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\title{\bf On stacked triangulated manifolds}


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\author{Basudeb Datta\thanks{Supported by DIICCSRTE, Australia (project AISRF06660) \& DST, India (DST/INT/AUS/P-56/2013 (G)) under
the Australia-India Strategic Research Fund and by the UGC Centre for Advanced Studies (F.510/6/CAS/2011 (SAP-I)).}\\
\small Department of Mathematics\\[-0.8ex]
\small Indian Institute of Science\\[-0.8ex]
\small Bangalore 560\,012, India\\
\small\tt dattab@iisc.ac.in\\
\and
Satoshi Murai\thanks{Supported by JSPS KAKENHI 26400043.}
\\
\small Department of Pure and Applied Mathematics\\[-0.8ex]
\small Graduate School of Information Science and Technology, Osaka University\\[-0.8ex]
\small Toyonaka, Osaka, 560-0043, Japan\\
\small\tt s-murai@ist.osaka-u.ac.jp
}




% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Jun 1, 2016}{Sep 28, 2017}{Oct 6, 2017}\\
\small Mathematics Subject Classifications: 57Q15, 57R20, 05C40}

\begin{document}

\maketitle



\begin{abstract}
We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $\Delta$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $\Delta$ is a stacked triangulation of a manifold. These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Stacked manifolds; Triangulations of 3-manifolds; Tight triangulations
\end{abstract}

\section{Introduction}

Stacked triangulations of spheres are of fundamental interest, in particular in the study of convex polytopes and triangulations of spheres. Recently, the notion of stackedness was extended to triangulations of manifolds in \cite{MN}. In this paper, we prove two results on stacked triangulations of manifolds.

A triangulation of a connected closed $d$-manifold is said to be {\em stacked} if it is the boundary of a triangulation of a $(d+1)$-manifold, and all of its interior faces have dimension $\geq d$. A triangulation of a closed $d$-manifold is said to be {\em locally stacked} if each vertex link is a stacked triangulation of a sphere. Kalai \cite{Ka} proved that, for $d \geq 4$, every locally stacked triangulation of a connected closed $d$-manifold can be obtained from the boundary of a $(d+1)$-simplex by certain combinatorial operations. This
result does not hold for $3$-manifolds since there are triangulations of $3$-manifolds which are locally stacked but cannot be obtained by these operations (see e.g.\ \cite[Example 6.2]{BDS1}). On the other hand, since the notions of stackedness and local stackedness are equivalent in dimension $\geq 4$ \cite{BDkst,MN}, Kalai's result also
characterizes stacked triangulations of connected closed manifolds of dimension $\geq 4$. We give a similar
characterization for stacked triangulations of manifolds with boundary (Theorem
\ref{theo:main.3.12}). As a consequence, we generalize the result of Kalai to stacked triangulations of closed
manifolds of dimension $\geq 2$ (Corollary \ref{cor:main.3.13}). This result and a recent result of Bagchi
\cite{Bag} solve a question posed  by Novik and Swartz \cite[Problem 5.3]{NS}.


Our second result is about an equivalence of tightness and tight-neighborliness.
For a  field $\FF$, a  simplicial complex $\Delta$ is said to be {\em $\FF$-tight} if it is
connected and, for any induced subcomplex $\Gamma$ of $\Delta$, the natural
map on reduced homology groups $\widetilde H_i(\Gamma;\FF) \to \widetilde
H_i(\Delta;\FF)$ induced by the inclusion map is injective for all $i\geq 0$.
We simply say that a simplicial complex is {\em tight} if it is $\FF$-tight for some field $\FF$.
See \cite{Ku,KL} for background and motivations of
tightness.
On the other hand,
an $n$-vertex triangulation $\Delta$ of a closed manifold of dimension $d \geq 3$ is said to be {\em
tight-neighborly} if $\binom{n -d -1}{2} = \binom{d+ 2}{2} \beta_1(\Delta; \FF)$, where $\beta_i(\Delta; \FF)$ is the $i$th Betti number computed over a field $\FF$. This condition is known to
be equivalent to saying that $\Delta$ is stacked and neighborly, in particular, is independent of the choice of $\FF$ (cf.\ Section 5). Tight-neighborliness was introduced by Lutz, Sulanke and Swartz. They conjectured that
tight-neighborly triangulations are $(\ZZ/2 \ZZ)$-tight \cite[Conjecture 13]{LSS}. The conjecture was solved by
Effenberger \cite[Corollary 4.4]{Ef} in dimension $\geq 4$ and by Burton, Datta, Singh and Spreer \cite[Corollary
1.3]{BDSS} in dimension $3$.
Moreover, it was recently proved in \cite{BDS2} that, for triangulations of $3$-manifolds, tightness is equivalent to tight-neighborliness.
On the other hand, Effenberger \cite[Question 4.5]{Ef} asked if $(\ZZ/2 \ZZ)$-tight triangulations of connected sums of $S^{d-1}$-bundles over $S^1$ are tight-neighborly when $d \geq 4$.


We answer Effenberger's question affirmatively.
More generally, we prove that, in dimension $d \geq 4$, every tight, closed, orientable, $\FF$-homology $d$-manifold with $\beta_i(\Delta;\FF)=0$ for $1 < i <d-1$, is tight-neighborly (Corollary \ref{cor4.4}). This result and the results mentioned above say
that, for triangulations of connected sums of $S^{d-1}$-bundles over $S^1$ with $d \geq 3$, tightness is
equivalent to tight-neighborliness. Also, since tight-neighborly triangulations are vertex minimal
triangulations, the result solves a special case of a conjecture of K\"uhnel and Lutz \cite[Conjecture 1.3]{KL}
which states that every tight combinatorial triangulation is vertex minimal.



This paper is organized as follows. In Section 2, we give a few basic definitions.
In Section 3, we define an analogue of a combinatorial handle addition for homology manifolds with boundary and study its basic properties. In Section 4, we present a
combinatorial characterization of stacked triangulated manifolds with and without boundary. In Section 5, we study the stackedness of tight triangulations.



\section{Preliminaries}









Recall that a {\em simplicial complex} is a collection of finite sets (sets of {\em vertices}) such
that every subset of an element is also an element. All simplicial complexes here are finite and non-empty.  For
$i \geq 0$, the elements of size $i+1$ are called the {\em $i$-faces} (or {\em faces of
dimension $i$}) of the complex. The empty set $\emptyset$ is a face (of dimension $-1$) of every simplicial
complex. For a simplicial complex $\Delta$, let $f_i(\Delta)$ be the number of $i$-faces of $\Delta$. The maximum
$k$ such that $\Delta$ has a $k$-simplex is called the {\em dimension} of $\Delta$ and is denoted by
$\dim(\Delta)$. A maximal face (under inclusion) in $\Delta$ is called a {\em facet} of $\Delta$. If $\sigma$ is
a face of $\Delta$ then the {\em link} of $\sigma$ in $\Delta$ is the subcomplex
\begin{align}
\lk_\Delta(\sigma) & =\{ \tau \setminus \sigma : \sigma \subseteq \tau \in \Delta\}. \nonumber
\end{align}
For any subset $W$ of the vertex set of $\Delta$, we write $\Delta[W] = \{\alpha \in \Delta: \alpha \subseteq W\}$ for the subcomplex of $\Delta$ \textit{induced} by $W$.
A simplicial complex is said to be {\em neighborly} if each pair of vertices forms an edge.


