# The moment polytope of the abelian polygon space

Navnath Daundkar  Chennai Mathematical Institute, India<EMAIL_ADDRESS>and
Priyavrat Deshpande  Chennai Mathematical Institute, India
<EMAIL_ADDRESS>
###### Abstract.

The moduli space of $n$ chains in the plane with generic side lengths that
terminate on a fixed line is a smooth, closed manifold of dimension $n-1$.
This manifold is also equipped with a locally standard action of
$\mathbb{Z}_{2}^{n-1}$. The orbit space of this action is a simple polytope
called the moment polytope. Interestingly, this manifold is also the fixed
point set of an involution on a toric manifold known as the abelian polygon
space. In this article we show that the moment polytope of the moduli space of
chains is completely characterized by the combinatorial data, called the
_short code_ of the length vector. We also classify aspherical chain spaces
using a result of Davis, Januszkiewicz and Scott.

###### Key words and phrases:

abelian polygon space, real toric variety, moment polytope, aspherical
manifolds

###### 2010 Mathematics Subject Classification:

52B05, 52C25, 55R80, 58D28

Both the authors are partially supported by a grant from the Infosys
foundation. The project is also supported by the second author’s MATRICS grant
funded by the Science and Engineering Board, Government of India.

##  1\. Introduction

A _length vector_ $\alpha:=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$ is an
$m$-tuple of positive real numbers. The _spatial polygon space_
$\mathrm{N}_{\alpha}$ parameterizes all piecewise linear paths in
$\mathbb{R}^{3}$, whose side-lengths are prescribed by $\alpha$, up to rigid
motions. Formally,

$\mathrm{N}_{\alpha}=\\{(v_{1},v_{2},\dots,v_{m})\in(S^{2})^{m}:\displaystyle\sum_{i=1}^{m}\alpha_{i}v_{i}=0\\}/\mathrm{SO}_{3},$

where $\mathrm{SO}_{3}$ acts diagonally. If we choose $\alpha$ such that
$\sum_{i=1}^{m}\pm\alpha_{i}\neq 0$ then the corresponding
$\mathrm{N}_{\alpha}$ is a complex manifold of dimension $2(m-3)$. Such length
vectors are called _generic_. In this article we consider only generic length
vectors. The map $(x,y,z)\mapsto(x,y,-z)$ induces an involution on
$\mathrm{N}_{\alpha}$. Let us denote the fixed point set of this involution by
$\overline{\mathrm{M}}_{\alpha}$. It can also be written as:

$\overline{\mathrm{M}}_{\alpha}=\\{(v_{1},v_{2},\dots,v_{m})\in(S^{1})^{m}:\displaystyle\sum_{i=1}^{m}\alpha_{i}v_{i}=0\\}/\mathrm{O}_{2}.$

The space $\overline{\mathrm{M}}_{\alpha}$ is also called the moduli space of
planar polygons (or _planar polygon space_); it is a smooth, closed manifold
of dimension $m-3$.

Polygon spaces (both spatial and planar) have been studied extensively in past
few decades. For example, the integer cohomology ring of $\mathrm{N}_{\alpha}$
and the mod-$2$ cohomology ring of $\overline{\mathrm{M}}_{\alpha}$ computed
by Hausmann and Knutson in [9]. In [10], the same authors described
$\mathrm{N}_{\alpha}$ as a symplectic quotient of Grasmmannian
$G(2,\mathbb{C}^{m})$ of $2$-planes in $\mathbb{C}^{m}$.

A $2n$-dimensional symplectic, smooth manifold is called a toric manifold if
it admits a Hamiltonian action of an $n$-dimensional torus. Note that a toric
manifold is equivariantly diffeomorphic to a non-singular toric variety (see
[5, Theorem 7.5]) It turns out that for some choices of $\alpha$ the manifold
$\mathrm{N}_{\alpha}$ is a toric manifold. The half-dimensional torus action
is given by bending flows [14]. Hausmann and Roudrigue [12, Proposition 6.8]
provided a (combinatorial) sufficient condition for $\mathrm{N}_{\alpha}$ to
be a toric manifold.

A smooth $2n$-dimensional (respectively $n$-dimensional) manifold $M$ is
called a quasi-toric (respectively small cover) if it has a locally standard
action of $n$-dimensional torus (respectively $\mathbb{Z}_{2}^{n}$) such that
the orbit space can be identiﬁed with a simple $n$-polytope. These manifolds
were introduced by Davis and Januszkiewicz in [5]. They showed that many
topological invariants of these spaces are encoded in the combinatorics of the
associated quotient polytope. In the same paper they prove that the small
cover can be realized as a fixed point set of an involution on a quasi-toric
manifold. Note that in this article we will refer to the quotient polytope as
the moment polytope even in the context of small covers.

Since the Hamiltonion action of a torus is locally standard (see [5, Section
7.3] for a proof) and the image of the moment map is a simple convex polytope,
hence toric manifolds are quasi-toric. Moreover, the fixed point set of an
anti-symplectic involution on a toric manifold is a small cover. This is true
since the image under the restriction of the moment map to the fixed point set
is again the same moment polytope. It can be seen that
$\overline{\mathrm{M}}_{\alpha}$ is the fixed point set of an anti-symplectic
involution on $\mathrm{N}_{\alpha}$. Therefore it has the structure of a small
cover whenever $\mathrm{N}_{\alpha}$ is a toric manifold. We refer the
interested reader to the paper of Hausmann and Knutson [10] for the
terminologies related to symplectic structure that are not defined here.

Whether or not a small cover is aspherical is also determined by the moment
polytope, as given by the following result of Davis, Januszkiewicz and Scott
[4, Theorem 2.2.5].

######  Theorem 1.1.

Let $M$ be a small cover and $P$ be the associated quotient polytope. Then the
following statements are equivalent:

  1. (1)

$M$ is aspherical.

  2. (2)

The boundary complex of $P$ is dual to a flag complex.

  3. (3)

The dual cubical subdivision of $M$ is nonpositively curved.

Our focus in this paper is to determine which planar polygonal spaces are
aspherical; in order to do that we consider the polygonal spaces that are
small covers. In fact, we describe the face poset of the associated moment
polytope in terms of the length vector information.

Some of the planar polygonal spaces that are small covers can also be
described as moduli spaces of chains.

######  Definition 1.2.

(Chain space) Let $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$ be a
length vector. The chain space corresponding to $\alpha$ is defined as :

$\mathrm{Ch}(\alpha)=\\{(v_{1},v_{2},\dots,v_{m-1})\in(S^{1})^{m-1}:\displaystyle\sum_{i=1}^{m-1}\alpha_{i}x_{i}=\alpha_{m}\\}/\mathbb{Z}_{2},$

where $v_{i}=(x_{i},y_{i})$ and the group $\mathbb{Z}_{2}$ acts diagonally.

Note that $\mathrm{Ch}(\alpha)\neq\emptyset$ if
$\alpha_{m}\leq\sum_{i=1}^{m-1}\alpha_{m-1}$. In fact, if $\alpha$ is generic
then $\mathrm{Ch}(\alpha)$ is a smooth, closed manifold of dimension $m-2$.
The elements of a chain space can be thought of as a piece-wise linear path
with side lengths $\alpha_{1},\alpha_{2},\dots,$ $\alpha_{m-1}$ terminating at
the line $x=\alpha_{m}$, considered up to reflection across the $X$-axis.

The spatial version of the chain space for a generic length vector was
introduced by Hausmann and Knutson; it was referred to as the _abelian polygon
space_. It is a _toric manifold_ (see [9, Section 1] for a proof). An element
of this spatial version can be viewed as a piece-wise linear path with side
lengths $\alpha_{1},\alpha_{2},\dots,\alpha_{m-1}$ terminating on the plane
$x=\alpha_{m}$, modulo the rotations about the $X$-axis. It is easy to see
that $\mathrm{Ch}(\alpha)$ is a fixed point set of an anti-symplectic
involution on its spatial version. Thus $\mathrm{Ch}(\alpha)$ is a small
cover.

In this article we characterize those $\alpha$ for which $\mathrm{Ch}(\alpha)$
is aspherical. In order to do that we introduce a combinatorial object called
the _short code_ of a length vector (see, Definition 3.1). We first show that
the short code of a (generic) length vector completely determines the moment
polytope of the associated chain space. Then we establish a combinatorial
condition to determine whether or not the chain space is aspherical. In
particular, we prove the following :

######  Theorem 1.3.

Let $\alpha$ be a generic length vector. Then the corresponding chain space is
aspherical if and only if the short code of $\alpha$ is one of the following :

  1. (1)

$<\\{1,2,\dots,m-3,m\\}>$,

  2. (2)

$<\\{1,2,\dots,m-2,m\\}>$,

  3. (3)

$<\\{1,2,\dots,m-4,m-2,m\\}>$,

  4. (4)

$<\\{1,2,\dots,m-4,m-1,m\\}>$.

The paper is organized as follows : In Section 2, we first state the
motivation for the results in this paper. We also explain why it is difficult
to investigate whether or not of general planar polygon space is aspherical
using techniques of Davis-Januszkiewicz-Scott. In Section 3, we introduce the
notion of _short codes_. We then show that a chain space is diffeomorphic to
some planar polygon space. In Section 4, we describe the moment polytope of a
chain space and explicitly describe it in some special cases. Next we
determine its number of facets in terms of the short subset information. We
then introduce the poset of _admissible subsets_ corresponding to a generic
length vector and prove that it is isomorphic to the face poset of the
corresponding moment polytope. In Section 5, we prove our main result Theorem
1.3.

##  2\. Motivation

The results in our paper are motivated by the techniques developed by Davis,
Januszkiewicz and Scott to conclude that the (real part of) moduli space of
certain point configurations is aspherical. In particular, the authors
consider the following situation: $M_{\mathbb{C}}$ is an $n$-dimensional
complex manifold and $D_{\mathbb{C}}$ is a smooth divisor. Assume that there
is a smooth involution defined on $M_{\mathbb{C}}$ which is locally isomorphic
to complex conjugation on $\mathbb{C}^{n}$. This situation provides a ‘real
version’ of the pair $(M_{\mathbb{C}},D_{\mathbb{C}})$, which we denote by
$(M,D)$, where $M$ is the fixed point set of the involution and
$D=D_{\mathbb{C}}\cap M$. In such a case $D$ is a union of codimension-one
smooth submanifolds which is locally isomorphic to an arrangement of
hyperplanes. Consequently, the complement of $D$ is a disjoint union of cells,
called _chambers_ , which are combinatorially equivalent to simple convex
polytopes. The cell structure induced by $D$ has a cubical subdivision (i.e.,
one can subdivide the polytopal cells to obtain a tiling by cubes). For
(smooth) manifolds equipped such a cell structure the authors established a
combinatorial condition to check whether or not the manifold is aspherical. In
the remaining of this section we formally state their results that are useful
to us and also show that the planar polygonal spaces posses similar structure.

######  Definition 2.1.

An $n$-dimensional convex polytope is called _simple_ if each vertex is an
intersection of $n$ codimension-$1$ faces (also called _facets_). A simple
polytope is called a flagtope if every collection of its pairwise intersecting
facets has a nonempty intersection.

Flagtopes have many interesting combinatorial properties. For example, an
$n$-dimensional flagtope has at least $2n$ facets. In fact, an $n$-cube is the
only flagtope with the minimum possible facets.

