Let $\Delta$ be a spherical building.
There are order-preserving maps
\[ \Gamma: \mathrm{sd} \PD(\Delta) \to {\rm CB}(\Delta),\]
\[ \phi :\OPD(\Delta) \to \Delta * \Delta,\]
such that for every group $H$ acting on $\Delta$ by simplicial automorphisms, $\Gamma$ and $\phi$ are $H$-equivariant and induce homotopy equivalences between the fixed point subposets:
\[ \Gamma_H : \PD(\Delta)^H \to {\rm CB}(\Delta)^H,\]
\[ \phi_H :\OPD(\Delta)^H \to \Delta^H * \Delta^H.\]
In particular, $\Gamma$ and $\phi$ induce homotopy equivalences and equivariant isomorphisms in (co)homology.

\begin{definition}
\label{def:decompBuilding}
Let $\Delta$ be a spherical building.
We denote by $\D(\Delta)$ the poset of all non-empty Levi spheres, with order induced by reverse inclusion.
\end{definition}

\begin{proposition}
\label{prop:lowLeviIntervalPD}
Let $L\in \D(\GG,k)$.
Then $\PD(\GG,k)_{\prec L} \simeq \Delta(\GG,k)^L * \PD(L,k)$.
Moreover, in homology, we have an isomorphism of $L(k)$-modules
\begin{equation}
\label{eq:isoHomologyIntervalsPDLevi}
\widetilde{H}_*\big(\PD(\GG,k)_{\prec L},R \big) \cong \widetilde{H}_*\big(\, \Delta(\GG,k)^L *\PD(L,k), R \, \big) \cong \widetilde{H}_*\big(\,\PD(L,k), R \, \big),
\end{equation}
provided that $RL(k)$ is semisimple.
\end{proposition}

\begin{theorem}
\label{thm:homologyWedge}
Suppose that $G$ is a finite group, and $f:X\to Y$ is a $G$-equivariant order-preserving map between finite-dimensional $G$-posets.
Let $R$ be a ring such that $RG$ is semisimple.
Then, under the conditions of Theorem \ref{thm:wedgeDecomposition}, for all $m\geq 0$ we have an isomorphism of $RG$-modules:
\[ \widetilde{H}_{m}(X, R) \cong_G \widetilde{Y}_m(Y, R) \oplus \bigoplus_{\overline{y} \in Y/G} \bigoplus_{i+j=m-1} \Ind_{\Stab_G(y)}^G \big( \widetilde{H}_i(f^{-1}(Y_{\leq y})) \otimes \widetilde{H}_j(Y_{> y})   \big). \]
\end{theorem}

Here we are denoting by $\overline{y}$ the image of an element $y\in Y$ in the orbit poset $Y/G$.

\subsection{Buildings}
We work with spherical buildings in the sense of \cite{AB}, so we do not assume that our buildings are thick.
From now on, $\Delta$ will denote a (spherical) building.
By the Solomon-Tits theorem, $\Delta$ is spherical in the sense that it has the homotopy type of a wedge of spheres of dimension $\dim \Delta$ \cite[Theorem 4.73]{AB}.
On the other hand, for every simplex $\sigma\in \Delta$, its link $\Lk_{\Delta}(\sigma)$ is a building of dimension $\codim_{\Delta}(\sigma)$ \cite[Proposition 4.9]{AB}.
In particular, buildings are Cohen-Macaulay.

We write $\A(\Delta)$ for the complete system of apartments of $\Delta$.
From now on, when we speak of an apartment of $\Delta$ we mean an
element of $\A(\Delta)$.
If $\sigma_1,\ldots,\sigma_r\in \Delta$ are simplices, we denote by $\A(\Delta,\sigma_1,\ldots,\sigma_r)$ the set of apartments $\Sigma\in \A(\Delta)$ such that $\sigma_i\in \Sigma$ for all $i$.
Recall that, since we work with spherical buildings, apartments are finite Coxeter complexes.

\subsubsection{Opposition}
\label{sub:opposition}
Let $\Sigma$ be a (finite) Coxeter complex.
Two chambers of $\Sigma$ are called opposite if their distance coincides with the diameter of $\Sigma$.
It is well known that every chamber of $\Sigma$ has a unique opposite, giving rise to an involutory bijection $\cham \Sigma\to \cham \Sigma$ that extends uniquely to an involutory automorphism $\op_{\Sigma}:\Sigma\to\Sigma$ \cite[2.39]{Tits1}.
Thus, two simplices $\sigma,\sigma' \in \Sigma$ are opposite if $\op_{\Sigma}(\sigma) = \sigma'$.
By convention, the opposite of the empty simplex is the empty simplex.

Two simplices of $\Delta$ are called opposite if they are opposite in some apartment (and hence in every apartment that contains both of them).
It follows that two opposite chambers lie in a unique apartment (see \cite[Lemma 4.69]{AB} and \cite[3.25]{Tits1}).
Note that a simplex may have multiple opposites in $\Delta$.

There is a bijection between apartments containing two given opposite simplices and apartments in the link of one of these simplices:

\begin{lemma}
\label{lm:bijectionApartmentsLinkOpposite}
Let $\Delta$ be a spherical building, and let $\sigma,\sigma'$ be two opposite simplices.
Then we have a bijection:
\[ \Sigma\in \A(\Delta, \sigma,\sigma') \longmapsto \Lk_{\Sigma}(\sigma)\in \A(\Lk_{\Delta}(\sigma)).\]
The inverse of this map is given as follows.
If $\widetilde{\Sigma}\in \A(\Lk_{\Delta}(\sigma))$ and $c,c'\in \widetilde{\Sigma}$ are two opposite chambers there, then $\widetilde{\Sigma} = \Lk_{\Sigma}(\sigma)$ where $\Sigma$ is the convex hull in $\Delta$ of the opposite chambers $c$ and $\proj_{\sigma'}(c')$. Thus we map $\widetilde{\Sigma}$ to $\Sigma$.
\end{lemma}

\begin{proof}
This is part of Proposition 2.1 in \cite{vH}.
\end{proof}

Here, $\proj_{\sigma}(\tau)$ denotes the projection of a simplex $\tau$ to $\sigma$, in the sense of Tits \cite[2.30]{Tits1} (see next subsection for the definition).

\subsubsection{Convexity}
\label{sub:convexity}
Now we look at convex subcomplexes of $\Delta$, in the sense we defined above.
Our notion of convexity coincides with Tits' definition \cite[1.5]{Tits1}, and it is stronger than the notion given in \cite[Definition 4.120]{AB}.
See also \cite[Remark 4.122]{AB}.
Moreover, we will mostly work with convex subcomplexes of $\Delta$ that are contained in some apartment (from the complete system of apartments).

Suppose first that $\Sigma$ is a Coxeter complex.
A root of $\Sigma$ is the image of a folding \cite[1.8]{Tits1}.
If $\alpha$ is a root of $\Sigma$, we write $-\alpha$ for its opposite root, and $\partial \alpha =\alpha \cap (-\alpha)$ is (by definition) a wall (see discussion at the top of page 11 in \cite{Tits1}).
Note that $-\alpha = \op_{\Sigma}(\alpha)$.
Write $\roots(\Sigma)$ for the set of roots of $\Sigma$.
Recall that a panel is a codimension-one simplex.
Given a chamber $\sigma\in \Sigma$ and a panel $\tau\subseteq \sigma$, there exists a unique root $\alpha\in \roots(\Sigma)$ such that $\tau\in \partial \alpha$ and $\sigma\in \alpha$.
Every root (and hence every wall) is a convex subcomplex of $\Sigma$.
Indeed, by \cite[2.19]{Tits1}, convex subcomplexes of $\Sigma$ are exactly those obtained as intersections of roots.
If $K$ is a subcomplex of $\Sigma$, we write $\roots_{\Sigma}(K)$ for the set of roots $\alpha\in \roots(\Sigma)$ that contain $K$.
Thus, $K$ is convex if and only if
\[ K = \bigcap_{\alpha\in \roots_{\Sigma}(K)} \alpha.\]

The convex hull of two opposite simplices $\sigma,\sigma'\in \Sigma$ is termed a Levi sphere of the Coxeter complex $\Sigma$.
Indeed, Levi spheres are exactly the subcomplexes of $\Sigma$ obtained as intersections of walls.
This terminology is
borrowed from the work in \cite{Serre} of Serre, who defines a Levi sphere as an intersection of $|\Sigma|$, viewed as the classical Euclidean sphere, with sets of reflecting hyperplanes of the underlying Coxeter group.
This concept generalizes the notion of Levi subgroups in algebraic groups to the context of buildings, as we will explain later in Lemma \ref{lm:LeviSubgroupsAndLeviSpheres}.

Now we go back to $\Delta$.
Let $S$ be a set of simplices of $\Delta$.
If $S \subseteq \Sigma$ for some apartment $\Sigma$, then the convex hull of $S$ is the same whether it is taken in $\Sigma$ or in $\Delta$, that is, $\Conv_{\Sigma}(S) = \Conv_{\Delta}(S)$.
If $\sigma,\tau\in \Delta$ are two simplices, then they lie in some apartment $\Sigma$.
The projection of $\tau$ onto $\sigma$, denoted by $\proj_{\sigma}(\tau)$ is the unique maximal simplex containing $\sigma$ in the convex hull $\Conv_{\Delta}(\sigma,\tau)=\Conv_{\Sigma}(\sigma,\tau)$ (see \cite[2.30, 3.19]{Tits1}).

A Levi sphere of $\Delta$ is the convex hull of two opposite simplices, which is then a Levi sphere of any apartment containing these simplices.
If $\sigma,\sigma'$ are opposite chambers, then $\Conv_{\Delta}(\sigma,\sigma')$ is the unique apartment that contains them and hence a $\dim \Delta$-sphere.
If $\sigma,\sigma'$ are opposite vertices, then $\Conv_{\Delta}(\sigma,\sigma') = \{\sigma,\sigma'\}$ is a $0$-sphere.

More generally, Levi spheres are always spheres (see also Remarques 2
on the bottom of page 200 in \cite{Serre}).

\begin{lemma}
\label{lm:convexIsleviSphere}
Let $\Sigma$ be a finite Coxeter complex, and let $K\subseteq \Sigma$ be a convex subcomplex.
Then $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\sigma_1,\sigma_2$ with dimension $\dim \sigma_i = \dim K$.
In such a case, $K$ is the convex hull of $\sigma_1,\sigma_2$ and $|K|$ is a sphere of dimension $\dim K$.
\end{lemma}

\begin{proof}
Suppose that $K$ contains a pair of opposite simplices $\sigma_1,\sigma_2$ of dimension $\dim K$.
Then, in the terminology of \cite[Théorème 2.1]{Serre}, $K$ must be completely reducible. This means that every point of $K$ has an opposite.
But every point of $K$ has at most one opposite as $K$ lies in the sphere $\Sigma$, so $K$ must be exactly the Levi sphere spanned by $\sigma_1$ and $\sigma_2$.
In particular, $K$ is the convex hull of $\sigma_1,\sigma_2$, and it triangulates a sphere of dimension $\dim K$.

The converse of this statement is clear (see \cite{Serre}).
\end{proof}

A root (resp. a wall) of $\Delta$ is a root (resp. a wall) of some apartment.
We write $\roots(\Delta)$ for the set of roots of $\Delta$, so
\[ \roots(\Delta) = \bigcup_{\Sigma \in \A(\Delta)} \roots(\Sigma).\]
Similarly, if $K\subseteq \Delta$, $\roots_{\Delta}(K)$ denotes the set of roots $\alpha\in \roots(\Delta)$ that contain $K$.
If $K$ is a convex subcomplex of $\Delta$ that is contained in some apartment, then $K$ is a convex subcomplex of any apartment containing it.
In particular, if $K\subseteq \Sigma\in \A(\Delta)$, then
$K = \bigcap_{\alpha \in \roots_{\Sigma}(K)} \alpha$, and more generally,
\[ K  =\bigcap_{\alpha\in \roots_{\Delta}(K)} \alpha.\]

\subsubsection{The CAT(1) metric}
\label{subsub:cat1metric}
The geometric realization of a spherical building $\Delta$ admits a canonical metric $\dcat$ that makes it a complete CAT(1) space.
See \cite[II.10 Theorem 10A.4]{BH} and \cite[Example 12.39]{AB}.
With this metric, an apartment becomes isometric to the unit sphere $\SS^d$ of the vector space on which the underlying reflection group acts, where  $d = \dim \Delta$.
In particular, the opposition map $\op_{\Sigma}:\Sigma\to\Sigma$ of an apartment $\Sigma$ gives rise to the involution $-\id_{\SS^d}:\SS^d\to \SS^d$ of the unit sphere.
Hence, two simplices $\sigma,\sigma'\in \Sigma$ are opposite if and only if their barycenters are opposite points when regarded in $\SS^d$ via this identification.