We say that a simplicial complex $\Delta$ is a {\em triangulation} of a manifold $M$ if its geometric carrier $|\Delta|$ is homeomorphic to $M$. We are mainly interested in triangulations of manifolds, but we actually consider slightly more general objects called homology manifolds. Let $\widetilde H_i(\Delta; \FF)$ be the $i$th reduced homology group of a topological space (or a simplicial complex) $\Delta$ with coefficients in a field $\FF$. The number $\beta_i(\Delta; \FF):= \dim_\FF \widetilde H_i(\Delta; \FF)$ is called
the {\em $i$th Betti number} of $\Delta$ with respect to $\FF$. For a field $\FF$, a simplicial complex $S$ of dimension $d$ is said to be an {\em $\FF$-homology $d$-sphere} if,
for each face $\sigma$ of dimension $i \geq -1$, $\lk_S(\sigma)$ has the same $\FF$-homology groups as the $(d-i-1)$-sphere. A simplicial complex $B$ of dimension $d$ is said to be an {\em $\FF$-homology $d$-ball} if (i) $B$ has trivial reduced $\FF$-homology groups, (ii) for each face $\sigma$ of dimension $i \leq d-1$, the reduced $\FF$-homology groups of $\lk_B(\sigma)$ are trivial or the same as those of the ($d-i-1)$-sphere and (iii) the
boundary
\begin{align}
\label{boundary}
\partial B =\{\sigma \in B \, : -1<\dim(\sigma) <d \mbox{ and } \widetilde H_{d-\dim(\sigma)- 1}(\lk_B(\sigma); \FF)=0 \}\cup \{\emptyset\}
\end{align}
is an $\FF$-homology $(d-1)$-sphere. A simplicial complex is said to be an {\em $\FF$-homology $d$-manifold} if each vertex link is either an $\FF$-homology $(d-1)$-sphere or an $\FF$-homology $(d-1)$-ball. Note that a triangulation of a $d$-manifold is an $\FF$-homology $d$-manifold for every field $\FF$. In the rest of the paper, we fix a field $\FF$ and by a {\em homology manifold/ball/sphere} we shall mean an $\FF$-homology manifold/ball/sphere.



We define the {\em boundary} $\partial \Delta$ of a homology $d$-manifold $\Delta$ in the same way as in \eqref{boundary}. If $\partial \Delta=\{\emptyset\}$, then $\Delta$ is called a {\em closed homology $d$-manifold} (or a {\em homology $d$-manifold without boundary}), otherwise $\Delta$ is called a {\em homology $d$-manifold with boundary}. If $\Delta$ is a homology $d$-manifold with boundary, then $\partial \Delta$ becomes a closed homology
$(d-1)$-manifold. We say that a connected, closed, $\FF$-homology $d$-manifold $\Delta$ is {\em $\FF$-orientable} (or simply, {\em orientable}) if $\widetilde H_d(\Delta;\FF)$ $\cong\FF$. Similarly, a connected homology $d$-manifold $\Delta$ with boundary is \textit{orientable} if $H_d(\Delta,\partial \Delta;\FF) \cong \FF$. We note the following easy fact.

\begin{lemma} \label{lemma:2.1}
Let $\Delta$ be an orientable homology $d$-manifold with boundary. If $\Delta$ has trivial reduced homology groups then $\Delta$ is a homology $d$-ball.
\end{lemma}

\begin{proof}
It is clear that $\Delta$ satisfies conditions (i) and (ii) of homology balls. The fact that $\partial \Delta$ is
a homology $(d-1)$-sphere follows from the long exact sequence of the pair $(\Delta,\partial \Delta)$ and the
Poincar\'{e}--Lefschetz duality \cite[Theorem 6.2.19]{Sp}.
\end{proof}


We also recall combinatorial manifolds.
A \textit{combinatorial $d$-ball} (resp.\ \textit{$d$-sphere}) is a simplicial complex which is PL homeomorphic to the $d$-simplex (resp.\ the boundary of the $(d+1)$-simplex).
A simplicial complex is said to be a \textit{combinatorial $d$-manifold} if each vertex link is either a combinatorial $(d-1)$-ball or a combinatorial $(d-1)$-sphere. Any combinatorial
manifold is a triangulation of an actual PL manifold.


 An $\FF$-homology $d$-manifold with non-empty boundary is said to be {\em stacked} if all its interior faces have dimension $\geq d-1$. An $\FF$-homology $d$-manifold  is said to be {\em stacked} if it is the boundary of a stacked triangulation of an $\FF$-homology $(d+1)$-manifold. An $\FF$-homology $d$-manifold is said to be {\em locally stacked} if each vertex link is
stacked.
Clearly, a stacked homology manifold is locally stacked. Since any stacked homology
ball (resp., sphere) is a combinatorial ball (resp., sphere), it follows that every (locally) stacked homology manifold is a combinatorial manifold.
Thus we simply call them (\textit{locally$)$ stacked manifolds} (\textit{with or without boundary}).



Next, we recall Walkup's class $\mathcal H^d$. Let $\Delta$ be a connected, closed, homology manifold and let
$\sigma$ and $\tau$ be facets of $\Delta$. We say that a bijection $\psi: \sigma \to \tau$ is {\em admissible} if
$\lk_\Delta(v) \cap \lk_\Delta(\psi(v))=\{\emptyset\}$ for each vertex $v \in \sigma$. Note that, for the
existence of an admissible bijection $\psi:\sigma \to \tau$, $\sigma$ and $\tau$ must be disjoint. For an
admissible bijection $\psi : \sigma \to \tau$, let $\Delta^\psi$ be the simplicial complex obtained from
$\Delta\setminus\{ \sigma, \tau\}$ by identifying $v$ and $\psi(v)$ for all $v \in \sigma$. The simplicial
complex $\Delta^\psi$ is said to be obtained from $\Delta$ by a {\em combinatorial handle addition}.



\begin{definition}[Walkup's class $\mathcal H^d$] \label{def:Walkup}
Let $d \geq 3$ be an integer. We recursively define the class $\mathcal H^d(k)$ as follows.
\begin{enumerate}
\item[{\rm (a)}] $\mathcal H^d(0)$ is the set of stacked triangulations of the $(d-1)$-sphere.
\item[{\rm (b)}] A simplicial complex $\Delta$ is in $\mathcal H^d(k +1)$ if it is obtained from a member of $\mathcal
H^d(k)$ by a combinatorial handle addition.
\end{enumerate}
Walkup's class $\mathcal H^d$ is the union $\mathcal H^d=\bigcup_{k \geq 0} \mathcal H^d(k)$.
\end{definition}

Kalai \cite[Corollary 8.4]{Ka} proved that every connected, locally stacked, closed,  $d$-manifold
is a member of Walkup's class $\mathcal{H}^{d+1}$ when $d \geq 4$, and as a consequence it follows that
$\mathcal{H}^{d+1}$ is the
set of all (locally) stacked, connected, closed, $d$-manifolds for $d\geq 4$. Although Kalai's
result is not true for $d=3$ (see e.g.\ \cite[Example 6.2]{BDS1}), we prove that $\mathcal{H}^{d+1}$ is the set
of all connected, stacked, closed, $d$-manifolds  for $d\geq 2$.



