# On the geometry and bounded cohomology of racks and quandles

Jarek Kędra  University of Aberdeen and University of Szczecin
<EMAIL_ADDRESS>
###### Abstract

We introduce and investigate a natural family of metrics on connected
components of a rack. The metrics are closely related to certain bi-invariant
metrics on the group of inner automorphisms of the rack. We also introduce a
bounded cohomology of racks and quandles, relate them to the above metrics and
prove a vanishing result for racks and quandles with amenable group of inner
automorphisms.

##  1 Introduction

Racks and quandles are algebraic structures resembling conjugation in a group
or crossing relations in a knot diagram. They are also invertible solutions of
the Yang-Baxter equation [4]. Quandles were first observed in nature by Joyce
[10] and Matveev [13] and since then have been spreading through various
branches of mathematics [9, 16, 17]. Their most successful applications so far
have been in knot theory, where they are used to classify knots with not so
many crossings. A rack is a slightly more general structure than a quandle, so
that every quandle is a rack but not vice versa. Every rack admits a maximal
quandle quotient.

Although they are usually defined algebraically, as binary operations on a
set, in this paper we propose a more geometric approach. First of all, we
mostly make use of a more geometric definition, which to every element $x$ of
a set $X$ associates a bijection of $X$ plus a certain compatibility axiom
(Definition 2.3). Secondly, we observe that every rack is equipped with a
family of natural metrics on its connected components (Definition 2.5). In
other words, a rack is a disjoint union of metric spaces and the automorphism
group of the rack acts on it by isometries (Proposition 2.9).

Our first result establishes a relation between the rack metric and a certain
bi-invariant metric on the group of inner automorphism of the rack (Theorem
3.1). As an application of this relatively simple observation together with
the general knowledge of bi-invariant metrics on groups, we identify families
of racks and quandles for which their metrics on all connected components have
finite diameter. Here is a sample result (see Corollary 3.5 and 3.7).

######  Theorem 1.1.

Let $(X,\triangleright)$ be a rack. If the group of its inner automorphisms is
either an S-arithmetic Chevalley group of higher rank or a semi-simple Lie
group with finite centre then the diameter of every connected component of
$(X,\triangleright)$ is finite.

On the other hand, we show that all connected components of a free product of
quandles (satisfying a mild hypothesis) have infinite diameter (Proposition
3.14). This holds, for example, for free nontrivial racks and quandles
(Corollary 3.16). We also show in Example 3.17 that the quandle metric on the
quandle of a nontrivial knot has infinite diameter.

In the second part of the paper we introduce the bounded cohomology of racks
and quandles with real coefficients. First, we relate the second bounded
cohomology to the geometry of the above metric (Corollary 5.5 and Proposition
5.6).

######  Theorem 1.2.

A rack or a quandle $(X,\triangleright)$ is unbounded (Definition 3.3) if and
only if the comparison map $H^{2}_{b}(X;{\mathbf{R}})\to
H^{2}(X;{\mathbf{R}})$ has nontrivial kernel.

Having introduced bounded cohomology, it is natural to test it under a
suitable amenability hypothesis. Functions that are constant on the connected
components of a rack give rise to obvious nontrivial bounded classes. If the
group of inner automorphisms of a rack is bounded and amenable then this is,
in fact, all (see Theorem 6.3 for more precise statement and Remark 6.4 for
the statement for quandles).

######  Theorem 1.3.

Let $(X,\triangleright)$ be a rack with bounded, amenable group of inner
automorphisms. Then there is an isomorphism

$\operatorname{Fun}_{b}(\pi_{0}(X,\triangleright)^{k})\cong
H^{k}_{b}(X;{\mathbf{R}}),$

where $\operatorname{Fun}_{b}(\pi_{0}(X,\triangleright)^{k},{\mathbf{R}})$
denotes the set of bounded functions on the $k$-fold product of the space
$\pi_{0}(X,\triangleright)$ of connected components of the rack.

Examples of amenable bounded groups include semisimple compact Lie groups [11]
and affine Coxeter groups [12, 14]. Since
$\operatorname{Fun}_{b}(\pi_{0}(X,\triangleright)^{k},{\mathbf{R}})\cong
H^{k}_{b}(\pi_{0}(X,\triangleright))$, where $\pi_{0}(X,\triangleright)$ is a
trivial rack (see Example 4.4, 4.5), the above theorem is indeed a result
about the triviality of the bounded cohomology. The main difference between
the above result and the corresponding one in group theory is the boundedness
hypothesis here. The above theorem is an extension of a result by Etingof and
Graña, who proved an analogous statement for finite racks [4, Theorem 4.2].

##### Acknowledgements.

I would like to thank Markus Szymik for introducing me to racks and quandles
and for answering my questions.

##  2 Preliminaries

###  2.1 Racks, quandles and their metrics

######  Definition 2.2 (Algebraic).

A quandle is a non-empty set $X$ together with a binary operation
$\triangleright\colon X\times X\to X$ satisfying the following axioms:

  1. A0

$x\triangleright-\colon X\to X$ is a bijection for every $x\in X$;

  2. A1

$x\triangleright(y\triangleright z)=(x\triangleright
y)\triangleright(x\triangleright z)$ for every $x,y,z\in X$;

  3. A2

$x\triangleright x=x$ for every $x\in X$.

If $(X,\triangleright)$ satisfies the first two axioms only then it is called
a rack.

######  Definition 2.3 (Geometric).

Let $X$ be a non-empty set and let $\operatorname{Sym}(X)$ denotes the group
of all bijections of $X$. A quandle is a map $\psi\colon
X\to\operatorname{Sym}(X)$ such that

  1. G1

$\psi_{\psi_{x}(y)}=\psi_{x}\circ\psi_{y}\circ\psi_{x}^{-1}$ for all $x,y\in
X$;

  2. G2

$\psi_{x}(x)=x$ for all $x\in X$.

If $\psi$ satisfies the first axiom only then it is called a rack. The
equivalence of the above definitions is given by the formula

$\psi_{x}(y)=x\triangleright y$

and a straightforward verification that suitable axioms are equivalent.

######  Definition 2.4.

Let $(X,\triangleright)$ be a rack. Two elements $x,y\in X$ are equivalent if
there exist $x_{1},\ldots,x_{n}\in X$ such that

$y=\left(\psi_{x_{1}}^{\pm 1}\circ\dots\circ\psi_{x_{n}}^{\pm 1}\right)(x).$

The equivalence class with respect to this equivalence relation is called a
connected component of $(X,\triangleright)$. The set of connected components
of a rack is denoted by $\pi_{0}(X,\triangleright)$.

######  Definition 2.5.

Let $X_{0}\subseteq X$ be a connected component of a rack. For $x,y\in X_{0}$
define

$d_{0}(x,y)=\min\left\\{n\in{\mathbf{N}}\ |\ y=\left(\psi_{x_{1}}^{\pm
1}\circ\dots\circ\psi_{x_{n}}^{\pm 1}\right)(x)\right\\}.$

It defines a metric on $X_{0}$ called the rack metric.

######  Remark 2.6.

The above metric is the graph metric on the Cayley graph of the rack
$(X,\triangleright)$ associated with the presentation of $(X,\triangleright)$
given by the multiplication table. In other words, with presentation of
$(X,\triangleright)$ in which the whole $X$ is a generating set.

###  2.7 Automorphisms of a rack

A bijective map $\alpha\colon X\to X$ is called an automorphism of a rack
$(X,\triangleright)$ if

$\alpha(x\triangleright y)=\alpha(x)\triangleright\alpha(y).$

The group of all automorphisms of a rack is denoted by
$\operatorname{Aut}(X,\triangleright)$. Observe that for every $x\in X$ the
bijection $\psi_{x}$ is an automorphism. The subgroup of
$\operatorname{Aut}(X,\triangleright)$ generated by $\psi_{x}$ for all $x\in
X$ is called the group of inner automorphisms of $(X,\triangleright)$ and it
is denoted by $G_{\psi}$ or by $\operatorname{Inn}(X,\triangleright)$. Notice
that a connected component of a rack is an orbit with respect to the action of
the inner automorphism group. For this reason a connected component is
sometimes called an orbit.

######  Lemma 2.8.

The subset $\psi(X)\subseteq\operatorname{Aut}(X,\triangleright)$ is invariant
under conjugation.

###### Proof.

Let $x,y\in X$ and let $\alpha\in\operatorname{Aut}(X,\triangleright)$. We
have that

$\displaystyle\left(\alpha\circ\psi_{x}\circ\alpha^{-1}\right)(y)$
$\displaystyle=\alpha\left(\psi_{x}\left(\alpha^{-1}(y)\right)\right)$
$\displaystyle=\alpha\left(x\triangleright\alpha^{-1}(y)\right)$
$\displaystyle=\alpha(x)\triangleright y$ $\displaystyle=\psi_{\alpha(x)}(y),$

which shows that $\alpha\circ\psi_{x}\circ\alpha^{-1}=\psi_{\alpha(x)}$. This
proves that $\psi(X)$ is invariant under conjugations by all automorphisms.
Notice that this is a strengthening of Axiom G1. ∎

######  Proposition 2.9.

The automorphism group of a rack acts by isometries. More precisely, if
$\alpha\in\operatorname{Aut}(X,\triangleright)$ and $x,y\in X_{s}$ then

$d_{s}(x,y)=d_{\alpha(s)}(\alpha(x),\alpha(y)).$

###### Proof.

Let $x,y\in X_{s}$ be such that $d_{s}(x,y)=n$. It means that there are
$x_{1},\ldots,x_{n}\in X$ such that

$y=\left(\psi_{x_{1}}^{\pm 1}\dots\psi_{x_{n}}^{\pm 1}\right)(x)$

It follows from Lemma 2.8 that
$\alpha\circ\psi_{z}=\psi_{\alpha(z)}\circ\alpha$. Applying this identity
repeatedly yields the following computation.

$\displaystyle\alpha(y)$ $\displaystyle=\alpha\left[\left(\psi_{x_{1}}^{\pm
1}\dots\psi_{x_{n}}^{\pm 1}\right)(x)\right]$
$\displaystyle=\left(\psi_{\alpha(x_{1})}^{\pm
1}\dots\psi_{\alpha(x_{n})}^{\pm 1}\right)(\alpha(x)),$

which shows that $d_{\alpha(s)}(\alpha(x),\alpha(y))\leq n$. Since $\alpha$ is
invertible we get an equality which finishes the proof. ∎

###  2.10 Representability of racks

######  Example 2.11.

Let $G$ be a group, $S\subseteq G$ a subset and $\\{H_{s}\leq G\ |\ s\in S\\}$
a family of subgroups such that $H_{s}\leq Z(s)$, where $Z(s)\leq G$ denotes
the centraliser of $s\in S$. Let $X=\bigsqcup_{s\in S}G/H_{s}$. The rack
operation on $X$ is defined by

$xH_{s}\triangleright yH_{t}=xsx^{-1}yH_{t}.$

It is straightforward to verify the axioms. If, moreover, $s\in H_{s}$ then
the above operation defines a quandle. The rack $(X,\triangleright)$ defined
above is denoted by $(G,S,\\{H_{s}\\})$. $\diamondsuit$

######  Lemma 2.12 (Joyce ).

Every rack $(X,\triangleright)$ is isomorphic to a rack of the form
$(G,S,\\{H_{s}\\})$. ∎

######  Remark 2.13.

Joyce proved his theorem for quandles only. The proof for racks is analogous.