######  Definition 2.2.

If all the cells of a regular cell complex are combinatorially equivalent to a
simple polytope then it is called a simple cell complex.

An interesting property of simple cell complexes is that it is possible to
subdivide each cell into cubes, thus turning it into a cubical complex. Recall
that a zonotope is a polytope all of whose faces are centrally symmetric (see
[16, Chapter 7] for more details).

######  Definition 2.3.

A regular cell complex is called zonotopal if every cell is combinatorially
isomorphic to a zonotope.

We can now state an important consequence of Gromov’s lemma that provides a
combinatorial condition to check whether the given cubical complex is non-
positively curved or not. The details of the following result can be found in
[4, Section 1.6]

######  Theorem 2.4.

Suppose that $\mathrm{K}$ is a simple cell complex structure on an
$n$-dimensional smooth manifold such that its dual cell complex is zonotopal.
If each $n$-cell $P$ of $K$ is combinatorially isomorphic to a flagtope then
$K$ is aspherical.

We now show that the planar polygon spaces have a natural simple cell
structure (such that the dual is zonotopal). However, it will turn out that in
most cases the top-cells are not flagtopes.

######  Definition 2.5.

Let $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$ be a generic length
vector. The moduli space $\mathrm{M}_{\alpha}$, of planar polygons is a
collection of all closed piece-wise linear paths in the plane up to
orientation preserving isometries with side lengths
$\alpha_{1},\alpha_{2},\dots,\alpha_{m}$, i.e.,

$\mathrm{M}_{\alpha}=\\{(v_{1},v_{2},\dots,v_{m})\in(S^{1})^{m}:\displaystyle\sum_{i=1}^{m}\alpha_{i}v_{i}=0\\}/\mathrm{SO}_{2}.$

It is known that (see, [7, Theorem 1.3]) the moduli space
$\mathrm{M}_{\alpha}$ is a closed, smooth manifold of dimension $m-3$. The
planar polygon space $\mathrm{M}_{\alpha}$ admits an involution $\tau$ defined
by

(2.1)
$\tau(v_{1},v_{2},\dots,v_{m})=(\bar{v}_{1},\bar{v}_{2},\dots,\bar{v}_{m}),$

where $\bar{v}_{i}=(x_{i},-y_{i})$ and $v_{i}=(x_{i},y_{i})$. Geometrically,
$\tau$ maps a polygon to its reflected image across the $X$-axis. Since the
length vector is generic, the involution $\tau$ does not have fixed points. It
is easy to see that $\mathrm{M}_{\alpha}$ is a double cover of
$\overline{\mathrm{M}}_{\alpha}$.

We denote the set $\\{1,\dots,m\\}$ by $[m]$. An ordered partition of $[m]$
into $k$-blocks is a tuple $(J_{1},J_{2},\dots,J_{k})$, where $J_{i}$’s are
pairwise disjoint subsets of $[m]$ whose union is $[m]$. The set of all
ordered partitions of $[m]$ forms a poset under the refinement order relation.
We consider now a special class of partitions, called a cyclically ordered
partition, which will enable us to define a regular cell structure on
$\mathrm{M}_{\alpha}$.

######  Definition 2.6.

A cyclically ordered partition of $[m]$ is an ordered partition
$(J_{1},J_{2},\dots,J_{k})$ which is equivalent to any ordered partition
obtained from it by a cyclic permutation of its blocks, i.e.,
$(J_{1},J_{2},\dots,J_{k})$, $(J_{2},J_{3},\dots,J_{k},J_{1})$,
$\dots,(J_{k},J_{1},\dots,J_{k-1})$ are all equivalent.

######  Definition 2.7.

Given a length vector $\alpha$ a subset $I\subset[m]$ is called
$\alpha$-_short_ if

$\sum_{i\in I}\alpha_{i}<\sum_{j\not\in I}\alpha_{j}.$

A cyclically ordered partition $(J_{1},J_{2},\dots,J_{k})$ of the set $[m]$ is
said to be $\alpha$-admissible if $J_{i}$ is $\alpha$-short for all $1\leq
i\leq k$.

Panina [15] described a regular cell structure on $\mathrm{M}_{\alpha}$. The
$k$-cells of this complex correspond to $\alpha$-admissible partitions of
$[m]$ into $k+3$ blocks. The boundary relations on the cells are described by
the partition refinement. Let us denote this cell structure by
$\mathrm{K}_{\alpha}$.

It is proved [15, Proposition 2.12] that the complex $\mathrm{K}_{\alpha}$ is
a simple cell complex.

######  Proposition 2.8.

For a generic $\alpha$ the dual of $\mathrm{K}_{\alpha}$ is zonotopal.

###### Proof.

We refer the reader to [15, Section 2] where it is proved that each dual cell
is a product of permutohedra (it is a particular type of simple zonotope). ∎

######  Remark 2.9.

It is clear that the involution $\tau$ (Equation 2.1) defined on
$\mathrm{M}_{\alpha}$ is cellular; the cell labeled by
$(I_{1},I_{2},\dots,I_{k})$ mapped to the cell labeled by
$(I_{k},I_{k-1},\dots,I_{2},I_{1})$. Therefore,
$\overline{\mathrm{M}}_{\alpha}$ has a simple cell structure with cells
labeled by bi-cyclically ordered $\alpha$-admissible partitions. We denote
this cell structure on $\overline{\mathrm{M}}_{\alpha}$ by
$\overline{\mathrm{K}}_{\alpha}$. Note that the dual of this cell complex is
zonotopal.

We now know that for a generic $\alpha$ the complex
$\overline{\mathrm{K}}_{\alpha}$ satisfies the premise of Theorem 2.4. So we
find out whether or not the top-cells of $\overline{\mathrm{K}}_{\alpha}$ are
flagtopes. We do this analysis by considering the number of sides. If $m=3$
then there is only one possibility, $\overline{\mathrm{K}}_{\alpha}$ is always
a point. If $m=4$ then again $\overline{\mathrm{K}}_{\alpha}\cong S^{1}$ for
any generic $\alpha$. If $m=5$ then $\overline{\mathrm{K}}_{\alpha}$ is either
a torus or the nonorientable surfaces of genus $1,2,3,4$ and $5$. It is not
difficult to verify that in case of genus $5$ all of the $12$ top-cells are
pentagonal, hence they are flagtopes. However, in all other cases the cell
structure contains at least one triangular top-cell, see [15, Section 2] for
details. So we can’t appeal to Theorem 2.4 in this case. In fact, for the same
reason we can’t use this theorem for the larger values of $m$.

######  Proposition 2.10.

Let $m\geq 6$ and $\alpha$ be a generic length vector with $m$ components.
Then, the cell structure $\overline{\mathrm{K}}_{\alpha}$ contains no top-cell
isomorphic to a flagtope.

###### Proof.

It is straightforward to see that any top-dimensional (i.e.
$(m-3)$-dimensional) cell of $\overline{\mathrm{K}}_{\alpha}$ has at most $m$
facets. Therefore, the only possible values of $m$ for which the number of
facets are greater than or equal to $2(m-3)$ are $m=3,4,5$ and $6$. Therefore,
if $m\geq 7$ then none of the top-dimensional cells of
$\overline{\mathrm{K}}_{\alpha}$ can be combinatorially isomorphic to a
flagtope. The cases $m=3,4,5$ are dealt above, so here we deal with the case
$m=6$, where the top-dimensional cells of $\overline{\mathrm{K}}_{\alpha}$ may
have $6$ facets.

From [1, Theorem 3] we know that there is only one $3$-dimensional flagtope
with $6$ facets and that is a $3$-cube. We now show that it is impossible for
the corresponding $\overline{\mathrm{K}}_{\alpha}$ to have each top-
dimensional cell of $\overline{\mathrm{K}}_{\alpha}$ isomorphic to a $3$-cube.
To see this assume the contrary that a cell denoted by $\sigma=(1,2,3,4,5,6)$
is combinatorially isomorphic to a $3$-cube. We write $\\{i,j\\}$ as $ij$ for
short. Then $\sigma$ have following $6$ facets,

$(12,3,4,5,6),(1,23,4,5,6),(1,2,34,5,6),(1,2,3,45,6),(1,2,3,4,56),(2,3,4,5,16)$

which are $2$-cubes. Consider the following $2$-faces and their edges.

  1. (1)

The possible faces of $(2,3,4,5,16)$ :

$(23,4,5,16),(2,34,5,16),(2,3,45,16),(2,3,4,156)\hskip
2.84544pt\text{or}\hskip 2.84544pt(3,4,5,126)$

  2. (2)

The possible faces of $(1,2,34,5,6)$ :

$(12,34,5,6),(1,2,34,56),(2,34,5,16),(1,234,5,6)\hskip
2.84544pt\text{or}\hskip 2.84544pt(1,2,345,6)$

  3. (3)

The possible faces of $(1,2,3,45,6)$ :

$(12,3,45,6),(1,23,45,6),(2,3,45,16),(1,2,345,6)\hskip
2.84544pt\text{or}\hskip 2.84544pt(1,2,3,456)$

Note that $156$ cannot be short. Otherwise both $(1,2,34,5,6)$ and
$(1,2,3,45,6)$ will be isomorphic to $2$-simplex, because then $234$, $345$
and $456$ will be long subsets. Since we assumed $(2,3,4,5,16)$ is a $2$-cube,
$126$ must be short. But then a $2$-face $(1,2,3,45,6)$ will be isomorphic to
a $2$-simplex. Which is contradiction. Therefore, if $m=6$, it is impossible
to have each top-dimensional cell of $\overline{\mathrm{K}}_{\alpha}$
isomorphic to a $3$-cube. ∎

Although the natural cell structure of $\overline{\mathrm{K}}_{\alpha}$ is
simple we cannot apply the Davis-Januszkiewicz-Scott schema as the top-cells
are not flagtopes. However, some of these polygon spaces are small covers and
the Davis-Januszkiewicz-Scott theorem can be applied to this situation since
they have a natural cell structure tiled by the moment polytope. If the moment
polytope is a flagtope then the corresponding small cover is aspherical (see
[4, Theorem 2.2.5]).

Hausmann and Roudrigue [12, Proposition 6.8] establish a sufficient condition
for spatial polygon space $\mathrm{N}_{\alpha}$ to be a toric manifold, which
we now state.

######  Theorem 2.11.

Let $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$ be a generic length
vector with $\alpha_{m}\geq\sum_{i=1}^{m-5}\alpha_{i}$ then
$\mathrm{N}_{\alpha}$ is diffeomorphic to a toric manifold.

Recall that $\overline{\mathrm{M}}_{\alpha}$ is a fixed point set of an anti-
syplectic involution on $\mathrm{N}_{\alpha}$. Hence when $\alpha$ satisfies
the condition of Theorem 2.11 corresponding polygon space is a small cover.

######  Remark 2.12.

For a generic length vector $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$,
the half-dimensional torus action on $\mathrm{N}_{\alpha}$ is given by bending
flows (see [10, Section 5], [12, Section 6]). They described the moment
polytope using triangle inequalities.

However, in general it is hard to characterize the face poset of the moment
polytope using those inequalities. For example, we cannot determine the number
of facets of the moment polytope, one of the important information to
determine whether a simple polytope is flagtope or not. So, whenever
$\mathrm{N}_{\alpha}$ is a toric manifold, it is not straightforward to use
Theorem 1.1 to conclude whether $\overline{\mathrm{M}}_{\alpha}$ is aspherical
or not.