\section{Simplicial handle addition}

In this section, we define an analogue of combinatorial handle additions for homology manifolds with boundary.


All homology groups are with coefficients in an arbitrary field $\FF$, which is fixed throughout,
and $\widetilde{H}_i(\Delta;\FF)$ and $\beta_i(\Delta;\FF)$ will be denoted by $\widetilde{H}_i(\Delta)$ and $\beta_i(\Delta)$, respectively.
Let $\Delta$
be a homology $d$-manifold with boundary on the vertex set $V$ and let $\sigma$ and $\tau$ be facets of $\partial
\Delta$. We say that a bijection $\psi: \sigma \to \tau$ is {\em admissible} if, for every vertex $v \in \sigma$,
$\lk_\Delta(v) \cap \lk_\Delta(\psi(v))=\{\emptyset\}$.
For an admissible bijection $\psi: \sigma \to \tau$, let $\Delta^{\overline \psi}$ be the simplicial complex obtained from $\Delta$ by identifying $v$ and $\psi(v)$
for all $v \in \sigma$.
(The main difference between $\Delta^\psi$ and $\Delta^{\overline \psi}$ is that we do not remove $\sigma$ and $\tau$ for the definition of $\Delta^{\overline \psi}$.)
Thus, if we define  $\psi_+$ to be the map  from $V$ to $V \setminus \sigma$ by $\psi_+(v)=\psi(v)$ if $v \in \sigma$ and $\psi_+(v)=v$ otherwise, then we can consider $\Delta^{\overline \psi}$ on the vertex set $V \setminus \sigma$ as
\begin{align}
\Delta^{\overline \psi} & =\{ \psi_+(\alpha): \alpha \in \Delta\}. \nonumber
\end{align}
If $\Delta$ is connected, then we say that $\Delta^{\overline \psi}$ is obtained from
$\Delta$ by a {\em simplicial handle addition}. If $\Delta$ has two connected components $\Delta_1$ and
$\Delta_2$ and if $\sigma \in \Delta_1$ and $\tau \in \Delta_2$, then we write $\Delta^{\overline \psi}=\Delta_1
\cup_{\psi} \Delta_2$ and call it a {\em simplicial connected union} of $\Delta_1$ and $\Delta_2$. Below we give
some basic properties of $\Delta^{\overline \psi}$.
For $\sigma \in \Delta$, we write $\overline{\sigma}$ for the simplicial complex having a single facet $\sigma$.


\begin{lemma} \label{lemma:3.1}
Let $\Delta$ and $\Gamma$ be two homology $d$-balls. If $\Delta \cap \Gamma = \partial\Delta\cap \partial\Gamma =
\overline{\alpha}$, where $\alpha$ is a $(d-1)$-simplex,  then $\Delta \cup \Gamma$ is a homology $d$-ball.
\end{lemma}

\begin{proof}
We use induction on $d$. The statement is obvious when $d=1$. Suppose $d >1$. Since $\Delta \cap \Gamma=
\overline \alpha$, the exactness of the Mayer--Vietoris sequence implies that $\Delta \cup \Gamma$ has trivial
reduced homology groups. Let $v$ be a vertex of $\Delta \cup \Gamma$. If $v \not\in \alpha$ then $\lk_{\Delta \cup
\Gamma}(v)$ is equal to either $\lk_{\Delta}(v)$ or $\lk_{\Gamma}(v)$ and hence a homology $(d-1)$-sphere or
$(d-1)$-ball. If $v\in \alpha$ then $v\in \partial \Delta \cap \partial \Gamma$ and hence $\lk_{\Delta}(v)$ and
$\lk_{\Gamma}(v)$ are homology $(d-1)$-balls and $\lk_{\Delta}(v) \cap \lk_{\Gamma}(v)= \overline{\alpha\setminus
\{v\}}$. Since $\lk_{\Delta \cup \Gamma}(v)= \lk_{\Delta}(v) \cup \lk_{\Gamma}(v)$, $\lk_{\Delta \cup \Gamma}(v)$
is a homology $(d-1)$-ball by the induction hypothesis. The lemma now follows from Lemma $\ref{lemma:2.1}$.
\end{proof}

It follows from Lemma \ref{lemma:3.1} that the simplicial connected union of two homology $d$-balls is a homology $d$-ball. Note that the same property holds for combinatorial manifolds.


\begin{lemma} \label{lemma:3.2}
For $d \geq 2$, let $\Delta$ be a $($not necessary connected$)$ homology $d$-manifold with boundary.
Let $\sigma$ and $\tau$ be two facets of $\partial \Delta$. If $\psi:\sigma \to \tau$ is  an admissible bijection then
\begin{enumerate}
\item[{\rm (i)}] $\Delta^{\overline \psi}$ is a homology $d$-manifold with boundary,

\item[{\rm (ii)}] $(\beta_0(\Delta^{\overline \psi}), \beta_1(\Delta^{\overline \psi})) =(\beta_0(\Delta), \beta_1(\Delta)+1)$ or $(\beta_0(\Delta)-1, \beta_1(\Delta))$ and

\item[{\rm (iii)}] $\Delta^{\overline \psi}$ is stacked if and only if $\Delta$ is stacked.
\end{enumerate}
\end{lemma}


\begin{proof}
(i) For every $\alpha \in \Delta^{\overline \psi}$ with $\alpha \not \subseteq \tau$, there is a unique face
$\gamma \in \Delta$ such that $\alpha=\psi_+(\gamma)$ and $\lk_{\Delta^{\overline \psi}}(\alpha)$ is
combinatorially isomorphic to $\lk_{\Delta}(\gamma)$. Thus, to prove the statement, it is enough to show that,
for every $\alpha \subseteq \tau$, $\lk_{\Delta^{\overline \psi}}(\alpha)$ is either a homology $(d-
\dim(\alpha)-1)$-sphere or $(d-\dim(\alpha)-1)$-ball. It is clear that $|\lk_{\Delta^{\overline \psi}}(\tau)|
\cong S^0$. For a proper face $\alpha$ of $\tau$, a straightforward computation implies
\begin{align}
\lk_{\Delta^{\overline \psi}}(\alpha) & = \lk_\Delta(\alpha) \cup_{\psi'} \lk_\Delta(\psi^{-1}(\alpha)), \nonumber
\end{align}
where $\psi': \psi^{-1}(\tau \setminus \alpha ) \to \tau\setminus \alpha$ is the restriction of $\psi$ to
$\psi^{-1}(\tau \setminus \alpha)$. By Lemma \ref{lemma:3.1}, $\lk_{\Delta^{\overline \psi}}(\alpha)$ is a
homology $(d-\dim(\alpha)-1)$-ball.


