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\title{Connectivity of Pseudomanifolds}


\author[ ]{Alexandru Papiu}

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\vspace{-1em}


\date{}

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\section*{Preliminaries and History}\label{preliminaries-and-history}

Given a polytope or a triangulated manifold it is natural to try to
understand the structure of its 1-dimensional skeleton. This was first
done by Steinitz in 1922 where he solved the problem in the
3-dimensional case: the graphs of 3-polytopes are exactly the
3-connected planar graphs. Balinksi extended these results to any
dimension by proving that the graphs of
\textbackslash{}(d\textbackslash{})-polytopes are
\textbackslash{}(d\textbackslash{})-connected. This was generalized
further by Barnette who showed that the graph of every d-dimensional
polyhedral pseudomanifold is (d + 1)-connected.\\

More recently these results have been sharpened in cases where more is
known about the structure of the simplical complex: Athanasiadis
\cite{athana} proved better connectivity bounds for
flag pseudomanifolds and Bjorner and Vorwerk
\cite{bjorner} and Adiprasito, Goodarzi and Verbaro
\cite{adiprasito} exteneted the results for banner
complexes.\\

Here we present a straightforward approach rooted in combinatorial
topology that extends and simplifies previous approaches. Here is the
basic idea: Let \textbackslash{}(\textbackslash{}Delta\textbackslash{})
be a triangulation of a (pseudo)manifold from which we remove a subset
of vertices \textbackslash{}(W\textbackslash{}) together with the
induced subcomplex
\textbackslash{}(\textbackslash{}Delta\_\{W\}\textbackslash{}). We will
analyze how the number of connected components of the 1-skeleton
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) on
\textbackslash{}(V\textbackslash{}setminus W\textbackslash{}) vertices
relates to
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{}).
In some cases one completely determines the other, in other cases the
way
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{})
''sits'' inside
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta)\textbackslash{})
matters. For pseudomanifolds we obtain that if
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{})
is trivial removing \textbackslash{}(W\textbackslash{}) does not
disconnect the graph and use this to prove lower bounds on connectivity
of different classes of complexes.\\

?maybe more background on simplicial complexes here?\\

Let \textbackslash{}(\textbackslash{}Delta\textbackslash{}) be a
simplicial complex. The underlying graph (or 1-skeleton)
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) of
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) is the graph
obtained by restricting
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) to faces of
cardinality at most two. A graph \textbackslash{}(G\textbackslash{}) is
\textbf{k-connected} if it has at least
\textbackslash{}(k\textbackslash{})-vertices and removing any
\textbackslash{}(k-1\textbackslash{}) vertices does not disconnect
\textbackslash{}(G\textbackslash{}). All simplicial complexes we
consider will be pure. We will work with homology over
\textbackslash{}(\textbackslash{}mathbb\{Z\}\_\{2\}\textbackslash{}).\\

A \textbackslash{}(d\textbackslash{})-dimensional simplicial complex
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) is a
\textbf{weak pseudomanifold} if every
\textbackslash{}(d-1\textbackslash{}) face is contained in exactly two
\textbackslash{}(d\textbackslash{}) faces. If in addition the link of
each face is connected we call
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) a \textbf{normal
pseudomanifold}. A \textbf{pseudomanifold} is a weak pseudomanifold in
which the facet graph is connected. It's not hard to show that every
normal pseudomanifold is indeed a pseudomanifold.\\

? maybe more background on pseudomanifolds - they're minimal cycles ?

\section*{Main Connectivity Bound}\label{main-connectivity-bound}

We will introduce the notion of strong connected components and use it
to prove the main theorem on the connectivity of manifolds. Let
\textbackslash{}(W\textbackslash{}) be a subset of the vertex set of
\textbackslash{}(\textbackslash{}Delta\textbackslash{}). We define the
following relation on the facets of
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) not contained in
\textbackslash{}(W\textbackslash{}):
\textbackslash{}(F\_\{1\}\textbackslash{}sim F\_\{n\}\textbackslash{})
if there is a sequence of facets
\textbackslash{}(F\_\{1\},F\_\{2\},\ldots{},F\_\{n\}\textbackslash{})
such that \textbackslash{}(F\_\{i\}\textbackslash{}cap
F\_\{i+1\}\textbackslash{}) has co-dimension 1 and is not in
\textbackslash{}(W\textbackslash{}). It's easy to see that this is an
equivalence relation and we will call the equivalence classes thus
obtained \textbf{strong components of
\textbackslash{}(\textbackslash{}Delta/W\textbackslash{})}. We will
denote the number of such classes by
\textbackslash{}(S(\textbackslash{}Delta/W)\textbackslash{}).