The diameter of $|\Delta|$, which is $\pi$ with this metric, is also the diameter of any apartment, which equals the distance between two opposite points.
In particular, for two points $x,y\in |\Delta|$ at distance $\dcat(x,y) < \pi$, there exists a unique geodesic from $x$ to $y$.
Recall that a subspace $X$ of a CAT(1) space is convex if for every two points $x,y\in X$ at distance $<\pi$, the unique geodesic segment joining $x,y$ is completely contained in $X$.
It follows that a convex subcomplex $K$ of $\Delta$ gives rise to a convex subspace of $|\Delta|$.
Notice that not every convex subspace of $|\Delta|$ arises in this way.
Now, if $\sigma,\sigma'$ are two non-opposite simplices in $\Delta$, then there is a unique geodesic (in the geometric realization) that joins their barycenters.

\subsubsection{Automorphism group}
\label{subsub:autDelta}
We denote by $\Aut(\Delta)$ the group of simplicial automorphisms of $\Delta$.
These automorphisms might not be type-preserving.
By a group acting on $\Delta$, we mean a group inducing simplicial automorphisms on $\Delta$.

Any simplicial automorphism on $\Delta$ gives rise to an isometry of $|\Delta|$ with the metric $\dcat$.
Therefore, if $H$ is a group acting simplicially on $\Delta$, and $x,y\in |\Delta|^H$ are two points at distance $<\pi$, then $H$ must fix the unique geodesic joining $x$ and $y$.
In particular, if $\sigma,\sigma'$ are non-opposite simplices of $\Delta$ that are invariant under the action of $H$, then $H$ fixes the unique geodesic in $|\Delta|$ that joins their barycenters.

From these observations, we get the following lemma.

\begin{lemma}
\label{lm:contractibleIntersectionApartments}
Let $\Delta$ be a spherical building, and let $H$ be a group acting on $\Delta$ by simplicial automorphisms.
Let $\tau\in \Delta^H$ be a non-empty simplex fixed by $H$, and let $\S \subseteq \A(\Delta,\tau)$ be a set of apartments containing $\tau$.

If $\bigcap_{\Sigma\in \S} (\Sigma^H)$ does not contain an opposite of $\tau$, then it is contractible.
\end{lemma}

\begin{proof}
Let $X := \bigcap_{\Sigma\in \S} \Sigma^H$, and note that $|X| =  \bigcap_{\Sigma\in \S} |\Sigma'|^H$.
Assume that there is no opposite of $\tau$ in $X$.
Then every point $x\in X$ is at distance $ < \pi$ from the barycenter of $\tau$ (say, $x_0$).
Hence, by the discussion in Subsection \ref{subsub:autDelta}, $H$ fixes the unique geodesic joining $x$ with $x_0$.
Thus, we can contract $X$ to $x_0$ using these geodesics.
\end{proof}

\section{Posets and simplicial complexes in the case of vector spaces}
\label{sec:vectorspaces}

In this section we take a closer look at the poset of partial decompositions, the poset of ordered partial decompositions
and the common bases complex for vector spaces.
We demonstrate how building-related constructions show up.
The results then motivate and guide the more general definitions for buildings in Sections \ref{sec:CBBuildings}, \ref{sec:leviSpheres} and
\ref{sec:opd}.

Let $V$ be a finite-dimensional vector space defined over a field $k$.
Write $T(V)$ for the poset of proper non-zero subspaces of $V$, ordered by inclusion.
We denote by $\Delta(V)$ the order complex of $T(V)$, which is also the building associated with the group $\GL(V)$.

\begin{definition}
\label{def:DandPDVectorSpaces}
Let $V$ be a finite-dimensional vector space defined over a field $k$.
A partial decomposition of $V$ is a subset $\{S_1,\ldots,S_r\} \subseteq T(V)$ such that
\[ \gen{S_1,\ldots,S_r} \cong S_1\oplus \cdots \oplus S_r.\]

We denote by $\PD(V)$ the poset of partial decompositions of $V$ other than $\emptyset$ and $\{V\}$, with order given by refinement; that is
for $d_1,d_2\in \PD(V)$
\[ d_1\leq d_2 \text{ if for all } S\in d_1 \text{ there is } T\in d_2 \text{ such that } S\leq T.\]

A full decomposition $V$ is a partial
decomposition $\{S_1,\ldots, S_r\}$ such that $V \cong S_1 \oplus\cdots \oplus S_r$. The poset of full decompositions of $V$ is the subposet
$\D(V)$ of $\PD(V)$ on the set of full decompositions of $V$.
\end{definition}

Note that we are not including $\{V\}$ in the poset $\D(V)$
and that
\[ d \text{ is a partial decomposition } \Leftrightarrow
\ \dim \gen{S\tq S\in d} = \sum_{S\in d} \dim S.\]

We will also work with the ordered versions of $\PD(V)$ and $\D(V)$:

\begin{definition}
\label{def:ODandOPDVectorSpaces}
Let $V$ be a finite-dimensional vector space defined over a field $k$.
The poset of ordered partial decompositions of $V$, denoted by $\OPD(V)$ consists of tuples of distinct subspaces $(S_1,\ldots,S_r)$ such that $\{S_1,\ldots,S_r\}\in \PD(V)$.
The ordering in $\OPD(V)$ is given by refinement that preserves the order of the elements of the tuples.
That is, if $d_1 = (S_1,\ldots,S_r)$ and $d_2 = (W_1,\ldots,W_t)$, then $d_1\leq d_2$ if for all $1\leq i\leq j\leq r$ there are $1\leq k\leq l\leq t$ such that $S_i\leq W_k$ and $S_j\leq W_l$.

The poset $\OD(V)$ of ordered full decompositions of $V$ is the subposet of
$\OPD(V)$ on the set of $(S_1,\ldots,S_r)$ such that $\{S_1,\ldots,S_r\}\in \D(V)$.
\end{definition}

As observed first in \cite{LR}, the poset $\OD(V)$ is naturally isomorphic to the poset of opposite pairs of the building $\Delta(V)$.

\begin{remark}
[{Opposite simplices}]
\label{rk:ODlinearCase}
An ordered full decomposition $d = (S_1,\ldots,S_r)\in \OD(V)$ determines the pair $\big(\,P(d),Q(d)\,\big)$ of flags given by
\[ P(d) = (\,S_1 < S_1 \oplus S_2 < \cdots < S_1\oplus\cdots \oplus S_{r-1}\,),\]
and
\[ Q(d) = (\, S_r < S_r\oplus S_{r-1} < \cdots < S_2\oplus\cdots \oplus S_r\,).\]
In the language of buildings, this means that $P(d),Q(d)$ are opposite simplices of $\Delta(V)$.
For $\Delta(V)$ the notion of opposition
of simplices from Subsection \ref{sub:opposition} translates into
the following. Two simplices $\sigma$ and $\tau$ from $\Delta(V)$ are opposite if they have the same dimension and for all $S\in \sigma$ there exists a unique $T\in \tau$ such that $V = \gen{S,T} \cong S\oplus T$.

Therefore, the poset $\OD(V)$ can be alternatively described as the poset of pairs of opposite simplices of the building $\Delta(V)$,
where the ordering is given by coordinate-wise reverse inclusion.
The isomorphism is given by $d\mapsto \big(\,P(d), Q(d)\,\big)$.

Another description of $\OD(V)$ is in terms of the Charney poset.
Recall that the Charney poset $\Ch(V)$ consists of pairs $(S,T)$ of proper non-zero subspaces of $V$ such that $S\oplus T = V$.
The ordering in $\Ch(V)$ is given by zig-zag containment:
\[ (S_1,T_1) \leq (S_2,T_2) \  \Leftrightarrow \ S_1\leq S_2 \text{ and } T_1\geq T_2.\]
Then it is not hard to see that
\[ \OD(V) = \X\big(\,\K(\,\Ch(V)\,)\,\big)^{\op}.\]
The Charney poset was introduced by R. Charney \cite{Charney} in the context of free modules over Dedekind domains $R$, where it was used to establish homological stability results for the linear groups $\GL_n(R)$.
\end{remark}

We have seen that $\OD(V)$ has an intrinsic description in terms of building properties.
Our next theorem describes $\D(V)$ in terms of Levi subgroups, hinting at a possible definition of the full decomposition poset for arbitrary buildings that arise from the BN-pair of the $k$-points of a connected reductive algebraic group.

In what follows, suppose that $\overline{k}$ is the algebraic closure of a field $k$.
We say that $L$ is a $k$-Levi subgroup of $\GL_n(k)$ if $L$ is an algebraic group defined over $k$ and it is the Levi complement in a parabolic subgroup of $\GL_n$ that is also defined over $k$.

\begin{theorem}
\label{thm:decompAndLeviVectorSpaces}
Let $\L(\GL_n,k)$ denote the poset of (proper) $k$-Levi subgroups of $\GL_n$ ordered by inclusion.
Then we have a $\GL_n(k)$-equivariant poset isomorphism $\L(\GL_n,k)\groupiso_{\GL_n(k)} \D(k^n)$.
\end{theorem}

\begin{definition}
\label{def:ODandOPDbuildings}
Let $\Delta$ be a spherical building.
We define the posets:
\begin{itemize}
\item $\OD(\Delta) := \Opp(\Delta) = \{ (\sigma,\sigma')\tq \sigma,\sigma'$ are opposite simplices$\}$, with order relation
$(\sigma,\sigma') \leq (\tau,\tau')$ if $\tau\subseteq \sigma$ and $\tau'\subseteq \sigma'$.
\item $\OPD(\Delta) = \X(\Delta) \cup \Opp(\Delta)$ with order relation $\preceq$ defined as follows. Among elements of $\X(\Delta)$ and
among elements of $\Opp(\Delta)$ the order $\preceq$ is inherited from the order on the respective posets. If $\sigma \in \X(\Delta)$ and
$(\tau,\tau') \in \Opp(\Delta)$ then we set
$\sigma \prec (\tau,\tau')$ if there is an apartment containing $\sigma,\tau$, and $\tau'$.
\end{itemize}
We call $\OD(\Delta)$ the ordered decomposition poset associated with $\Delta$, and $\OPD(\Delta)$ the ordered partial decomposition poset.
\end{definition}

\begin{definition}
\label{def:PDbuildings}
Let $\Delta$ be a spherical building.
We define $\PD(\Delta)$ as the poset on the disjoint union of
$\X(\Delta)$ and ${\D(\Delta)}$ with order relation $\preceq$
defined as follows.
Among elements of $\X(\Delta)$ and among elements of $\D(\Delta)$ the
order $\preceq$ is the one inherited from the order on the
respective posets.
If $\sigma \in \Delta$ and $S\in {\D(\Delta)}$, then we set $\sigma \prec S$ if and only if there exists an apartment $\Sigma$ of $\Delta$ such that $\sigma\in \Sigma$ and $S$ is a Levi sphere of $\Sigma$.
\end{definition}

\begin{definition}
\label{def:commonBasis}
Let $\Delta$ be a spherical building.
We write $\CB(\Delta)$ for the simplicial complex on the same
vertex set as $\Delta$ whose maximal simplices are
$\displaystyle{\bigcup_{\sigma \in \Sigma}} \sigma$ for
$\Sigma \in \A(\Delta)$.

\end{definition}

\label{sec:intro}

The order complex of the lattice of non-trivial subspaces of a finite-dimensional vector space is well known to be the spherical building of type A (see e.g., \cite{Tits1}).
Recently, simplicial complexes defined
on the set of non-trivial subspaces
via constraints on common bases (see e.g., \cite{Rognes,MPW}) or the relative position of the subspaces were studied (see e.g., \cite{LR,BPW24}). Such complexes are often order complexes of partially ordered sets.
The goals of this paper are to provide definitions of such
simplicial complexes and posets that are independent of Lie type, and to provide results on the associated homology groups
and homotopy types. In some cases, the complexes we construct in type A are the same as the previously studied ones.  In other cases, our type A complexes are different than those previously studied but have the same equivariant homotopy type. We start by defining the motivating posets and simplicial complexes and reviewing the relevant literature.

