalent approach to complexity via Matveev’s theory of special spines [4]. Let $\mathtt{d}(P)$ and $\mathtt{v}(P)$ denote the number of 2-components and the number of true vertices of the special polyhedron $P$, respectively. Since the singular graph of $P$ is 4-regular, it contains exactly $2\mathtt{v}(P)$ edges, and we see that $\chi(P) = \mathtt{d}(P) - \mathtt{v}(P)$. Let $\beta_k(X,\mathbb{Z}_2)$ denote the $k$th Betti number of the space $X$ with coefficients in $\mathbb{Z}_2$. Lemma. For every connected special polyhedron $P$ the following relations hold: (i) $\mathtt{d}(P) \geqslant \beta_2(P, \mathbb{Z}_2)+1$; (ii) $\mathtt{d}(P)-(\beta_2(P, \mathbb{Z}_2)+1)= \mathtt{v}(P)-\beta_1(P, \mathbb{Z}_2)$. Proof. The polyhedron $P$ has a natural cell complex structure, which consists of true vertices, edges, and 2-components. Consider the cellular chain complex of $P$:
$$
\begin{equation*}
0\to C_{2}\xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0\to 0
\end{equation*}
\notag
$$
with coefficients in $\mathbb{Z}_2$. We have We denote the edges and $2$-components of $P$ by $\gamma_1,\dots,\gamma_{2\mathtt{v}(P)}$ and $\alpha_1,\dots,\alpha_{\mathtt{d}(P)}$, respectively. It is an easy consequence of the definition of the special polyhedron that Hence $\dim(\operatorname{Im} \partial_2) \geqslant 1$, and statement (i) follows.
Statement (ii) is immediate since $P$ is connected, so that $\chi(P)=1-\beta_1(P, \mathbb{Z}_2)+ \beta_2(P, \mathbb{Z}_2)=\mathtt{d}(P)-\mathtt{v}(P)$. $\Box$ Proof of Theorem 1. Let $P$ be the special polyhedron dual to a minimal ideal triangulation of $M$ (see [4]). Then $c_{\Delta}(M)=\mathtt{v}(P)$. Because $\partial M \ne \varnothing$, the polyhedron $P$ is homotopy equivalent to $M$; thus, $\beta_1(P, \mathbb{Z}_2)=\beta_1(M, \mathbb{Z}_2)$. Then the lemma implies that $c_{\Delta}(M) \geqslant \beta_1(M, \mathbb{Z}_2)$. $\Box$
It was shown in [5] that the number of edges in any ideal triangulation of a compact 3-manifold $M$ with boundary is bounded below by the number of connected components of $\partial M$, denoted by $|\partial M|$. This provides the following lower complexity bound. Theorem 2. Let $M$ be a connected compact $3$-manifold with boundary. Then $c_{\Delta}(M) \geqslant |\partial M|-\chi(M)$. Proof. Let $P$ be the special polyhedron dual to a minimal ideal triangulation $\mathcal{T}$ of $M$. Then $\chi(M)=\chi(P)$, $c_{\Delta}(M)=\mathtt{v}(P)$, and $\mathtt{d}(P)$ equals the number of edges in $\mathcal{T}$. By [5], Lemma 2.1, we have $\mathtt{d}(P) \geqslant |\partial M|$. Hence $c_{\Delta}(M)=\mathtt{d}(P)-\chi(P) \geqslant |\partial M|-\chi(M)$. $\Box$
It turns out that the lower bound found in Theorem 1 is stronger than the one in Theorem 2. Theorem 3. Let $M$ be a compact $3$-manifold with boundary. Then $\beta_1(M, \mathbb{Z}_2) \geqslant |\partial M|-\chi(M)$. Proof. Consider the long exact sequence
$$
\begin{equation*}
\cdots\to H_1(M,\partial M;\mathbb{Z}_2) \xrightarrow{\varphi} H_0(\partial M;\mathbb{Z}_2) \xrightarrow{\psi}H_0(M; \mathbb{Z}_2) \to \cdots
\end{equation*}
\notag
$$
of relative homology groups with coefficients in $\mathbb{Z}_2$. The exactness of this sequence implies that Lefschetz duality provides a natural isomorphism $H_1(M, \partial M; \mathbb{Z}_2) \cong H^2(M; \mathbb{Z}_2)$. Since the group $H_2(M; \mathbb{Z}_2)$ is finitely generated, the vector spaces $H_2(M; \mathbb{Z}_2)$ and $H^2(M; \mathbb{Z}_2)$ are finite-dimensional and mutually dual. In particular, they have the same dimension. Hence $\beta_1(M,\partial M;\mathbb{Z}_2)=\beta_2(M;\mathbb{Z}_2)$. Summing up, we deduce that In upcoming papers we will describe the class $\mathcal{M}_h$ of connected compact 3- manifolds for which the lower bound in Theorem 1 is attained. We will show that $\mathcal{M}_h$ is infinite and prove that all manifolds in $\mathcal{M}_h$, except six manifolds of small complexity, are hyperbolic with totally geodesic boundary.
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\Bibitem{NigFom23} |
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