\textbf{.}

\emph{Let \textbackslash{}(\textbackslash{}Delta\textbackslash{}) be a
d-pseudomanifold and \textbackslash{}(W\textbackslash{}) a subset of
vertices. If
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta\_\{W\})=0\textbackslash{})
then removing \textbackslash{}(W\textbackslash{}) does not disconnect
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}).}

\emph{Proof.}

We will in fact prove the stronger statement:

\begin{equation}
dimH_{0}(\Delta_{V-W})\leq S(\Delta/W)\leq dimH_{d-1}(\Delta_{W})+1\\
 \end{equation}

Let \textbackslash{}(K\textbackslash{}) be a strong component in
\textbackslash{}(\textbackslash{}Delta/W\textbackslash{}). Let
\textbackslash{}(v\_\{1\},v\_\{2\}\textbackslash{}) be two vertices
contained in \textbackslash{}(v\_\{i\}\textbackslash{}in
F\_\{i\}\textbackslash{}) for \textbackslash{}(i=1,2\textbackslash{}).
We can then build a path between
\textbackslash{}(v\_\{1\},v\_\{2\}\textbackslash{}) in
\textbackslash{}(\textbackslash{}Delta\_\{V-W\}\textbackslash{}) by
following the sequence of facets connecting
\textbackslash{}(F\_\{1\}\textbackslash{}) and
\textbackslash{}(F\_\{2\}\textbackslash{}) and at each step choosing a
point in \textbackslash{}(F\_\{i\}\textbackslash{}cap
F\_\{i+1\}\textbackslash{}) which is not in W. This proves the first
inequality.

For the second inequality let
\textbackslash{}(K\_\{1\},K\_\{2\},\ldots{},K\_\{n\}\textbackslash{}) be
the strong components of
\textbackslash{}(\textbackslash{}Delta/W\textbackslash{}). These are
elements in
\textbackslash{}(C\_\{d\}(\textbackslash{}Delta)\textbackslash{}) and
\textbackslash{}(\textbackslash{}partial(K\_\{i\})\textbackslash{}in
C\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{}) since
otherwise one could extend the strong component
\textbackslash{}(K\_\{i\}\textbackslash{}) over a
\textbackslash{}(d-2\textbackslash{}) face not in contained in
\textbackslash{}(W\textbackslash{}). Since
\textbackslash{}(\textbackslash{}partial(\textbackslash{}partial(K\_\{i\})=0\textbackslash{})
we can think of
\textbackslash{}({[}\textbackslash{}partial(K\_\{i\}){]}\textbackslash{})
as elements in
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{}).
Now assume some of the strong components satisfy a linear relation in
\textbackslash{}(H\_\{d\_\{1\}\}(\textbackslash{}Delta\_\{W\})\textbackslash{})
so their sum will be equal to
\textbackslash{}(\textbackslash{}partial(\textbackslash{}sigma)\textbackslash{})
with \textbackslash{}(\textbackslash{}sigma\textbackslash{}) in
\textbackslash{}(C\_\{d\}(\textbackslash{}Delta)\textbackslash{}). This
implies that

\begin{equation}
\partial(\sum K_{i}+\sigma)=0\nonumber \\
 \end{equation}

However since \textbackslash{}(\textbackslash{}Delta\textbackslash{}) is
a pseudomanifold, the top homology of
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) must be
supported on the entire complex and this can only happen if all the
\textbackslash{}(K\_\{i\}\textbackslash{}) are in the sum. Thus we get
that any \textbackslash{}(n-1\textbackslash{}) strong components are
linearly independent so
\textbackslash{}(S(\textbackslash{}Delta/W)=n\textbackslash{}leq
dimH\_\{d-1\}(\textbackslash{}Delta\_\{W\})+1\textbackslash{}) [?]