It follows from the above definition that $G$ acts on
$(X,\triangleright)=(G,S,\\{H_{s}\\})$ by automorphisms. Moreover, the map
$\psi\colon X\to\operatorname{Aut}(X,\triangleright)$ factors through $G$ as

$\psi_{gH_{s}}=gsg^{-1}.$

Consequently, the group of inner automorphisms $G_{\psi}$ is a subgroup of $G$
normally generated by the subset $S$.

######  Lemma 2.14.

The group $G_{\psi}$ of inner automorphisms of a rack $(G,S,\\{H_{s}\\})$ is
normally generated by the subset $S$. In particular, if $S$ normally generates
$G$ then $G=G_{\psi}$. ∎

###  2.15 The enveloping group of a rack

######  Definition 2.16.

Let $(X,\triangleright)$ be a rack. The group $G_{X}$ defined by the
presentation

$G_{X}=\langle X\ |\ x\triangleright y=xyx^{-1},\ x,y\in X\rangle$

is called the enveloping group of the rack $(X,\triangleright)$. Notice that
the constant function $X\to\\{1\\}\subseteq{\mathbf{Z}}$ defines a surjective
homomorphism $G_{X}\to{\mathbf{Z}}$ for any rack. Consequently, the enveloping
group is always infinite.

######  Lemma 2.17.

The projection $\pi\colon G_{X}\to G_{\psi}$ defined by $\pi(x)=\psi_{x}$ is a
central extension. If, moreover, the natural map $X\to G_{X}$ is injective
then $\ker\pi=Z(G_{X})$.

###### Proof.

The projection $\pi$ is obviously surjective. So we need to check that its
kernel is contained in the centre of $G_{X}$. Let $g=x_{1}^{\pm 1}\dots
x_{n}^{\pm 1}\in\ker\pi$ which means that

$\psi_{x_{1}}^{\pm 1}\circ\dots\circ\psi_{x_{n}}^{\pm 1}=\operatorname{Id}.$

Let $x\in X$. Observe that the defining relation can be written as

$\psi_{x_{i}}(x)=x_{i}xx_{i}^{-1}.$

It follows that

$\displaystyle gxg^{-1}$ $\displaystyle=\left(x_{1}^{\pm 1}\dots x_{n}^{\pm
1}\right)\cdot x\cdot\left(x_{n}^{\mp 1}\dots x_{1}^{\mp 1}\right)$
$\displaystyle=\left(\psi_{x_{1}}^{\pm 1}\circ\dots\circ\psi_{x_{n}}^{\pm
1}\right)(x)$ $\displaystyle=x,$

which shows that $g\in Z(G_{X})$ since $x\in X$ was an arbitrary generator.

Let $X\to G_{X}$ be injective. Let $g=x_{1}^{\pm 1}\dots x_{n}^{\pm 1}\in
Z(G_{X})$. Then we have that $\left(\psi_{x_{1}}^{\pm
1}\circ\dots\circ\psi_{x_{n}}^{\pm 1}\right)(x)=x$ for every $x\in X$ which
implies that $\left(\psi_{x_{1}}^{\pm 1}\circ\dots\circ\psi_{x_{n}}^{\pm
1}\right)=\operatorname{Id}$ due to the injectivity of $X\to G_{X}$. ∎

######  Remark 2.18.

The map $X\to G_{X}$ is never injective for racks that are not quandles.
Indeed, if $\psi_{x}(x)=y\neq x$ then we have that

$y=x\triangleright x=xxx^{-1}=x$

in $G_{X}$. It, moreover, follows that $\psi_{x}=\psi_{\psi_{x}^{n}(x)}$ for
every $n\in{\mathbf{Z}}$.

###  2.19 Conjugation-invariant norms on groups

Let $G$ be a group and let $S\subset G$ be a generating set. The word norm
associated with $S$ is defined by

$\|g\|_{S}=\min\left\\{n\in{\mathbf{N}}\ |\ g=s_{1}^{\pm 1}\dots s_{n}^{\pm
1},\ s_{i}\in S\right\\}.$

If the subset $S\subseteq G$ is invariant under conjugations then the norm is
conjugation-invariant, that is,

$\|hgh^{-1}\|_{S}=\|g\|_{S},$

holds for all $g,h\in G$. The associated metric is defined by
$d_{S}(g,h)=\|g^{-1}h\|_{S}$. It is left-invariant and if the norm is
conjugation-invariant then the metric is bi-invariant. That is, both left and
right multiplications are isometries of the metric.

A group $G$ is called bounded [2] if every bi-invariant metric on $G$ has
finite diameter. If a group is generated by a union of finitely many conjugacy
classes then its boundedness is equivalent to the boundedness of the
associated word metric [11]. Examples of bounded groups include S-arithmetic
Chevalley groups of higher rank [7], semisimple Lie groups with finite centre,
diffeomorphism groups of compact manifolds and many others [11].

Let $G$ be equipped with a bi-invariant word metric associated with a normally
generating set $S$. Let $H\leq G$ be a subgroup. The quotient metric on the
quotient $G/H$ is defined by the distance in $G$ between the cosets.
Equivalently,

$\displaystyle d_{S}(xH,yH)$ $\displaystyle=\min\\{d_{S}(xh_{1},yh_{2})\ |\
h_{1},h_{2}\in H\\}$ $\displaystyle=\min\\{d_{S}(x,yh)\ |\ h\in H\\}$
$\displaystyle=\min\\{\|x^{-1}yh\|_{S}\ |\ h\in H\\}$
$\displaystyle=d_{S}(H,x^{-1}yH).$

In particular, $G$ acts on $G/H$ by isometries.

######  Example 2.20.

It follows from Lemma 2.8 that $\psi(X)$ generates $G_{\psi}$ and is invariant
under conjugations by elements of ${\rm Aut}(X,\triangleright)$. Consequently,
the associated word norm $\|g\|_{\psi}$ on $G_{\psi}$ is
$\operatorname{Aut}(X,\triangleright)$-invariant (in particular, conjugation-
invariant). $\diamondsuit$

##  3 The geometry of racks

#### A characterisation of the rack metric

With the preparations from the previous section the proof of the following
theorem is fairly obvious.

######  Theorem 3.1.

Let $(X,\triangleright)=(G,S,\\{H_{s}\\})$ be a rack such that $S\subseteq G$
normally generates $G$. Let $X_{s}=G/H_{s}$ be a connected component of
$(X,\triangleright)$. The rack metric $d_{s}$ on $X_{s}$ is equivalent to the
quotient metric $d_{S}$ on $G/H_{s}$.

###### Proof.

Let $x\in G$.

$\displaystyle d(xH_{s},H_{s})$ $\displaystyle=\min\left\\{n\in{\mathbf{N}}\
|\ xH_{s}=\left(\psi_{x_{1}H_{s_{1}}}^{\pm 1}\dots\psi_{x_{n}H_{s_{n}}}^{\pm
1}\right)(H_{s})\right\\}$ $\displaystyle=\min\left\\{n\in{\mathbf{N}}\ |\
xH_{s}=x_{1}{s_{1}}^{\pm 1}x_{1}^{-1}\dots x_{n}s_{n}^{\pm
1}x_{n}^{-1}H_{s}\right\\}$ $\displaystyle=\min\left\\{\|g\|_{S}\ |\
xH_{s}=gH_{s}\right\\}$ $\displaystyle=\min\left\\{\|xh\|_{S}\ |\ h\in
H\right\\}$ $\displaystyle=d_{S}(xH_{s},H_{s})$

Since both metrics are $G$-invariant, the above computation proves the
statement. ∎

#### A canonical rack-quandle extension

Let $(X,\triangleright)$ be a rack and let $y=\psi_{x}(x)$. We have that

$\psi_{y}=\psi_{\psi_{x}(x)}=\psi_{x}\cdot\psi_{x}\cdot\psi_{x}^{-1}=\psi_{x}.$

It follows that if $y=\psi_{x}^{n}(x)$ then $\psi_{y}=\psi_{x}$ for any
$n\in{\mathbf{Z}}$. Call $x$ and $y$ equivalent if $y=\psi_{x}^{n}(x)$ for
some $n\in{\mathbf{Z}}$. It is straightforward to verify that it is an
equivalence relation. Let $\underline{X}=X/\\!\approx$ be the quotient. The
rack product descends to the quotient. That is,

$[x]\triangleright[y]=[x\triangleright y]$

is well defined and defines a quandle structure on $\underline{X}$. We call
$X\to\underline{X}$ the canonical rack-quandle extension. Notice that this
extension maps connected components of the rack to connected components of the
quandle.

######  Lemma 3.2.

The canonical rack-quandle extension is $1$-Lipschitz. That is,

$d_{\underline{X}}([x],[y])\leq d_{X}(x,y),$

for every $x,y\in X$.

###### Proof.

Suppose that $d_{X}(x,y)=n$. Then there elements exist $x_{1},\ldots,x_{n}\in
X$ such that $y=\left(\psi_{x_{1}}^{\pm 1}\dots\psi_{x_{n}}^{\pm
1}\right)(x)$. It follows that

$[y]=\left(\psi_{[x_{1}]}^{\pm}\dots\psi_{[x_{n}]}^{\pm 1}\right)[x],$

which implies that $d_{\underline{X}}([x],[y])\leq n$. ∎

#### Bounded racks

######  Definition 3.3.

If the diameter of the rack metric on each connected component of a rack
$(X,\triangleright)$ is finite then $(X,\triangleright)$ is called bounded.
Otherwise, it is called unbounded.

In the rack language this means that there exists a number $N>0$ such that for
every $x,y\in X_{s}$ there exist $x_{1},\ldots,x_{n}\in X$ with $n\leq N$ such
that

$y=x_{n}\triangleright(x_{n-1}\triangleright(\dots(x_{1}\triangleright
x)\dots)).$

It follows from Lemma 3.2 that if the rack in the canonical rack-quandle
extension is bounded then so is the underlying quandle. Conversely, if the
underlying quandle is unbounded then so is the rack.

######  Corollary 3.4.

Let $(X,\triangleright)=(G,S,\\{H_{s}\\})$ be a rack. If the group $G$ is
bounded then the rack $(X,\triangleright)$ is bounded. ∎

In what follows we specify the above corollary to classes of groups that are
well known to be bounded.

######  Corollary 3.5.

Let $(X,\triangleright)=(\Gamma,S,\\{H_{s}\\})$, where $\Gamma$ is an
$S$-arithmetic Chevalley group of rank at least $2$, $S\subseteq\Gamma$ is a
set of root elements normally generating $\Gamma$ (such a set can be chosen
finite) and $H_{s}\subseteq Z(s)$. Then the rack $(X,\triangleright)$ is
bounded. ∎

######  Example 3.6.

Consider a rack

$\left(\operatorname{SL}(n,{\mathbf{Z}}),E_{12},Z(E_{12})\right),$

where $n\geq 3$ and $E_{12}$ denotes the elementary matrix with entries
$e_{ii}=e_{12}=1$ and zero otherwise. Notice that the above rack is a quandle.
Since the conjugacy class of $E_{12}$ generates
$\operatorname{SL}(n,{\mathbf{Z}})$ the above quandle is connected. It is
bounded, since $\operatorname{SL}(n,{\mathbf{Z}})$ is bounded for $n\geq 3$.
$\diamondsuit$

######  Corollary 3.7.

Let $(X,\triangleright)=(G,S,\\{H_{s}\\})$, where $G$ is a semisimple Lie
group with finite centre, $S$ is a normal generating set (which can be chosen
finite) and $H_{s}\leq Z(s)$. Then the rack metric on each connected component
of $(X,\triangleright)$ has finite diameter.