Let $V$ be an $n$-dimensional vector space over a field $k$.
A partial decomposition of $V$ is a set $\{V_1,\ldots,V_r\}$ of non-zero and proper subspaces of $V$ such that $\langle V_1,\ldots, V_r \rangle \cong V_1\oplus\cdots\oplus V_r$;
that is, the $V_1,\ldots, V_r$ are in internal direct sum.
The set $\PD(V)$ of all non-empty partial decompositions of $V$ is a poset with order given by refinement; i.e., $\{V_1,\ldots, V_r\}\leq \{W_1,\ldots, W_s\}$ if for all $1\leq i \leq r$ there is $1\leq j\leq s$ such that $V_i\leq W_j$.

In \cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.
In particular, $\PD(V)$ is spherical (see Section \ref{sec:preliminaries} for definitions).
More recently, in \cite{BPW24}, it was proved that for any field $k$,\,$\PD(V)$ has the homotopy type of the common bases complex ${\rm CB}(V)$ of $V$, as defined by J. Rognes \cite{Rognes}.
Concretely, ${\rm CB}(V)$ is the simplicial complex whose simplices are sets $\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\sigma$ is generated by some subset of $B$.
In this case, we say that the elements of $\sigma$ have a common basis.
Rognes showed that, although ${\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\leq 2n-3$, and he conjectured that ${\rm CB}(V)$ must indeed be $(2n-4)$-connected.
In \cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.
In particular, via \cite{BPW24} this implies that $\PD(V)$ is spherical for any field.

Considering the ordered version $\OPD(V)$ of $\PD(V)$
appears to be equally natural.
By definition, $\OPD(V)$ is the poset whose elements are tuples
$(V_1,\ldots,V_r)$, $r\geq 1$, of distinct subspaces of $V$ such that $\{V_1,\ldots,V_r\} \in \PD(V)$.
The ordering in $\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \ref{def:ODandOPDVectorSpaces}).
We call the elements of $\OPD(V)$ ordered partial decompositions.
It follows from \cite[Theorem 6.8]{PW25} that $\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.
If $V$ is a finite vector space then \cite[Theorem 6.10]{PW25} implies
that $\OPD(V)$ has the homotopy type of the two-fold join $\Delta * \Delta$ of the Tits building $\Delta$ of $V$.
Using GAP we also verified in some small examples that the top homology group of
$\OPD(V)$, as a $\GL(V)$-module, is the tensor square of the Steinberg module.

Later, we found an alternative approach to determining the
homotopy type of $\OPD(V)$ using closely related constructions, which inspired the work in this paper.
Let $\OD(V)$ denote the subposet of $\OPD(V)$ consisting of full ordered decompositions $(V_1,\ldots, V_r)$; i.e., those decompositions satisfying
$V \cong V_1\oplus \cdots \oplus V_r$.
Let $X = \Delta \cup \OD(V)$ be the poset on the disjoint union of
$\Delta$ (regarded as a poset without the empty simplex) and $\OD(V)$ with the following
ordering.
We keep the inclusion ordering among elements of $\Delta$, and the refinement ordering in $\OD(V)$.  No element of $\OD(V)$ lies below any element of $\Delta$. If $\sigma\in \Delta$ and $d\in \OD(V)$, we set $\sigma \prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\sigma$.
In Theorem \ref{thm:PDandPDBuildingVectorSpace} we show that
$\OPD(V) \simeq X$.
The poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \cite{MPW}, which is shown therein to have
the homotopy type of $\Delta * \Delta$.
This again allows us to conclude that $\OPD(V) \simeq \Delta * \Delta$ for any finite-dimensional vector space $V$.
Since all homotopy equivalences can be seen to be $\GL(V)$-equivariant, it now follows that the top homology group of
$\OPD(V)$ carries the tensor square of the Steinberg representation.

The poset $\OD(V)$ already appears in work {\cite{LR} of  G.I. Lehrer and L.J. Rylands, who observe that
$\OD(V)$ can be identified with poset of pairs of opposite parabolics of $\GL(V)$.
\footnote{This observation is motivated by previous work by R. Charney \cite{Charney}, and $\OD(V)$ is commonly known as the Charney complex. See also \cite[Proposition 4.15]{PW25}.}
Recall that two parabolic subgroups are called opposite if their intersection is a Levi complement in both of them.
Indeed, two parabolic subgroups are opposite if and only if they are opposite as simplices of the building $\Delta$ of $V$, which is a notion intrinsically defined for every building (see Subsection \ref{sub:opposition}).
Thus, $\OD(V)$ can be described in terms of simplices of $\Delta$, and we define
\[\, \OD(\Delta) = \big\{\,(\sigma,\sigma') \tq \sigma,\sigma'\in \Delta \text{ are (non-empty) opposite simplices}\,\big\},\]
where the ordering of this poset is inclusion-reversing in each coordinate (so a pair of opposite maximal simplices is an ordered frame of $V$, which is a minimal element of the poset).
Using this identification the ordering between crossed terms in $X = \Delta \cup \OD(\Delta)$ becomes $\sigma\prec (\sigma_1,\sigma_2)$ if there is a basis of $V$ spanning the subspaces from $\sigma\cup \sigma_1\cup \sigma_2$.
This condition is equivalent to  saying that $\sigma,\sigma_1,\sigma_2$ lie in a common apartment of $\Delta$. This shows
that
the poset $\Delta\cup \OD(\Delta)$ can be defined in terms of intrinsic combinatorial properties of the building.

These observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.
In this paper, we study the following questions and answer all of them positively.
Let $\Delta$ be a (spherical) building.
\begin{enumerate}
    \item Can we define a poset $\OPD(\Delta)$ that, up to homotopy, coincides with $\OPD(V)$ when $\Delta$ is the building of the vector space $V$?
    \item Can we similarly define $\PD(\Delta)$? Is it highly connected?
    \item Do we have a common basis complex ${\rm CB}(\Delta)$ for $\Delta$ that coincides with ${\rm CB}(V)$ when $\Delta$ is the building of the vector space $V$?
    \item If questions (2) and (3) have a positive answer, do we have ${\rm CB}(\Delta) \simeq \PD(\Delta)$?
    \item Is $\OPD(\Delta)\simeq \Delta * \Delta$?
\end{enumerate}

While the definitions of $\OPD(\Delta)$ and ${\rm CB}(\Delta)$ arise naturally, a suitable definition of $\PD(\Delta)$ is less obvious.
Indeed, as in the case of $\OPD$, Theorem \ref{thm:PDandPDBuildingVectorSpace} shows that $\PD(V)\simeq \Delta \cup \D(V)$, where $\D(V)$ is the poset of full decompositions of $V$, and the ordering between crossed terms in $\Delta \cup \D(V)$ is defined as in the ordered case: if $\sigma\in \Delta$ and $d\in \D(V)$, then we set $\sigma\prec d$ if $\sigma\cup d\in {\rm CB}(V)$ ($\sigma$ and $d$ have a common basis).
Our next observation is, that $\D(V)$ can be identified with the poset of (split) Levi subgroups of $\GL(V)$.
In the language of buildings, split Levi subgroups correspond to Levi spheres, as introduced by J.P. Serre in \cite{Serre} (see Subsection \ref{sub:convexity} and Section \ref{sec:leviSpheres}).
If we identify a full decomposition $d\in \D(V)$ with a Levi sphere, then the condition $\sigma \prec d$ becomes ``$\sigma$ and the Levi sphere $d$ lie in a common apartment".
Therefore, for a spherical building $\Delta$, we propose:

\begin{itemize}
    \item ${\rm CB}(\Delta) = \{\sigma \tq \sigma$ is a subset of the vertex set of some apartment of $\Delta\}$ (Definition \ref{def:commonBasis})
    \item $\D(\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \ref{def:decompBuilding}).
    \item $\PD(\Delta) = \Delta\cup \D(\Delta)$, with crossed-term ordering $\sigma\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\sigma$ (Definition \ref{def:PDbuildings}).
    \item $\OD(\Delta) = \{(\sigma_1,\sigma_2)\tq \sigma_1,\sigma_2$ are non-empty opposite simplices of $\Delta\}$, with coordinate-wise reverse-inclusion ordering (Definition \ref{def:ODandOPDbuildings}).
    \item $\OPD(\Delta) = \Delta \cup \OD(\Delta)$, with crossed-term ordering $\sigma\prec (\sigma_1,\sigma_2)$ if there is an apartment containing $\sigma,\sigma_1,\sigma_2$ (Definition \ref{def:ODandOPDbuildings}).
\end{itemize}

The main theorems of the paper are the following:

In \cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.
In particular, $\PD(V)$ is spherical (see Section \ref{sec:preliminaries} for definitions).
More recently, in \cite{BPW24}, it was proved that for any field $k$,\,$\PD(V)$ has the homotopy type of the common bases complex ${\rm CB}(V)$ of $V$, as defined by J. Rognes \cite{Rognes}.
Concretely, ${\rm CB}(V)$ is the simplicial complex whose simplices are sets $\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\sigma$ is generated by some subset of $B$.
In this case, we say that the elements of $\sigma$ have a common basis.
Rognes showed that, although ${\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\leq 2n-3$, and he conjectured that ${\rm CB}(V)$ must indeed be $(2n-4)$-connected.
In \cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.
In particular, via \cite{BPW24} this implies that $\PD(V)$ is spherical for any field.

Considering the ordered version $\OPD(V)$ of $\PD(V)$
appears to be equally natural.
By definition, $\OPD(V)$ is the poset whose elements are tuples
$(V_1,\ldots,V_r)$, $r\geq 1$, of distinct subspaces of $V$ such that $\{V_1,\ldots,V_r\} \in \PD(V)$.
The ordering in $\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \ref{def:ODandOPDVectorSpaces}).
We call the elements of $\OPD(V)$ ordered partial decompositions.
It follows from \cite[Theorem 6.8]{PW25} that $\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.
If $V$ is a finite vector space then \cite[Theorem 6.10]{PW25} implies
that $\OPD(V)$ has the homotopy type of the two-fold join $\Delta * \Delta$ of the Tits building $\Delta$ of $V$.
Using GAP we also verified in some small examples that the top homology group of
$\OPD(V)$, as a $\GL(V)$-module, is the tensor square of the Steinberg module.

Later, we found an alternative approach to determining the
homotopy type of $\OPD(V)$ using closely related constructions, which inspired the work in this paper.
Let $\OD(V)$ denote the subposet of $\OPD(V)$ consisting of full ordered decompositions $(V_1,\ldots, V_r)$; i.e., those decompositions satisfying
$V \cong V_1\oplus \cdots \oplus V_r$.
Let $X = \Delta \cup \OD(V)$ be the poset on the disjoint union of
$\Delta$ (regarded as a poset without the empty simplex) and $\OD(V)$ with the following
ordering.
We keep the inclusion ordering among elements of $\Delta$, and the refinement ordering in $\OD(V)$. No element of $\OD(V)$ lies below any element of $\Delta$. If $\sigma\in \Delta$ and $d\in \OD(V)$, we set $\sigma \prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\sigma$.
In Theorem \ref{thm:PDandPDBuildingVectorSpace} we show that
$\OPD(V) \simeq X$.
The poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \cite{MPW}, which is shown therein to have
the homotopy type of $\Delta * \Delta$.
This again allows us to conclude that $\OPD(V) \simeq \Delta * \Delta$ for any finite-dimensional vector space $V$.
Since all homotopy equivalences can be seen to be $\GL(V)$-equivariant, it now follows that the top homology group of
$\OPD(V)$ carries the tensor square of the Steinberg representation.

These observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.
In this paper, we study the following questions and answer all of them positively.
Let $\Delta$ be a (spherical) building.
\begin{enumerate}
 \item Can we define a poset $\OPD(\Delta)$ that, up to homotopy, coincides with $\OPD(V)$ when $\Delta$ is the building of the vector space $V$?
 \item Can we similarly define $\PD(\Delta)$? Is it highly connected?
 \item Do we have a common basis complex ${\rm CB}(\Delta)$ for $\Delta$ that coincides with ${\rm CB}(V)$ when $\Delta$ is the building of the vector space $V$?
 \item If questions (2) and (3) have a positive answer, do we have ${\rm CB}(\Delta) \simeq \PD(\Delta)$?
 \item Is $\OPD(\Delta)\simeq \Delta * \Delta$?
\end{enumerate}

\begin{itemize}
 \item ${\rm CB}(\Delta) = \{\sigma \tq \sigma$ is a subset of the vertex set of some apartment of $\Delta\}$ (Definition \ref{def:commonBasis})
 \item $\D(\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \ref{def:decompBuilding}).
 \item $\PD(\Delta) = \Delta\cup \D(\Delta)$, with crossed-term ordering $\sigma\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\sigma$ (Definition \ref{def:PDbuildings}).
 \item $\OD(\Delta) = \{(\sigma_1,\sigma_2)\tq \sigma_1,\sigma_2$ are non-empty opposite simplices of $\Delta\}$, with coordinate-wise reverse-inclusion ordering (Definition \ref{def:ODandOPDbuildings}).
 \item $\OPD(\Delta) = \Delta \cup \OD(\Delta)$, with crossed-term ordering $\sigma\prec (\sigma_1,\sigma_2)$ if there is an apartment containing $\sigma,\sigma_1,\sigma_2$ (Definition \ref{def:ODandOPDbuildings}).
\end{itemize}

The main theorems of the paper are the following:

Later, we found an alternative approach to determining the
homotopy type of $\OPD(V)$ using closely related constructions, which inspired the work in this paper.
Let $\OD(V)$ denote the subposet of $\OPD(V)$ consisting of full ordered decompositions $(V_1,\ldots, V_r)$; i.e., those decompositions satisfying
$V \cong V_1\oplus \cdots \oplus V_r$.
Let $X = \Delta \cup \OD(V)$ be the poset on the disjoint union of
$\Delta$ (regarded as a poset without the empty simplex) and $\OD(V)$ with the following
ordering.
We keep the inclusion ordering among elements of $\Delta$, and the refinement ordering in $\OD(V)$. No element of $\OD(V)$ lies below any element of $\Delta$. If $\sigma\in \Delta$ and $d\in \OD(V)$, we set $\sigma \prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\sigma$.
In Theorem \ref{thm:PDandPDBuildingVectorSpace} we show that
$\OPD(V) \simeq X$.
The poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \cite{MPW}, which is shown therein to have
the homotopy type of $\Delta * \Delta$.
This again allows us to conclude that $\OPD(V) \simeq \Delta * \Delta$ for any finite-dimensional vector space $V$.
Since all homotopy equivalences can be seen to be $\GL(V)$-equivariant, it now follows that the top homology group of
$\OPD(V)$ carries the tensor square of the Steinberg representation.

The main theorems of the paper are the following:

Here, $\CB(\Delta)$ and $\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.

If $\Delta$ is the building of a finite-dimensional vector space $V$, then $\CB(V) = \CB(\Delta)$, $\PD(\Delta)\simeq \PD(V)$ and $\OPD(\Delta)\simeq \OPD(V)$.
In particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \cite{MPW}.
Our proof is independent of that of \cite{MPW}, and it is based on geometric properties of buildings.
To prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.
Key ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible." For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.
This implies that $K$ is contractible by \cite{Serre}.
We extend this idea to prove that certain fixed-point subposets are contractible.

\begin{proof}
First, we show that $\Gamma$ is well-defined.
Note that if $c\in \PD(\Delta)'$, then there is an apartment $\Sigma$ containing all the vertices involved in $c$ by definition of $\PD(\Delta)$.
In particular, the convex hull of $c$ is a convex subcomplex of $\Sigma$, so it lies in $Y(\Delta)$.
If in addition the maximal element of $c$, say $x$, satisfies $x\prec S$, we can take $\Sigma$ containing $S$ by definition of the ordering in $\PD(\Delta)$, and hence $\Gamma(c) = \Conv_{\Delta}(c,S)\in Y(\Delta)_{\supseteq S}$.
Thus $\Gamma$ is an order-preserving map between posets.
Also, as pointed out at the end of Subsection \ref{subsub:chambercomplexes}, $\Gamma$ is $H$-equivariant.

Next, we prove that $\OPD(\Delta)$ has the homotopy type of the simplicial join $\Delta*\Delta$, and that it carries the tensor-square Steinberg representation in homology.
For a poset $X$, recall that $X^{(i)}$ denotes the subposet of elements of height at most $i$.
If $X = \X(K)$ where $K$ is a Cohen-Macaulay simplicial complex, then $X$ is Cohen-Macaulay as a poset.
Also, every rank selection of a Cohen-Macaulay poset is spherical (see Theorem 6.2 of \cite{Baclawski}).
In particular, rank selections of the face poset of a (spherical) building are Cohen-Macaulay.

\begin{theorem}
\label{thm:OPDandDeltaDelta}
Let $\Delta$ be a spherical building of dimension $m$ and let $H$ be a group acting on $\Delta$ by simplicial automorphisms.
Write $D_1 = \X(\Delta)$ and $D_2 = \X(\Delta)^{\op}$.
Note that $\OPD(\Delta) = D_1 \cup \OD(\Delta)$.

Let $\phi:\OPD(\Delta)\to D_1 \ojoin D_2$ be the following map:
\begin{align*}
 \phi(\sigma,\sigma') & = \sigma \in D_2 \quad \text{ if } (\sigma,\sigma')\in \OD(\Delta) \subset \OPD(\Delta),\\
 \phi(\sigma) & = \sigma \in D_1 \quad \text{ if } \sigma\in D_1 \subset \OPD(\Delta).
\end{align*}
Then $\phi$ is an $H$-equivariant order-preserving map.
Moreover, the following hold:
\begin{enumerate}
\item The restriction
\[ \phi_H: (\OPD(\Delta)^{(i+m+1)})^H \to (D_1 \ojoin (D_2^{(i)}))^H\]
is a homotopy equivalence for all $0 \leq i \leq m$.
Thus, $\OPD(\Delta)^{(i+m+1)} \simeq D_1 \ojoin D_2^{(i)}$ is spherical of dimension $m + i + 1$.
\item In particular, $\phi$ gives rise to a homotopy equivalence
\[\OPD(\Delta) \simeq \Delta * \Delta,\]
which is $H$-equivariant if $H$ is a compact Lie group, and to an $H$-equivariant isomorphism
\[ \widetilde{H}_{*}(\OPD(\Delta)) \cong_H \widetilde{H}_*(\Delta*\Delta).\]
\end{enumerate}
\end{theorem}

\begin{theorem}
\label{thm:PDandPDBuildingVectorSpace}
Let $V$ be an $n$-dimensional vector space over a field $k$.
Let $\sigma \in \OPD(V)'$, and write $\sigma = \sigma_0\cup \sigma_1$ where $\sigma_0$ is the set of ordered partial decompositions in $\sigma$ that span a proper subspace of $V$, and $\sigma_1$ is the set of full ordered decompositions in $\sigma$.
Then, for $\sigma \in \OPD(V)'$, we set
\[ \phi(\sigma) = \begin{cases}
\{ \gen{s} \tq s\in \sigma_0\} & \sigma_1 = \emptyset,\\
\max \sigma_1 & \sigma_1\neq\emptyset.
\end{cases}\]
Then $\phi$ is a $\GL(V)$-equivariant poset map $\phi: \OPD(V)' \to \OKtwo{V}$ such that for all $H\leq \GL(V)$, the induced map $\phi_H: (\OPD(V)')^H \to \OKtwo{V}^H$ is a homotopy equivalence.

We get analogous conclusions for the corresponding map $\PD(V)'\to \Ktwo{V}$.
\end{theorem}

Here, ${\rm CB}(\Delta)$ and $\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.

There is a canonical map $F:\OD(\Delta)\to \D(\Delta)$ that maps a pair of opposite simplices $(\sigma_1,\sigma_2)\in\OD(\Delta)$ to the Levi sphere spanned by $\sigma_1,\sigma_2$ (which is actually their convex hull, see Subsection \ref{sub:convexity}).

In \cite{vH}, A. von Heydebreck showed that $\OD(\Delta)$ is Cohen-Macaulay of dimension $\dim\Delta$.
Using the map $F$, we conclude:

\begin{corollary}
\label{coro:DDeltaCM}
Let $\Delta$ be a spherical building.
Then the poset of Levi spheres $\D(\Delta)$ is Cohen-Macaulay of dimension $\dim \Delta$.
\end{corollary}

We mention that Corollary \ref{coro:DDeltaCM} can potentially be applied to compute the homotopy type of Quillen's $p$-subgroup posets \cite{Qui78} of connected reductive algebraic groups $G$ over finite fields $\GF{q}$, where $p$ is a ``good" prime for which $q$ has order $1$ modulo $p$.
This is based on the work by D. Rossi \cite{Rossi}, where it is proved that for such a good prime $p$, the $p$-subgroup poset of the finite group $G(\GF{q})$ has the homotopy type of the poset of $\GF{q}$-Levi subgroups, which coincides with the poset $\D(\Delta(G,\GF{q}))$.
Here $\Delta(G,\GF{q})$ is the building of $G(\GF{q})$.

We will write $|S|$ to denote the geometric realization of the Levi sphere $S$, which is a sphere of dimension $\dim S$ (see Lemma \ref{lm:convexIsleviSphere}).

The map $F$ naturally extends to an order-preserving map $F:\OPD(\Delta)\to \PD(\Delta)$ as the identity on the $\Delta$-part.
Studying fibers of this map, we prove:

\begin{theorem}
Let $\Delta$ be a spherical building.
Then we have a homotopy equivalence
\begin{equation}
\OPD(\Delta) \simeq \PD(\Delta) \bigvee_{ T \in {\D(\Delta)}} |T| * |T|*\PD(\Lk_{\Delta}(\sigma_T)),
\end{equation}
where $\sigma_T\in T$ is some maximal simplex.
In particular, $\PD(\Delta)$ is spherical of dimension $2\dim\Delta + 1$, and ${\rm CB}(\Delta)$ has the homotopy type of a wedge of spheres of dimension $2\dim\Delta + 1$.

Moreover, for a Levi sphere $S$ and a pair of opposite maximal simplices $\sigma_1,\sigma_2\in S$, we have
\begin{align*}
\PD(\Delta)_{\prec S} & \simeq |S| * \PD(\Lk_{\Delta}(\sigma_1))\\
\OPD(\Delta)_{\prec (\sigma_1,\sigma_2)}  & \simeq |S| * \OPD(\Lk_{\Delta}(\sigma_1)).
\end{align*}
\end{theorem}

Here we are using the bar notation for Levi spheres specifically to distinguish between $S$ as a sphere (via $|S|$) and $S$ as an element of the poset $\D(\Delta)$.

If $\Delta$ is the building of a finite-dimensional vector space $V$, then ${\rm CB}(V) = {\rm CB}(\Delta)$, $\PD(\Delta)\simeq \PD(V)$ and $\OPD(\Delta)\simeq \OPD(V)$.
In particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \cite{MPW}.
Our proof is independent of that of \cite{MPW}, and it is based on geometric properties of buildings.
To prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.
Key ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible."  For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.
This implies that $K$ is contractible by \cite{Serre}.
We extend this idea to prove that certain fixed-point subposets are contractible.

\subsection*{Organization of the article}

The paper is organized as follows.
In Section \ref{sec:preliminaries} we introduce the notation from algebra, combinatorics, geometry, and topology that we will use throughout the paper. We also recall results that will play an important role in the rest of the paper. In particular,  we outline the results of the theory of Tits buildings, mostly following \cite{AB} and sometimes \cite{Tits1}.
In Section \ref{sec:vectorspaces}, we study the motivating type A situation.
More precisely, we give exact definitions of the ordered and unordered (partial) decomposition posets and the common basis complex for vector spaces. We also provide the proof that the ordered partial vector space decompositions are equivariantly homotopy equivalent to the poset of ordered partial decompositions of the corresponding building.

In Sections \ref{sec:CBBuildings}, \ref{sec:leviSpheres} and \ref{sec:opd}, we define common basis complexes, ordered and unordered decomposition, and partial decomposition posets for arbitrary buildings.
We provide results about homotopy types and homotopy equivalences, including equivariant versions and fixed points. Some of them extend
facts from the linear case discussed in
Section \ref{sec:vectorspaces}.

Finally, in Section \ref{sec:algebraicgroups} we specialize these results to the case of rational points of connected reductive algebraic groups, obtaining explicit descriptions of the posets and complexes in terms of rational parabolics and Levi subgroups.
In particular, as an application of the equivariance of our maps and the connectivity results, in Proposition \ref{prop:lowLeviIntervalPD} we provide a long exact sequence in terms of Steinberg-square modules of rational Levi subgroups for the rational points of a connected reductive algebraic group, ending in the Steinberg module of such a group.

Computer calculations were performed with GAP \cite{GAP4}
and software package \cite{posets}.

\subsection*{Acknowledgments}
We thank Bernhard Mühlherr and Richard Weiss for helpful and motivating conversations.
The first author was supported by the FWO grant 12K1223N.

To prove the stated homotopy equivalences, the authors say they “use the notion of convexity and complete reducibility introduced by Serre.” The “key ingredients” are statements that “a certain (convex) subcomplex $K$ of the building is contractible.” Their method is: “we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$,” and then “this implies that $K$ is contractible by \cite{Serre}.” They also note that they “extend this idea to prove that certain fixed-point subposets are contractible.”