\section*{Stronger Results for Normal
Pseudomanifolds}\label{stronger-results-for-normal-pseudomanifolds}

One would expect better estimates on the connectivity of
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) if we
restrict ourselves to spaces without singularities, say triangulated
manifolds. In fact a much more lax condition is necessary, namely
requiring that the links be connected, to force the first inequality in
(1) to become an equality.\\

\textbf{.}

\emph{If \textbackslash{}(\textbackslash{}Delta\textbackslash{}) is a
normal pseudomanifold then
\textbackslash{}(dim(H\_\{d\}(\textbackslash{}Delta,\textbackslash{}Delta\_\{W\}))=dimH\_\{0\}(\textbackslash{}Delta\_\{V-W\})=S(\textbackslash{}Delta/W)\textbackslash{})\\
}

\emph{Proof.}

For the first inequality: Let \textbackslash{}(v,w\textbackslash{}) be
in the same connected component of
\textbackslash{}(G/W\textbackslash{}). We now need to show is that any
two facets \textbackslash{}(F\textbackslash{}) and
\textbackslash{}(G\textbackslash{}) with the first containing
\textbackslash{}(v\textbackslash{}) and the second
\textbackslash{}(w\textbackslash{}) are in the same strong component of
\textbackslash{}(\textbackslash{}Delta/W\textbackslash{}). The way we
will build the facet sequence is by following the path between
\textbackslash{}(v\textbackslash{}) and
\textbackslash{}(w\textbackslash{}) and taking advantage of the facet
connectivity of the links. Say
\textbackslash{}(v\_\{i\},v\_\{i+1\}\textbackslash{}) are two
consecutive vertices in the path from
\textbackslash{}(v\textbackslash{}) to
\textbackslash{}(w\textbackslash{}) and
\textbackslash{}(F\_\{i\},F\_\{i+1\}\textbackslash{}) facets with
\textbackslash{}(v\_\{i\}\textbackslash{}in F\_\{i\}\textbackslash{})
and \textbackslash{}(v\_\{i+1\}\textbackslash{}in
F\_\{i+1\}\textbackslash{}) Now the link of
\textbackslash{}(v\_\{i\}\textbackslash{}) is easily checked to also be
a normal peseudomanifold and thus facet connected. Let
\textbackslash{}(F\_\{i,i+1\}\textbackslash{}) be a face in the link of
\textbackslash{}(\textbackslash{}\{v\_\{i\},v\_\{i+1\}\textbackslash{}\}\textbackslash{}).
We can find a sequence of facets in
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) going from
\textbackslash{}(F\_\{i\}\textbackslash{}) to
\textbackslash{}(F\_\{i,1+1\}\textbackslash{}) using the facet
connectivity of lk\textbackslash{}(v\_\{i\}\textbackslash{}).
Furthermore any two such facets have
\textbackslash{}(v\_\{i\}\textbackslash{}) in common so this is a strong
sequence in \textbackslash{}(\textbackslash{}Delta/W\textbackslash{}).
Analogously we can find a sequence from
\textbackslash{}(F\_\{i,1+1\}\textbackslash{}) to
\textbackslash{}(F\_\{i\}\textbackslash{}) so
\textbackslash{}(F\_\{i\}\textbackslash{}) and
\textbackslash{}(F\_\{i+1\}\textbackslash{})are in the same strong
component of \textbackslash{}(\textbackslash{}Delta/W\textbackslash{})
Following the path between \textbackslash{}(v\textbackslash{}) to
\textbackslash{}(w\textbackslash{}) and applying the procedure above one
gets that \textbackslash{}(F\textbackslash{}) and
\textbackslash{}(G\textbackslash{}) are in the same strong component of
\textbackslash{}(\textbackslash{}Delta/W\textbackslash{}).\\