######  Example 3.8.

Let ${\mathcal{P}}\subseteq\operatorname{PSL}(2,{\mathbf{C}})$ be the quandle
consisting of all parabolic elements. It can be represented as
$(\operatorname{PSL}(2,{\mathbf{C}}),S,\\{Z(s)\\})$, where
$S\subseteq\operatorname{PSL}(2,{\mathbf{C}})$ is a set of representatives of
conjugacy classes of parabolic elements. Inoue and Kabaya [9] used the
cohomology of this quandle to compute the complex volume of hyperbolic links.
It follows from Theorem 3.1 and boundedness of
$\operatorname{PSL}(2,{\mathbf{C}})$ that the quandle ${\mathcal{P}}$ is
bounded. $\diamondsuit$

######  Example 3.9.

Let $G$ be a bounded simple group and let $1\neq g\in G$. Since the conjugacy
class of $g$ generates $G$, due to simplicity, we get that the rack
$(G,g,H_{g})$, where $H_{g}\subseteq Z(g)$, is bounded. This, for example,
holds for simple Lie groups or (the commutator subgroups of) Higman-Thompson
groups [6, 11]. $\diamondsuit$

All the examples above rely of the fact that the group of inner automorphisms
of the rack is bounded. However, for the boundedness of a connected component
$G/H_{s}$ is is enough that the embedding $H_{s}\subseteq G$ is coarsely
surjective. The latter means that there exists a number $N>0$ such that for
every $g\in G$ there exists $h\in H_{s}$ with $d_{S}(g,h)\leq N$.

######  Example 3.10.

Let
${\mathbf{Z}}\to\widetilde{\operatorname{Sp}}(2n;{\mathbf{Z}})\to\operatorname{Sp}(2n;{\mathbf{Z}})$
be a nontrivial central extension of the integral symplectic group. This
extension is unbounded and the inclusion of the centre is coarsely surjective
[7]. Consequently, every connected component
$\widetilde{\operatorname{Sp}}(2n;{\mathbf{Z}})/H_{s}$ of a rack
$\left(\widetilde{\operatorname{Sp}}(2n;{\mathbf{Z}}),S,\\{H_{s}\\}\right)$
such that $H_{s}$ contains the centre of
$\widetilde{\operatorname{Sp}}(2n;{\mathbf{Z}})$ has finite diameter.

$\diamondsuit$

#### Unbounded racks and quandles

######  Example 3.11.

Let $({\mathbf{Z}},\triangleright)$ be a rack defined by
$k\triangleright\ell=\ell+1$. It is connected and isometric to ${\mathbf{Z}}$
with the standard metric, hence unbounded. $\diamondsuit$

######  Example 3.12.

Let ${\mathbf{F}}_{2}=\langle x,y\rangle$ be the free group on two generators.
Let $(X,\triangleright)=({\mathbf{F}}_{2},\\{x,y\\},\\{Z(x),Z(y)\\})$ be a
quandle. It is a union of conjugacy classes of $x$ and of $y$ and each
conjugacy class is a connected component. Since $Z(x)=\langle x\rangle$ is the
cyclic subgroup generated by $x$ its inclusion into ${\mathbf{F}}_{2}$ is not
coarsely surjective (for example $d_{\\{x,y\\}}(Z(x),y^{n})=n$) and hence the
quandle is unbounded. $\diamondsuit$

######  Proposition 3.13.

Let ${\mathbf{F}}_{n}$ be the free group of rank $n\geq 2$ and let
$(X,\triangleright)=({\mathbf{F}}_{n},S,\\{H_{s}\\})$, where
$S\subseteq{\mathbf{F}}_{n}$ is finite normally generating subset. Then each
connected component $X_{s}$ of $(X,\triangleright)$ has infinite diameter.

###### Proof.

It is enough to show that the embedding the cetraliser $Z(g)$ of any
nontrivial element $g\in{\mathbf{F}}_{n}$ cannot be coarsely surjective. First
observe that the centraliser $Z(g)$ is a cyclic subgroup containing $g$. The
inclusion $Z(g)\to{\mathbf{F}}_{n}$ is never coarsely surjective which can be
seen as follows.

Let $\pi\colon{\mathbf{F}}_{n}\to{\mathbf{Z}}^{n}$ be the abelianisation. The
projection is Lipschitz with constant $1$, provided ${\mathbf{Z}}^{n}$ is
equipped with the word metric associated with the (finite) generating set
$\pi(S)$. This metric is Lipschitz equivalent to the standard word metric. Let
$s_{1},s_{2}\in S$ be two generators such that their images $\pi(s_{1})$ and
$\pi(s_{2})$ generate a free abelian subgroup of rank $2$. The image
$\pi(Z(g))\leq{\mathbf{Z}}^{n}$ is cyclic and hence the distance of either
$\pi(s_{1}^{k})$ or $\pi(s_{2}^{k})$ from $\pi(Z(g))$ grows linearly with
$k\in{\mathbf{N}}$. Since the projection is Lipschitz the same is true for the
distance between $Z(g)$ and $s_{1}$ or $s_{2}$ in ${\mathbf{F}}_{n}$. ∎

The following proposition deals with free products of quandles. See [1,
Section 7] for a definition of a free product of quandles as well as their
presentations. Notice, for example, that the free quandle on $n$ generators is
the free product of $n$ copies of the trivial quandle.

######  Proposition 3.14.

Let $(X_{i},\triangleright)$ for $i=1,2$ be quandles with finitely many
connected components and such that the maps $X_{i}\to G_{X_{i}}$ are
injective. Let $A_{i}\subseteq X_{i}$ be the set of representatives of
connected components of $X_{i}$. Then every connected component of the free
product $X_{1}*X_{2}$ has infinite diameter.

###### Proof.

Recall that the enveloping group $G_{X_{i}}$ is infinite (see Definition
2.16). It follows from [1, Theorem 7.2] that the free product $X_{1}*X_{2}$ is
isomorphic to the quandle $(G_{X_{1}}*G_{X_{2}},A_{1}\cup A_{2})$; the latter
denotes a quandle defined in [1, Section 4]. Moreover, $G=G_{X_{1}}*G_{X_{2}}$
is the enveloping group of $X_{1}*X_{2}$ by [1, Lemma 7.1]. Furthermore,
Proposition 4.3 of the same paper implies that this quandle is isomorphic to
$(G_{X_{1}}*G_{X_{2}},A_{1}\cup A_{2},\\{Z_{G}(a_{1}),Z_{G}(a_{2})\\})$. Since
both $G_{X_{i}}$ are infinite, their free product has trivial centre and we
have an isomorphism $G_{X_{1}}*G_{X_{2}}\cong\operatorname{Inn}(X_{1}*X_{2})$.
Thus in order to prove the statement it suffices to show that the inclusion of
no centraliser $Z_{G}(a_{1})$ or $Z_{G}(a_{2})$, where $a_{i}\in A_{i}$, is
coarsely surjective.

To see this, notice that $Z_{G}(a_{i})\leq G_{X_{i}}\leq G_{X_{1}}*G_{X_{2}}$
and consider the surjective homomorphism $\varphi\colon
G_{X_{1}}*G_{X_{2}}\to{\mathbf{F}}_{2}=\langle u_{1},u_{2}\rangle$ defined by
$x_{i}\to u_{i}$. Notice that this homomorphism is Lipschitz with respect to
the bi-invariant metrics associated with $A_{1}\cup A_{2}$ and
$\\{u_{1},u_{2}\\}$, due to the finiteness of $A_{i}$. If the inclusion of the
centraliser $Z_{G}(a_{i})$ was coarsely surjective then the composition with
$\varphi$ would be coarsely surjective. However, $\varphi(Z(a_{i}))\leq\langle
u_{i}\rangle$ and hence it is not coarsely surjective. This shows that every
connected component of $X_{1}*X_{2}$ has infinite diameter. ∎

######  Example 3.15.

Let $T_{1}=\\{x_{i}\\}$ and $T_{2}=\\{x_{2}\\}$ be trivial quandles. Each has
one connected component represented by $x_{i}$. The enveloping group
$G_{T_{i}}$ of $T_{i}$ is infinite cyclic $G_{T_{i}}\cong{\mathbf{Z}}$ and
hence the free product

$T_{1}*T_{2}=({\mathbf{F}}_{2}=\langle
x_{1},x_{2}\rangle,\\{x_{1},x_{2}\\},\\{Z(x_{1}),Z(x_{2})\\}).$

Notice that this is the quandle considered in Example 3.12. It follows from
Proposition 3.14 that both connected components have infinite diameter.
$\diamondsuit$

######  Corollary 3.16.

Every connected component of a free quandle $\operatorname{FQ}(X)$ has
infinite diameter. Consequently, the same is true for free racks due to Lemma
3.2. ∎

######  Example 3.17.

Let $K\subseteq{\mathbf{S}}^{3}$ be a non-trivial knot and let
$G_{K}=\pi_{1}({\mathbf{S}}^{3}\setminus K)$ be the fundamental group of its
complement. Let $Q_{K}$ be the associated quandle. Then
$Q_{K}=(G_{K},\\{s\\},P)$, where $s\in G_{K}$ is the element represented by
the meridian of $K$ and $P$ is the image of $\pi_{1}(\partial U)\to G_{K}$,
where $U\subseteq{\mathbf{S}}^{3}$ is a tubular neighbourhood of $K$ [10,
Corollary 16.2]. Notice that $Q_{K}$ is connected.

Fujiwara [5, Theorem 1.6] proved that the second bounded cohomology of $G_{K}$
is infinite dimensional. Since $P$ is abelian, it follows that there is a non-
trivial homogeneous quasi-morphism $q\colon G_{K}\to{\mathbf{R}}$ vanishing on
$P$. Let $g\in G_{K}$ be an element such that $q(g)>0$. Then for any $h\in P$
we have that

$d(g^{n},h)=\|g^{n}h^{-1}\|\geq C(q(gh^{-1}))\geq C(nq(g)-D)$

is arbitrarily large for a large $n\in{\mathbf{N}}$. We used here the fact
that quasimorphisms are Lipschitz with respect to conjugation-invariant norms
on normally finitely generated groups and $C>0$ above is the Lipschitz
constant; $D\geq 0$ is the defect of $q$. This shows that the inclusion
$P\subseteq G_{K}$ is not coarsely surjective and hence the quandle metric has
infinite diameter. $\diamondsuit$

##  4 Bounded cohomology of racks and quandles

######  Remark 4.1.

In this paper we consider only the cohomology with real coefficients,
considered as a trivial module. For a general definition of quandle or rack
cohomology see, for example, [3, 4].