In view of this discussion, we now focus on a particular sub-class of planar
polygon spaces, called chain spaces (or abelian polygon spaces), because they
admit a torus action and it is possible to explicitly describe their moment
polytope.

##  3\. The short code of a chain space

As stated above we now focus solely on chain spaces (i.e., real abelian
polygon spaces). In this section, we introduce the notion of a combinatorial
object associated with generic length vector, called the short code. The short
code of a generic length vector is closely related to the _genetic code_
defined by Hausmann in [11, Section 1.5] in the context of polygon spaces. We
also show that chain spaces are planar polygon spaces, up to diffeomorphism.

Since the diffeomorphism type of a chain space does not depend on the ordering
of its side lengths, we assume that our (generic) length vector
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m-1},\alpha_{m})$ satisfies
$\alpha_{1}\leq\alpha_{2}\leq\dots\leq\alpha_{m-1}$. Note that the only
restriction on $\alpha_{m}$ is that it is less than the sum
$\sum_{i=1}^{m-1}\alpha_{i}$.

######  Definition 3.1.

For a generic length vector $\alpha$, we define the following collection of
subsets of $[m]$:

$S_{m}(\alpha)=\\{J\subset[m]:m\in J\hskip 2.84526pt\text{and}\hskip
2.84526ptJ\hskip 2.84526pt\text{is}\hskip 2.84526pt\alpha-\text{short}\\}.$

A partial order $\leq$ is defined on $S_{m}(\alpha)$ by declaring $I\leq J$ if
$I=\\{i_{1},\dots,i_{t}\\}$ and $\\{j_{i},\dots,j_{t}\\}\subseteq J$ with
$i_{s}\leq j_{s}$ for $1\leq s\leq t$. The _short code_ of $\alpha$ is the set
of maximal elements of $S_{m}(\alpha)$ with respect to this partial order. If
$A_{1},A_{2},\dots,A_{k}$ are the maximal elements of $S_{m}(\alpha)$, we
denote the short code as $<A_{1},\dots,A_{k}>$.

######  Example 3.2.

The short codes of length vectors $(1,1,3,3,3)$ and $(1,2,2,5,3)$ are
$<\\{1,2,5\\}>$ and $<\\{1,3,5\\}>$ respectively.

Given a generic length vector
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m-1},\alpha_{m})$ of a chain
space and a positive real number $\delta>\sum_{i=1}^{m-1}\alpha_{i}$, define a
new generic length vector (for a polygon space) as

(3.1)
$\alpha^{\prime}=(\alpha_{1},\alpha_{2},\dots,\alpha_{m-1},\delta,\alpha_{m}+\delta).$

It was shown in [9] that the spatial polygon space
$\mathrm{N}_{\alpha^{\prime}}$ is a toric manifold. In particular, the authors
proved that the moment polytope of the abelian polygon space corresponding to
$\alpha$ and that of $\mathrm{N}_{\alpha^{\prime}}$ are isomorphic [9,
Proposition 1.3]. Hence the two spaces are diffeomorphic. Here we prove the
planar version of their result by providing an explicit diffeomorphism.

######  Proposition 3.3.

Let $\alpha$ be a generic length vector and $\alpha^{\prime}$ be the vector
defined in Equation 3.1. Then the corresponding chain space
$\mathrm{Ch}(\alpha)$ is diffeomorphic to the planar polygon space
$\overline{\mathrm{M}}_{\alpha^{\prime}}$.

###### Proof.

Given a chain with side lengths $\alpha_{1},\alpha_{2},\dots,\alpha_{m-1}$ we
can associate unique polygon with side lengths
$\alpha_{1},\alpha_{2}\dots\alpha_{m-1},\delta,\alpha_{m}+\delta$ in the
following way:

Figure 1.

Let $(v_{1},v_{2},\dots,v_{m-1})\in\mathrm{Ch}(\alpha)$ and let
$\vec{v}=\sum_{i=1}^{m-1}v_{i}$. We have following inequalities :

$|\vec{v}|+(\delta+\alpha_{m})>\delta,\hskip
2.84544pt\delta+(\delta+\alpha_{m})>|\vec{v}|,\hskip
2.84544pt|\vec{v}|+\delta>\delta+\alpha_{m}$

as $\vec{v}=(\alpha_{m},y)$ for some $y\in\mathbb{R}$. Therefore, the sides
lengths $|\vec{v}|$, $\delta$ and $\delta+\alpha_{m}$ satisfy the triangle
inequalities. In fact there exist unique triangle (see Figure 1) with
direction vectors $\vec{v}$, $\vec{w}$ and $\vec{x}$ up to isometries.
Consequently, we have an $m+1$-gon
$(v_{1},v_{2},\dots,v_{m-1},\vec{w},\vec{x})$ with side length
$\alpha_{1},\alpha_{2},\dots,\alpha_{m-1},\delta,$ $\alpha_{m}+\delta$. Its
not hard to see that the map

$\phi:\mathrm{Ch}(\alpha)\longrightarrow\overline{\mathrm{M}}_{\alpha^{\prime}}$

defined by

$\phi((v_{1},v_{2},\dots,v_{m-1}))=(v_{1},v_{2},\dots,v_{m-1},\vec{w},\vec{x})$

is a diffeomorphism. ∎

######  Remark 3.4.

If the short code of a generic length vector
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m-1},\alpha_{m})$ is
$<A_{1}\cup\\{m\\},A_{2}\cup\\{m\\},\dots,A_{k}\cup\\{m\\}>$ where
$A_{i}\subseteq[m-1]$ for all $1\leq i\leq k$ then
$<A_{1}\cup\\{m+1\\},A_{2}\cup\\{m+1\\},\dots,A_{k}\cup\\{m+1\\}>$ is the
genetic code of the length vector $\alpha^{\prime}$ defined in Equation 3.1.

######  Proposition 3.5.

Let $\alpha$ and $\beta$ be two generic length vectors with the same short
code. Then the corresponding chain spaces $\mathrm{Ch}(\alpha)$ and
$\mathrm{Ch}(\beta)$ are diffeomorphic.

###### Proof.

Consider $\alpha^{\prime}$ and $\beta^{\prime}$ be two generic length vectors
defined as in Equation 3.1. Note that the genetic codes of $\alpha^{\prime}$
and $\beta^{\prime}$ coincide, since the short code of $\alpha$ and $\beta$
are same. Consequently [12, Lemma 4.2] and [11, Lemma 1.2 ] give that the the
planar polygon spaces $\overline{\mathrm{M}}_{\alpha^{\prime}}$ and
$\overline{\mathrm{M}}_{\beta^{\prime}}$ corresponding to $\alpha^{\prime}$
and $\beta^{\prime}$ are diffeomorphic. It follows from the Proposition 3.3
that $\mathrm{Ch}(\alpha^{\prime})$ and $\mathrm{Ch}(\beta^{\prime})$ are
diffeomorphic. ∎

######  Corollary 3.6.

Let $<\\{1,2,\dots,m-2,m\\}>$ be the short code of a length vector $\alpha$.
Then the corresponding chain space $\mathrm{Ch}(\alpha)$ is diffeomorphic to
the $(m-2)$-dimensional torus $T^{m-2}$.

###### Proof.

Let $\alpha^{\prime}$ be the new length vector defined as in Equation 3.1.
Using Remark 3.4 we get the genetic code of $\alpha^{\prime}$; which is
$<\\{1,2,\dots,m-2,m+1\\}>$. By [13, Theorem 1] we infer that
$\overline{\mathrm{M}}_{\alpha^{\prime}}$ is diffeomorphic to $T^{m-2}$. Now
the corollary follows from Proposition 3.3. ∎

##  4\. The face poset of the moment polytope

Recall that the chain space $\mathrm{Ch}(\alpha)$ is the real part of a toric
manifold (see the paragraph after Definition 1.2). Our aim is to understand
the topology of this space via the combinatorics of the associated moment
polytpoe. In this section we describe the face poset of the moment polytope.
We show that it is completely determined by the short code of the
corresponding length vector. The moment polytopes corresponding to two
different length vectors are strongly isomorphic if they have the same short
code. In particular, the results in this section imply that if the short codes
are same then the equations defining the corresponding moment polytopes differ
by a constant (equivalently their facets are parallel).

Given a generic length vector $\alpha$, the moment polytope of the
corresponding chain space was first described in [9] as an intersection of a
parallelepiped with a hyperplane. We begin with the description of the moment
map:

$\mu:\mathrm{Ch}(\alpha)\longrightarrow\mathbb{R}^{m-1}$

such that

$\mu(v_{1},\dots,v_{m-1})=(\alpha_{1}x_{1},\dots,\alpha_{m-1}x_{m-1})$

where $v_{i}=(x_{i},y_{i})$. We have

$\mathrm{P}(\alpha):=\mu(\mathrm{Ch}(\alpha))=\\{(w_{1},w_{2},\dots,w_{m-1})\in\prod_{i=1}^{m-1}[-\alpha_{i},\alpha_{i}]:\sum_{i=1}^{m-1}w_{i}=\alpha_{m}\\}.$

Let

$\mathrm{C}^{m-1}(\alpha)=\displaystyle\prod_{i=1}^{m-1}[-\alpha_{i},\alpha_{i}]$

and

$\mathrm{H}(\alpha)=\\{(w_{1},w_{2},\dots,w_{m-1})\in\mathbb{R}^{m-1}:\displaystyle\displaystyle\sum_{i=1}^{m-1}w_{i}=\alpha_{m}\\}.$

Hence the moment polytope
$\mathrm{P}(\alpha)=\mathrm{C}^{m-1}(\alpha)\cap\mathrm{H}(\alpha)$.

It is clear that the facets of $\mathrm{P}(\alpha)$ are given by intersections
of the facets of $\mathrm{C}^{m-1}(\alpha)$ with the hyperplane
$\mathrm{H}(\alpha)$. Note that the facets of $\mathrm{C}^{m-1}(\alpha)$ are
described by the equations $x_{j}=\pm\alpha_{j}$. We call the facets described
by equations $x_{j}=\alpha_{j}$ and $x_{j}=-\alpha_{j}$ as _positive facets_
and _negative facets_ respectively.

Corollary 3.6 says that, if the short code of a length vector is
$<\\{1,2,\dots,m-2,m\\}>$ then the corresponding chain space is diffeomorphic
to $(m-2)$-dimensional torus. So it is clear that the corresponding moment
polytope must be a $(m-2)$-cube. Here we explicitly describe it in terms of
intersections of half-spaces.

######  Lemma 4.1.

Let $<\\{1,2,\dots,m-2,m\\}>$ be the short code of a length vector $\alpha$.
Then the moment polytope $\mathrm{P}(\alpha)\cong I^{m-2}$, the
$(m-2)$-dimensional cube.