They further indicate a structural approach via the canonical map $F:\OD(\Delta)\to \D(\Delta)$ (sending an opposite pair $(\sigma_1,\sigma_2)$ to the Levi sphere they span), which “naturally extends to an order-preserving map $F:\OPD(\Delta)\to \PD(\Delta)$ as the identity on the $\Delta$-part,” and that “studying fibers of this map” yields the wedge-type homotopy decomposition stated in the later theorem.

Question: Let \(\Delta\) be a spherical building. Define the common-basis complex \({\rm CB}(\Delta)\) to be the simplicial complex whose simplices are subsets of the vertex set of some apartment of \(\Delta\). Let \(\mathcal D(\Delta)\) be the poset of nonempty Levi spheres ordered by reverse inclusion, and let \(\mathrm{PD}(\Delta)=\Delta\cup \mathcal D(\Delta)\), with cross relation \(\sigma\prec S\) when there is an apartment containing both the simplex \(\sigma\) and the Levi sphere \(S\). Let \(\mathrm{OPD}(\Delta)\) be the associated ordered partial-decomposition poset, let \(\mathrm{sd}\,\mathrm{PD}(\Delta)\) denote the barycentric subdivision, and let \(\Delta*\Delta\) be the simplicial join. For a group \(H\) acting on \(\Delta\) by simplicial automorphisms, write \(X^H\) for the fixed-point subposet or subcomplex. Which statement is valid for these objects?

There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences on fixed points, namely \(\Gamma_H:\mathrm{PD}(\Delta)^H\to {\rm CB}(\Delta)^H\) and \(\phi_H:\mathrm{OPD}(\Delta)^H\to \Delta^H*\Delta^H\). In particular, \(\Gamma\) and \(\phi\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology.

Choices:
B. There exist order-preserving maps \(\Gamma:\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences on fixed points, namely \(\Gamma_H:\mathrm{PD}(\Delta)^H\to {\rm CB}(\Delta)^H\) and \(\phi_H:\mathrm{OPD}(\Delta)^H\to \Delta^H*\Delta^H\). In particular, \(\Gamma\) and \(\phi\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology.
C. There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that \(\Gamma\) and \(\phi\) are homotopy equivalences. In particular, they induce isomorphisms in homology and cohomology.
D. There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences on fixed points, namely \(\Gamma_H:\mathrm{PD}(\Delta)^H\to {\rm CB}(\Delta)^H\) and \(\phi_H:\mathrm{OPD}(\Delta)^H\to (\Delta*\Delta)^H\). In particular, \(\Gamma\) and \(\phi\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology.
E. There exist order-preserving maps \(\Gamma:\mathrm{sd}\,\mathrm{PD}(\Delta)\to {\rm CB}(\Delta)\) and \(\phi:\mathrm{OPD}(\Delta)\to \Delta*\Delta\) such that, for every group \(H\) acting on \(\Delta\) by simplicial automorphisms, both \(\Gamma\) and \(\phi\) are \(H\)-equivariant and induce homotopy equivalences between the underlying spaces. Moreover, for every such \(H\), the fixed-point subposets \(\mathrm{PD}(\Delta)^H\) and \(\mathrm{OPD}(\Delta)^H\) are contractible.

barycentric-subdivision requirement for the PD-to-CB map

fixed-point equivariance and fixed-point-level homotopy equivalences for every H

identification of the fixed-point target as \(\Delta^H*\Delta^H\), not \((\Delta*\Delta)^H\)

contractibility applies only to certain convex fibers/fixed-point subposets used in the proof, not to all fixed-point subposets themselves


\label{t.main}
	Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. Then $$e(n, s) = \max\big\{|\mathcal{P}(s, \ell)|, |\mathcal{P}'(\ell)|, |\mathcal{Q}(s, \ell)|, |\mathcal{W}(s, \ell)|\big\}.$$ Moreover, if $s \geq 3$, ${\mathcal F}\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\mathcal F}|=e(n,s)$, then ${\mathcal F}$ must coincide with one of the families above.

\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |\pp(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
    &\mathrm{(ii)}\ \ \  m=1,\\
     &\mathrm{(iii)}  \ \  s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}

\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |\pp(s,m,\ell)|.
\end{equation}
\end{gypo}

Let $[n] := \{1,2,\ldots, n\}$ and, more generally, $[a,b]=\{a,a+1,\ldots, b\}$. For a set $X$ and an integer $k$, let  $2^{X}$, ${X\choose k}$ and ${X\choose \geq k}$ stand for the power set of $X$, the set of its $k$-element subsets and the set of its subsets with size at least $k$, respectively. Any collection of sets is called a {\it family.} A {\it matching} is a collection of pairwise disjoint sets. An {\it $s$-matching} is a matching of size $s$. Given a family ${\mathcal F},$ its {\it matching number}
$\nu(\mathcal F)$ is the size of the largest matching in ${\mathcal F}$.

One of the classical topics in extremal set theory is the study of {\it intersecting} families, that is, families with matching number $1$. Erd\H os, Ko and Rado~\cite{EKR} showed that the largest intersecting family ${\mathcal F}\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\ge 2k$ the largest intersecting family ${\mathcal F}\subset {[n]\choose k}$ has size ${n-1\choose k-1}.$ In the several years that followed, Erd\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.
\begin{align*}
    e(n,s)&=\max\big\{|{\mathcal F}|: {\mathcal F}\subset 2^{[n]}, \nu({\mathcal F})<s\big\},\\
    e_k(n,s)&=\max\Big\{|{\mathcal F}|: {\mathcal F}\subset {[n]\choose k}, \nu({\mathcal F})<s\Big\}.
\end{align*}
\subsection{The uniform case}
The value of $e_k(n,s)$ is the subject of the famous Erd\H os Matching Conjecture. Let us define the families $\aaa_i^{(k)}(n,s):$
\begin{equation}\label{eq0011} \aaa_i^{(k)}(n,s-1) := \Bigl\{F\in {[n]\choose k}:|F\cap [si-1]|\ge i\Bigr\}, \ \ \ 1\le i\le k.\end{equation}
It is not difficult to see that $\nu(\aaa_i^{(k)}(n,s-1)\le s-1$ for any $n,s$.
\begin{gypo}[Erd\H os Matching Conjecture \cite{E}]\label{conj2} For $n\ge sk$
\begin{equation}\label{eq008} e_k(n,s) = \max\bigl\{|\aaa_1^{(k)}(n,s-1)|,|\aaa_k^{(k)}(n,s-1)|\bigr\}.
\end{equation}
\end{gypo}
Let us comment on the statement of the conjecture. First, the condition $n\ge sk$ is required in order for ${[n]\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\choose k}$ \cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$

The conjecture (\ref{eq008}) is known to be true for $k= 2$ \cite{EG}, and $k=3$ \cite{LM,F11}. In fact, an elegant proof of Frankl of the $k=2$ case of the conjecture, initially proven by Erd\H os and Gallai, will play an important role in this paper. We will recite it later.

The conjecture was proved for $n\ge n_0(s,k)$ by Erd\H os \cite{E}. Many researchers worked on extending the range in which the conjecture is valid:  \cite{E}, \cite{BDE}, \cite{HLS}, \cite{FLM},  \cite{F4}, \cite{FK21}. Notably, in the last two papers, it is proved that
\begin{equation}\label{eq009} e_k(n,s) = |\aaa_1|={n\choose k}-{n-s+1\choose k} \end{equation}
for $n\ge (2s+1)k-s$ and for ($n\ge \frac 53sk$, $s\ge s_0$), respectively. In \cite{F7} and \cite{KoKu}, the authors studied the problem from the other end and showed that $\aaa_k$ is extremal for ($n\le s(k+1/100k)$, $s\ge ck^3$ with some absolute $c>0$). We refer the reader to \cite{aletal, FK21} for the connections of the Erd\H os Matching Conjecture and other questions, such as Dirac thresholds and small deviations in probability theory. In \cite{HLS}, \cite{K49}, the multi-family variant of the EMC was addressed.  In \cite{FK6}, a Hilton--Milner type stability result for the EMC is obtained.

\subsection{The non-uniform case}
The study of $e(n,s)$ was also initiated by Erd\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\ ({\rm mod\ } s)$.  Answering a question of Erd\H os, Kleitman proved the following theorem.
\begin{thm}[Kleitman \cite{Kl}]\label{thmkl}
\begin{align}\label{eq001} e(sm-1,s) &= \sum_{t=m}^{sm-1}{sm-1\choose t},\\
\label{eq002} e(sm,s) &= {sm-1\choose m}+\sum_{t=m+1}^{sm}{sm\choose t}.
\end{align}
\end{thm}
The matching example for the first case is the family ${[n]\choose \ge m}$ of all subsets of $[n]$ of size at least $m$. It is also  not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\ge 2e(n,s)$ because of the {\it doubling} construction. Given a family ${\mathcal F}\subset 2^{[n]}$ with $\nu({\mathcal F})<s$, we may construct the {\it doubling} ${\mathcal F}'$ of ${\mathcal F}$ as follows: ${\mathcal F}' = \{F\subset [n+1]: F\cap [n]\in {\mathcal F}\}$. It is easy to see that $\nu({\mathcal F}')<s$ as well.

In \cite{FK9}, Frankl and the first author put forth the following conjecture.
\begin{opr}\label{def3} Let $n = sm+s-\ell$, $0<\ell\le s$. Set
\[{\mathcal P}(s,m,\ell) := \bigl\{P\subset 2^{[n]}: |P|+|P\cap [\ell-1]|\ge m+1\bigr\}.\]
\end{opr}
It is not difficult to check (cf. \cite{FK9}) that $\nu( \mathcal P(s,m,\ell))<s.$
\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|.
\end{equation}
\end{gypo}
They confirmed the conjecture for a variety of cases:
\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
    &\mathrm{(ii)}\ \ \  m=1,\\
     &\mathrm{(iii)}  \ \  s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}
The case of $\ell=2,s=3$ was solved much earlier by Quinn \cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \cite{FK6}.

Let us return to the definition of $\mathcal P(s,m,\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\ell-1]$, i.e., the family $\aaa_1^{(m)}(n,\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\ell$. We should note here that the sets in $\mathcal P(s,m,\ell)\cap {[n]\choose \le m}$, a bulk of which is  $\aaa_1^{(m)}(n,\ell-1)$, could be replaced by any other family in ${[n]\choose m}$ with no matching of size $\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.

Here, one sees that the problems of determining $e(n,s)$ and $e_k(n,s)$ are, in fact, closely connected. In particular, let us give  the reason for the constraint $\ell\le s/2$ in Conjecture~\ref{conj1}: in this regime, $n\ge 2\ell m,$ and the result of Frankl \cite{F4} guarantees that the largest family in ${[n]\choose m}$ with no matching of size $\ell$ is indeed $\aaa_1^{(m)}(n,\ell-1)$.

Incidentally, one could also interpret the extremal family for $n=sm = sm+s-s$ in the same way: the extremal example consists of all sets of size at least $m+1$, as well as of the family of sets of size $m$ with no $s$-matching. The construction for the sets of uniformity $m$ is, however, different: it is $\aaa_k^{(m)}(n,s-1)$. Actually, the situation is even more complicated, as is shown in \cite{FK9}: already for $n = sm+1$ there is a construction in which some  $(m+1)$-element sets are missing and that is bigger than constructions with all $(m+1)$-element sets present. Namely, it is the doubling construction.

Concluding the discussion, the authors of \cite{FK9} state that it is hard to formulate a general conjecture, and that they had already struggled with the case $n=2s+t$ for some small values of $t$.
\subsection{Main result}
In this paper, we find the value of $e(n,s)$ for all $n,s$, provided $n\le 3s.$ The easy case $n<2s$ is solved in Theorem~\ref{thmfk}. The cases $n=2s$ and $n=3s$ are answered in \eqref{eq002}. Thus, we are left to deal with the cases $2s<n<3s$. For the rest of the paper, unless otherwise stated, we use notation $n=2s+c=3s-\ell$, where $c, \ell\in[s-1]$, $c+\ell=s$.

In order to state the result, we describe the prospective extremal families. We have already seen the first example.
$$\mathcal{P}(s, \ell) := \mathcal P(s,2,\ell)=\big\{P \in 2^{[n]} : |P| + |P \cap [\ell - 1]| \geq 3\big\}.$$
The second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\ell$-matching: a clique on $2\ell-1$ vertices.
$$\mathcal{P}'(s, \ell) := {[n] \choose \geq 3} \cup {[2\ell-1] \choose 2}.$$
The third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\mathcal{Q}(s, \ell) := {[n] \choose \geq 3} \cup {[s+\ell-1] \choose 2} \setminus {[s+\ell, n] \choose 3}.$$
Finally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.
$$\mathcal{W}(s, \ell) := \big\{P \in 2^{[n]} : |P \cap [2s - 1]| \geq 2\big\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.