#### Definition of bounded cohomology

Let $(X,\triangleright)$ be a quandle. Recall that the rack cochain complex
$C^{*}(X;{\mathbf{R}})$ with real coefficients is a complex in which
$C^{k}(X;{\mathbf{R}})$ consist of functions $f\colon X^{k}\to{\mathbf{R}}$
with the differential is given by

$\displaystyle\delta f(x_{1},x_{2},\ldots,x_{k+1})$
$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}f(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{k+1})$
$\displaystyle-\sum_{i=1}^{k}(-1)^{i-1}f(x_{1},\ldots,x_{i-1},x_{i}\triangleright
x_{i+1},\ldots,x_{i}\triangleright x_{k+1}).$

The cohomology $H^{*}(X;{\mathbf{R}})$ of the above complex is called the real
rack cohomology of $(X,\triangleright)$. The subcomplex
$C^{*}_{b}(X;{\mathbf{R}})\subseteq C^{*}(X;{\mathbf{R}})$ consisting of
bounded functions defines the (real) rack bounded cohomology of
$(X,\triangleright)$ denoted by $H^{*}_{b}(X;{\mathbf{R}})$. The inclusion
$C^{*}_{b}(X;{\mathbf{R}})\subseteq C^{*}(X;{\mathbf{R}})$ induces a
homomorphism

$H^{*}_{b}(X;{\mathbf{R}})\to H^{*}(X;{\mathbf{R}}),$

called the comparison map. The quandle (bounded) cohomology is defined as
above for the subcomplex consisting of (bounded) functions $f\colon
X^{k}\to{\mathbf{R}}$ satisfying the additional condition that

$f(x_{1},\ldots,x_{k})=0\text{ if }x_{i}=x_{i+1}\text{ for some
}i=1,\ldots,k-1.$ (4.1)

The notation for both the rack and quandle cohomology is for simplicity the
same with the convention that if $(X,\triangleright)$ is a rack (quandle) then
$H^{*}_{b}(X;{\mathbf{R}})$ is the rack (quandle) cohomology.

######  Remark 4.2.

The reason that the quandle cochain complex is smaller is to discard redundant
cohomology arising from inclusions of singletons which, in the case of
quandles, are retracts. More precisely, if $x\in X$ then $p\colon X\to\\{x\\}$
is a morphism of quandles that has a section $i\colon\\{x\\}\to X$. The rack
cohomology of the trivial rack $\\{x\\}$ is isomorphic to ${\mathbf{R}}$ in
each degree. Taking into account the retracts for every point of the quandle
makes its rack cohomology unnecessarily enormous; see also [16, Remark 2.6].

##### Convention:

In order to make the paper less cumbersome, in what folllows, we present
examples, arguments and proofs mostly for racks. All arguments carry over
almost verbatim for quandles and their bounded cohomology. If necessary, a
separate statement will be given for quandles.

######  Example 4.3.

A (bounded) one-cocycle $f\colon X\to{\mathbf{R}}$ is constant on each
connected component of $(X,\triangleright)$. Indeed, the defining property
yields

$0=\delta f(x,y)=f(x)-f(y\triangleright x),$

which implies that
$f(x)=f\left(\left(\psi_{x_{1}}\dots\psi_{x_{n}}\right)(x)\right)$ for every
$x,x_{1},\ldots,x_{n}\in X$. Consequently,
$H^{1}_{b}(X;{\mathbf{R}})=\operatorname{Fun}_{b}(\pi_{0}(X,\triangleright))$
and $H^{1}(X;{\mathbf{R}})=\operatorname{Fun}(\pi_{0}(X,\triangleright))$.
Notice, that a one-cocycle is a rack homomorphism, where ${\mathbf{R}}$ is
considered as a trivial rack. $\diamondsuit$

######  Example 4.4.

Let $(X,\triangleright)$ be the trivial rack. That is $x\triangleright y=y$
for all $x,y\in X$ or, equivalently, $\psi\colon X\to\operatorname{Sym}(X)$ is
constant equal to the identity. It is immediate to verify that the
differential in the rack cochain complex is identically zero which implies
that $H^{k}_{b}(X;{\mathbf{R}})$ is isomorphic to the space
$\operatorname{Fun}_{b}(X^{k};{\mathbf{R}})$ of bounded functions on $X^{k}$.
If $(X,\triangleright)$ is a quandle then its quandle bounded cohomology of
degree $k$ is isomorphic to the space
$\underline{\operatorname{Fun}}_{b}(X^{k},{\mathbf{R}})$ of bounded functions
which are zero on the elements $(x_{1},\ldots,x_{k})$ such that
$x_{i}=x_{i+1}$ for some $i=1,\ldots,k-1$. $\diamondsuit$

######  Example 4.5.

If $(X,\triangleright)$ is a rack then the set $\pi_{0}(X,\triangleright)$ of
its connected component is a trivial rack with respect to the operation
induced from $(X,\triangleright)$. That is,

$\pi(x)\triangleright\pi(y)=\pi(x\triangleright y)=\pi(y).$

The latter also means that the projection $\pi\colon
X\to\pi_{0}(X,\triangleright)$ is a morphism of racks. Hence, it induces a
homomorphism

$\pi^{*}\colon
H^{k}_{b}(\pi_{0}(X);{\mathbf{R}})=\operatorname{Fun}_{b}(\pi_{0}(X)^{k},{\mathbf{R}})\to
H^{k}_{b}(X;{\mathbf{R}}).$

There is an analogous homomorphism on quandles defined on
$\underline{\operatorname{Fun}}_{b}(X^{k},{\mathbf{R}})$. $\diamondsuit$

The space of cochains $C^{k}_{b}(X;{\mathbf{R}})$ is an
$\operatorname{Aut}(X,\triangleright)$-module. Indeed, let $f\colon
X^{k}\to{\mathbf{R}}$ be a cochain. If
$\alpha\in\operatorname{Aut}(X,\triangleright)$ is an automorphism of
$(X,\triangleright)$ then let $f\cdot\alpha\colon X^{k}\to{\mathbf{R}}$ be
defined by

$(f\cdot\alpha)(x_{1},\ldots,x_{k})=f(\alpha(x_{1}),\ldots,\alpha(x_{k})).$

Thus $C^{k}(X;{\mathbf{R}})$ is a right
$\operatorname{Aut}(X,\triangleright)$-module.

######  Lemma 4.6.

The differential $\delta\colon C^{k}(X;{\mathbf{R}})\to
C^{k+1}(X;{\mathbf{R}})$ is a map of
$\operatorname{Aut}(X,\triangleright)$-modules.

###### Proof.

We apply the identity $\alpha\circ\psi_{x}=\psi_{\alpha}\circ\alpha$ from
Lemma 2.8 in the following computation.

$\displaystyle(\delta(f\cdot\alpha))(x_{1},\ldots,x_{k+1})$
$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}f(\alpha(x_{1}),\ldots,\alpha(x_{i-1}),\alpha(x_{i+1}),\ldots\alpha(x_{k+1}))$
$\displaystyle-\sum_{i=1}^{k}(-1)^{i-1}f(\alpha(x_{1}),\ldots,\alpha(x_{i-1}),\alpha(\psi_{x_{i}}(x_{i+1})),\ldots\alpha(\psi_{x_{i}}(x_{k+1})))$
$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}f(\alpha(x_{1}),\ldots,\alpha(x_{i-1}),\alpha(x_{i+1}),\ldots\alpha(x_{k+1}))$
$\displaystyle-\sum_{i=1}^{k}(-1)^{i-1}f(\alpha(x_{1}),\ldots,\alpha(x_{i-1}),\psi_{\alpha(x_{i})}(\alpha(x_{i+1}))),\ldots\psi_{\alpha(x_{i})}(\alpha(x_{k+1}))))$
$\displaystyle=((\delta f)\cdot{\alpha})(x_{1},\ldots,x_{k+1}).$

∎

######  Corollary 4.7.

The $\operatorname{Aut}(X,\triangleright)$-invariant cochains form a
subcomplex $C^{*}_{b}(X;{\mathbf{R}})^{\rm Aut}$.∎

By restricting the action to the inner automorphism group
$G_{\psi}\leq\operatorname{Aut}(X,\triangleright)$ we also obtain an
subcomplex of $G_{\psi}$-invariant cochains and its homology will be called
the invariant bounded cohomology of $(X,\triangleright)$. It will be denoted
by $H^{*}_{b,\rm inv}(X;{\mathbf{R}})$. The inclusion of the complex induces
the homomorphism

$H^{*}_{b,\rm inv}(X;{\mathbf{R}})\to H^{*}_{b}(X;{\mathbf{R}}).$

######  Lemma 4.8.

If $f\colon X^{k}\to{\mathbf{R}}$ is a cocycle then $f$ and $f\cdot g$ are
cohomologous for every $g\in G_{\psi}$. Consequently,
$H^{*}_{b}(X;{\mathbf{R}})$ is a trivial $G_{\psi}$-module.

###### Proof.

Given $z\in X$, let $f_{z}\in C^{k-1}(X;{\mathbf{R}})$ be defined by

$f_{z}(x_{1},\ldots,x_{k-1})=f(z,x_{1},\ldots,x_{k-1}).$

The following equalities are straightforward to verify. The second one follows
because $f$ is a cocycle.

$\displaystyle\delta(f_{z})(x_{1},\ldots,x_{k})$
$\displaystyle=(f-f\cdot{\psi_{z}})(x_{1},\ldots,x_{k})-\delta
f(z,x_{1},\ldots,x_{k})$
$\displaystyle=(f-f\cdot{\psi_{z}})(x_{1},\ldots,x_{k}).$

Since $\psi_{z}$ generate $G_{\psi}$, we have

$\displaystyle f-f\cdot g$
$\displaystyle=f-f\cdot\psi_{x_{1}}\dots\psi_{x_{n}}$
$\displaystyle=f-f\cdot\psi_{x_{1}}+f\cdot\psi_{x_{1}}-f\cdot\psi_{x_{1}}\psi_{x_{2}}+f\cdot\psi_{x_{1}}\psi_{x_{2}}-\dots-f\cdot\psi_{x_{1}}\dots\psi_{x_{n}}$
$\displaystyle=\delta(f_{x_{1}})+\delta\left((f\cdot\psi_{x_{1}})_{x_{2}}\right)+\dots+\delta\left((f\cdot\psi_{x_{1}}\dots\psi_{x_{n-1}})_{x_{n}}\right)$

which proves the statement holds for every $g\in G_{\psi}$. ∎

##  5 Rack quasimorphisms and unboundedness

######  Definition 5.1.

A rack quasimorphism is a function $f\colon X\to{\mathbf{R}}$ for which there
exists $D\geq 0$ such that

$|f(y)-f(x\triangleright y)|\leq D,$

for all $x,y\in X$. If $(X,\triangleright)$ is a quandle then $f$ will be
called a quandle quasimorphism (definition is the same).

######  Lemma 5.2.

If $f\colon X\to{\mathbf{R}}$ is a rack quasimorphism then $\delta f$ is a
two-cocycle. The class $[\delta f]\in H^{2}_{b}(X;{\mathbf{R}})$ is in the
kernel of the comparison map. If $f$ is unbounded on a connected component of
$(X,\triangleright)$ then $\delta f$ is nontrival.