(ii) It is clear that $\beta_0(\Delta^{\overline \psi})= \beta_0(\Delta)-1$ if $\sigma$ and $\tau$ belong to
different connected components and $\beta_0(\Delta^{\overline \psi})= \beta_0(\Delta)$ if $\sigma$ and $\tau$ are
in the same connected component. Observe that $\widetilde H_i(|\Delta^{\overline \psi}|) \cong \widetilde
H_i(|\Delta^{\overline \psi}|,|\tau|) \cong \widetilde H_i(|\Delta|,|\sigma| \cup |\tau|)$ for all $i$. Then the
desired statement follows from the following exact sequence of pairs
\begin{eqnarray*}
\begin{array}{lllllllllllll}
 0=  \widetilde H_1(|\sigma| \cup |\tau|) &
\longrightarrow & \widetilde H_1(|\Delta|) & \longrightarrow &
\widetilde H_1(|\Delta|,|\sigma| \cup |\tau|) &  &\\
\longrightarrow  \widetilde H_0(|\sigma| \cup |\tau|) &
\longrightarrow & \widetilde H_0(|\Delta|) & \longrightarrow &
\widetilde H_0(|\Delta|,|\sigma| \cup |\tau|) & \longrightarrow & 0.
\end{array}
\end{eqnarray*}

(iii) This statement follows from the proof of (i) since it says that the interior faces of $\Delta^{\overline
\psi}$ are $\tau$ and $\psi_+(\alpha)$ for all interior faces $\alpha$ of $\Delta$.
\end{proof}

The proof of Lemma \ref{lemma:3.2}\,(i) also says that if $\Delta$ is connected then $\partial(\Delta^{\overline
\psi})=(\partial \Delta)^\psi$. Also $|\partial(\Delta_1 \cup_\psi \Delta_2)|$ is a connected sum of $|\partial
\Delta_1|$ and $|\partial \Delta_2|$.


We can consider the inverse of the construction of $\Delta^{\overline{\psi}}$, which we call a simplicial handle deletion. Indeed, we prove the following statement.


\begin{theorem} \label{theo:3.7}
Let $\Delta$ be a connected homology $d$-manifold with boundary.
If $\Delta$ has an interior $(d-1)$-face $\sigma$ with $\partial \overline{\sigma} \subseteq \partial \Delta$,
then there exists a homology $d$-manifold with boundary $\widetilde \Delta$
such that $\Delta$ is obtained from $\widetilde \Delta$ by a simplicial handle addition or by taking a simplicial connected union.
%$\widetilde \Delta^\sigma$
% and an admissible bijection $\psi$ between $(d-1)$-faces of $\partial (\widetilde \Delta)$ such that
%$\Delta=(\widetilde \Delta)^{\overline \psi}$.
\end{theorem}

We say that $\widetilde \Delta$ in the above theorem can be obtained from $\Delta$ by a \textit{simplicial handle deletion} over $\sigma$.
Intuitively, the simplicial complex $\widetilde \Delta$ can be considered as the simplicial complex obtained from $\Delta$ by cutting it along an interior face $\sigma$ of codimension one. On the other hand, one needs to be careful with this intuitive understanding since it is not clear for us if the above theorem holds in the class of triangulated (or combinatorial) manifolds.
For example, we are not sure if the converse of Lemma \ref{lemma:3.1} holds for combinatorial balls, that is, if $\Delta$ and $\Gamma$ are combinatorial balls when $\Delta \cup \Gamma$ is a combinatorial $d$-ball and $\Delta  \cap \Gamma$ is a $(d-1)$-simplex.


In the rest of the section, we will prove Theorem \ref{theo:3.7} by giving
an explicit construction of $\widetilde \Delta$.
To do this, we need the following two technical lemmas.
Recall that, for any homology $d$-manifold $\Delta$ and its interior $(d-1)$-face $\sigma$, $\lk_\Delta(\sigma)$ consist of two vertices since it has the same homology groups as $S^0$.


\begin{lemma} \label{lemma:3.3}
Let $B$ be a homology $d$-ball with vertex set $V$, $\sigma$ an interior $(d-1)$-face of $B$ with $\partial
\overline \sigma \subseteq \partial B$.
Then,
$B[{V\setminus \sigma}]$ contains exactly two connected components, say $C_1$ and $C_2$.
Moreover, If $\lk_B(\sigma)=\{x,y\}$, then one of $x$ and $y$ is in $C_1$ and the other is in $C_2$.
% Then the following statements hold.
%\begin{enumerate}[{\rm (i)}]
%\item $B[{V\setminus \sigma}]$ contains exactly two connected components, say $C_1$ and $C_2$.
%\item $B=B_1 \cup B_2$ and $B_1 \cap B_2 = \overline \sigma$, where $B_i := B[V(C_i) \cup \sigma]$ for $i=1,2$.
% \item If $\lk_B(\sigma)=\{x,y\}$, then one of $x$ and $y$ is in $B_1$ and the other is in $B_2$.
%\end{enumerate}
%Moreover, if $\lk_B(\sigma)=\{x,y\}$, then $x$ and $y$ are different components in $B[{V\setminus \sigma}]$.
\end{lemma}

\begin{proof}
Let $v \ast \partial B=\partial B \cup \{ \{v\} \cup \alpha : \alpha \in \partial B\}$ be the cone over $\partial
B$, where $v$ is a new vertex. It is easy to see that $S= B \cup (v \ast \partial B)$ is a homology $d$-sphere. Then
\begin{align}
\widetilde H_0(S[{V\setminus \sigma}]) &
\cong
\widetilde H_{d-1}(S[{\sigma \cup \{v\}}])
\cong
\widetilde H_{d-1}(S[{\sigma \cup \{v\}}], (v \ast \partial B)[{\sigma \cup \{v\}}])), \nonumber
\end{align}
where the first isomorphism follows from the Alexander duality \cite[Theorem 6.2.17]{Sp} and the second
isomorphism follows from the long exact sequence of pairs since $\widetilde H_i((v\ast \partial B)[{\sigma \cup
\{v\}}])=0$ for all $i$. Since $B[{V\setminus \sigma}]= S[{V\setminus \sigma}]$ and since
\begin{align}
\widetilde H_{d-1}(S[{\sigma \cup \{v\}}], (v \ast \partial B)[{\sigma \cup \{v\}}])) &
\cong
\widetilde H_{d-1}(B[\sigma], (\partial B)[\sigma]))
=
\widetilde H_{d-1}(\overline \sigma, \partial \overline \sigma) \cong \FF, \nonumber
\end{align}
$B[{V\setminus\sigma}]$ has exactly two connected components. This proves the first statement.