The main result of the paper is the following theorem.

One of the classical topics in extremal set theory is the study of {\it intersecting} families, that is, families with matching number $1$. Erd\H os, Ko and Rado~\cite{EKR} showed that the largest intersecting family ${\mathcal F}\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\ge 2k$ the largest intersecting family ${\mathcal F}\subset {[n]\choose k}$ has size ${n-1\choose k-1}.$ In the several years that followed, Erd\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.
\begin{align*}
 e(n,s)&=\max\big\{|{\mathcal F}|: {\mathcal F}\subset 2^{[n]}, \nu({\mathcal F})<s\big\},\\
 e_k(n,s)&=\max\Big\{|{\mathcal F}|: {\mathcal F}\subset {[n]\choose k}, \nu({\mathcal F})<s\Big\}.
\end{align*}
\subsection{The uniform case}
The value of $e_k(n,s)$ is the subject of the famous Erd\H os Matching Conjecture. Let us define the families $\aaa_i^{(k)}(n,s):$
\begin{equation}\label{eq0011} \aaa_i^{(k)}(n,s-1) := \Bigl\{F\in {[n]\choose k}:|F\cap [si-1]|\ge i\Bigr\}, \ \ \ 1\le i\le k.\end{equation}
It is not difficult to see that $\nu(\aaa_i^{(k)}(n,s-1)\le s-1$ for any $n,s$.
\begin{gypo}[Erd\H os Matching Conjecture \cite{E}]\label{conj2} For $n\ge sk$
\begin{equation}\label{eq008} e_k(n,s) = \max\bigl\{|\aaa_1^{(k)}(n,s-1)|,|\aaa_k^{(k)}(n,s-1)|\bigr\}.
\end{equation}
\end{gypo}
Let us comment on the statement of the conjecture. First, the condition $n\ge sk$ is required in order for ${[n]\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\choose k}$ \cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$

\subsection{The non-uniform case}
The study of $e(n,s)$ was also initiated by Erd\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\ ({\rm mod\ } s)$. Answering a question of Erd\H os, Kleitman proved the following theorem.
\begin{thm}[Kleitman \cite{Kl}]\label{thmkl}
\begin{align}\label{eq001} e(sm-1,s) &= \sum_{t=m}^{sm-1}{sm-1\choose t},\\
\label{eq002} e(sm,s) &= {sm-1\choose m}+\sum_{t=m+1}^{sm}{sm\choose t}.
\end{align}
\end{thm}
The matching example for the first case is the family ${[n]\choose \ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\ge 2e(n,s)$ because of the {\it doubling} construction. Given a family ${\mathcal F}\subset 2^{[n]}$ with $\nu({\mathcal F})<s$, we may construct the {\it doubling} ${\mathcal F}'$ of ${\mathcal F}$ as follows: ${\mathcal F}' = \{F\subset [n+1]: F\cap [n]\in {\mathcal F}\}$. It is easy to see that $\nu({\mathcal F}')<s$ as well.

In \cite{FK9}, Frankl and the first author put forth the following conjecture.
\begin{opr}\label{def3} Let $n = sm+s-\ell$, $0<\ell\le s$. Set
\[{\mathcal P}(s,m,\ell) := \bigl\{P\subset 2^{[n]}: |P|+|P\cap [\ell-1]|\ge m+1\bigr\}.\]
\end{opr}
It is not difficult to check (cf. \cite{FK9}) that $\nu( \mathcal P(s,m,\ell))<s.$
\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|.
\end{equation}
\end{gypo}
They confirmed the conjecture for a variety of cases:
\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
 &\mathrm{(ii)}\ \ \ m=1,\\
 &\mathrm{(iii)} \ \ s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}
The case of $\ell=2,s=3$ was solved much earlier by Quinn \cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \cite{FK6}.

Let us return to the definition of $\mathcal P(s,m,\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\ell-1]$, i.e., the family $\aaa_1^{(m)}(n,\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\ell$. We should note here that the sets in $\mathcal P(s,m,\ell)\cap {[n]\choose \le m}$, a bulk of which is $\aaa_1^{(m)}(n,\ell-1)$, could be replaced by any other family in ${[n]\choose m}$ with no matching of size $\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.

In order to state the result, we describe the prospective extremal families. We have already seen the first example.
$$\mathcal{P}(s, \ell) := \mathcal P(s,2,\ell)=\big\{P \in 2^{[n]} : |P| + |P \cap [\ell - 1]| \geq 3\big\}.$$
The second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\ell$-matching: a clique on $2\ell-1$ vertices.
$$\mathcal{P}'(s, \ell) := {[n] \choose \geq 3} \cup {[2\ell-1] \choose 2}.$$
The third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\mathcal{Q}(s, \ell) := {[n] \choose \geq 3} \cup {[s+\ell-1] \choose 2} \setminus {[s+\ell, n] \choose 3}.$$
Finally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.
$$\mathcal{W}(s, \ell) := \big\{P \in 2^{[n]} : |P \cap [2s - 1]| \geq 2\big\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.

The main result of the paper is the following theorem.

The main result of the paper is the following theorem.

\begin{figure}

\begin{theorem} \label{t.truncated_lattice}
 Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. If $\mathcal{F} \subset {[n] \choose \leq 3}$ has no $s$-matching, then
 $$|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)^{(\leq 3)}|, |\mathcal{P}'(s, \ell)^{(\leq 3)}|, |\mathcal{Q}(s, \ell)^{(\leq 3)}|, |\mathcal{W}(s, \ell)^{(\leq 3)}|\big\}.$$
\end{theorem}

\begin{theorem} \label{t.Erdos_Gallai}
 Let $n,s$ be positive integers, $n\geq 2s$. Suppose that $\mathcal{F} \subset 2^{[n]}$ satisfies $\nu(\mathcal{F}^{(2)}) < s$. Then $y(2) \geq \min({n - s + 1 \choose 2}, {n \choose 2} - {2s-1 \choose 2})$.
\end{theorem}
\begin{proof}
 As we have mentioned, we may assume that $\ff$ is shifted, cf.~\cite{F3}.
 Consider the sets $\{i, 2s+1-i\}$, $i\in [s]$. These sets form an $s$-matching, and thus at least one of them is not in $\mathcal{F}$. By Lemma \ref{l.EG}, we get $$y(2) \geq \frac{(n+2s+1-3i)(n-2s+i)}{2}$$ for some $i \in [1, s]$. The right-hand side is upward convex as a function of $i$. Thus, its minimum in the interval $[1, s]$ is attained at one of its ends. We get the bound
 $$y(2) \geq \min\Big\{{n - s + 1 \choose 2}, \frac{(n+2s-2)(n-2s+1)}{2}\Big\}.$$ This is the claimed bound since ${n \choose 2} - {2s-1 \choose 2} = \frac{(n+2s-2)(n-2s+1)}{2}$.
\end{proof}

\begin{thm}\label{t.relaxed} Let $n,s,\ell$ be non-negative integers such that $n = 3s-\ell$ and $0\le \ell\le s$. Suppose that $\mathcal{F} \subset 2^{[n]}$ is a shifted up-set with $\nu(\mathcal{F}) < s$ and $\mathcal{F} \cap {[n] \choose 1} = \emptyset$. Then $|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)|, |\mathcal{P}'(s, \ell)|, |\mathcal{Q}(s, \ell)|, |\mathcal{W}(s, \ell)|\big\}$. Moreover, equality is achieved only if $\mathcal{F}$ coincides with one of these families.
\end{thm}
\begin{proof}[Proof of Theorem~\ref{t.main} using Theorem~\ref{t.relaxed}]
We prove Theorem~\ref{t.main} by induction on $\ell$. For $\ell \in \{0, 1\}$ the statement is given by Theorem~\ref{thmkl}, together with uniqueness that is proven in \cite{Kl}. In what follows, we assume that $\ell\ge 2.$

If $\{1\} \notin \mathcal{F}$ then by shiftedness $\mathcal{F} \cap {[n] \choose 1} = \emptyset$ and we are done.
If $\{1\} \in \mathcal{F}$ then we apply induction. Let $\mathcal{F'} = \mathcal{F} \cap 2^{[2, n]}$. The family $\m F'$ is a family of subsets of an $(n-1)$-element set, $\nu(\mathcal{F'}) < s-1$, and we have $n-1 =3(s-1)-(\ell-2).$ Note that $\ell-2\ge 0$ and $\ell-2<s-1$, and thus we may apply induction to $\m F'$, getting that
$$|\overline{\mathcal{F'}}| \geq \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}|, |\overline{\mathcal{P}(s-1, \ell-2)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}|, |\overline{\mathcal{W}(s-1, \ell-2)}|\big\},$$
where the complement is taken with respect to $2^{[2,n]}.$ Clearly, $|\overline{\mathcal{F}}|\ge |\overline{\mathcal{F'}}|$, and so Theorem~\ref{t.main} is implied by the following claim.
\begin{restatable}{cla}{claimeleven} We have
 \begin{multline*}
 \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}|, |\overline{\mathcal{P}(s-1, \ell-2)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}|, \\|\overline{\mathcal{W}(s-1, \ell-2)}|\big\} >\min\big\{|\overline{\mathcal{P'}(s, \ell)}|, |\overline{\mathcal{P}(s, \ell)}|, |\overline{\mathcal{Q}(s, \ell)}|, |\overline{\mathcal{W}(s, \ell)}|\big\}.
 \end{multline*}
 Moreover,
 \begin{multline*}
 \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}^{(\leq 3)}|, |\overline{\mathcal{P}(s-1, \ell-2)}^{(\leq 3)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}^{(\leq 3)}|, \\|\overline{\mathcal{W}(s-1, \ell-2)}^{(\leq 3)}|\big\} >\min\big\{|\overline{\mathcal{P'}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{P}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{Q}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{W}(s, \ell)}^{(\leq 3)}|\big\}.
 \end{multline*}
\label{c.no_siggletons}
\end{restatable}

The case of odd $d$ requires a more careful analysis. In this case, we use the inequality \eqref{eqd2c} which states that $d(\mathcal{F}) \leq 2c$.
\begin{lemma} \label{l.odd_d_to_y2}
 Let $d$ be a positive odd integer, $d \leq 2c$. If $\mathcal{F} \subset 2^{[n]}$ is a shifted family with $d(\mathcal{F}) = d$, then {\small \begin{equation}\label{eqy22}y(2) \geq \min\Big\{\frac{(4\ell+3c+d-2)(3c-d+1)}{2}, \frac{(\ell+3c-\frac{d-1}{2})(\ell+3c-\frac{d+1}2)}{2}\Big\}.\end{equation}} Moreover, equality is achieved only if $\mathcal{F}^{(2)} = {[2\ell+d-1] \choose 2}$ or $\mathcal{F}^{(2)} = \{F \in {n \choose 2 }: F\cap[\ell+\frac{d-1}{2}] \neq \emptyset\}$.
\end{lemma}

\subsection{$c \in \{3, 4\}$}
\begin{lemma} \label{l.c_eq_3}
 Let $n = 2s+3$ and $\mathcal{F} \subset 2^{[n]}$ is a shifted up-set with $\nu(\mathcal{F}) < s$ and $\mathcal{F} \cap {[n] \choose 1} = \emptyset$. Then $|\mathcal{F}| \leq \max(|\mathcal{P}(s, \ell)|, |\mathcal{P}'(s, \ell)|, |\mathcal{Q}(s, \ell)|)$. Moreover, equality is achieved only if $\mathcal{F}$ is one of the families $\mathcal{P}(s, \ell), \mathcal{P}'(s, \ell), \mathcal{Q}(s, \ell)$.
\end{lemma}

\begin{figure}

\centering

\includegraphics[width=0.8\linewidth]{extremal_families.png}

\caption{Extremal families. $\mathcal{W}$ is one of the extremal families for $c = 1, s \geq 5$.}

\label{fig1}

\end{figure}

\begin{figure}

\centering

\includegraphics[width=0.6\linewidth]{extremal_families_2.png}

\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\ell = 1$, that is, $s = c + 1$, the family $\mathcal{P}'$ coincides with $\mathcal{P}$ (both are equal to ${n \choose \geq (m + 1)}$) and thus is formally extremal. When $\mathcal{P}' \neq \mathcal{P}$, for $c \leq 9$ $\mathcal{P}'$ cannot be extremal.}

\label{fig2}

\end{figure}
On Figures~\ref{fig1} and~\ref{fig2} we show, which families are extremal for different regimes of the parameters $s,c$. For some values we get that three different families are extremal at the same time.