###### Proof.

Let $F_{m-1}$ and $\bar{F}_{m-1}$ be the two opposite facets of
$\mathrm{C}^{m-1}(\alpha)$ represented by the equations $x_{m-1}=\alpha_{m-1}$
and $x_{m-1}=-\alpha_{m-1}$ respectively. Note that the set

$\\{(\pm\alpha_{1},\pm\alpha_{2},\dots,\pm\alpha_{m-2},\alpha_{m-1})\\}$

forms the vertices of $F_{m-1}$ and the set

$\\{(\pm\alpha_{1},\pm\alpha_{2},\dots,\pm\alpha_{m-2},-\alpha_{m-1})\\}$

forms the vertices of $\bar{F}_{m-1}$. Since $\\{m-1\\}$ is a long subset we
have the following inequality,

(4.1)
$-\alpha_{m-1}+\sum_{i=1}^{m-2}\pm\alpha_{i}<\alpha_{m}<\alpha_{m-1}+\sum_{i=1}^{m-2}\pm\alpha_{i}.$

The rightmost inequality above implies that, the hyperplane
$\mathrm{H}(\alpha)$ does not intersect the facet $\bar{F}_{m-1}$. Similarly
the leftmost inequality says that $\mathrm{H}(\alpha)$ does not intersect the
facet $F_{m-1}$. Therefore, the hyperplane $\mathrm{H}(\alpha)$ passes through
$\mathrm{C}^{m-1}(\alpha)$ dividing it into two isomorphic polytopes which are
combinatorially isomorphic to $(m-1)$-cubes. We conclude that
$C^{m-1}(\alpha)\cap\mathrm{H}(\alpha)\cong I^{m-2}$, being a facet of both
the divided parts. ∎

Since our aim is to classify the aspherical chain spaces, henceforth we
discard length vectors whose short code is $<\\{1,2,\dots,m-2,m\\}>$, i.e., we
discard the length vector where
$\alpha_{m-1}>\sum_{i=1}^{m-2}\alpha_{i}+\alpha_{m}$. In particular we assume
that $\alpha_{i}<\sum_{j\neq i}\alpha_{j}$.

######  Proposition 4.2.

Let $\alpha=(\alpha_{1},\dots,\alpha_{m})$ be a length vector with
$\alpha_{i}<\sum_{j\neq i}\alpha_{j}$. Then the hyperplane
$\mathrm{H}(\alpha)$ intersects all the positive facet of
$\mathrm{C}^{m-1}(\alpha)$.

###### Proof.

We argue by contradiction. Suppose that the hyperplane $\mathrm{H}(\alpha)$
does not intersect all the positive facets. In particular, let it miss the
facet given by the equation $x_{j_{0}}=\alpha_{j_{0}}$. The coordinate sum of
the elements of $\mathrm{H}(\alpha)$ (which is $\alpha_{m}$) is then strictly
less than the coordinate sum of points on the missed facet, i.e.,

$\alpha_{m}<\alpha_{j_{0}}-\sum_{i\neq j_{o}}\alpha_{i}$

equivalently

$\alpha_{m}+\sum_{i\neq j_{o}}\alpha_{i}<\alpha_{j_{0}}.$

This is impossible since $\alpha$ is generic. ∎

The Proposition 4.2 describes $m-1$ many facets of the moment polytope. We now
characterize length vectors $\alpha$ when $\mathrm{P}(\alpha)$ is a simplex.

######  Lemma 4.3.

Let $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$ be a generic length
vector. The hyperplane $\mathrm{H}(\alpha)$ intersects exactly $m-1$ facets of
$\mathrm{C}^{m-1}(\alpha)$ given by an equations $x_{j}=\alpha_{j}$ for $1\leq
j\leq m-1$ if and only if $\\{m,j\\}$ is long subset for all $1\leq j\leq
m-1$. In particular $\mathrm{P}(\alpha)\cong\bigtriangleup^{m-2}$, an
$(m-2)$-simplex.

###### Proof.

Consider a vertex $v=(\alpha_{1},\dots,\alpha_{m-1})$ of
$\mathrm{C}^{m-1}(\alpha)$ and

$\\{v(j)=(\alpha_{1},\dots,-\alpha_{j},\dots,\alpha_{m-1}):1\leq j\leq m-1\\}$

be its neighboring vertices. We can observe that the hyperplane
$\mathrm{H}(\alpha)$ cannot intersects $v$ since $\alpha$ is generic. Note
that $\mathrm{H}(\alpha)$ intersects exactly $m-1$ positive facets of
$\mathrm{C}^{m-1}(\alpha)$ if and only if the coordinate sum (which is
$\alpha_{m}$) of an element of $\mathrm{H}(\alpha)$ is greater than or equal
the coordinate sum of adjacent vertices of $v(j)$. Let $\beta_{j}=\sum_{i\neq
j}\alpha_{i}-\alpha_{j}$ be the coordinate sum of $v(j)$ for $1\leq j\leq
m-1$. The hyperplane $\mathrm{H}(\alpha)$ intersects exactly $m-1$ positive
facets of $\mathrm{C}^{m-1}(\alpha)$ if and only if

$\alpha_{m}\geq\beta_{j}=\sum_{i\neq j}\alpha_{i}-\alpha_{j},$

for all $1\leq j\leq m-1$. Which gives

$\alpha_{m}+\alpha_{j}\geq\sum_{i\neq j}\alpha_{i}.$

This proves the lemma. ∎

The next result determines the remaining facets of the moment polytope. It
shows that these facets can be determined using the short subset information.

######  Lemma 4.4.

Let $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$ be a generic length
vector. The hyperplane $\mathrm{H}(\alpha)$ intersects a facet of
$\mathrm{C}^{m-1}(\alpha)$ given by an equation $x_{j}=-\alpha_{j}$ for some
$1\leq j\leq m-1$ if and only if $\\{m,j\\}$ is an $\alpha$-short subset.

###### Proof.

Let $v(j)=(\alpha_{1},\dots,-\alpha_{j},\dots,\alpha_{m-1})$ and
$w=(-\alpha_{1},\dots,+\alpha_{j},\dots,-\alpha_{m-1})$ be the two (extreme)
vertices of the facet supported by $x_{j}=-\alpha_{j}$. The coordinate sum of
$v(j)$ is

$\sum_{i\neq j}\alpha_{i}-\alpha_{j}.$

The hyperplane $\mathrm{H}(\alpha)$ intersects the facet supported by
$x_{j}=-\alpha_{j}$ if and only if the sum of the coordinates of an element of
$\mathrm{H}(\alpha)$ is between the coordinate sums of points $v(j)$ and $w$,
i.e., $-\sum_{i}^{m-1}\alpha_{i}\leq\alpha_{m}\leq\sum_{i\neq
j}\alpha_{i}-\alpha_{j}$, equivalently

$\alpha_{m}+\alpha_{j}\leq\sum_{i\neq j}\alpha_{i}.$

This proves the lemma. ∎

Both the Lemma 4.3 and Lemma 4.4 give the exact count of the facets in terms
of the short subset information.

######  Lemma 4.5.

Let $\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{m})$ be a generic length
vector and $k$ be the number of two element $\alpha$-short subsets containing
$m$. Then $\mathrm{P}(\alpha)$ has $m-1+k$ many facets.

###### Proof.

Follows from the Lemma 4.3 and Lemma 4.4. ∎

We now describe the face poset of the moment polytope $\mathrm{P}(\alpha)$ in
terms of certain subsets of $[m]$. We further show that this face poset is
completely determined by the short code.

Define $[m-1]^{+}\coloneqq\\{1,2,\dots,m-1\\}$ and
$[m-1]^{-}\coloneqq\\{\bar{1},\bar{2},\dots,\overline{m-1}\\}$ where
$\bar{i}=-i$. Denote by $[m-1]^{\pm}$ the union $[m-1]^{+}\cup[m-1]^{-}$. Let
$J_{1}\subseteq[m-1]^{+}$ and $J_{2}\subseteq[m-1]^{-}$. We write
$\alpha_{J_{1}}$ for $\sum_{j_{i}\in J_{1}}\alpha_{j_{i}}$ and
$\alpha_{J_{2}}$ for $\sum_{j_{s}\in-J_{2}}\alpha_{j_{s}}$.

######  Definition 4.6.

Let $J\subseteq[m-1]^{\pm}$, such that $J=J_{1}\cup J_{2}$ where
$J_{1}\subseteq[m-1]^{+}$, $J_{2}\subseteq[m-1]^{-}$ and $J_{1}\cap-
J_{2}=\emptyset$. Then $J$ is called an _admissible subset_ of $[m-1]^{\pm}$
if the shorter length vector

$\alpha(J):=(\alpha_{j_{1}},\alpha_{j_{2}},\dots,\alpha_{j_{k}},|\alpha_{m}+\alpha_{J_{1}}-\alpha_{J_{2}}|)$

is generic, where $(J_{1}\cup-J_{2})^{c}=\\{j_{1},\dots,j_{k}\\}$.

We denote the set of all admissible subsets of $[m-1]^{\pm}$ by
$\mathrm{Ad}(\alpha)$. We have a natural partial order on
$\mathrm{Ad}(\alpha)$, $J\leq S$ if and only if $S\subseteq J$. With this
partial order $\mathrm{Ad}(\alpha)$ becomes a poset.

######  Theorem 4.7.

The poset $\mathrm{Ad}(\alpha)$ is isomorphic to the face poset of
$\mathrm{P}(\alpha)$.

###### Proof.

We start by associating a face of the moment polytope to an admissible subset
of $[m-1]^{\pm}$. Let $J$ be an admissible subset of $[m-1]^{\pm}$. Define

$F_{J}\coloneqq\\{(x_{1},x_{2},\dots,x_{m-1})\in\mathrm{P}(\alpha):x_{j_{i}}=\alpha_{j_{i}},j_{i}\in
J_{1},x_{j_{s}}=-\alpha_{j_{s}},j_{s}\in-J_{2}\\}$

Let $F_{i}$ and $\bar{F}_{i}$ be the facets of $\mathrm{C}^{m-1}(\alpha)$
given by an equations $x_{i}=\alpha_{i}$ and $x_{i}=-\alpha_{i}$. Note that

$F_{J}=\displaystyle\bigcap_{j_{i}\in
J_{1}}F_{j_{i}}\bigcap\displaystyle\bigcap_{j_{s}\in-
J_{2}}\bar{F}_{j_{s}}\bigcap\mathrm{H}(\alpha).$

In fact it is easy to see that any face of $\mathrm{P}(\alpha)$ looks like
$F_{J}$ for some admissible subset of $[m-1]^{\pm}$. The map

$\phi:\mathrm{Ad}(\alpha)\longrightarrow\mathrm{P}(\alpha)$

defined by

$\phi(J)=F_{J},$

is a poset isomorphism. ∎

######  Remark 4.8.

We observe straightforwardly that for an admissible subset $J$,
$\mathrm{dim}(F_{J})=m-2-|J|$. Moreover, let $\alpha$ and $\beta$ be two
generic length vectors with an isomorphism between $\mathrm{Ad}(\alpha)$ and
$\mathrm{Ad}(\beta)$. Any such isomorphism restricts to an isomorphism between
$S_{m}(\alpha)$ and $S_{m}(\beta)$. Therefore, the short code of $\alpha$
determines the poset $\mathrm{Ad}(\alpha)$.

######  Proposition 4.9.

Let $<\\{k-1,m\\}>$ and $<\\{k,m\\}>$ be short codes of generic length vectors
$\alpha$ and $\beta$ respectively. Then the moment polytope
$\mathrm{P}(\beta)$ is obtained by truncating a vertex of
$\mathrm{P}(\alpha)$.