Let $B_i=B[V(C_i)\cup \sigma]$ for $i=1,2$,
where $V(C_i)$ is the vertex set of $C_i$. Clearly $B_1 \cap B_2 =\overline \sigma$. Also, any facet of $B$ belongs either $B_1$ or $B_2$.
Indeed, if $\alpha$ is a facet of $B$, then $\alpha \setminus \sigma \in B[{V\setminus\sigma}]$ is contained
either in $C_1$ or in $C_2$, which implies that $\alpha$ is either in $B_1$ or in $B_2$.
Let $u \in V(C_1)$ and $w \in V(C_2)$,
and let $\alpha$ and $\gamma$ be facets of $B$
with $v \in \alpha$ and $w \in \gamma$.
Then $\alpha\in B_1$ and $\gamma \in B_2$.
Since $B$ is a homology $d$-ball, it is a $d$-dimensional
pseudomanifold and hence there is a sequence $\alpha=\alpha_0,\alpha_1,\dots, \alpha_k =\gamma$ of facets such that
$\alpha_{i-1} \cap \alpha_i$ has dimension $d-1$ for $1\leq i\leq k$
(see \cite[p.\ 150 and 278]{Sp}).
Let $j$ be a
number such that $\alpha_{j-1} \in B_1$ and $\alpha_j \in B_2$. Then $\alpha_{j-1} \cap \alpha_{j}$ must be
$\sigma$. Since $\{x \} \cup \sigma$ and $\{y \} \cup \sigma$ are the only facets containing $\sigma$, they must
be $\alpha_{j-1}$ and $\alpha_j$. This proves the second statement.
\end{proof}

%\begin{remark}
%In a previous version of this article, we showed that $B_1$ and $B_2$ defined in Lemma \ref{lemma:3.3} are homology balls. Since we are not using this fact, we have not included this in this version.
%\end{remark}

We say that $B_i=B[V(C_i)\cup \sigma]$ in Lemma \ref{lemma:3.3} is the {\em $x$-component} (resp.\ {\em $y$-component}) of $B$ with
respect to $\sigma$ if it contains $x$ (resp.\ $y$).
We later see in the proof of Theorem \ref{theo:3.7}
that $B_1$ and $B_2$ are homology balls.


Let $\Delta$ be a homology $d$-manifold with boundary. Suppose that $\Delta$ has an interior $(d-1)$-face
$\sigma=\{z_1,\dots,z_d\}$ with $\partial \overline \sigma \subseteq \partial \Delta$. Let $\{x \} \cup \sigma$
and $\{y\} \cup \sigma$ be the facets of $\Delta$ containing $\sigma$. Consider
\begin{align}
R & =\{\alpha \in \Delta: \alpha \cap \sigma \ne \emptyset,\ \alpha \not \subseteq \sigma\}. \nonumber
\end{align}
Observe that, for each $\tau \in \partial \overline{\sigma}$, $\lk_\Delta(\tau)$ is a homology ball satisfying the assumption
of Lemma \ref{lemma:3.3} in the sense that $\sigma \setminus \tau$ is an interior face of $\lk_{\Delta}(\tau)$
with $\partial(\overline{\sigma \setminus \tau}) \subseteq \partial (\lk_\Delta(\tau))$.
Consider the collection of simplices (which is not a subcomplex of $\Delta$)
\begin{align*}
R_x(k) & =\{ \alpha  \cup \{z_k\} : \alpha \in R \mbox{ and $\alpha$ is in the $x$-component of }
\lk_\Delta(z_k) \mbox{ w.r.t.\ } \sigma \setminus \{z_k\}\}.
\end{align*}
Also, define $R_y(k)$ similarly.
Let
\begin{align*}
X & =\bigcup_{k=1}^d R_x(k) \mbox{ and } Y=\bigcup_{k=1}^d R_y(k).
\end{align*}
Note that $R=X \cup Y$.

\begin{lemma} \label{lemma:3.5}
If $\sigma=\{z_1,\dots,z_d\}$, $R_x(k)$, $R_y(k)$, $X$ and $Y$ are as above, then $X \cap Y=\emptyset$.
Also, $\overline {\sigma \setminus \{z_k\}} \cup \{\alpha \setminus\{z_k\}: z_k \in \alpha \in X\}$ is the $x$-component of $\lk_\Delta(z_k)$ and $\overline {\sigma \setminus \{z_k\}} \cup  \{\alpha \setminus\{z_k\}: z_k \in \alpha \in Y\}$ is the $y$-component of $\lk_\Delta(z_k)$ for $1\leq k\leq d$.
%Also, $\{\alpha \in X : z_k \in \alpha\}=R_x(k)$ and $\{\alpha \in Y: z_k \in \alpha\}=R_y(k)$ for $1\leq k\leq d$.
\end{lemma}

\begin{proof}
To prove the first statement, we must prove that $R_x(k) \cap R_y(\ell) = \emptyset$ for all $k \ne \ell$.
Suppose to the contrary that $\alpha \in R_s(k) \cap R_y(\ell)$ for some $k \neq \ell$. Then $\alpha\setminus
\{z_k, z_{\ell}\}$ is in the $x$-component and the $y$-component of $\lk_\Delta(\{z_k,z_\ell\})$ with respect to
$\sigma \setminus \{z_k,z_\ell\}$ and hence $\alpha \subseteq \sigma$, a contradiction since $\alpha \in R$.

We prove the second statement only for the $x$-components (the proof for $y$-components is similar).
Since
$\overline {\sigma \setminus \{z_k\}} \cup \{\alpha \setminus\{z_k\}: \alpha \in R_x(k)\}$ is the $x$-component of $\lk_\Delta(z_k)$,
to prove the statement,
it is enough to show $\{\alpha \in X : z_k \in \alpha\}=R_x(k)$.
%and $\{\alpha \in Y : z_k \in \alpha\}=R_y(k)$.
Let $\alpha \in X$ with $z_k \in \alpha$.
Then $\alpha \in R_x(\ell)$ for some $\ell$. If $\ell=k$ then $\alpha
\in R_x(k)$. Otherwise, $\alpha\setminus \{z_k, z_{\ell}\}$ and $x$ are in the same component of
$\lk_{\Delta}(\{z_k, z_\ell\})$. Since $\lk_{\Delta}(z_k) \supseteq \lk_{\Delta}(\{z_k,z_\ell\})$, we have $\alpha
\in R_x(k)$. These prove that $\{\alpha \in X : z_k \in \alpha\}=R_x(k)$.
\end{proof}


\begin{definition} \label{def:handel.deletion}
Let $\Delta$ be a homology $d$-manifold with boundary and let $\sigma=\{z_1,\dots,z_d\}$ be an interior $(d-
1)$-face of $\Delta$ with $\partial \overline \sigma \subseteq \partial \Delta$. Let $R$, $R_x(k)$, $R_y(k)$, $X$
and $Y$ be as above. Let $z_1^+,\dots,z_d^+$ be new vertices and $\sigma^+=\{z_1^+,\dots,z_d^+\}$. For $\alpha
=\alpha^{\,\prime} \cup\{z_{i_1},\dots,z_{i_\ell}\} \in X$ with $\alpha^{\,\prime}\cap \sigma=\emptyset$, define
$\alpha^+=\alpha^{\,\prime} \cup\{z_{i_1}^+,\dots,z_{i_\ell}^+\}$. Consider the simplicial complex
\begin{align}
\widetilde \Delta^\sigma & = \{\alpha \in \Delta: \alpha \not \in X\} \cup \{\alpha^+ : \alpha \in X\} \cup
\overline {\sigma^+}. \nonumber
\end{align}
\end{definition}

The above construction is a simplified version of the construction in \cite[Lemma 3.3]{BDlow}. Also, a similar construction for manifolds without boundary was considered by Walkup \cite{Wa}.
Note that if $\Delta$ is a homology ball, then $\widetilde \Delta^\sigma$ is the disjoint union of its $x$-component and $y$-component.
We now prove Theorem \ref{theo:3.7}.

\begin{proof}[Proof of Theorem \ref{theo:3.7}]
We claim that $\widetilde \Delta=\widetilde \Delta^\sigma$ satisfies the desired conditions by induction on the dimension $d$.
It is clear that we have $\Delta=( \widetilde
\Delta^\sigma)^{\overline \psi}$ if $\widetilde
\Delta^\sigma$ is a homology manifold and we define $\psi: \sigma^+ \to \sigma$ by $\sigma(z_i^+)=z_i$ for all $i$.
Thus it is enough to prove that $\widetilde \Delta^\sigma$ is a homology manifold.