We define shifted families in the next section. We should note that actually there are rather natural examples of families with no $s$-matching interpolating between $\mathcal{P}'(s, l)$ and $\mathcal{Q}(s, l)$ in a somewhat similar way as $\aaa_i$ interpolate between $\aaa_0$ and $\aaa_k$, but, as in the case of the EMC, there is a certain convexity that leads to the fact that it is the endpoints that must be extremal.

In the proof we will work only with sets of size $3$ or less. Therefore, any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family. We thus get the following theorem about the truncated boolean lattice, confirming a conjecture of Frankl and the first author \cite{FK9} in our regime of the parameters.

\begin{theorem} \label{t.truncated_lattice}
    Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. If $\mathcal{F} \subset {[n] \choose \leq 3}$ has no $s$-matching, then
    $$|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)^{(\leq 3)}|, |\mathcal{P}'(s, \ell)^{(\leq 3)}|, |\mathcal{Q}(s, \ell)^{(\leq 3)}|, |\mathcal{W}(s, \ell)^{(\leq 3)}|\big\}.$$
\end{theorem}

Note that a similar statement about $2$ first layers of boolean lattice is obviously false. Indeed, one of the families $\aaa_1^{(2)}(n,s-1), \aaa_2^{(2)}(n,s-1)$ has a larger cardinality than families $\mathcal{P}(s, \ell)^{(\leq 2)}, \mathcal{P}'(s, \ell)^{(\leq 2)}, \mathcal{Q}(s, \ell)^{(\leq 2)}, \mathcal{W}(s, \ell)^{(\leq 2)}$.

In Section~\ref{sec2}, we prove several easy facts and make the necessary preparations for the proof of the main theorem. In Section~\ref{sec3} we describe the strategy of the proof of the main theorem.

To prove Theorem~\ref{t.main}, the authors note that they will "work only with sets of size $3$ or less." Hence, "any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family." This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\ref{t.truncated_lattice}, which bounds $|\mathcal F\subset {[n]\choose \le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\mathcal P,\mathcal P',\mathcal Q,\mathcal W$. They also mention that while there are "natural examples" interpolating between $\mathcal P'$ and $\mathcal Q$, "there is a certain convexity" implying "it is the endpoints that must be extremal." Finally, they indicate the paper structure: in Section~\ref{sec2} they "prove several easy facts and make the necessary preparations" and in Section~\ref{sec3} they "describe the strategy of the proof of the main theorem."

To prove the main theorem, the authors note that they will "work only with sets of size $3$ or less." Hence, "any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family." This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\ref{t.truncated_lattice}, which bounds $|\mathcal F\subset {[n]\choose \le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\mathcal P,\mathcal P',\mathcal Q,\mathcal W$. They also mention that while there are "natural examples" interpolating between $\mathcal P'$ and $\mathcal Q$, "there is a certain convexity" implying "it is the endpoints that must be extremal." Finally, they indicate the paper structure: next they "prove several easy facts and make the necessary preparations" and later they "describe the strategy of the proof of the main theorem."

Classification or Bijection
Equality or Bound

Question: Let [n] = {1,2,...,n}, and define e(n,s) = max{|F| : F ⊂ 2^[n] and ν(F) < s}, where ν(F) is the maximum size of a pairwise disjoint subfamily of F. A family F ⊂ 2^[n] is shifted if whenever A ∈ F, j ∈ A, i ∉ A, and 1 ≤ i < j ≤ n, then (A \ {j}) ∪ {i} ∈ F. Suppose n, s, ℓ, c are positive integers satisfying n = 2s + c = 3s - ℓ with c, ℓ ∈ [s - 1]. Let P(s,ℓ) = {P ⊂ [n] : |P| + |P ∩ [ℓ - 1]| ≥ 3}, and let P'(s,ℓ), Q(s,ℓ), and W(s,ℓ) be the other three explicit candidate extremal families on 2^[n] from the same setup. Which statement is valid for e(n,s) and for shifted extremal families with no s-matching?

e(n,s) = max{|P(s,ℓ)|, |P'(s,ℓ)|, |Q(s,ℓ)|, |W(s,ℓ)|}. Moreover, if s ≥ 3 and F ⊂ 2^[n] is shifted, satisfies ν(F) < s, and has |F| = e(n,s), then F must be exactly one of the four families P(s,ℓ), P'(s,ℓ), Q(s,ℓ), or W(s,ℓ).

Choices:
B. e(n,s) = \max\{|P(s,\ell)|, |P'(s,\ell)|, |Q(s,\ell)|, |W(s,\ell)|\}. Moreover, if s \ge 3 and F \subset 2^{[n]} is shifted, satisfies \nu(F) < s, and has |F| = e(n,s), then F must be exactly one of the four families P(s,\ell), P'(s,\ell), Q(s,\ell), or W(s,\ell) up to changing its members of size at least 4.
C. e(n,s) \ge \max\{|P(s,\ell)|, |P'(s,\ell)|, |Q(s,\ell)|, |W(s,\ell)|\}. Moreover, if s \ge 3, each of the four families P(s,\ell), P'(s,\ell), Q(s,\ell), and W(s,\ell) satisfies \nu(F) < s.
D. e(n,s) = \max\{|P(s,\ell)|, |P'(s,\ell)|, |Q(s,\ell)|, |W(s,\ell)|\}. Moreover, if s \ge 3 and F \subset 2^{[n]} satisfies \nu(F) < s and has |F| = e(n,s), then F must be exactly one of the four families P(s,\ell), P'(s,\ell), Q(s,\ell), or W(s,\ell).
E. e(n,s) = \max\{|P(s,\ell)|, |P'(s,\ell)|, |Q(s,\ell)|, |W(s,\ell)|\}. Moreover, if s \ge 3 and F \subset 2^{[n]} is shifted, satisfies \nu(F) < s, and has |F| = e(n,s), then F must be exactly one of the two endpoint families P'(s,\ell) or Q(s,\ell).


\begin{definition}\label{definition:nonsplit}
 Let $r \geq 3$ be an integer. A \emph{Shimura curve of Mumford type} in $\Agr$ (see \cite{viehwegzuo-2004}) is a proper Shimura curve $C  \subset \Agr$ of the following form. Consider a polarized abelian scheme $B_1 \to C_1$ as defined above.
Pick an étale cover $C_2 \to C_1$ with pull-back $B_2 \to C_2$, so that there is a principally polarized abelian scheme $A_2 \to C_2$ isogenous to the polarized abelian scheme $B_2 \to C_2$, such that $A_2 \to C_2$ carries a symplectic level $r$ structure. This gives a morphism $C_2 \to \ca A_{g,[r]}$, and we let $C \subset \ca A_{g,[r]}$ denote its image.

Let us say that the Shimura curve of Mumford type $C$ is \emph{split} (resp.\ \emph{non-split}) if we are in case \eqref{case:i} (resp.\ \eqref
{case:ii}) of Lemma \ref{lemma:cases}.
\end{definition}

\begin{theorem} \label{theorem:special}
    Let $g=2^d$ with $d \geq 4$. Then there exists an abelian variety $A$ of dimension $g$ over $\overline{\mathbb{Q}(t)}$ with no power $A^n$ isogenous to a Jacobian.
\end{theorem}

Every abelian variety $A$ over an algebraically closed field is dominated by a Jacobian, but not always isogenous to one: over $\mathbb{C}$ this follows from a countability argument, over $\overline{\Q}$ this was proven by Chai--Oort and Tsimerman \cite{chaioort, tsimerman-jacobian}, and over $\overline{\F_p(t)}$ this was proven
by the second named author and Tsimerman
\cite{shankartsimerman-Fptbar}.\footnote{The case of finite fields is expected to be different as suggested by heuristics, independently offered by \cite{shankar-tsimerman-heuristics} and Poonen.}

This leaves open the question whether some power of $A$ is isogenous to a Jacobian.
In \cite{dGFSchreieder-2025}, the first named author and Schreieder
showed that the answer is again no in general: for any $g \geq 4$,
there exists a
complex abelian variety of dimension $g$ with no power isogenous to a Jacobian, and for $g=5$ one can find intermediate Jacobians of smooth cubic threefolds among such examples. The authors ask whether one can find
abelian varieties
over $\overline{\Q}$ with no power isogenous to a Jacobian, see \cite[Remark 1.6]{dGFSchreieder-2025}.

The first main result of this paper says that if $g \geq 16$ is a power of $2$, then such examples exist at least over $\smash{\overline{\Q(t)}}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.

The first main result of this paper says that if $g \geq 16$ is a power of $2$, then such examples exist at least over $\smash{\Qtbar}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.

Let $\Ag$ be the moduli space of principally polarized abelian varieties of dimension $g$. A point $x \in \Ag$ is called \emph{Hodge generic} if $x$ is not contained in a special subvariety of $\Ag$ of positive codimension---equivalently, the associated Mumford--Tate group is $\GSp_{2g}$.
In \cite{tsimerman-jacobian}, Tsimerman showed that one can find CM abelian varieties of dimension $g \geq 4$ which are not isogenous to a Jacobian; in \cite{masser-zannier} and \cite{shankartsimerman-Fptbar}, the authors (using different methods) showed that there are also Hodge generic $\Qbar$ points of $\Ag$ with that property.

\begin{theorem} \label{theorem:generic}
Let $g = 2^d \geq 16$ and $N \geq 1$. Then there exists a Hodge generic point $x = [A] \in \Ag(\Qtbar)$ such that the corresponding abelian variety $A$ over $\Qtbar$ has no power $A^n$ with $n \leq N$ which is isogenous to a Jacobian.
\end{theorem}

The situation may be depicted in the following diagram, in which the three curves $C', \tilde C$ and $C$ are smooth,
the morphism $\tilde C \to C'$ is finite, the morphism $\tilde C \to C$ is finite étale,
and the two squares are cartesian:
\[
\xymatrix{
&&\tilde Q\ar[dl] \ar[dr]&&\tilde A^n \ar[dr]\ar[ll] \ar[dl]& &\\
&Q'\ar[dr]&& \tilde C\ar[dl] \ar[dr] && A^n \ar[dl] &
\\
 \ca A_{ng,[s]}&& C' \ar@{_{(}->}[ll] & & C \ar@{^{(}->}[rr]& &\ca A_{g,[r]}.
}
\]
Consider the abelian schemes $f \colon A \to C, g \colon A^n \to C$, $\tilde h \colon \tilde Q \to \tilde C$, and $h' \colon Q' \to C'$, and define $L_f, L_g, L_{\tilde h}$ and $L_{h'}$ as the respective Hodge line bundles of these families.
As $C \subset \Agr$ is a smooth Shimura curve of Mumford type, we have by Corollary \ref{corollary:equality} that $\deg(L_f) = (g/2) \cdot \deg(\Omega_C)$.
By Lemmas \ref{lemma:inequality} and \ref{lemma:higgs-power}, we therefore obtain:
\begin{align} \label{first}
\deg(L_{\tilde h}) = \frac{ng}{2} \cdot \deg(\Omega_{\tilde C}).
\end{align}
The geometric generic fibers of $\tilde Q \to \tilde C$ and $Q' \to C'$ are canonically isomorphic as principally polarized abelian varieties, and by construction isomorphic to the Jacobian $JX$ of the smooth projective connected curve $X$ over $\smash{\overline{\mathbb{Q}(t)}}$. Therefore, the smooth curve $C' \subset \ca A_{ng,[s]}$ is contained in the closed Torelli locus, and intersects the open Torelli locus non-trivially. As $g = 2^d>11$, this implies by \cite[Theorem 1.4]{luzuo-2019} that
\begin{align}\label{align:inequality-proof}
\deg(L_{h'}) < \frac{ng}{2} \cdot \deg(\Omega_{C'}).
\end{align}
The morphism
$
\tilde C \to C'
$
is a finite morphism of smooth curves. Therefore, by Lemma \ref{lemma:inequality}, the strict inequality \eqref{align:inequality-proof} implies that we have a strict inequality
$
\deg(L_{\tilde h}) < (ng/2) \cdot \deg(\Omega_{\tilde C})$, which contradicts \eqref{first}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theorem:special}]
We must provide an abelian variety of dimension $g = 2^d \geq 16$ over $\smash{\overline{\mathbb{Q}(t)}}$ with no power isogenous to a Jacobian. Let $r \geq 3$ be an integer and $C \subset \Agr$ a non-split Shimura curve of Mumford type (cf.\ Definition \ref{definition:nonsplit}); such a pair $(r,C)$ exists by Remark \ref{remark}. Since $C \subset \Agr$ is a Shimura curve, it is defined over $\Qbar$. Let $A \to C$ be the natural abelian scheme. The geometric generic fiber $A_{\bar \eta}$ is a $g$-dimensional abelian variety over $\smash{\overline{\mathbb{Q}(t)}}$ which, by Theorem \ref{theorem:shimuracurve}, has no power isogenous to a Jacobian.
\end{proof}