For the second inequality: Let
\textbackslash{}(\textbackslash{}sigma\textbackslash{}) be a relative
cycle in
ker\textbackslash{}(\textbackslash{}partial\_\{d\}\textbackslash{}),
\textbackslash{}(F\textbackslash{}) be a facet of
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) contained in
\textbackslash{}(\textbackslash{}sigma\textbackslash{}) and
\textbackslash{}(K\_\{F\}\textbackslash{}) the corresponding strong
component. Asume there is a facet
\textbackslash{}(F\^{}\{\textbackslash{}prime\}\textbackslash{}) in
\textbackslash{}(K\_\{F\}\textbackslash{}) that is not in
\textbackslash{}(\textbackslash{}sigma\textbackslash{}). There will thus
exist \textbackslash{}(F\_\{i\},F\_\{i+1\}\textbackslash{}) in the facet
sequence connecting \textbackslash{}(F\textbackslash{}) and
\textbackslash{}(F\^{}\{\textbackslash{}prime\}\textbackslash{}) in such
that \textbackslash{}(F\_\{i\}\textbackslash{}cap
F\_\{i+1\}=G\textbackslash{}not\textbackslash{}subset W\textbackslash{})
with
\textbackslash{}(F\_\{i\}\textbackslash{}in\textbackslash{}sigma\textbackslash{})
and
\textbackslash{}(F\_\{i+1\}\textbackslash{}not\textbackslash{}in\textbackslash{}sigma\textbackslash{})
.But since \textbackslash{}(G\textbackslash{}) is contained in exactly
two facets we get that \textbackslash{}(G\textbackslash{}) is in
\textbackslash{}(\textbackslash{}partial\_\{d\}(\textbackslash{}sigma)\textbackslash{}subset
W\textbackslash{}) - a contradiction. So every relative cycle is a sum
of the strong compoenents. It follows that the strong components of
\textbackslash{}(\textbackslash{}Delta/W\textbackslash{}) are a basis
for the kernel and the result follows.\\

[?]

\textbf{.}

\emph{Let \textbackslash{}(\textbackslash{}Delta\textbackslash{}) be a
normal pseudomanifold and let \textbackslash{}(i\textbackslash{}) be the
inclusion map
\textbackslash{}(i:H\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{}rightarrow
H\_\{d-1\}(\textbackslash{}Delta)\textbackslash{}). Then
\textbackslash{}(dimH\_\{0\}(\textbackslash{}Delta\_\{V-W\})\textbackslash{})=
dim (ker \textbackslash{}(i\textbackslash{}))-1.}

\emph{Proof.}

We will be using the fact that
\textbackslash{}(H\_\{d\}(\textbackslash{}Delta,\textbackslash{}Delta\_\{W\})\textbackslash{})
fits into the exact long sequence

\( \)

By using the exact sequence above we get
\textbackslash{}(S(\textbackslash{}Delta/W)=dim(H\_\{d\}(\textbackslash{}Delta,\textbackslash{}Delta\_\{W\})=\textbackslash{})dim(ker
\textbackslash{}(i\textbackslash{}))\textbackslash{}(-1\textbackslash{})
and combined with Lemma 3.1 the result follows. [?]

\textbf{Remarks:}

For normal pseudomanifolds we get that
\textbackslash{}(H\_\{0\}(\textbackslash{}Delta\_\{V-W\})\textbackslash{}cong
H\_\{d\}(\textbackslash{}Delta,\textbackslash{}Delta\_\{W\})\textbackslash{})
which can be intepreted as a weak form of Poinacere-Lefschetz Duality
for normal pseudomanifolds.\\

\textbf{Corollary} : If
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) is a normal
d-pseudomanifold with
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta)=0\textbackslash{})
then \textbackslash{}(v(\textbackslash{}Delta/W)=\textbackslash{})
dim\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta\_\{W\})-1\textbackslash{})
and
\textbackslash{}(k(G(\textbackslash{}Delta))=\textbackslash{})min\textbackslash{}(\textbackslash{}\{\textbar{}W\textbar{}:H\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{}neq
0\textbackslash{}\}\textbackslash{})

\section*{Applications}\label{applications}

Let \textbackslash{}(C(i,d)\textbackslash{}) the class of simplicial
complexes with
\textbackslash{}(H\_\{d\}(\textbackslash{}Delta)\textbackslash{}neq
0\textbackslash{}) and no missing faces of dimension greater than
\textbackslash{}(i\textbackslash{}). These complexes were introduced in
\cite{nevo} by Nevo For given
\textbackslash{}(i,d\textbackslash{}) there exist unique integers
\textbackslash{}(0\textbackslash{}leq r\textbackslash{}) and
\textbackslash{}(1\textbackslash{}leq r\textbackslash{}leq
i\textbackslash{}) such that \textbackslash{}(d+1=qi+r\textbackslash{}).
Define