###### Proof.

Let
$Z=\\{\overline{1},\overline{2},\dots,\overline{k-1},\overline{k+1},\dots,\overline{m-1}\\}$
be an admissible subset corresponding to $\alpha$. Note that $Z$ represents a
vertex of $\mathrm{P}(\alpha)$. Let
$Z_{i}=Z\setminus\\{\overline{i}\\}\cup\\{k\\}$ for all
$i\in\\{1,2,\dots,k-1,k+1,\dots,m-1\\}$. Each $Z_{i}$ is an admissible subset
corresponding to $\beta$ and represents a vertex of $\mathrm{P}(\beta)$. Let
$\mathrm{Ad}(\alpha)$ and $\mathrm{Ad}(\beta)$ be the posets of an admissible
subsets corresponding to the length vector $\alpha$, $\beta$ respectively.
Then we have,

(4.2) $\mathrm{Ad}(\beta)=\mathrm{Ad}(\alpha)\setminus
Z\bigsqcup\\{\\{k\\},J:J\subseteq Z_{i}\\},$

where $i\in\\{1,2,\dots,k-1,k+1,\dots,m-1\\}$. Note that
$\\{\\{k\\},J:J\subseteq Z_{i}\\}$ represents an $(m-3)$-simplex in
$\mathrm{P}(\beta)$. The Equation 4.2 implies the vertex $Z$ of
$\mathrm{P}(\alpha)$ is replaced by an $(m-3)$-simplex represented by the
admissible subset $\\{k\\}$ thus proving the proposition. ∎

######  Example 4.10.

If the short code of a generic length vector is $<\\{5\\}>$ then the
corresponding moment polytope is a $3$-simplex. On the other hand, if the
short code is $<\\{1,5\\}>$ then the corresponding moment polytope is obtained
by truncating a vertex of the $3$-simplex.

Figure 2. Vertex truncation

Now we show that the poset of admissible subsets $\mathrm{Ad}(\alpha)$ is
determined by the short code of the corresponding length vector.

######  Theorem 4.11.

Let $\alpha=(\alpha_{1},\dots,\alpha_{m})$ be a generic length vector. Let
$J=J_{1}\cup J_{2}$ where $J_{1}\subseteq[m-1]^{+}$ and
$J_{2}\subseteq[m-1]^{-}$ with $J_{1}\cap J_{2}=\emptyset$. Then $J$ is an
admissible subset of $[m-1]^{\pm}$ if and only if $J_{1}\cup\\{m\\}$ and
$-J_{2}$ are $\alpha$-short subsets of $[m]$.

###### Proof.

Suppose $J=J_{1}\cup J_{2}$ is an admissible subset of $[m-1]^{\pm}$. i.e.

(4.3) $|\alpha_{m}+\sum_{i\in J_{1}}\alpha_{i}-\sum_{j\in-
J_{2}}\alpha_{j}|<\sum_{k\in(J_{1}\cup-J_{2})^{c}}\alpha_{k}.$

The above equation gives us two inequalities, the first of which is

(4.4) $\alpha_{m}+\sum_{i\in J_{1}}\alpha_{i}-\sum_{j\in
J_{2}}\alpha_{j}<\sum_{k\in(J_{1}\cup-J_{2})^{c}}\alpha_{k}$

and the second one is

(4.5) $-\sum_{k\in(J_{1}\cup-J_{2})^{c}}\alpha_{k}<\alpha_{m}+\sum_{i\in
J_{1}}\alpha_{i}-\sum_{j\in J_{2}}\alpha_{j}.$

Equation 4.4 gives

$\alpha_{m}+\sum_{i\in J_{1}}\alpha_{i}<\sum_{k\in(J_{1}\cup-
J_{2})^{c}}\alpha_{k}+\sum_{j\in J_{2}}\alpha_{j},$

i.e., $J_{1}\cup\\{m\\}$ is a short subset. Equation 4.5 gives

$\sum_{j\in J_{2}}\alpha_{j}<\alpha_{m}+\sum_{i\in
J_{1}}\alpha_{i}+\sum_{k\in(J_{1}\cup-J_{2})^{c}}\alpha_{k},$

i.e., $-J_{2}$ is a short subset.

Conversely we can obtain the Equation 4.3 using the inequalities in Equation
4.4 and Equation 4.5. It proves the converse since the length vector
$\alpha(J)$ is generic. As the short code determines the collection of short
subsets, it automatically determines the poset of admissible subsets. ∎

Recall the definition of _small covers_ from the Introduction. Note that these
manifolds are a topological generalization of real toric varieties as proved
by Davis and Januszkiewicz in [5]. One of their important result specifies how
to build a small cover from the quotient polytope (see [5, Section 1.5] for
details). It says that there is a regular cell structure on the manifold
consisting of $2^{n}$ copies of the quotient polytope as the top-dimensional
cells. We describe their construction briefly.

Consider an $n$-dimensional simple polytope $P$ with the facet set
$\mathcal{F}=\\{F_{1},\dots,F_{m}\\}$. A function
$\chi:\mathcal{F}\to\mathbb{Z}_{2}^{n}$ is called characteristic for $P$ if
for each vertex $v=F_{i_{1}}\cap\cdots\cap F_{i_{n}}$, the $n\times n$ matrix
whose columns are $\chi(F_{i_{1}}),\dots,\chi(F_{i_{n}})$ is unimodular. Given
the pair $(P,\chi)$ the corresponding small cover $X(P,\chi)$ is constructed
as follows:

$X(P,\chi):=\left((\mathbb{Z}_{2})^{n}\times
P\right)/\\{(t,p)\sim(u,q)\\}\quad\text{if~{}}p=q\text{~{}and~{}}t^{-1}u\in\mathrm{stab}(F_{q})$

where $F_{q}$ is the unique face of $P$ containing $q$ in its relative
interior.

In the remaining section we explicitly determine the entries of the
characteristic matrix for chain spaces. The moment polytope
$\mathrm{P}(\alpha)$ is $(m-2)$-dimensional but described as a subset of
$\mathbb{R}^{m-1}$, so project it onto an affinely isomorphic polytope
$\mathrm{Q}(\alpha)$. This new polytope is embeded in $\mathbb{R}^{m-2}$ and
we determine its outward normals, which determine the characteristic function.

Given a generic length vector
$\alpha=(\alpha_{1},\dots,\alpha_{m-1},\alpha_{m})$, we define the hyperplanes
in $\mathbb{R}^{m-2}$ as follows:

$\bar{H}_{m-1}(\alpha)=\\{(x_{1},x_{2},...,x_{m-2})\in\mathbb{R}^{m-2}:\sum_{i=1}^{m-2}x_{i}=\alpha_{m}-\alpha_{m-1}\\}$

and

$H_{m-1}(\alpha)=\\{(x_{1},x_{2},...,x_{m-2})\in\mathbb{R}^{m-2}:\sum_{i=1}^{m-2}x_{i}=\alpha_{m}+\alpha_{m-1}\\}.$

Let $\bar{H}_{m-1}^{\geq 0}$ and $H_{m-1}^{\leq 0}$ be a positive and negative
part of $\bar{H}_{m-1}$ and $H_{m-1}$ respectively.

######  Theorem 4.12.

Define an $(m-2)$-dimensional polytope as follows

$\mathrm{Q}(\alpha)=\begin{cases}\displaystyle\prod_{i=1}^{m-2}[-\alpha_{i},\alpha_{i}]\cap\bar{H}_{m-1}^{\geq
0},&\text{if $\\{m-1,m\\}$ is long subset,}\\\
\displaystyle\prod_{i=1}^{m-2}[-\alpha_{i},\alpha_{i}]\cap\bar{H}_{m-1}^{\geq
0}\cap H_{m-1}^{\leq o},&\text{if $\\{m-1,m\\}$ is short subset.}\end{cases}$

Then $\mathrm{P}(\alpha)$ is affinely isomorphic to $\mathrm{Q}(\alpha)$.

###### Proof.

Let

$\pi:\mathbb{R}^{m-1}\longrightarrow\mathbb{R}^{m-2}$

be the projection defined by

$\pi(x_{1},x_{2},\dots,x_{m-1})=(x_{1},x_{2},\dots,x_{m-2}).$

Note that $\pi$ restricts to an isomorphism on $\mathrm{H}(\alpha)$. Since
$\mathrm{P}(\alpha)\subseteq\mathrm{H}(\alpha)$, $\pi$ gives a bijection
between $\mathrm{P}(\alpha)$ and $\pi(\mathrm{P}(\alpha))$. We show that
$\mathrm{Q}(\alpha)=\pi(\mathrm{P}(\alpha))$. Suppose $\\{m-1,m\\}$ is a long
subset. Then $\mathrm{H}(\alpha)$ does not intersects a facet of
$\mathrm{C}^{m-1}(\alpha)$ given by $x_{m-1}=-\alpha_{m-1}$. Also $H_{m-1}$
does not intersect $\mathrm{C}^{m-2}(\alpha)$. Let
$(x_{1},x_{2},\dots,x_{m-1})\in\mathrm{P}(\alpha)$. Then the following
inequality is clear:

$\sum_{i=1}^{m-2}x_{i}=\alpha_{m}-x_{m-1}\geq\alpha_{m}-\alpha_{m-1}.$

This gives us $\pi(\mathrm{P}(\alpha))\subseteq\mathrm{Q}(\alpha)$. Now we
show the other inclusion. Let
$(y_{1},y_{2},\dots,y_{m-2})\in\mathrm{Q}(\alpha)$. Then it follows that
$\displaystyle\sum_{i=1}^{m-2}y_{i}\geq\alpha_{m}-\alpha_{m-1}$. Let
$a=\alpha_{m}-\displaystyle\sum_{i=1}^{m-2}y_{i}$ and
$\bar{y}=(y_{1},y_{2},\dots,y_{m-2},a)$. Note that
$\bar{y}\in\mathrm{P}(\alpha)$ as
$\displaystyle\sum_{i=1}^{m-2}y_{i}+a=\alpha_{m}$ and $|a|\leq\alpha_{m-1}$.
Since $\pi(\bar{y})=(y_{1},y_{2},\dots,y_{m-2})$,
$(y_{1},y_{2},\dots,y_{m-2})\in\pi(\mathrm{P}(\alpha))$. Which gives
$\mathrm{Q}(\alpha)\subseteq\pi(\mathrm{P}(\alpha))$. We conclude that
$\mathrm{Q}(\alpha)=\pi(\mathrm{P}(\alpha))$. Similar arguments works when
$\\{m-1,m\\}$ is a short subset. ∎

Note that the facets of $\mathrm{Q}(\alpha)$ are given by following equations.

  * •

$\mathrm{F}_{i}:x_{i}=\alpha_{1}$ for $1\leq i\leq m-2$.

  * •

$\overline{\mathrm{F}}_{m-1}:\displaystyle\sum_{i=1}^{m-2}x_{i}=\alpha_{m}-\alpha_{m-1}$
when $\\{m-1,m\\}$ is long subset.

  * •

$\mathrm{F}_{i}:x_{i}=-\alpha_{i}$ when $\\{i,m\\}$ is short subset.

  * •

$\mathrm{F}_{m-1}:\displaystyle\sum_{i=1}^{m-2}x_{i}=\alpha_{m}+\alpha_{m-1}$
when $\\{m-1,m\\}$ is short subset.