###### Proof.

The first two statements are obvious. Notice that if $(X,\triangleright)$ is a
quandle then $\delta f$ is a quandle cocycle, since $\delta
f(x,x)=f(x)-f(\psi_{x}(x))=0$. Suppose that $f$ is unbounded on a connected
component of $(X,\triangleright)$. If $[\delta f]=0$ in
$H^{2}_{b}(X;{\mathbf{R}})$ then $\delta f=\delta\beta$ for some bounded
cochain $\beta\colon X\to{\mathbf{R}}$. It follows that $\delta(f-\beta)=0$,
that is $f-\beta$ is an ordinary one-cocycle and hence it has to be constant
on each connected component of $(X,\triangleright)$, which is impossible,
because $f$ is unbounded and $\beta$ is bounded. ∎

######  Example 5.3.

Let ${\mathbf{F}}_{2}=\langle x,y\rangle$ be a free group on two generators.
Let $\varphi\colon{\mathbf{F}}_{2}\to{\mathbf{R}}$ be a non-trivial
homogeneous group quasimorphism. Let
$(X,\triangleright)=({\mathbf{F}}_{2},\\{x,y\\},\\{Z(x),Z(y)\\})$ be a free
quandle on two generators and let $\widehat{\varphi}\colon X\to{\mathbf{R}}$
be defined by

$\widehat{\varphi}(gxg^{-1})=\varphi(g^{\prime}),$

where $g=g^{\prime}x^{k}$ and $g^{\prime}$ is a reduced word finishing with
$y$. Define $\widehat{\varphi}(gyg^{-1})$ analogously.

Since $\varphi$ is a homogeneous quasimorphism on ${\mathbf{F}}_{2}$, it is
constant on conjugacy classes and hence bounded, say by $B>0$, on $C(x^{\pm
1})\cup C(y^{\pm 1})$. Let $s\in{\mathbf{F}}_{2}$ be a conjugate of a
generator. We thus have that

$\displaystyle|\widehat{\varphi}(s\triangleright
gxg^{-1})-\widehat{\varphi}(gxg^{-1})|$
$\displaystyle=|\widehat{\varphi}(sgxg^{-1}s^{-1})-\varphi(g^{\prime})|$
$\displaystyle=|\varphi(sg^{\prime})-\varphi(g^{\prime})|$
$\displaystyle\leq|\varphi(s)|+D$ $\displaystyle\leq B+D,$

which shows that $\widehat{\varphi}$ is an unbounded quandle quasimorphism (a
similar computation is done for $gyg^{-1}$). $\diamondsuit$

######  Proposition 5.4.

A rack quasimorphism $f\colon X\to{\mathbf{R}}$ is Lipschitz with respect to
the rack metric. More precisely, there exists a constant $C>0$ such that

$|f(x)-f(y)|\leq Cd(x,y)$

for any $x,y\in X_{s}$, where $X_{s}\subseteq X$ is a connected component.

###### Proof.

Let $x,y\in X_{s}$ be such that $d(x,y)=n$. It means that there exist
$x_{1},\ldots,x_{n}\in X$ such that
$y=\left(\psi_{x_{1}}\dots\psi_{x_{n}}\right)(x)$. By applying inductively the
defining property $n$ times we get the following estimate which proves the
statement.

$\displaystyle|f(y)-f(x)|$
$\displaystyle=\left|f\left(\left(\psi_{x_{1}}\dots\psi_{x_{n}}\right)(x)\right)-f(x)\right|$
$\displaystyle=\left|f\left(\left(\psi_{x_{1}}\dots\psi_{x_{n}}\right)(x)\right)-f\left(\left(\psi_{x_{2}}\dots\psi_{x_{n}}\right)(x)\right)+f\left(\left(\psi_{x_{2}}\dots\psi_{x_{n}}\right)(x)\right)-f(x)\right|$
$\displaystyle\leq
D+\left|f\left(\left(\psi_{x_{2}}\dots\psi_{x_{n}}\right)(x)\right)-f(x)\right|$
$\displaystyle\leq\dots\leq Dn=Dd(x,y).$

∎

######  Corollary 5.5.

If a rack $(X,\triangleright)$ admits a quasimorphism that is unbounded on a
connected component $X_{s}$ then this component is unbounded. Equivalently, if
the kernel of the comparison map $H^{2}_{b}(X;{\mathbf{R}})\to
H^{2}(X;{\mathbf{R}})$ is nontrivial then $(X,\triangleright)$ is unbounded. ∎

######  Proposition 5.6.

If $(X,\triangleright)$ is unbounded then the kernel of the comparison map
$H^{2}_{b}(X;{\mathbf{R}})\to H^{2}(X;{\mathbf{R}})$ is nontrivial. In
particular, $H^{2}_{b}(X;{\mathbf{R}})\neq 0$.

###### Proof.

Let $x_{s}\in X_{s}$ be a basepoint fixed for each connected component $X_{s}$
of $(X,\triangleright)$. Let $f\colon X\to{\mathbf{R}}$ be defined by

$f(x)=d(x_{s},x)$

if $x\in X_{s}$. It is unbounded according to the hypothesis. Then

$|\delta f(x,y)|=|f(x)-f(\psi_{y}(x))|=|d(x_{s},x)-d(x_{s},\psi_{y}(x)|\leq
1.$

The cocycle is non-zero for a non-trivial rack. Its bounded cohomology class
is non-zero by an argument analogous to the one in Lemma 5.2. ∎

##  6 Amenability

This section is motivated by the result in group theory that states that the
bounded cohomology of an amenable group is trivial [8, Section 3.0]. We prove
that the bounded cohomology of a rack or a quandle $(X,\triangleright)$ with
amenable inner automorphism group is as trivial as if $(X,\triangleright)$
were finite. Our proof follows almost verbatim the proof of the analogous
statement finite racks by Etingof-Graña [4, Theorem 4.2].

######  Definition 6.1.

A group $G$ is called amenable if there exist a functional ${\bf
m}\colon\ell^{\infty}(G)\to{\mathbf{R}}$ such that

  1. 1.

${\mathbf{m}}(1)=1$;

  2. 2.

if $\varphi\geq 0$ then ${\bf m}(\varphi)\geq 0$;

  3. 3.

${\mathbf{m}}(\varphi\circ R_{h})={\mathbf{m}}(\varphi)$ for every $h\in G$.

Such a functional is called a right-invariant mean on $G$.

Let $(X,\triangleright)$ be a rack and assume that the group $G_{\psi}$ of its
inner automorphism is amenable. Let $P\colon C^{k}_{b}(X;{\mathbf{R}})\to
C^{k}_{b,\rm inv}(X;{\mathbf{R}})$ be defined by

$P(f)(x_{1},\ldots,x_{k})={\mathbf{m}}(g\mapsto f(g(x_{1}),\ldots,g(x_{k})).$

######  Lemma 6.2.

Let $(X,\triangleright)$ be a rack with amenable $G_{\psi}$. The projection
$P$ is a morphism of complexes and it induces a surjective homomorphism
$P\colon H^{*}_{b}(X;{\mathbf{R}})\to H^{*}_{b,\rm inv}(X;{\mathbf{R}})$. The
map $\iota\colon H^{*}_{b,\rm inv}(X;{\mathbf{R}})\to
H^{*}_{b}(X;{\mathbf{R}})$ induced by the inclusion of complexes is the right
inverse of $P$.

###### Proof.

First observe that $P(f)$ is an invariant cochain.

$\displaystyle(Pf)(hx_{1},\ldots,hx_{k})$ $\displaystyle={\mathbf{m}}(g\mapsto
f(ghx_{1},\ldots,ghx_{k}))$ $\displaystyle={\mathbf{m}}(g\mapsto
f(gx_{1},\ldots,gx_{k}))$ $\displaystyle=(Pf)(x_{1},\ldots,x_{k}),$

because the mean ${\mathbf{m}}$ is right-invariant. Secondly, the projection
$P$ commutes with differential.

$\displaystyle P(\delta f)(x_{1},\ldots,x_{k+1})$
$\displaystyle={\mathbf{m}}(g\mapsto\delta f(gx_{1},\ldots,gx_{k+1}))$
$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}{\mathbf{m}}(g\mapsto
f(gx_{1},\ldots,gx_{i-1},gx_{i+1},\ldots,gx_{k+1})))$
$\displaystyle-\sum_{i=1}^{k}(-1)^{i-1}{\mathbf{m}}(g\mapsto
f(gx_{1},\ldots,gx_{i-1},\psi_{gx_{i}}gx_{i+1},\ldots,\psi_{gx_{i}}gx_{k+1})))$
$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}{\mathbf{m}}(g\mapsto
f(gx_{1},\ldots,gx_{i-1},gx_{i+1},\ldots,gx_{k+1})))$
$\displaystyle-\sum_{i=1}^{k}(-1)^{i-1}{\mathbf{m}}(g\mapsto
f(gx_{1},\ldots,gx_{i-1},g\psi_{x_{i}}x_{i+1},\ldots,g\psi_{x_{i}}x_{k+1})))$
$\displaystyle=\sum_{i=1}^{k}(-1)^{i-1}Pf(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{k+1})$
$\displaystyle-\sum_{i=1}^{k}(-1)^{i-1}Pf(x_{1},\ldots,x_{i-1},\psi_{x_{i}}x_{i+1},\ldots,\psi_{x_{i}}x_{k+1})$
$\displaystyle=\delta(Pf)(x_{1},\ldots,x_{k}).$

The third equality follows from the identity $\psi_{gx}=g\psi_{x}g^{-1}$.

If $f$ is an invariant cocycle then it is immediate that $P(f)=f$ and hence
$P\circ\iota=\operatorname{Id}$ which implies the surjectivity of $P$ and the
last statement. ∎

######  Theorem 6.3.

Let $(X,\triangleright)$ be a rack with amenable group $G_{\psi}$ of inner
automorphisms. If the conjugation-invariant word norm on $G_{\psi}$ associated
with $\psi(X)$ has finite diameter then:

  1. 1.

the projection

$P\colon H^{*}_{b}(X;{\mathbf{R}})\to H^{*}_{b,\rm inv}(X;{\mathbf{R}})$

is an isomorphism;

  2. 2.

the homomorphism (from Example 4.5)

$\pi^{*}\colon\operatorname{Fun}_{b}(\pi_{0}(X)^{k},{\mathbf{R}})\to
H^{k}_{b}(X;{\mathbf{R}})$

is an isomorphism.

Consequently we have isomorphisms

$\operatorname{Fun}_{b}(\pi_{0}(X)^{k},{\mathbf{R}})\cong H^{k}_{b,\rm
inv}(X;{\mathbf{R}})\cong H^{k}_{b}(X;{\mathbf{R}}).$

######  Remark 6.4.

The statement for quandles is the same except that the space bounded functions
is replaced by the subspace
$\underline{\operatorname{Fun}}_{b}(\pi_{0}(X,\triangleright)^{k},{\mathbf{R}})$
from Example 4.4.