\( \)

where
\textbackslash{}(\textbackslash{}partial\textbackslash{}sigma\^{}\{i\}\textbackslash{}),
the boundary of the \textbackslash{}(i\textbackslash{})-simplex, appears
\textbackslash{}(q\textbackslash{})-times in the join. We have the
following lemma.\\

\textbf{.}

\emph{(Nevo \cite{nevo}) If
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) is in
\textbackslash{}(C(i,d)\textbackslash{}) then
\textbackslash{}(f\_\{0\}(\textbackslash{}Delta)\textbackslash{}geq
f\_\{0\}(C(i,d))\textbackslash{})= q(i+1)+(r+1) = d +1 + q + 1\\
}

\textbf{.}

\emph{Let \textbackslash{}(\textbackslash{}Delta\textbackslash{}) be a
\textbackslash{}(d\textbackslash{}) - pseudomanifold with no missing
faces of dimension higher than i then
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) is
\textbackslash{}(d+q-1\textbackslash{}) - connected.\\
}

\emph{Proof.}

Assume removing the subset of vertices
\textbackslash{}(W\textbackslash{}) disconnects
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}). By Theorem 1
we get that
\textbackslash{}(H\_\{d-1\}(\textbackslash{}Delta\_\{W\})\textbackslash{}neq
0\textbackslash{}) and since the complex
\textbackslash{}(\textbackslash{}Delta\_\{W\}\textbackslash{}) is
induced it cannot have any missing faces of dimension higher than
\textbackslash{}(i\textbackslash{}). It follows that
\textbackslash{}(\textbackslash{}Delta\_\{W\}\textbackslash{}in
C(i,d-1)\textbackslash{}) so
\textbackslash{}(f\_\{0\}(\textbackslash{}Delta\_\{W\})\textbackslash{}geq
f\_\{0\}(S(i,d-1)\textbackslash{}geq d+q-1\textbackslash{}).

[?]

Note that the two previous results of Barnette (in the simplicial case)
and Athanasiadis follow from Theorem 4.2:\\

\textbf{Corollary} : Let
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) be a
pseudomanifold then
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) is
\textbackslash{}(d+1\textbackslash{}) connected. Furthermore if
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) is a flag
complex then \textbackslash{}(G(\textbackslash{}Delta)\textbackslash{})
is \textbackslash{}(2d\textbackslash{}) connected.\\

\textbf{Remark} : Results similar to Theorem 2 have been published in
\cite{adiprasito} using methods from algebraic topology
similar to ours but also commutative algebra. The results there are for
a class of complexes called banner complexes which are a generalization
of flag complexes. We will show however that the connectivity bounds we
get are tighter.

We illustrate this by an example: Let
\textbackslash{}(\textbackslash{}Delta=C\_\{3\}*C\_\{4\}\textbackslash{})
be the join of 3-cycle and a 4-cycle. This is a 3-sphere.
\textbackslash{}(\textbackslash{}Delta\textbackslash{}) is not flag
since it contains an empty triangle, however it is easily seen that
\textbackslash{}(\textbackslash{}Delta\textbackslash{}in
C(2,3)\textbackslash{}) since no empty tetrahedra are introduced by the
join construction. Thus by Theorem 4.2 we get that
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) is
\textbackslash{}(5\textbackslash{})-connected and notice that this is
the best bound possible - it is easily checked that the connectivity of
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) is 5. However
\textbackslash{}(b(\textbackslash{}Delta)\textbackslash{}) as defined in
\cite{adiprasito} page 3 will be 2. This is because the
link of any vertex in the 4-cycle
\textbackslash{}(C\_\{4\}\textbackslash{}) is the boundary of a
triangular bipyramid which is not banner. Thus by Theorem 12 in
\cite{adiprasito} we only get that
\textbackslash{}(G(\textbackslash{}Delta)\textbackslash{}) is 4 -
connected which is true but weaker than our result.

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