Let $\mathscr{F}(\mathrm{Q}(\alpha))$ be the collection of facets of
$\mathrm{Q}(\alpha)$. We define a map

$\chi_{\alpha}:\mathscr{F}(\mathrm{Q}(\alpha))\longrightarrow\mathbb{Z}_{2}^{m-2}$

as follows:

$\chi_{\alpha}(\overline{\mathrm{F}}_{i})=\begin{dcases}-e_{i},&1\leq i\leq
m-2,\\\
\displaystyle\sum_{i=1}^{m-2}e_{i},&\text{~{}if~{}}\\{m-1,m\\}\hbox{~{}is
long~{}},\end{dcases}$

and

$\chi_{\alpha}(\mathrm{F}_{i})=\begin{dcases}e_{i},&\hbox{~{}if~{}}\\{i,m\\}\hbox{~{}is
short~{}},\\\
-\displaystyle\sum_{i=1}^{m-2}e_{i},&\hbox{~{}if~{}}\\{m-1,m\\}\hbox{~{}is
short~{}}.\end{dcases}$

The following result is clear.

######  Lemma 4.13.

The function $\chi_{\alpha}$ is characteristic for $\mathrm{Q}(\alpha)$.

We can now state the main equivalence.

######  Theorem 4.14.

With the notation as above the chain space $\mathrm{Ch}(\alpha)$ is the small
cover corresponding to the pair $(\mathrm{Q}(\alpha),\chi_{\alpha})$.

###### Proof.

The natural action of $\mathbb{Z}_{2}^{m-1}$ on $(S^{1})^{m-1}$ descends to
$\mathrm{Ch}(\alpha)$. However, this action is not effective. The element
$(r,\dots,r)$ of $\mathbb{Z}_{2}^{m-1}$ (where $r$ generates a copy of
$\mathbb{Z}_{2}$) fixes every element of $\mathrm{Ch}(\alpha)$. So we get the
effective action by dividing by the diagonal subgroup. The remaining details
are easy to verify hence omitted here. ∎

######  Remark 4.15.

Note that the above charcterisitc vectors are normal to the corresponding
facets. Hence, if all the $\alpha_{i}$’s are rational then the moment polytope
is a lattice polytope and the corresponding small cover is a non-singular,
real toric variety. We refer the reader to [5, Section 7] for more on the
characteristic functions of non-singular toric varieties.

######  Example 4.16.

Let $\alpha=(1,2,2,2)$ $\beta=(1,1,2,1)$ and $\gamma=(2,2,2,1)$ be generic
length vectors. The shaded regions denotes the corresponding moment polytopes.
The corresponding characterstic functions are given by matrices
$\chi_{\alpha}=\begin{bmatrix}1&0&1&1\\\ 0&1&0&1\end{bmatrix}$,
$\chi_{\beta}=\begin{bmatrix}1&0&1&1&0\\\ 0&1&0&1&1\end{bmatrix}$ and
$\chi_{\gamma}=\begin{bmatrix}1&1&0&1&1&0\\\ 0&1&1&0&1&1\end{bmatrix}.$

Figure 3. Moment polytopes

##  5\. The proof of Theorem 1.3

A polytope is _triangle-free_ if it does not contain a triangular face of
dimension $2$. It is easy to see that flagtopes are triangle-free. We state
the important result due to G. Blind and R. Blind, which characterize those
triangle-free polytopes that are also flagtopes. Recall that $I^{k}$ stands
for the $k$-cube.

######  Theorem 5.1.

[1, Theorem 3] If ${P}$ is a triangle-free convex polytope of dimension n then
$f_{i}(P)\geq f_{i}(I^{n})$ for $i=0,\dots,n$. In particular, such a polytope
has at least $2n$ facets. Furthermore, if ${P}$ is simple then

  1. (1)

$f_{n-1}(P)=2n$ implies that ${P}=I^{n}$ ;

  2. (2)

$f_{n-1}(P)=2n+1$ implies that $P=P_{5}\times I^{n-2}$ where $P_{5}$ is a
pentagon;

  3. (3)

$f_{n-1}(P)=2n+2$ implies that $P=P_{6}\times I^{n-2}$ or $P=Q\times I^{n-3}$
or $P=P_{5}\times P_{5}\times I^{n-4}$ where $P_{6}$ is a hexagon and
$\mathrm{Q}$ is the $3$-polytope obtained from a pentagonal prism by
truncating one of the edges forming a pentagonal facet.

The above theorem helps us classify length vectors $\alpha$’s for which the
moment polytope ${P}(\alpha)$ is a flagtope.

######  Theorem 5.2.

Let $\alpha$ be a generic length vector then the moment polytope
$\mathrm{P}(\alpha)$ is a flagtope if and only if the short code of $\alpha$
is one of the following:

  1. (1)

$<\\{1,2,\dots,m-3,m\\}>$

  2. (2)

$<\\{1,2,\dots,m-4,m-2,m\\}>$

  3. (3)

$<\\{1,2,\dots,m-4,m-1,m\\}>$

###### Proof.

First we describe the idea of the proof, which is the same in each case. It
follows from Lemma 4.5 that, the number of facets of $\mathrm{P}(\alpha)$
corresponding to each short code mentioned above, in that order, are $2(m-2)$,
$2(m-2)+1$ and $2(m-2)+2$, respectively. Therefore, to show that the polytope
$\mathrm{P}(\alpha)$ is triangle-free, we need to make sure that each
$2$-dimensional face is not a $2$-simplex. Note that the $2$-dimensional faces
of $\mathrm{P}(\alpha)$ correspond to admissible subsets of cardinality $m-4$.
Therefore, in each case we determine these subsets and show that the
corresponding $2$-dimensional faces have at least $4$ edges. Then we use
Theorem 5.1 to conclude $\mathrm{P}(\alpha)$ is a flagtope. We now analyze
each short code.

Case 1. _The short code of $\alpha$ is $<\\{1,2,\dots,m-3,m\\}>$._

Let $J=J_{1}\cup J_{2}$ be an admissible subset of $[m-1]^{\pm}$ where
$J_{1}\subset[m-1]^{+}$, $J_{2}\subset[m-1]^{-}$ with $J_{1}\cap-
J_{2}=\emptyset$ and $|J|=m-4$. By Theorem 4.11 we have that
$J_{1}\cup\\{m\\}$ and $-J_{2}$ are $\alpha$-short subsets. Moreover
$J_{1}\subset[m-3]$ as short code of $\alpha$ is $<\\{1,2,\dots,m-3,m\\}>$.
Note that $\\{m-2,m-1\\}$ cannot be a subset of $-J_{2}$ since it is long.
There are following three possibilities for $-J_{2}$.

  1. (1)

We have _$-J_{2}\subset[m-3]$_ : Note that the facets of $F_{J}$ correspond to
admissible subsets of cardinality $m-3$ containing $J$. We have,
$\\{i,m-2,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ for $i\in[m-3]$. Clearly, the
subsets $\\{i\\}\cup J_{1}\cup\\{m\\}$ and $\\{i\\}\cup-J_{2}$ are
$\alpha$-short. Moreover, the subsets $\\{m-2\\}\cup-J_{2}$ and
$\\{m-1\\}\cup-J_{2}$ are $\alpha$-short, since their complements contains
long subsets, $\\{m-1,m\\}$ and $\\{m-2,m\\}$ respectively. Therefore, the
subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{-(m-2)\\}\cup
J,\hskip 2.84526pt\\{-(m-1)\\}\cup J$

are admissible. Note that these admissible subsets represents the facets of
$F_{J}$. Clearly $F_{J}\cong\mathrm{I}^{2}$.

  2. (2)

We have _$m-2\in-J_{2}$ but $m-1\not\in-J_{2}$_: In this case we have
$\\{i,j,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where $\\{i.j\\}\subseteq[m-3]$.
Note that the subsets,

$\\{i\\}\cup J_{1}\cup\\{m\\},\hskip 2.84526pt\\{j\\}\cup
J_{1}\cup\\{m\\},\hskip 2.84526pt\\{i\\}\cup-J_{2},\hskip
2.84526pt\\{j\\}\cup-J_{2}$

are $\alpha$-short. Therefore, the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible. Therefore, $2$-dimensional faces $F_{J}$ have at least four
facets.

  3. (3)

We have _$m-1\in-J_{2}$ but $m-2\not\in-J_{2}$_: This is exactly similar to
the earlier case.

As a consequence of Lemma 4.5 we have, $f_{m-3}(\mathrm{P}(\alpha))=2(m-2)$.
Therefore, using Theorem 5.1 we get $\mathrm{P}(\alpha)\cong I^{m-2}$, the
$(m-2)$-cube.

Case 2. _The short code of $\alpha$ is $<\\{1,2,\dots,m-4,m-2,m\\}>$._

Note that $f_{m-3}(\mathrm{P}(\alpha))=2(m-2)+1$. The short code information
gives following possibilities for $J_{1}$ and $J_{2}$.

  * •

_Possibilities for $J_{1}$_:

    1. (1)

$J_{1}\subset[m-3]$

    2. (2)

$m-2\in J_{1}$ and $J_{1}\setminus\\{m-2\\}\subset[m-4]$

  * •

_Possibilities for $J_{2}$_:

    1. (1)

$-J_{2}\subset[m-2]$

    2. (2)

$-J_{2}=S\cup\\{m-1\\}$ where $S\subseteq[m-4]$ as $\\{m-3,m-1\\}$ and
$\\{m-2,m-1\\}$ are long subsets.

We now consider all four possible combinations of $J_{1}$ and $J_{2}$.

  1. (1)

_$J_{1}\subset[m-3]$ and $-J_{2}\subset[m-2]$_ :

Note that $\\{i,j,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$\\{i.j\\}\subseteq[m-2]$. Consider the following possibilities for
$\\{i.j\\}$ :

    1. (a)

$\\{i,j\\}\subset[m-3]$ :

Observe that $S\cup\\{m\\}$ is $\alpha$-short if $S\subset[m-3]$. Therefore,
$\\{i\\}\cup J_{1}\cup\\{m\\}$ and $\\{j\\}\cup J_{1}\cup\\{m\\}$ are
$\alpha$-short subsets. Since $\\{m-1,m\\}$ is long, any subset of $[m-2]$ is
$\alpha$-short. Consequently, $\\{i\\}\cup-J_{2}$ and $\\{j\\}\cup-J_{2}$ are
$\alpha$-short subsets. Hence

$\\{i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible subsets. Therefore, the $2$-dimensional faces $F_{J}$ have at
least four edges.

    2. (b)

_$i\in[m-3]$ and $j=m-2$_ :

In this case we have $\\{i,m-2,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$. Consider
the following three sub-cases :

      1. (i)

_If $i\in[m-4]$ and $m-3\in J_{1}$ :_

Note that $\\{i\\}\cup J_{1}\cup\\{m\\}$, $\\{i\\}\cup-J_{2}$ and
$\\{m-2\\}\cup-J_{2}$ are short subsets. Since $m-3\in J_{1}$, $m-3\notin-
J_{2}$. Therefore, $-J_{2}\subset[m-4]$. Hence, a subset $\\{m-1\\}\cup-J_{2}$
is $\alpha$-short. Therefore,

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{-(m-2)\\}\cup
J,\hskip 2.84526pt\\{-(m-1)\\}\cup J$

are admissible subsets.