###### Proof.

We first prove that the projection $P\colon H^{*}_{b}(X;{\mathbf{R}})\to
H^{*}_{b,\rm inv}(X;{\mathbf{R}})$ is an isomorphism. Since the associated
word metric on $G_{\psi}$ has finite diameter, there exists $N\in{\mathbf{N}}$
such that for every $g\in G_{\psi}$ we have that
$g=\psi_{x_{1}}\dots\psi_{x_{n}}$ for some $x_{1},\ldots,x_{n}\in G_{\psi}$
and $n\leq N$. Let $f\colon X^{k}\to{\mathbf{R}}$ be a bounded cocycle. We
know from Lemma 4.8 that $f-f\circ\psi_{x}=\delta(f_{x})$. Applying this
identity inductively, we get that

$f\cdot
g=f-\delta\left((f_{x_{1}})+(f\cdot\psi_{x_{1}})_{x_{2}}+\dots+(f\cdot\psi_{x_{1}}\dots\psi_{x_{k-1}})_{x_{k}}\right).$

Let
$\alpha_{g}=(f_{x_{1}})+(f\cdot\psi_{x_{1}})_{x_{2}}+\dots+(f\cdot\psi_{x_{1}}\dots\psi_{x_{k-1}})_{x_{k}}$
(notice that there is a choice of a decomposition of $g$ embedded in this
definition). Since $f$ is bounded, so is the cochain $\alpha_{g}$, and there
exists a bound uniform with respect to $g$. We claim that $P(f)$ and $f$ are
homologous. Indeed,

$\displaystyle P(f)(x_{1},\ldots,x_{k})$
$\displaystyle={\mathbf{m}}(g\mapsto(f\cdot g)(x_{1},\ldots,x_{k}))$
$\displaystyle={\mathbf{m}}(g\mapsto((f-\delta\alpha_{g})(x_{1},\ldots,x_{k}))$
$\displaystyle=f(x_{1},\ldots,x_{k})-{\mathbf{m}}(g\mapsto\delta\alpha_{g}(x_{1},\ldots,x_{k}))$
$\displaystyle=f(x_{1},\ldots,x_{k})-\delta{\mathbf{m}}(g\mapsto\alpha_{g}(x_{1},\ldots,x_{k})).$

Since $\alpha_{g}$ is uniformly bounded, the cochain
$(x_{1},\ldots,x_{k})\mapsto{\mathbf{m}}(g\mapsto\alpha_{g}(x_{1},\ldots,x_{k}))$
is well defined and bounded. This shows that if $P[f]=0$ then, since $P(f)$
and $f$ are homologous, $[f]=0$ which shows the injectivity of $P$ on
cohomology. The surjectivity was shown in Lemma 6.2.

Now we prove the second statement. We start with the injectivity of $\pi^{*}$
and the proof is by induction with respect to the degree. The statement is
clear in degree zero. Assume it is true for all degrees smaller than $k$ and
let $f\in\operatorname{Fun}_{b}(\pi_{0}(X,\triangleright)^{k},{\mathbf{R}})$
be such that $\pi^{*}(f)=\delta\alpha$ for some $\alpha\in
C^{k-1}_{b}(X;{\mathbf{R}})$. It follows from the first part that we can take
$\alpha$ to be an invariant cochain which implies that

$(\pi^{*}f)_{x}=(\delta\alpha)_{x}=-\delta(\alpha_{x}),$

for every $x\in X$. However, $(\pi^{*}f)_{x}=\pi^{*}(f_{[x]})$, where
$f_{[x]}\colon\pi_{0}(X,\triangleright)^{k-1}\to{\mathbf{R}}$ is defined
analogously. The induction hypothesis implies that $f_{[x]}=0$ for every $x\in
X$ which proves the statement.

In order to prove surjectivity assume that the mean ${\mathbf{m}}$ on
$G_{\psi}$ is bi-invariant. Such a mean always exists on an amenable group $G$
[15, Exercise 1.26] which can be seen by taking the left-right action of
$G\times G$ on the compact convex set of means on $G$. Since $G\times G$ is
amenable as well, this action has a fixed point.

The second ingredient of the proof is the cochain complex decomposition

$C^{*}_{b}(X;{\mathbf{R}})=C_{b,\rm
inv}^{*}(X;{\mathbf{R}})\oplus(1-P)C^{*}_{b}(X;{\mathbf{R}});$

notice that the second summand is acyclic.

Let $[f]\in H^{k}_{b}(X;{\mathbf{R}})$ be any element. We can assume it is
represented by an invariant cocycle $f$. The invariance of $f$ implies that
$f_{x}$ is a cocycle for every $x\in X$. Consider the decomposition

$f_{x}=P(f_{x})+(1-P)(f_{x})$

for every $x\in X$. Let $f^{+},f^{-}\in C_{b}^{k}(X;{\mathbf{R}})$ be cochains
such that

$f^{+}_{x}=P(f_{x})\qquad\text{and}\qquad f^{-}_{x}=(1-P)(f_{x}),$

for every $x\in X$. The aim is to show that $f^{+}$ is an invariant cocycle
homologous to $f$. Let’s show the invariance first and let’s start with the
following computation. Let $h\in G_{\psi}$ and $x\in X$ be any elements.

$\displaystyle f^{+}_{hx}(x_{2},\ldots,x_{k})$
$\displaystyle=P(f_{hx})(x_{2},\ldots,x_{k})$
$\displaystyle={\mathbf{m}}(g\mapsto f_{hx}(gx_{2},\ldots,gx_{k}))$
$\displaystyle={\mathbf{m}}(g\mapsto f_{hx}(hgx_{2},\ldots,hgx_{k}))$ by left-
invariance of ${\mathbf{m}}$ $\displaystyle={\mathbf{m}}(g\mapsto
f(hx,hgx_{2},\ldots,hgx_{k}))$ $\displaystyle={\mathbf{m}}(g\mapsto
f(x,gx_{2},\ldots,gx_{k}))$ by invariance of $f$
$\displaystyle=P(f_{x})(x_{2},\ldots,x_{k})$
$\displaystyle=f^{+}_{x}(x_{2},\ldots,x_{k}).$

It follows that

$\displaystyle(f^{+}\cdot h)_{x}$ $\displaystyle=f^{+}_{hx}\cdot h$ obvious
$\displaystyle=f^{+}_{x}\cdot h$ by the previous computation
$\displaystyle=P(f_{x})\cdot h$ $\displaystyle=P(f_{x})$ by the invariance of
$P(f_{x})$ $\displaystyle=f^{+}_{x}.$

The invariance of $f^{+}$ is now clear:

$\displaystyle f^{+}(hx_{1},\ldots,hx_{k})$
$\displaystyle=f^{+}_{hx_{1}}(hx_{2},\ldots,hx_{k})=f^{+}_{x_{1}}(hx_{2},\ldots,hx_{k})$
$\displaystyle=(f^{+}_{x_{1}}\cdot
h)(x_{2},\ldots,x_{k})=f^{+}_{x_{1}}(x_{2},\ldots,x_{k})=f^{+}(x_{1},\ldots,x_{k}).$

Next observe that $f^{+}$ is a cocycle:

$\displaystyle\delta f^{+}(x,x_{1},\ldots,x_{k})$
$\displaystyle=f^{+}(x_{1},\ldots,x_{k})-f^{+}(\psi_{x}x_{1},\ldots,\psi_{x}x_{k})-\delta
f^{+}_{x}(x_{1},\ldots,x_{k})=0.$

The sum of the first two terms vanish by invariance of $f^{+}$ and the $\delta
f^{+}_{x}=\delta P(f_{x})=0$ because $f_{x}$ is a cocycle since $f$ is an
invariant one. We obtain that both summands in the decomposition
$f=f^{+}+f^{-}$ are invariant cocycles, since $f$ and $f^{+}$ are.

Next we show that $f^{-}$ is a coboundary. Let $\alpha\in
C^{k-1}(X;{\mathbf{R}})$ be such that $\delta(\alpha_{x})=f^{-}_{x}$ for every
$x\in X$. We have that

$\delta\left((\alpha\cdot g)_{x}\right)=\delta(\alpha_{gx}\cdot
g)=\delta(\alpha_{gx})\cdot g=f^{-}_{gx}\cdot g=(f^{-}\cdot g)_{x}=f^{-}_{x},$

for every $g\in G_{\psi}$ and $x\in X$. It follows that

$\displaystyle\delta\left((P\alpha)_{x}\right)(x_{1},\ldots,x_{k})$
$\displaystyle=(P\alpha)_{x}(x_{2},\ldots,x_{k})-(P\alpha)_{x}(\psi_{x_{1}}x_{2},\ldots,\psi_{x_{1}}x_{k})-\dots$
$\displaystyle=(P\alpha)(x,x_{2},\ldots,x_{k})-(P\alpha)(x,\psi_{x_{1}}x_{2},\ldots,\psi_{x_{1}}x_{k})-\dots$
$\displaystyle={\mathbf{m}}(g\mapsto\alpha(gx,gx_{2},\ldots,gx_{k})-\alpha(gx,g\psi_{x_{1}}x_{2},\ldots,g\psi_{x_{1}}x_{k})-\dots)$
$\displaystyle={\mathbf{m}}(g\mapsto(\alpha\cdot
g)(x,x_{2},\ldots,x_{k})-(\alpha\cdot
g)(x,\psi_{x_{1}}x_{2},\ldots,\psi_{x_{1}}x_{k})-\dots)$
$\displaystyle={\mathbf{m}}(g\mapsto(\alpha\cdot
g)_{x}(x_{2},\ldots,x_{k})-(\alpha\cdot
g)_{x}(\psi_{x_{1}}x_{2},\ldots,\psi_{x_{1}}x_{k})-\dots)$
$\displaystyle={\mathbf{m}}(g\mapsto\delta\left((\alpha\cdot
g)_{x}\right)(x_{1},\ldots,x_{k}))$ $\displaystyle={\mathbf{m}}(g\mapsto
f^{-}_{x}(x_{1},\ldots,x_{k}))=f^{-}_{x}(x_{1},\ldots,x_{k}).$

Since $P\alpha$ is invariant we get that

$(\delta(P\alpha))_{x}=\delta((P\alpha)_{x})=f^{-}_{x},$

for every $x\in X$ which implies that $\delta\alpha=f^{-}$. Hence we can
assume that the class $[f]$ is represented by $f^{+}$.

Since $f^{+}_{x}=P(f_{x})$ is an invariant $(k-1)$-cocycle for every $x\in X$,
we can consider $f^{+}$ as a cocycle of the form

$\sum_{s\in\pi_{0}(X,\triangleright)}I_{s}\otimes f^{+}_{s}\in
Z^{1}_{b}(X;{\mathbf{R}})\otimes Z^{k-1}_{b,\rm inv}(X;{\mathbf{R}})\subseteq
Z_{b,\rm inv}^{k}(X;{\mathbf{R}}),$

where $I_{s}$ is the indicator function of $\\{s\\}$, that is $I_{s}(t)=1$ if
$s=t$ and zero otherwise. Proceeding by induction we get that every cohomology
class in $H^{k}_{b}(X;{\mathbf{R}})$ is decomposed as a product of classes of
degree one and hence they are represented by cocycles from
$\operatorname{Fun}_{b}(\pi_{0}(X)^{k},{\mathbf{R}})$. ∎

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# Conic decomposition of a toric variety and its application to cohomology

Seonjeong Park  Department of Mathematics Education, Jeonju University, 303,
Cheonjam-ro, Wansan-gu, Jeonju-si, Jeollabuk-do, 55069, Republic of Korea
<EMAIL_ADDRESS>and  Jongbaek Song  School of Mathematics, KIAS, 85
Hoegiro Dongdaemun-gu, Seoul 02455, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of a _conic sequence_ of a convex polytope. It is a
way of building up a polytope starting from a vertex and attaching faces one
by one with certain regulations. We apply this to a toric variety to obtain an
iterated cofibration structure on it. This allows us to prove several
vanishing results in the rational cohomology of a toric variety and to
calculate Poincaré polynomials for a large class of singular toric varieties.