The assertion is clear when $d=1$.
Suppose $d>1$.
We first prove that
the $x$-component and the $y$-component of $\lk_\Delta(z_k)$ are homology balls for all $1 \leq k \leq d$,
by applying the induction hypothesis to $\lk_\Delta(z_k)$.
Let $\Gamma=\lk_\Delta(z_k)$, and let  $B_1$ and $B_2$ be the $x$-component and the $y$-component of the homology $(d-1)$-ball $\Gamma=\lk_\Delta(z_k)$ with respect to $\sigma \setminus \{z_k\}$, respectively.
They are homology $(d-1)$-manifolds since their disjoint union $\widetilde \Gamma ^{\sigma \setminus \{z_k\}}$
is a homology $(d-1)$-manifold by the induction hypothesis.
Also, since $B_1 \cup B_2=\lk_\Delta(z_k)$ and $B_1 \cap B_2= \overline{\sigma\setminus \{z_k\}}$, the exactness of the Mayer-Vietoris sequence says that they have trivial homology groups.
Then, since $B_1$ and $B_2$ are orientable as they are subcomplexes of the orientable homology $(d-1)$-manifold $\lk_\Delta(z_k)$,
$B_1$ and $B_2$ are homology balls by Lemma \ref{lemma:2.1}.


We now prove that $\widetilde \Delta$ is a homology manifold.
What we must prove is that each vertex link of $\widetilde \Delta$ is either a homology $(d-1)$-sphere or a $(d-1)$-ball.
Suppose $v \not \in \sigma^+ \cup \sigma$. Define the map $\varphi: \Delta \to \widetilde \Delta$ by
$\varphi(\alpha)=\alpha$ if $\alpha \not \in X$ and $\varphi(\alpha)=\alpha^+$ if $\alpha \in X$. Then $\varphi$
gives a bijection between $\Delta \setminus\overline \sigma$ and $\widetilde \Delta \setminus (\overline
{\sigma^+} \cup \overline \sigma)$, in particular, it gives a bijection between $\{\alpha: v \in \alpha \in
\Delta\}$ and $\{\alpha : v \in \alpha \in \widetilde \Delta\}$. Thus $\lk_{\widetilde \Delta}(v)$ is
combinatorially isomorphic to $\lk_\Delta(v)$, which implies the desired property. Suppose $v =z_k^+$ for some
$k$. Then
\begin{align*}
\lk_{\widetilde \Delta}(v) & =\lk_{\widetilde \Delta}(z_k^+)= \overline{(\sigma \setminus \{z_k\})^+} \cup \{
(\alpha \setminus \{z_k\})^+ : z_k \in \alpha \in X\} \end{align*}
is combinatorially isomorphic to the $x$-component of $\lk_{\Delta}(z_k)$ by Lemma \ref{lemma:3.5}, and therefore is a
homology $(d-1)$-ball. Finally, suppose $v =z_k$ for some $k$. Since $X \cap Y = \emptyset$,
\begin{align*}
\lk_{\widetilde \Delta}(v)
&= \overline{\sigma \setminus \{v\}} \cup \{ \alpha \setminus \{v\}: v \in \alpha \in R_y(k) \setminus X\}\\
& = \overline{\sigma \setminus \{z_k\}} \cup \{ \alpha \setminus \{z_k\}: \alpha
\in R_y(k)\} \nonumber
\end{align*}
is the $y$-component of $\lk_\Delta(v)$ w.r.t.\ $\sigma\setminus\{v\}$. Thus $\lk_{\widetilde{\Delta}}(v)$ is a homology $(d-1)$-ball.
%
%Finally, $\widetilde \Delta$ has a non-empty boundary since $\sigma \in \partial \widetilde{\Delta}$.
\end{proof}





\section{A characterization of stacked manifolds}

In this section, we present a characterization of stacked manifolds.
We first define an analogue of Walkup's class for manifolds with boundary.

\begin{definition} \label{def:Walkup.boundary}
Let $d \geq 2$ be an integer. We recursively define $ \overline{\mathcal H^d}(k)$ as follows.
\begin{enumerate}
\item[{\rm (a)}] $\overline{\mathcal H^d}(0)$ is the set of stacked triangulations of $d$-balls.

\item[{\rm (b)}] $\Delta$ is a member of $\overline{\mathcal H^d}(k+1)$ if it is obtained from a member of
$\overline{\mathcal H^d}(k)$ by a simplicial handle addition.
\end{enumerate}
Let $\overline{\mathcal H^d}=\bigcup_{k \geq 0} \overline{\mathcal H^d}(k)$.
\end{definition}

Note that every stacked triangulation of the $d$-ball is obtained from a $d$-simplex by taking a simplicial
connected union with a $d$-simplex repeatedly. See \cite[Lemma 2.1]{DS}. The classes  $\overline{\mathcal H^d}$ and
$\mathcal H^d$ have the following simple relation.

\begin{lemma}\label{lemma:3.9}
For all integers $d \geq 3$ and $k \geq 0$, one has $\mathcal H^d(k)= \{ \partial \Delta: \Delta \in
\overline{\mathcal H^d}(k)\}$.
\end{lemma}

\begin{proof}
The case when $k=0$ and the inclusion $\mathcal H^d(k) \supseteq \{ \partial \Delta: \Delta \in
\overline{\mathcal H^d}(k)\}$, for all $k\geq 0$, are obvious. For $k>0$, the converse inclusion follows by
induction on $k$. Indeed, if $\Gamma \in \mathcal H^{d}(k)$, then by induction we may assume that there is a
$\Delta \in \overline{\mathcal H^d}(k-1)$ such that $\Gamma= (\partial \Delta)^\psi$ for some admissible
bijection $\psi: \sigma \to \tau$ in $\partial \Delta$. Since, $\partial \Delta$ and $\Delta$ have the same
$1$-faces by Lemma \ref{lemma:3.2}(iii), the bijection $\psi$ is also admissible for $\Delta$, and $\Gamma=
\partial (\Delta^{\overline \psi}) \in \{ \partial \Delta: \Delta \in \overline{\mathcal H^d}(k)\}$.
\end{proof}


\begin{lemma} \label{lemma:3.10}
If $\Delta \in \overline{\mathcal H^d}(k)$ and $\Gamma \in \overline{\mathcal H^d}(\ell)$ then their simplicial
connected union belongs to $\overline{\mathcal H^d}(k+\ell)$.
\end{lemma}