      2. (ii)

_If $i\in[m-4]$ and $m-3\in-J_{2}$ :_ This case is exactly same as above case.

      3. (iii)

_If $i=m-3$ :_

Note that $\\{m-3,m-2,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$. It is easy to see
that the following subsets

$\\{m-3\\}\cup J,\hskip 2.84526pt\\{m-2\\}\cup J,\hskip
2.84526pt\\{-(m-3)\\}\cup J,\hskip 2.84526pt\\{-(m-2)\\}\cup J,\hskip
2.84526pt\\{-(m-1)\\}\cup J$

are admissible containing $J$ and represents the facets of $F_{J}$. Therefore,
$2$-dimensional faces $F_{J}$ have exactly five edges.

  2. (2)

_$J_{1}\subset[m-3]$ and $-J_{2}=S\cup\\{m-1\\}$ where $S\subset[m-4]$ :_

In this case we have $\\{i,j,m-2\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$\\{i.j\\}\subseteq[m-3]$. Consider the following possibilities for
$\\{i.j\\}$ :

    1. (a)

_Suppose $\\{i.j\\}\subset[m-4]$ :_

It is easy to see that the following subsets

$\\{i\\}\cup J_{1}\cup\\{m\\},\hskip 2.84526pt\\{j\\}\cup J_{1},\hskip
2.84526pt\\{j\\}\cup-J_{2},\hskip 2.84526pt\\{j\\}\cup-J_{2}$

are $\alpha$-short. Hence, the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible.

    2. (b)

_Suppose $i\in[m-4]$ and $j=m-3$ :_

In this case we have $\\{i,m-3,m-2\\}\subseteq(J_{1}\cup-J_{2})^{c}$. Since
$m-3\notin J_{1}$, $J_{1}\subset[m-4]$. Hence the subsets, $\\{m-3\\}\cup
J_{1}\cup\\{m\\}$ and $\\{m-2\\}\cup J_{1}\cup\\{m\\}$ are $\alpha$-short.
Note that, $\\{i\\}\cup J_{1}$ and $\\{i\\}\cup-J_{2}$ are also $\alpha$-short
subsets. Consequently,

$\\{m-3\\}\cup J,\hskip 2.84526pt\\{m-2\\}\cup J,\hskip 2.84526pt\\{i\\}\cup
J,\hskip 2.84526pt\\{-i\\}\cup J$

are admissible subsets.

  3. (3)

_$m-2\in J_{1}$ , $J_{1}\setminus\\{m-2\\}\subset[m-4]$ and
$-J_{2}\subset[m-2]$ :_

Observe that, $\\{i,j,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$\\{i.j\\}\subseteq[m-3]$. Consider the following possibilities for
$\\{i.j\\}$ :

    1. (a)

_If $\\{i,j\\}\subset[m-4]$ :_

Clearly the subsets,

$\\{i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible.

    2. (b)

_If $i\in[m-4]$ and $j=m-3$ :_

In this case we have $\\{i,m-3,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ . Since
$m-2\in J_{1}$, $m-2\notin-J_{2}$. On the other hand, since $m-3\notin-J_{2}$,
$-J_{2}\subset[m-4]$. Note that, $S\cup\\{m-1\\}$ is $\alpha$-short for
$S\subset[m-4]$. Therefore, $\\{m-1\\}\cup-J_{2}$ is $\alpha$-short subset.
Consequently $\\{-(m-1)\\}\cup J$ is $\alpha$-admissible subset. Hence,

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{-(m-3)\\}\cup
J,\hskip 2.84526pt\\{-(m-1)\\}\cup J$

are admissible subsets.

  4. (4)

_$m-2\in J_{1}$ , $J_{1}\setminus\\{m-2\\}\subset[m-4]$ and $J_{2}=S\cup[m-1]$
, $S\subset[m-4]$ :_

Here, we have $\\{i,j,m-3\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$\\{i,j\\}\subseteq[m-4]$. Clearly the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible subsets.

Finally we conclude that $\mathrm{P}(\alpha)$ is triangle-free if the short
code of $\alpha$ is $<\\{1,2,\dots,m-4,m-1,m\\}>$. Moreover using Theorem 5.1
and Lemma 4.5, we get $\mathrm{P}(\alpha)\cong P_{5}\times I^{m-4}$, where
$P_{5}$ is a pentagon.

Case 3. _The short code of $\alpha$ is $<\\{1,2,\dots,m-4,m-1,m\\}>$._

Note that the short code of $\alpha$ gives us the following possibilities of
$J_{1}$ and $J_{2}$ :

  * •

_Possibilities for $J_{1}$:_

    1. (1)

$J_{1}\subseteq[m-3]$

    2. (2)

$J_{1}\subseteq[m-4]$ and $m-2\in J_{1}$

    3. (3)

$J_{1}\subseteq[m-4]$ and $m-1\in J_{1}$

  * •

_Possibilities for $J_{2}$_

    1. (1)

$-J_{2}\subseteq[m-3]$

    2. (2)

$-J_{2}\subseteq[m-4]$ and $m-2\in-J_{2}$

    3. (3)

$-J_{2}\subseteq[m-4]$ and $m-1\in-J_{2}$

We consider the all nine possible combinations of $J_{1}$ and $J_{2}$. Let’s
begin with the fist combination:

  1. (1)

_$J_{1}\subseteq[m-3]$ and $-J_{2}\subseteq[m-3]$ :_

Observe that, $\\{i,m-2,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$i\in[m-3]$. Since $\\{m-2,m-1\\}$ is a long, the subsets $\\{i\\}\cup
J_{1}\cup\\{m\\}$ and $i\cup-J_{2}$ are $\alpha$-short. Consider the following
possibilities :

    1. (a)

_$m-3\in J_{1}$ :_

Since $J_{1}\cap-J_{2}=\emptyset$, $-(m-3)\notin J_{2}$. Hence
$-J_{2}\subseteq[m-4]$. Therefore, $\\{m-2\\}\cup-J_{2}$ and $\\{m-1\\}\cup-
J_{2}$ are $\alpha$-short subsets. Consequently, the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{-(m-2)\\}\cup
J,\hskip 2.84526pt\\{-(m-1)\\}\cup J$

are admissible.

    2. (b)

_$m-3\notin J_{1}$_ :

Therefore $J_{1}\subseteq[m-4]$. Hence, the subsets $\\{m-2\\}\cup
J_{1}\cup\\{m\\}$ and $\\{m-1\\}\cup J_{1}\cup\\{m\\}$ are $\alpha$-short.
Consequently, the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{m-2\\}\cup
J,\hskip 2.84526pt\\{m-1\\}\cup J$

are admissible.

  2. (2)

_$J_{1}\subseteq[m-3]$ and $-J_{2}\subseteq[m-4]$, $m-2\in-J_{2}$_:

Since $\\{m-3.m-2\\}$ is long, $m-3\notin-J_{2}$. In this case we have
$\\{i,j,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where $\\{i,j\\}\subseteq[m-3]$.
Therefore, the subsets $\\{i\\}\cup J_{1}\cup\\{m\\}$ and $\\{j\\}\cup
J_{1}\cup\\{m\\}$ are $\alpha$-short. Consider the following possibilities:

    1. (a)

_$m-3\in J_{1}$_ :

Hence $\\{i,j\\}\subseteq[m-4]$. Therefore, $\\{i\\}\cup-J_{2}$ and
$\\{j\\}\cup-J_{2}$ are $\alpha$-short subsets. Hence, the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible.

    2. (b)

_$m-3\notin J_{1}$ :_

Hence $J_{1}\subseteq[m-4]$. Therefore, $\\{m-1\\}\cup J_{1}\cup\\{m\\}$ is a
$\alpha$-short subset. Since $m-3\notin-J_{2}$ ,
$\\{i,m-3,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where $i\in[m-4]$. So
$\\{i\\}\cup-J_{2}$ is $\alpha$-short subset. Therefore, the subsets,

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{m-3\\}\cup
J,\hskip 2.84526pt\\{m-1\\}\cup J$

are admissible.

  3. (3)

_$J_{1}\subseteq[m-3]$ and $-J_{2}\subseteq[m-4]$, $m-1\in-J_{2}$ :_ This case
is exactly similar to the above case.

  4. (4)

_$J_{1}\subseteq[m-4]$ , $m-2\in J_{1}$ and $-J_{2}\subseteq[m-3]$ :_

In this case we have $\\{i,j,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$\\{i,j\\}\subseteq[m-3]$. Consider the following possibilities :

    1. (a)

_$m-3\in-J_{2}$ :_

Hence $\\{i,j\\}\subseteq[m-4]$. Therefore, the subsets,

$\\{i\\}\cup J_{1}\cup\\{m\\},\hskip 2.84526pt\\{j\\}\cup
J_{1}\cup\\{m\\},\hskip 2.84526pt\\{i\\}\cup-J_{2},\hskip
2.84526pt\\{j\\}\cup-J_{2}$

are $\alpha$-short. Hence the subsets,

$\\{i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible.

    2. (b)

_$m-3\notin-J_{2}$ :_

In this case we have $\\{i,m-3,m-1\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$i\in[m-4]$. Therefore,

$\\{i\\}\cup J_{1}\cup\\{m\\},\hskip 2.84526pt\\{i\\}\cup-J_{2},\hskip
2.84526pt\\{m-3\\}\cup-J_{2},\hskip 2.84526pt\\{m-1\\}\cup-J_{2}$

are $\alpha$-short subsets. Consequently, the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup J,\hskip 2.84526pt\\{-(m-3)\\}\cup
J,\hskip 2.84526pt\\{-(m-1)\\}\cup J$

are admissible.

  5. (5)

_$J_{1}\subseteq[m-4]$ , $m-2\in J_{1}$ and $-J_{2}\subseteq[m-4]$ and
$m-2\in-J_{2}$ :_

This case doesn’t not arise as $J_{1}\cap-J_{2}\neq\emptyset$.

  6. (6)

_$J_{1}\subseteq[m-4]$ , $m-2\in J_{1}$ and $-J_{2}\subseteq[m-4]$, $m-1\in-
J_{2}$ :_

In this case we have $\\{i,j,m-3\\}\subseteq(J_{1}\cup-J_{2})^{c}$ where
$\\{i.j\\}\subseteq[m-4]$. It is easy to see that the subsets,

$\\{i\\}\cup J_{1}\cup\\{m\\},\hskip 2.84526pt\\{j\\}\cup
J_{1}\cup\\{m\\},\hskip 2.84526pt\\{i\\}\cup-J_{2},\hskip
2.84526pt\\{j\\}\cup-J_{2}$

are $\alpha$-short subsets. Hence the subsets

$\\{i\\}\cup J,\hskip 2.84526pt\\{j\\}\cup J,\hskip 2.84526pt\\{-i\\}\cup
J,\hskip 2.84526pt\\{-j\\}\cup J$

are admissible.

  7. (7)

_$J_{1}\subseteq[m-4]$ , $m-1\in J_{1}$ and $-J_{2}\subseteq[m-3]$ :_ This
case is exactly similar to the case where, _$J_{1}\subseteq[m-4]$ , $m-2\in
J_{1}$ and $-J_{2}\subseteq[m-3]$_.