###### Key words and phrases:

toric variety, singular cohomology, Betti number, $f$-vector, $h$-vector,
Poincaré polynomial

###### 2020 Mathematics Subject Classification:

14M25, 55N10, 52B05, 52B11

##  1\. Introduction

Let $P$ be a convex polytope of dimension $n$ and
$f(P)\colonequals(f_{0},\dots,f_{n})$ its face vector, i.e., $f_{i}$ is the
number of $i$-dimensional faces of $P$. In literature, the vector $f(P)$ is
often called the _$f$ -vector_ or the _face vector_ of $P$. If $P$ is a simple
polytope, one can associate the _$h$ -vector_ $h(P)=(h_{0},\dots,h_{n})$
defined by the following equation

(1.1) $\sum_{k=0}^{n}f_{k}(x-1)^{k}=\sum_{k=0}^{n}h_{k}x^{k}.$

From the geometric point of view, if $P$ is a full dimensional lattice
polytope in $M\otimes_{\mathbb{Z}}\mathbb{R}$ for some $n$-dimensional lattice
$M$, one can associate a projective toric variety $X_{P}$ of complex dimension
$n$ whose orbit space with respect to the action of the $n$-dimensional
compact torus $T^{n}$ is combinatorially equivalent to $P$. In particular, if
$P$ is a simple lattice polytope, the corresponding toric variety $X_{P}$ is a
rationally smooth projective toric variety, which could be understood as a
_toric orbifold_ [DJ91, PS10, BSS17]. In this case, the singular cohomology of
$X_{P}$ with rational coefficients vanishes in odd degrees. Moreover, plugging
$t^{2}$ into $x$ in the right-hand side of (1.1), we get the Poincaré
polynomial $\operatorname{\mathsf{Poin}}(X_{P},t)$ of $X_{P}$, namely $h_{k}$
is the rank of the $2k$-th cohomology group. In other words, the Betti numbers
of a rationally smooth toric variety are completely determined by its
underlying combinatorics. This is one of the key observations in Stanley’s
proof of the $g$-conjecture, we refer to [McM71, Sta80].

If we turn our attention to more general lattice polytopes other than simple
polytopes, then the associated projective toric varieties are no longer
rationally smooth. In general, for a singular projective variety, we often
consider the intersection cohomology $\mathit{IH}^{\ast}(-)$ because it
carries geometric and topological properties of singular strata. Moreover,
there are certain similarities between $\mathit{IH}^{\ast}(-)$ and the
singular cohomology of a smooth variety, for instance Poincaré duality or the
hard Lefschetz property. Especially for a projective toric variety $X_{P}$,
the Betti numbers of $\mathit{IH}^{\ast}(X_{P})$ are combinatorial invariants.
Indeed, Stanley [Sta87] defines the _toric $h$-polynomial_ of $P$, which
agrees with (1.1) when $P$ is a simple polytope, and he shows that the
coefficients in the toric $h$-polynomial give the Betti numbers of
$\mathit{IH}^{\ast}(X_{P})$.

However, if we stick to the singular cohomology of a toric variety with
singularities beyond orbifold singularities, it is somehow more subtle to deal
with than the intersection cohomology because it is far from the Poincaré
duality, as well as its rational cohomology is not concentrated in even
degrees in general. For instance, the $6$-dimensional toric variety associated
with the $3$-dimensional cross polytope, that is the convex hull of $(\pm
1,0,0),(0,\pm 1,0)$ and $(0,0,\pm 1)$ in
$\mathbb{Z}^{3}\otimes_{\mathbb{Z}}\mathbb{R}$, has a nontrivial rational
cohomology class in degree $3$. We refer to [CLS11, Example 12.3.8] whose
computation is based on the results in [Fis92, Jor98]. Moreover, Barthel, et.
al. [BBFK96, Example 3.5] provide two $6$-dimensional toric varieties $X$ and
$X^{\prime}$ associated with combinatorially equivalent polytopes such that
$\operatorname{rank}H^{3}(X;\mathbb{Q})\neq\operatorname{rank}H^{3}(X^{\prime};\mathbb{Q})$.
This tells us that the Betti numbers are neither combinatorial invariants in
general, nor obtainable by the $h$-vector defined in (1.1).

The aim of this paper is to explore the category of toric varieties other than
toric orbifolds whose Betti numbers for the singular cohomology can be
calculated by the underlying combinatorics. Our main idea is to consider a
topological stratification

(1.2) $\\{\text{a point}\\}=X_{1}\subset X_{2}\subset\cdots\subset X_{P}$

on $X_{P}$ such that each stratum is a mapping cone, see (3.3). We call such a
stratification a _conic decomposition_ of $X_{P}$, which is named after the
notion of a conic decomposition of a map $f\colon X\to Y$ introduced by
Jeffrey A. Strom. In this case, we consider the map $\\{\text{a
point}\\}\hookrightarrow X_{P}$, regarding $X_{P}$ as a pointed space. A conic
decomposition of $X_{P}$ can be parametrized by a sequence of certain
subspaces in $P$, which we call a _conic sequence_ of $P$. Beginning with $P$
itself as the initial term, the sequence continues inductively by deleting
faces whose geometric realization agrees with a cone on a polytope if it has,
and the sequence stops otherwise. We discuss explicit combinatorial and
topological definitions of a conic sequence of $P$ in Section 2.

We emphasize again that each stratum $X_{i}$ in (1.2) is a mapping cone of a
map $\Phi_{i}\colon Y_{i}\to X_{i-1}$ for some topological space $Y_{i}$.
Hence, the topology of $X_{i}$ is dominated by the topology of $Y_{i}$
together with the attaching map $\Phi_{i}$. Accordingly, if a toric variety
$X_{P}$ admits a conic decomposition, then one can study the topology of
$X_{P}$ by analyzing each cofiber sequence

(1.3) $Y_{i}\xrightarrow{\Phi_{i}}X_{i-1}\to X_{i}$

successively.

In Section 3, we focus on the case where the conic sequence of $P$ is given by
iterated deletions of the cones of simple polytopes, where we denote these
simple polytopes by $C_{i}$’s throughout this paper. This is equivalent to say
that $Y_{i}$ in (1.3) is an orbifold $S^{1}$-fibration over a toric orbifold
corresponding to $C_{i}$ (Lemma 3.2). A particular example of this setup is
the case where each $C_{i}$ is given by a simplex, which has been introduced
in [BSS17] and further studied in subsequent papers [BSS19, BNSS21, SS21,
LMPS21], where the conic sequence of this particular type is referred as a
_retraction sequence_.

In Section 4, we use a conic decomposition of $X_{P}$ to calculate the
cohomology group of $X_{P}$ with rational coefficients. We first see how the
cohomology group of $Y_{i}$ affects the cohomology of $X_{i}$. Then we prove
several vanishing theorems for the cases discussed in Section 3. In
particular, we verify in Theorem 4.5 that a certain class of toric varieties
including all toric orbifolds enjoys (1.1) to obtain their Poincaré
polynomials.

### Acknowledgements

Some part of this paper is motivated by Jeffrey A. Strom’s talk at Fields
institute during the Thematic program on Toric Topology and Polyhedral
Products in 2020. The second named author is grateful to the organizers of the
program and to Jeffrey A. Strom for his nice presentation and communication
about the usage of some terminologies.

Park has been supported by Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT
(NRF-2020R1A2C1A01011045). Song has been supported by Basic Science Research
Program through the National Research Foundation of Korea (NRF) funded by the
Ministry of Education (NRF-2018R1D1A1B07048480) and a KIAS Individual Grant
(MG076101) at Korea Institute for Advanced Study. This research was supported
by the Research Grant of Jeonju University in 2021.

##  2\. Conic sequence of a convex polytope

In this section, we introduce the notion of a conic sequence of a convex
polytope. A polytopal complex $\mathcal{P}$ is a finite collection of
polytopes in an Euclidean space such that

  1. (i)

the empty polytope is in $\mathcal{P}$,

  2. (ii)

if $E\in\mathcal{P}$, then all the faces of $E$ are also in $\mathcal{P}$, and

  3. (iii)

the intersection $E\cap F$ of two polytopes $E,F\in\mathcal{P}$ is a face both
of $E$ and of $F$.

The dimension $\dim\mathcal{P}$ is the largest dimension of a polytope in
$\mathcal{P}$. The geometric realization of $\mathcal{P}$ is the point set
$|\mathcal{P}|\colonequals\bigcup_{E\in\mathcal{P}}E$. The combinatorial
structure of a polytopal complex $\mathcal{P}$ is captured by its face poset
$L(\mathcal{P}):=(\mathcal{P},\subseteq)$, that is, the finite set of
polytopes in $\mathcal{P}$, ordered by inclusion. We define two polytopal
complexes to be combinatorially equivalent if their face posets are isomorphic
as posets. We refer to [Zie95, Chapter 5] for more details. In what follows,
we assume that $\mathcal{P}$ is of dimension $n$ otherwise stated and we
denote by $\ell$ the number of all the vertices of $\mathcal{P}$.

We now define a sequence $\\{\mathcal{P}_{i}\\}_{1\leq i\leq\ell}$ of
polytopal subcomplexes $\mathcal{P}_{i}$ of $\mathcal{P}$ inductively as
follows. For technical reasons, we construct the sequence with decreasing
indices. We set the initial term $\mathcal{P}_{\ell}$ by $\mathcal{P}$. Given
$\mathcal{P}_{i}$ for $i\leq\ell$, we take a vertex $v_{i}$ of
$\mathcal{P}_{i}$ such that there exists a unique maximal element $E_{i}$ in
the subposet $\\{F\in L(\mathcal{P}_{i})\mid v_{i}\subseteq F\\}$. We call
this vertex $v_{i}$ a _cone vertex_ of $\mathcal{P}_{i}$. Then we define the
next term $\mathcal{P}_{i-1}$ by

(2.1) $\mathcal{P}_{i-1}\colonequals\mathcal{P}_{i}-[v_{i},E_{i}],$

where $[v_{i},E_{i}]\colonequals\\{F\in\mathcal{P}_{i}\mid v_{i}\subseteq
F\subseteq E_{i}\\}$. We call the sequence $\\{\mathcal{P}_{i}\\}_{1\leq
i\leq\ell}$ defined as above a _conic sequence_ of $\mathcal{P}$ if the
sequence ends up with $\mathcal{P}_{1}=\\{\emptyset,v_{1}\\}$ for some vertex
$v_{1}$ of $\mathcal{P}$. If in particular $\mathcal{P}$ is the complex
associated with a polytope $P$, namely the complex of all faces of $P$, then
we simply call the sequence $\\{\mathcal{P}_{i}\\}_{1\leq i\leq\ell}$ a conic
sequence of a polytope $P$.