\begin{proof}
We may assume $k\leq \ell$. We use induction on $k+\ell$. If $k +\ell=0$ then the assertion follows from Lemma
\ref{lemma:3.2}(iii). Suppose $k+\ell >0$. Then $\Gamma=\Sigma^{\overline \varphi}$ for some $\Sigma \in
\overline{\mathcal H^d}(\ell-1)$ and for some admissible bijection $\varphi$ between facets of $\partial \Sigma$.
Let $\psi$ be a bijection from a facet of $\partial \Delta$ to a facet of $\partial \Gamma$. Then $\Delta
\cup_\psi \Gamma$ is $(\Delta \cup_\psi \Sigma)^\varphi$ (by an appropriate identification of the vertices). By
induction hypothesis, we have $\Delta \cup_\psi \Sigma \in \overline{\mathcal H^d}(k+\ell-1)$ and hence $\Delta
\cup_\psi \Gamma \in \overline{\mathcal H^d}(k+\ell)$.
\end{proof}

\begin{remark}
A similar result for $\mathcal H^d$ was proved by Walkup \cite[Proposition 4.4]{Wa}.
\end{remark}

\begin{theorem} \label{theo:main.3.12}
For $d\geq 2$, let $\Delta$ be a connected homology $d$-manifold with boundary. Then $\Delta$ is
stacked if and only if $\Delta \in \overline{\mathcal H^d}$.
\end{theorem}

\begin{proof}
It is clear any member of $\overline{\mathcal H^d}$ is stacked.
We prove the `only if part'. The assertion  is obvious if $\Delta$ has one facet. Suppose that $\Delta$ has more than one facet. Then
$\Delta$ has an interior $(d-1)$-face $\sigma$. Since $\Delta$ is stacked, it has no interior faces of dimension
$\leq d-2$. Thus we have $\partial \overline \sigma \subseteq \partial \Delta$.
Then by Theorem \ref{theo:3.7}, $\Delta$ is a simplicial connected union of two connected stacked manifolds or
is obtained from a connected stacked manifold having a smaller first Betti number by a simplicial handle
addition.
Then the assertion follows from Lemma \ref{lemma:3.10} by induction on the number of interior $(d-1)$-faces.
\end{proof}


By Lemma \ref{lemma:3.9} and Theorem \ref{theo:main.3.12} we obtain the following.

\begin{corollary} \label{cor:main.3.13}
Let $\Delta$ be a connected, closed, homology manifold of dimension $d \geq 2$. Then $\Delta$ is stacked if and
only if $\Delta \in \mathcal H^{d+1}$.
\end{corollary}

Finally, we discuss a connection between Corollary \ref{cor:main.3.13} and a question posed by Novik and Swartz \cite{NS}. Novik and Swartz \cite[Theorem 5.2]{NS} gave the
following interesting characterization of members of Walkup's class $\mathcal{H}^{d+1}$ for $d\geq 4$.


\begin{proposition}[Novik--Swartz] \label{prop:NS2009}
Let $\Delta$ be a connected, closed, orientable, homology manifold of dimension $d \geq 3$. Then
\begin{align}
f_1(\Delta) -(d+1) f_0(\Delta) + \binom{d+2}{2} \geq \binom{d+2}{2} \beta_1(\Delta). \nonumber
\end{align}
Further, if $d\geq 4$ then $f_1(\Delta) -(d+1) f_0(\Delta) + \binom{d+2}{2} = \binom{d+2}{2} \beta_1(\Delta)$
if and only if $\Delta\in \mathcal{H}^{d+1}$.
\end{proposition}


It was asked by Novik and Swartz \cite[Problem 5.3]{NS} if the equality case of the last statement in Proposition
\ref{prop:NS2009} also holds in dimension $3$.
Corollary \ref{cor:main.3.13}  and the next result of Bagchi \cite[Theorem 1.14]{Bag}
answer this question.

\begin{proposition}[Bagchi] \label{prop:Ba2016}
Let $\Delta$ be a connected, closed, homology $3$-manifold.
Then $f_1(\Delta) - 4f_0(\Delta) + 10 = 10 \beta_1(\Delta)$ if and only if $\Delta$ is stacked.
\end{proposition}


\begin{corollary} \label{cor:NS2009.problem}
Let $\Delta$ be a connected, closed, homology $3$-manifold. The following conditions are equivalent.
\begin{enumerate}
\item[{\rm (i)}] $\Delta$ is a member of $\mathcal{H}^{4}$.

\item[{\rm (ii)}] $\Delta$ is stacked.

\item[{\rm (iii)}] $f_1(\Delta) -4 f_0(\Delta) + 10 = 10 \beta_1(\Delta)$.
\end{enumerate}
\end{corollary}



\begin{remark}
It is known that the topological type of a member of $\mathcal H^{d+1}$ is one of the following: (i) the
$d$-sphere $S^d$ (ii) connected sums of sphere product $S^{d-1} \times S^1$ (iii) connected sums of twisted
sphere product $S^{d-1} \mbox{$\times \hspace{-8.2pt} _-$} S^1$. See \cite[Section 3]{LSS}. Thus Corollary
\ref{cor:main.3.13} also gives a new restriction on the topological types of stacked manifolds in
dimensions 2 and 3.
\end{remark}


\begin{remark}
\label{nonorientable}
Recently, Proposition \ref{prop:NS2009} was extended to non-orientable homology manifolds (and even to normal pseudomanifolds) by the second author. See \cite[Theorem 5.3]{Mu}.
\end{remark}

\section{Tight triangulations and stackedness}


In this section, we study stackedness of tight triangulations.
For a simplicial complex $\Delta$ with vertex set $V$, a subset $\sigma \subseteq V$ of $k+1$
elements is called a \emph{missing $k$-face} of $\Delta$ if $\sigma \notin \Delta$ and all proper subsets of
$\sigma$ are faces of $\Delta$. If $\sigma$ is a missing $k$-face of $\Delta$, then we have $\widetilde
H_{k-1}(\Delta[\sigma]) \cong \FF$. The following lemma follows from the definition of tightness.

\begin{lemma} \label{lemma:4.1}
Let $\Delta$ be a tight simplicial complex on the vertex set $V$. Then
\begin{enumerate}
\item[{\rm (i)}] for all subsets $U \subseteq W$ of $V$, the natural map $\widetilde H_i(\Delta[U]) \to \widetilde
H_i(\Delta[W])$ induced by the inclusion is injective, and

\item[{\rm (ii)}] if $\beta_{k-1}(\Delta)=0$ then $\Delta$ has no missing $k$-faces.
\end{enumerate}
\end{lemma}


For a simplicial complex $\Delta$, we identify its {\em $1$-skeleton} ${\rm Skel}_1(\Delta) = \{\sigma \in \Delta
: \dim(\sigma) \leq 1\}$ with the simple graph whose vertex set is the set of the vertices of $\Delta$ and whose
edge set is the set of the edges (1-simplices) in $\Delta$. We say that a simple graph $G$ is {\em chordal} if it
has no induced cycle of length $\geq 4$. The following result is due to Kalai \cite[Theorem 8.5]{Ka}.


\begin{proposition}[Kalai] \label{prop:Ka1987}
Let $\Delta$ be a homology $(d-1)$-sphere with $d \geq 3$. Then $\Delta$ is stacked if and only if the
$1$-skeleton of $\Delta$ is chordal and $\Delta$ has no missing $k$-faces for $1 < k < d-1$.
\end{proposition}