  8. (8)

_$J_{1}\subseteq[m-4]$ , $m-1\in J_{1}$ and $-J_{2}\subseteq[m-4]$, $m-2\in-
J_{2}$ :_ This case is exactly similar to the case where,
_$J_{1}\subseteq[m-4]$ , $m-2\in J_{1}$ and $-J_{2}\subseteq[m-4]$, $m-1\in-
J_{2}$ :_

  9. (9)

_$J_{1}\subseteq[m-4]$ and $m-1\in J_{1}$ and $-J_{2}\subseteq[m-4]$ and
$m-1\in-J_{2}$ _ This case doesn’t arise as $J_{1}\cap-J_{2}\neq\emptyset$.

Moreover, using Theorem 5.1 and Lemma 4.5, we get $\mathrm{P}(\alpha)\cong
P_{6}\times I^{m-4}$, where $P_{6}$ is a hexagon.

Now we prove the converse. Recall that, the Remark 4.8 says that the poset of
an admissible subsets determine the short code of a length vector. In
particular, for generic length vectors $\alpha$ and $\beta$, if
$\mathrm{P}(\alpha)\cong\mathrm{P}(\beta)$ then the short codes of $\alpha$
and $\beta$ coincides. Suppose, $\mathrm{P}(\alpha)$ is a flagtope for a
generic length vector $\alpha$. Note that Lemma 4.5 give,

$2(m-2)\leq f_{m-3}(\mathrm{P}(\alpha))\leq 2(m-2)+2.$

Therefore, there are only three possibilities for the number of facets of
$\mathrm{P}(\alpha)$. Suppose that $f_{m-3}(\mathrm{P}(\alpha))=2(m-2)$.
Theorem 5.1 shows that $\mathrm{P}(\alpha)\cong I^{m-2}$, a $m-2$-cube. But we
have shown that $I^{m-2}$ is the moment polytope for short code
$<\\{1,2,\dots,m-3,m\\}>$. Therefore, the short code of $\alpha$ is
$<\\{1,2,\dots,m-3,m\\}>$ whenever $\mathrm{P}(\alpha)$ is flagtope and
$f_{m-3}(\mathrm{P}(\alpha))=2(m-2)$. Similar arguments works when
$f_{m-3}(\mathrm{P}(\alpha))=2(m-2)+1$ and
$f_{m-3}(\mathrm{P}(\alpha))=2(m-2)+2$. ∎

The proof of Theorem 1.3 now follows from Theorem 5.2 and Theorem 1.1.

######  Example 5.3.

Below are some examples.

  1. (1)

If $\alpha=(1,1,3,3,3)$ then the short code of $\alpha$ is $<\\{1,2,5\\}>$ and
$\mathrm{P}(\alpha)\cong I^{3}$.

Figure 4.

  2. (2)

If $\alpha=(1,2,2,5,3)$ then the short code of $\alpha$ is $<\\{1,3,5\\}>$ and
$\mathrm{P}(\alpha)\cong P_{5}\times I$.

Figure 5.

  3. (3)

If $\alpha=(1,3,3,3,1)$ then the short code of $\alpha$ is $<\\{1,4,5\\}>$ and
$\mathrm{P}(\alpha)\cong P_{6}\times I$.

Figure 6.

We now describe the homeomorphism type of these aspherical chain spaces.
However, before that we mention an important and relevant result about polygon
spaces. Let $<A_{1},\dots,A_{k}>$ be the genetic code of
$\alpha=(\alpha_{2},\dots,\alpha_{m})$. Consider another generic length vector
$\alpha^{+}$ whose genetic code is $<B_{1},\dots,B_{k}>$ where
$B_{i}=\\{a+1:a\in A_{i}\\}\cup\\{1\\}$. Note that $\alpha^{+}$ is an
$m$-tuple. Hausmann [11, Proposition 2.1] described a relationship between the
polygon spaces $\overline{\mathrm{M}}_{\alpha^{+}}$ and
$\overline{\mathrm{M}}_{\alpha}$.

######  Proposition 5.4.

There polygon space $\overline{\mathrm{M}}_{\alpha^{+}}$ is diffeomorphic to
the fibered product $S^{1}\times_{O(1)}\mathrm{M}_{\alpha}$, where $O(1)$ acts
diagonally.

Recall that

$q:\mathrm{M}_{\alpha}\longrightarrow\overline{\mathrm{M}}_{\alpha}$

is a double cover. This double cover helps us to define a natural map

$\Phi:\overline{\mathrm{M}}_{\alpha^{+}}\longrightarrow\overline{\mathrm{M}}_{\alpha}.$

It is easy to see that $\Phi$ is an $S^{1}$-fibration.

Now we return to aspherical chain spaces. Recall Proposition 3.3; it says that
given a generic length vector $\alpha=(\alpha_{1},\dots,\alpha_{m})$ we have
an isomorphism between

$\mathrm{Ch}(\alpha)\longrightarrow\overline{\mathrm{M}}_{\alpha^{\prime}}$

where

$\alpha^{\prime}=(\alpha_{1},\alpha_{2},\dots,\alpha_{m-1},\delta,\alpha_{m}+\delta)$

for some positive real number $\delta>\sum_{i=1}^{m-1}\alpha_{i}$.

Suppose the short code of $\alpha$ is $<1,2,\dots,m-3,m>$. Then the genetic
code of $\alpha^{\prime}$ is $<1,2,\dots,m-3,m+1>$. Let $\beta$ be a generic
length vector whose genetic code is $<1,2,\dots,m-4,m>$. Clearly,
$\beta^{+}=\alpha^{\prime}$. Let
$\alpha(1)=(\alpha_{2},\dots,\alpha_{m}+\alpha_{1})$ be a generic length
vector. Note that the short code of $\alpha(1)$ is $<1,2,\dots,m-4,m-1>$ with
$\alpha(1)^{\prime}=\beta$. We have an isomorphism
$\mathrm{Ch}(\alpha(1))\longrightarrow\overline{\mathrm{M}}_{\alpha(1)^{\prime}}$.
Using Hausmann’s $S^{1}$-fibration described in the proof of Proposition 5.4,
we have the following maps.

$\mathrm{Ch}(\alpha)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\overline{\mathrm{M}}_{\alpha^{\prime}}\stackrel{{\scriptstyle\Phi_{1}}}{{\longrightarrow}}\overline{\mathrm{M}}_{\beta}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathrm{Ch}(\alpha(1)).$

Clealry, the above composition of maps gives a $S^{1}$-fibration
$\tilde{\Phi}$ from

$\mathrm{Ch}(\alpha)\stackrel{{\scriptstyle\tilde{\Phi}_{1}}}{{\longrightarrow}}\mathrm{Ch}(\alpha(1)).$

Let $I_{j}=\\{1,2,\dots,j\\}$. By induction we get the chain of
$S^{1}$-fibrations

$\mathrm{Ch}(\alpha)\stackrel{{\scriptstyle\tilde{\Phi}_{1}}}{{\longrightarrow}}\mathrm{Ch}(\alpha(I_{1}))\stackrel{{\scriptstyle\tilde{\Phi}_{2}}}{{\longrightarrow}}\mathrm{Ch}(\alpha(I_{2}))\stackrel{{\scriptstyle\tilde{\Phi}_{3}}}{{\longrightarrow}}\dots\stackrel{{\scriptstyle\tilde{\Phi}_{m-3}}}{{\longrightarrow}}\mathrm{Ch}(\alpha(I_{m-3}))\stackrel{{\scriptstyle\tilde{\Phi}_{m-2}}}{{\longrightarrow}}\\{\star\\},$

where
$\alpha(I_{j})=(\alpha_{j+1},\dots,\alpha_{m-1},\alpha_{m}+\displaystyle\sum_{i=1}^{j}\alpha_{i})$.

Recall that the moment polytope is the $(m-2)$-cube. Hence,the above chain of
fibrations is a real Bott tower. One can easily check that the characterstic
matrix of $\mathrm{Ch}(\alpha)$ is

$\begin{bmatrix}\begin{matrix}\mathbf{I}_{m-2}&\vline&\mathbf{I}^{\prime}_{m-3}&\vline&\mathbf{1}\end{matrix}\end{bmatrix}$

where $\mathbf{I}_{m-2}$ is the block of $(m-2)\times(m-2)$ identity matrix,
$\mathbf{I}^{\prime}_{m-3}$ is the $(m-2)\times(m-3)$ block containing size
$(m-3)$ identity matrix with the last row of zeros, the last block
$\mathbf{1}$ is the column of $1$’s. The corresponding Bott matrix (that
encodes the Stiefel-Whitney class of the fibration at each stage) is

$\begin{bmatrix}\begin{matrix}{0}&\dots&0&{1}\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&\dots&0&1\\\
0&\dots&0&0\end{matrix}\end{bmatrix}.$

Similarly, the remaining three aspherical chain spaces are towers of
$S^{1}$-fibrations starting from a non-orientable surface.

## References

  * [1] G. Blind and R. Blind. Triangle-free polytopes with few facets. Arch. Math. (Basel) 58 (1992), no. 6, 599–605.
  * [2] M. R. Bridson, André Haefliger. Metric Spaces of Non-Positive Curvature Springer-Verlag.
  * [3] V. M. Buchstaber, Taras E. Panov. Toric Topology Mathematical Surveys and Monographs Volume 204, American Mathematical Society.
  * [4] M. Davis, T. Januszkiewicz and R. Scott . Nonpositive curvature of blow-ups. Sel. math., New ser. 4 (1998) 491 – 547.
  * [5] M. Davis and T. Januszkiewicz. Convex polytopes, Coxeter Orbifolds and Torus Action.Vol. 62, No. 2 Duke Mathematical Journal.
  * [6] M. Farber, D.Schütz. Homology of planar polygon spaces Geom. Dedicata 125, 75–92 (2007).
  * [7] M.Farber. Invitation to Topological Robotics EMS Zurich Lectures in Advanced Mathematics, 2008.
  * [8] M. Gromov. Hyperbolic groups in Essays in Group Theory (S. M. Gersten, ed.). M.S.R.I. Publ. 8. Springer-Verlag, New York, 1987.
  * [9] J. C. Hausmann, Allen Knutson. The cohomology ring of polygon spaces Annales de l’institut Fourier, tome 48, no 1 (1998), p. 281-321.
  * [10] J.-C. Hausmann, A. Knutson. Polygon spaces and Grassmannians, Enseign. Math. (2) 43 (1997), 281–321.
  * [11] J. C. Hausmann. Geometric descriptions of polygon and chain spaces. ”Topology and Robotics”, American Math. Soc. Contemporary Mathematics 438 (2007) 47-57.
  * [12] J. C. Hausmann and Eugenio Rodriguez. The Space of Clouds in Euclidean Space Experimental Mathematics, Vol. 13 (2004), No. 1.
  * [13] M. Kapovich, J. Millson. On the moduli space of polygons in the Euclidean plane J. Diff. Geom. 42, 430–464 (1995).
  * [14] M. Kapovich, J. Millson. The symplectic geometry of polygons in Euclidean space J. Diff. Geom. 44 (1996), 479-513.
  * [15] G. Panina. Moduli space of a planar polygonal linkage: a combinatorial description Arnold Math. J. Vol. 3 No. 3 (2017), 351–364.
  * [16] G.M. Ziegler. Lectures on Polytopes Graduate Texts in Math., Springer-Verlag, Berlin, 1994.