Let $\eta \in C$ be the generic point of $C$ and let $A \to C$ be the induced abelian scheme. Assume for a contradiction that for some integer $n$ with $1 \leq n \leq N$, we have an isogeny \begin{align} \label{align:isogeny:JX}A_{\bar \eta}^n \to JX, \end{align} where $X$ is a compact type curve over $\Qtbarsmash$. Let $\lambda_{A, \bar \eta}$ (resp.\ $\lambda_{JX}$) be the principal polarization of $A_{\bar \eta}$ (resp.\ $JX$), and let $\mu$ be the polarization on $A^n_{\bar \eta}$ obtained by pulling back $\lambda_{JX}$ to $A^n_{\bar \eta}$ along \eqref{align:isogeny:JX}. As $C$ has maximal monodromy, we have $\End(A_{\bar \eta}) = \Z$, see Lemma \ref{lemma:endo-mono}. By \cite[Lemma 5.2]{dGFSchreieder-2025}, this implies that there exists a symmetric positive definite matrix $\alpha \in \rm{M}_n(\Z)$ such that $\mu = \lambda_\alpha$, see Notation \ref{notation:alpha}. Consider the morphism of moduli spaces
$$
\Phi \colon \ca A_{g,[r]} \to \ca A_{ng, \delta,[r]}, \quad \quad (A, \lambda_A, \bar \phi) \mapsto (A^n, \lambda_\alpha, \bar \phi_\alpha),
$$
where
$
\delta = (\delta_1, \dotsc, \delta_{ng})$ is the type of the polarization $\lambda_\alpha$.
Consider the generic point $\eta \in C \subset \Agr$ of the curve $C$.
There exists an element $a \in \GSp_{2ng}(\QQ)_+$
and a $\Qtbar$ point $\xi \in \ca A_{ng,[r]}$, with $(JX, \lambda_{JX})$ as underlying principally polarized abelian variety over $\Qtbar$, such that with respect to the correspondence $\ca A_{ng,[r]} \xleftarrow{p_1} {\ca A}_a \xrightarrow{p_2} \ca A_{ng,\delta,[r]}$ of Definition \ref{def:a}, we have
\begin{align*}
\xi \in \tau_a(\Phi(\eta)) = p_1(p_2^{-1}(\Phi(\eta))).
\end{align*}
We define $S' = \Phi(S)$ and $C' = \Phi(C)$, so that
$
C' \subset S' \subset \ca A_{ng,\delta, [r]}.
$
Let $C^\# \subset \tau_a(C')$ be the irreducible component of $\tau_a(C')$ with $\xi \in C^\#$. By Corollary \ref{singlecurve-upgraded}, there is no irreducible curve $D \neq C$ with $D\subset S$ such that $C^\# \subset \tau_a(\Phi(D))$.
Therefore, by Theorem \ref{theorem:degree}, there exists an irreducible component $S^\# \subset \tau_a(S')$ with $C^\# \subset S^\#$, such that
 \begin{align}\label{align:ratio:proof}
 \frac{\deg(C^\#)}{\deg(S^\#)} = \frac{\deg(C)}{\deg(S)}.
 \end{align}
As $\xi \in \ca T_{ng,[r]}$, we have $C^\# \subset \ca T_{ng,[r]}$, hence \eqref{intersection} implies that
$
C^\# \subset H_{n, i}$ for each $ i \in I_n$.
By construction, $S \subset \ca A_{g,[r]}$ contains a Shimura curve of Mumford type, hence by Theorem \ref{theorem:shimuracurve}, we have $
S^\# \not \subset \ca T_{ng,[r]} = \bigcap_{n, i \in I_n} H_{n,i}$. Thus, $S^\# \not \subset H_{n,i}$ for some $i \in I_n$. Since the irreducible components of $S^\# \cap H_{n,i}$ have dimension at least one, they have dimension exactly one. In particular, the intersection $S^\# \cap H_{n,i} = S^\# \cap \overline H_{n,i}$ is transverse, so that $$\deg(S^\# \cap \overline H_{n,i} \subset \PP^{N_n}) = \deg(S^\# \subset \PP^{N_n}) \cdot \deg(\overline H_{n,i} \subset \PP^{N_n}).$$
As $C^\# \subset \overline H_{n,i}$, we have
$
C^\# \subset S^\# \cap \overline H_{n, i} \subsetneq S^\#$, hence $C^\#$ is an irreducible component of the purely one-dimensional scheme $S^\# \cap \overline H_{n, i}$. Consequently,
\begin{align*}
\begin{split}
\deg(C^\# \subset \PP^{N_n}) &\leq \deg( S^\# \cap \overline H_{n,i} \subset \PP^{N_n}) \\
& =
\deg(S^\# \subset \PP^{N_n}) \cdot \deg(\overline H_{n,i} \subset \PP^{N_n}),\end{split}
\end{align*}
and hence, by \eqref{align:deg:CS} and \eqref{align:def:d}, we get
\begin{align} \label{inequality:intersection}
\deg(C^\#) \leq k_n \cdot \deg(S^\#) \cdot \deg(\overline H_{n,i} \subset \PP^{N_n}) \leq d \cdot \deg(S^\#).
\end{align}
From \eqref{align:ratio:proof} and \eqref{inequality:intersection}, we deduce $
\deg(C) \leq d\cdot \deg(S)$, which contradicts \eqref{degree-strict}. Thus,
the $n$-th power $A_{\bar \eta}^n$ of
the geometric generic fiber $A_{\bar \eta}$ of the abelian scheme $A \to C$ is not isogenous to a Jacobian, for every $n \leq N$.
\end{proof}

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$. A point $x \in \mathcal{A}_g$ is called \emph{Hodge generic} if $x$ is not contained in a special subvariety of $\mathcal{A}_g$ of positive codimension---equivalently, the associated Mumford--Tate group is $\GSp_{2g}$.
In \cite{tsimerman-jacobian}, Tsimerman showed that one can find CM abelian varieties of dimension $g \geq 4$ which are not isogenous to a Jacobian; in \cite{masser-zannier} and \cite{shankartsimerman-Fptbar}, the authors (using different methods) showed that there are also Hodge generic $\overline{\Q}$ points of $\mathcal{A}_g$ with that property.

As the proof of Theorem \ref{theorem:special} shows, for $g = \smash{ 2^d} \geq 16$, there are $x\in \smash{\mathcal{A}_g(\overline{\Q(t)})}$, the geometric generic point of a special curve $C \subset \mathcal{A}_g$, such that $A_x$ has no power isogenous to a Jacobian. It is natural to ask whether there exist Hodge generic points $x \in \smash{\mathcal{A}_g(\overline{\Q(t)})}$ with that same property. Our second main result says that at least for bounded powers, the answer is yes.

\begin{theorem} \label{theorem:generic}
Let $g = 2^d \geq 16$ and $N \geq 1$. Then there exists a Hodge generic point $x = [A] \in \mathcal{A}_g(\overline{\Q(t)})$ such that the corresponding abelian variety $A$ over $\overline{\Q(t)}$ has no power $A^n$ with $n \leq N$ which is isogenous to a Jacobian.
\end{theorem}

\subsection{Outline of proof}
We now briefly describe our methods. Let $\mathcal A_{g,[r]}$ be the moduli space of principally polarized abelian varieties of dimension $g$ with symplectic level $r\geq 3$ structure. The proof of Theorem \ref{theorem:special} uses an Arakelov inequality due to Lu and Zuo \cite[Theorem 1.4]{luzuo-2019}, which says in particular that given a smooth proper curve $C \subset \mathcal A_{g,[r]}$ with induced abelian scheme $\pi \colon A \to C$, one has $\deg(\pi_\ast \Omega_{A/C}) < (g/2) \cdot \deg(\Omega_C)$ whenever $g \geq 12$ and $C$ is generically contained in the Torelli locus. On the other hand, this becomes an equality if $C$ is a Shimura curve of Mumford type (cf.\ Definition \ref{definition:nonsplit}), from which Lu and Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus if $g \geq 12$. We apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\bar \eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\eta \in C$ the generic point.

We prove the above for Shimura curves of Mumford type $C \subset \mathcal A_{g,[r]}$ which are not necessarily smooth. Such curves exist in $\mathcal A_{g,[r]}$ for each $r \geq 3$. For technical reasons, we proceed in the second part of the paper to assume that $r$ is prime. We pick a suitable surface $S\subset\mathcal A_{g,[r]}$ that contains $C$ as well as Hodge generic points. Theorem \ref{theorem:special} implies that the induced abelian scheme $B\rightarrow S$ has the property that the $n$-th power $B_{\bar \eta}^n$ is not isogenous to a Jacobian for every $n$, where now $\eta \in S$ denotes the generic point of $S$. Bounding $n$, we then use an intersection-theoretic argument to find a curve $C_{\textrm{gen}} \subset S$ over $\overline{\Q}$ that satisfies the conclusion of Theorem \ref{theorem:generic}.

\subsection{Acknowledgements}

We would like to thank Emiliano Ambrosi, Philip Engel, Stefan Schreieder and Jacob Tsimerman for stimulating discussions.

O.d.G.F.~has received funding from the ERC Consolidator Grant FourSurf \textnumero 101087365. A.S.~was partially supported by the NSF grant DMS-2338942, the Institute for advanced studies (via the NSF grant DMS-2424441), and a Sloan research fellowship.

The post-theorem discussion explains that the proof of Theorem~\ref{theorem:special} proceeds by applying an Arakelov inequality of Lu--Zuo \cite[Theorem 1.4]{luzuo-2019} to a (possibly singular) Shimura curve of Mumford type $C\subset \mathcal A_{g,[r]}$ with induced abelian scheme $\pi:A\to C$. The inequality gives, for $g\ge 12$ and $C$ generically contained in the Torelli locus, a strict bound
\[
\deg(\pi_*\Omega_{A/C}) < (g/2)\cdot \deg(\Omega_C),
\]
while for a Shimura curve of Mumford type this “becomes an equality”. Since Lu--Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus for $g\ge 12$, the authors “apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\bar\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\eta\in C$ the generic point,” yielding the conclusion for $A$ over $\overline{\mathbb Q(t)}$ when $g=2^d\ge 16$.

The post-theorem discussion explains that the proof of the main theorem proceeds by applying an Arakelov inequality of Lu--Zuo \cite[Theorem 1.4]{luzuo-2019} to a (possibly singular) Shimura curve of Mumford type $C\subset \mathcal A_{g,[r]}$ with induced abelian scheme $\pi:A\to C$. The inequality gives, for $g\ge 12$ and $C$ generically contained in the Torelli locus, a strict bound
\[
\deg(\pi_*\Omega_{A/C}) < (g/2)\cdot \deg(\Omega_C),
\]
while for a Shimura curve of Mumford type this “becomes an equality”. Since Lu--Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus for $g\ge 12$, the authors “apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\bar\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\eta\in C$ the generic point,” yielding the conclusion for $A$ over $\overline{\mathbb Q(t)}$ when $g=2^d\ge 16$.

Question: Let \(g=2^d\) with \(d\ge 4\), so \(g\ge 16\). Here a Jacobian means the Jacobian of a projective stable curve of compact type. Which statement holds about abelian varieties of dimension \(g\) over \(\overline{\mathbb{Q}(t)}\)?

There exists an abelian variety \(A\) of dimension \(g\) over \(\overline{\mathbb{Q}(t)}\) such that for every integer \(n\ge 1\), the power \(A^n\) is not isogenous to a Jacobian.

Choices:
B. There exists an abelian variety \(A\) of dimension \(g\) over \(\overline{\mathbb{Q}(t)}\) such that for every integer \(n\ge 1\), the power \(A^n\) is not isomorphic to a Jacobian.
C. There exists an abelian variety \(A\) of dimension \(g\) over \(\overline{\mathbb{Q}(t)}\) such that \(A\) itself is not isogenous to a Jacobian.
D. For every abelian variety \(A\) of dimension \(g\) over \(\overline{\mathbb{Q}(t)}\), and for every integer \(n\ge 1\), the power \(A^n\) is not isogenous to a Jacobian.
E. There exists an abelian variety \(A\) of dimension \(g\) over \(\overline{\mathbb{Q}(t)}\) such that for every integer \(1\le n\le g\), the power \(A^n\) is not isogenous to a Jacobian.