Let $\Delta$ be a closed homology manifold of dimension $d \geq 3$.
Recall that $\Delta$ is tight-neighborly if $\binom{f_0(\Delta) -d -1}{2}= \binom{d+2}{2} \beta_1(\Delta;\FF)$.
Since $\binom{f_0}{2}
-(d+1)f_0 + \binom{d+2}{2}= \binom{f_0 -d -1}{2}$, $\Delta$ is tight-neighborly if and only if $\Delta$ is
stacked and neighborly by Proposition \ref{prop:NS2009} (see Remark \ref{nonorientable} for the non-orientable case). Here we prove the following.

\begin{theorem} \label{theo:main.4.3}
Let $\Delta$ be a tight, connected, closed, homology manifold of dimension $d \geq 4$ such that $\beta_i(\Delta)=0$
for $1 <i < d-1$. Then $\Delta$ is locally stacked.
\end{theorem}


\begin{proof}
Let $v$ be a vertex of $\Delta$. We prove that $\lk_\Delta(v)$ is stacked.

We first claim that no induced subcomplex of $\lk_\Delta(v)$ can be a $1$-dimensional simplicial complex which
forms a cycle. Suppose to the contrary that $\lk_\Delta(v)[W]$ is a cycle for some $W$. Let $C=\lk_\Delta(v)[W]$
and $v\ast C=C \cup \{ \{v\} \cup \sigma: \sigma \in C\}$. Then we have $\Delta[{W \cup \{v\}}]= \Delta[W] \cup
(v\ast C)$ and $\Delta[W] \cap (v\ast C)=C$. Consider the Mayer--Vietoris exact sequence
\begin{align}
\widetilde H_2(\Delta[{W \cup \{v\}}]) & \longrightarrow \widetilde H_1(C) \longrightarrow \widetilde
H_1(\Delta[W]) \oplus \widetilde H_1(v\ast C) \stackrel \varphi \longrightarrow \widetilde H_1(\Delta[{W \cup
\{v\}}]). \nonumber
\end{align}
Since $\Delta$ is tight and $\beta_2(\Delta)=0$, we have $\widetilde H_2(\Delta[{W\cup\{v\}}])=0$. Then since
$\widetilde H_1(C) \ne 0$, the map $\varphi$ has a non-trivial kernel. However, since $\widetilde H_1(v\ast C)=0$,
this contradicts the tightness of $\Delta$ as it implies that $\varphi$ is injective by Lemma \ref{lemma:4.1}(i).
Hence no induced subcomplex of $\lk_\Delta(v)$ can be a cycle.

Now we prove the statement. By Lemma \ref{lemma:4.1}(ii), $\Delta$ has no missing $k$-faces for $2<k<d$. This
implies that $\lk_\Delta(v)$ has no missing $k$-faces for $2 < k <d-1$. Also, $\lk_\Delta(v)$ has no missing
$2$-faces since if it has a missing $2$-face $\sigma$ then $\lk_\Delta(v)[\sigma]$ is a cycle of length $3$.
Similarly, the $1$-skeleton of $\lk_\Delta(v)$ is a chordal graph since if it has an induced cycle of length
$\geq 4$ with the vertex set $W$, then $\lk_\Delta(v)[W]$ is a cycle. Thus, by Proposition \ref{prop:Ka1987},
$\lk_\Delta(v)$ is stacked.
\end{proof}


For any field, a tight (connected) homology manifold is orientable and neighborly (cf.\ \cite{BDtight}). From Theorem \ref{theo:main.4.3}
and all the known results, we have the following.

\begin{corollary}
\label{cor4.4}
Let $\Delta$ be a closed, orientable, homology manifold of
dimension $d\geq 4$. Then the following are equivalent.
\begin{enumerate}
\item[{\rm (i)}] $\Delta$ is tight-neighborly.

\item[{\rm (ii)}] $\Delta$ is a neighborly member of $\mathcal{H}^{d+1}$.

\item[{\rm (iii)}] $\Delta$ is neighborly and stacked.

\item[{\rm (iv)}] $\Delta$ is neighborly and locally stacked.

\item[{\rm (v)}] $\Delta$ is tight and $\beta_i(\Delta)=0$ for $1< i< d-1$.
\end{enumerate}
\end{corollary}

\begin{proof}
The equivalence (i) $\Leftrightarrow$ (ii) follows from Proposition  \ref{prop:NS2009}, (ii) $\Leftrightarrow$
(iii) follows from Corollary \ref{cor:main.3.13}, and (ii) $\Leftrightarrow$ (iv) follows from Kalai's result
\cite[Corollary 8.4]{Ka}. Now, (v) $\Rightarrow$ (iv) follows from Theorem \ref{theo:main.4.3}. Since $\Delta \in
\mathcal{H}^{d+1}$ implies $\beta_i(\Delta)=0$ for $1 < i < d-1$, (ii) \& (iv) $\Rightarrow$ (v) follows from
\cite[Theorem 3.11]{BDtight}. This completes the proof.
\end{proof}

From the equivalence of (i) and (v) in Corollary \ref{cor4.4} it follows that tight triangulations of connected
sums of $S^{d-1}$-bundles over $S^1$ are tight-neighborly for $d \geq 4$. This answers a question asked by
Effenberger \cite[Question 4.5]{Ef}.


It would be natural to ask if the results in this section hold in dimension $3$.
Very recently, Bagchi, Spreer and the first author \cite{BDS2}
proved the following result (which answers a question asked in a previous version of this paper).

\begin{proposition}[Bagchi--Datta--Spreer] \label{prop:BDS2016}
A closed triangulated $3$-manifold $M$ is $\FF$-tight if and only if $M$ is $\FF$-orientable, neighborly and
stacked.
\end{proposition}

Since every closed triangulated manifold is $(\mathbb Z/2 \mathbb Z)$-orientable,
as a consequence of Proposition \ref{prop:BDS2016} and Corollary
\ref{cor:NS2009.problem} we get the following (compare Corollaries 1.4 and 1.5 of \cite{BDS2}).

\begin{corollary} \label{cor:BDS2016}
Let $\Delta$ be %an $\FF$-orientable,
a closed triangulated $3$-manifold. The following conditions are equivalent.
\begin{enumerate}
\item[{\rm (i)}] $\Delta$ is tight-neighborly, that is,
$(f_0(\Delta)-4)(f_0(\Delta)-5) = 20\beta_1(\Delta; \FF)$.
\item[{\rm (ii)}] $\Delta$ is a neighborly member of $\mathcal{H}^{4}$.
\item[{\rm (iii)}] $\Delta$ is neighborly and stacked.
%\item $\Delta$ is $\FF$-tight.
\item[{\rm (iv)}] $\Delta$ is tight.
\end{enumerate}
\end{corollary}



\medskip


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
 The authors thank the anonymous referees for some useful comments. They thank Bhaskar Bagchi for many useful comments on a preliminary version of this paper. They also thank Eran Nevo for letting them know that the first part of Lemma \ref{lemma:3.3} follows from the Alexander duality.


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