######  Example 2.1.

Consider the square $P$ as a $2$-dimensional polytope. Its associated
polytopal complex $\mathcal{P}$ is given by
$\\{\emptyset,p_{1},p_{2},p_{3},p_{4},e_{14},e_{12},e_{23},e_{34},P\\}$, where
$p_{1},\dots,p_{4}$ are vertices of $P$ and each $e_{ij}$ is the edge
connecting $p_{i}$ and $p_{j}$. Recall that the initial term $\mathcal{P}_{4}$
of a conic sequence is $\mathcal{P}$ itself. Now, we may take $p_{1}$ as a
cone vertex of $\mathcal{P}_{4}$ because the subposet
$\\{F\in{L}(\mathcal{P})\mid p_{1}\subseteq F\\}$ has the unique maximal
element $P$. Hence, we have

$\mathcal{P}_{3}=\mathcal{P}-\\{p_{1},e_{12},e_{14},P\\}=\\{\emptyset,p_{2},p_{3},p_{4},e_{23},e_{34}\\}.$

Since the subposet $\\{F\in{L}(\mathcal{P}_{3})\mid p_{3}\subseteq F\\}$ has
two maximal elements $e_{23}$ and $e_{34}$, the vertex $p_{3}$ does not
satisfy the condition for a cone vertex. Instead, one can take either $p_{2}$
or $p_{4}$ as a cone vertex to construct the next term of a conic sequence.
Figure 1 shows an example of a conic sequence of the polytope $P$, where the
cone vertices are highlighted.

$P$$e_{14}$$e_{12}$$e_{23}$$e_{34}$$\boldsymbol{p_{1}}$$p_{2}$$p_{3}$$p_{4}$$\emptyset$${L}(\mathcal{P})={L}(\mathcal{P}_{4})$$e_{23}$$e_{34}$$\boldsymbol{p_{2}}$$p_{3}$$p_{4}$$\emptyset$${L}(\mathcal{P}_{3})$$e_{34}$$\boldsymbol{p_{3}}$$p_{4}$$\emptyset$${L}(\mathcal{P}_{2})$$p_{4}$$\emptyset$${L}(\mathcal{P}_{1})$
Figure 1. The face posets associated with a conic sequence of a square.

Note that not every convex polytope admits a conic sequence. For example, the
octahedron does not admit a conic sequence. In fact, any $n$-gonal bipyramid
does not admit a conic sequence for $n\geq 4$.

######  Definition 2.2.

A convex polytope $P$ is said to be _conic_ if it admits a conic sequence.

Since each $\mathcal{P}_{i}$ is a polytopal subcomplex of $\mathcal{P}_{i+1}$,
a conic sequence $\\{\mathcal{P}_{i}\\}_{1\leq i\leq\ell}$ of a polytopal
complex $\mathcal{P}$ associated with a conic polytope $P$ induces an
increasing filtration

(2.2) $P_{1}\hookrightarrow\cdots\hookrightarrow P_{i}\hookrightarrow
P_{i+1}\hookrightarrow\cdots\hookrightarrow P_{\ell-1}\hookrightarrow
P_{\ell}=P$

of $P$, where each $P_{i}$ is the geometric realization $|\mathcal{P}_{i}|$.
In particular, $P_{i}-P_{i-1}$ agrees with the union

$\bigcup_{F\in[v_{i},E_{i}]}\operatorname{relint}(F),$

where $\operatorname{relint}(F)$ means the relative interior of the face $F$
in $P$.

Let $C_{i}$ be the intersection of $E_{i}$ with an affine hyperplane
separating $v_{i}$ and $P_{i-1}\cap E_{i}$. Then we can characterize $P_{i}$
topologically by the mapping cone of the map

(2.3) $\phi_{i}\colon C_{i}\to P_{i-1}$

whose image $\phi(x)$ of $x\in C_{i}$ is defined by the intersection of
$P_{i-1}$ and the straight line passing through $x$ and $v_{i}$. See Figure 2
for a pictorial description of $\phi_{i}$ for the polytope given in Example
2.1, where we illustrate $\phi_{4}$ with respect to the first cone vertex
$v_{4}$. We finally notice that since there is only one vertex of $E_{i}$ not
contained in $P_{i-1}$, the dimension of $E_{i}$ is at most $\dim P_{i-1}+1$,
so we have $\dim C_{i}\leq\dim P_{i-1}$.

$x$$\phi_{i}(x)\in P_{3}$$C_{4}$$v_{4}$ Figure 2. A pictorial description of
$\phi_{i}$.

In order to keep the information of $C_{i}$’s for $2\leq i\leq\ell$, we extend
the filtration in (2.2) to the following diagram

(2.4)
${C_{2}}$${C_{i}}$${C_{i+1}}$${C_{\ell}}$${P_{1}}$${\cdots}$${P_{i-1}}$${P_{i}}$${\cdots}$${P_{\ell-1}}$${P_{\ell}=P,}$$\scriptstyle{\phi_{2}}$$\scriptstyle{\phi_{i}}$$\scriptstyle{\phi_{i+1}}$$\scriptstyle{\phi_{\ell}}$

and we summarize the information of (2.4) as follows:

  1. (1)

$P_{1}$ is a vertex of $P$;

  2. (2)

each $C_{i}$ for $2\leq i\leq\ell$ is a convex polytope with $\dim
C_{i}\leq\dim P_{i-1}$; and

  3. (3)

each $P_{i}$ for $2\leq i\leq\ell$ is the mapping cone of the map
$\phi_{i}\colon C_{i}\to P_{i-1}$ defined in (2.3).

For convenience, we denote by $\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ the
sequence $\\{C_{2},\dots,C_{\ell}\\}$ corresponding to the given conic
sequence $\\{\mathcal{P}_{i}\\}_{1\leq i\leq\ell}$ of a polytope $P$. If
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ consists of simplices, we call (2.4) a
_$\Delta$ -conic sequence_, which agrees with the definition of a retraction
sequence studied in [BSS17, BNSS21]. A polytope admitting a $\Delta$-conic
sequence is referred to as an _almost simple polytope_ in [SS21]. We note that
every simple polytope has at least one $\Delta$-conic sequence (see [BSS17,
Proposition 2.3]), which is not true for arbitrary convex polytope.

######  Example 2.3.

A triangular bipyramid $P$ is conic, but not $\Delta$-conic. Indeed, there is
a conic sequence $\\{\mathcal{P}_{i}\\}_{1\leq i\leq 5}$ of $P$ such that
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}=\\{\mathrm{pt},I,I,I^{2}\\}$, see Figure
3.

$P_{1}$${P_{2}\atop C_{2}=\text{pt}}$${P_{3}\atop C_{3}=I}$${P_{4}\atop
C_{4}=I}$${P_{5}\atop C_{5}=I^{2}}$ Figure 3. A conic sequence on a triangular
bipyramid.

In the following, we see the relationship between the face numbers of $P$ and
the face numbers of $C_{i}\in\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ for
$i=2,\dots,\ell$. Let $f_{i}(P)$ be the number of $i$-dimensional faces of a
polytope $P$, and let $\Phi_{P}(x)$ be the generating function of the face
numbers of $P$, i.e., $\Phi_{P}(x)\colonequals\sum_{j=0}^{\dim
P}f_{j}(P)x^{j}$.

######  Proposition 2.4.

Let $\mathscr{C}_{\\{\mathcal{P}_{i}\\}}=\\{C_{2},\dots,C_{\ell}\\}$ be the
sequence of polytopes corresponding to a conic sequence
$\\{\mathcal{P}_{i}\\}_{1\leq i\leq\ell}$ of $P$ as in (2.4). Then

$\Phi_{P}(x)=1+\sum_{i=2}^{\ell}\left(1+x\Phi_{C_{i}}(x)\right).$

###### Proof.

Applying (2.1) iteratively, we obtain

(2.5) $\Phi_{P}(x)=1+\sum_{i=2}^{\ell}\sum_{F\in[v_{i},E_{i}]}x^{\dim F}.$

Since there is a one-to-one correspondence between the $k$-dimensional faces
of $C_{i}$ and the $(k+1)$-dimensional faces of the interval $[v_{i},E_{i}]$
for each $0\leq k\leq\dim C_{i}$, we have

(2.6) $\sum_{F\in[v_{i},E_{i}]}x^{\dim F}=1+\sum_{k=0}^{\dim
C_{i}}f_{k}(C_{i})x^{k+1}=1+x\Phi_{C_{k}}(x).$

Hence the result follows by combining (2.5) and (2.6). ∎

######  Example 2.5.

Let $P$ be the following Bruhat interval polytope (see [TW15]):

$\mathsf{Q}_{1324,4231}\colonequals\text{conv}\\{(\sigma(1),\sigma(2),\sigma(3),\sigma(4))\in\mathbb{R}^{4}\mid\sigma\in\mathfrak{S}_{4},~{}(1324)\leq\sigma\leq(4231)\\},$

where $\leq$ denotes the Bruhat order on the permutation group
$\mathfrak{S}_{4}$ and $(i_{1}i_{2}i_{3}i_{4})$ is a permutation written in
the one-line notation. See Figure 4 for the description of
$\mathsf{Q}_{1324,4231}$. A conic sequence of $\mathsf{Q}_{1324,4231}$ is
illustrated in Figure 5. In this case, the corresponding sequence
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ and their generating functions are given
by

$C_{i}=\begin{cases}\mathrm{pt}&2\leq i\leq 5;\\\ I&6\leq i\leq 15;\\\
I^{2}&i=16,\end{cases}\quad\text{and}\quad\Phi_{k}(x)=\begin{cases}1&2\leq
k\leq 5;\\\ 2+x&6\leq k\leq 15;\\\ 4+4x+x^{2}&k=16.\end{cases}$

We apply Proposition 2.4 to get

$\displaystyle\begin{split}\sum_{k=0}^{3}f_{k}(\mathsf{Q}_{1324,4231})x^{k}&=1+4(1+x)+10(1+2x+x^{2})+(1+4x+4x^{2}+x^{3})\\\
&=16+28x+14x^{2}+x^{3}.\end{split}$

It is easy to see that $\mathsf{Q}_{1324,4231}$ does not admit a
$\Delta$-conic sequence, see [LMPS21, Section 5] for instance.

$(2,3,1,4)$$(2,4,1,3)$$(1,4,2,3)$$(1,3,2,4)$$(1,3,4,2)$$(1,4,3,2)$$(3,1,4,2)$$(4,2,1,3)$$(4,1,2,3)$$(3,1,2,4)$$(3,2,1,4)$$(1,2,3,4)$$(1,2,4,3)$$(2,1,4,3)$$(2,1,3,4)$$(4,3,2,1)$$(3,4,2,1)$$(3,4,1,2)$$(4,3,1,2)$$(2,3,4,1)$$(3,2,4,1)$$(4,2,3,1)$$(4,1,3,2)$$(2,4,3,1)$
Figure 4. The Bruhat interval polytope $\mathsf{Q}_{1324,4231}$. $(1,3,2,4)$
Figure 5. A conic decomposition of $\mathsf{Q}_{1324,4231}$.

We notice that the generating function $\Phi_{\Delta^{d-1}}(x)$ of
$(d-1)$-simplex $\Delta^{d-1}$ is $\sum_{k=1}^{d}{d\choose k}x^{k-1}$, which
means

$1+x\Phi_{\Delta^{d-1}}(x)=(x+1)^{d}.$

Hence Proposition 2.4 implies that if $P$ is $\Delta$-conic, then

(2.7) $\displaystyle\sum_{k=0}^{\dim P}f_{k}(P)x^{k}=\sum_{k=0}^{\dim
P}h_{k}(x+1)^{k},$

where $h_{0}=1$ and $h_{k}$ denotes the number of $(k-1)$-simplices in
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ for a $\Delta$-conic sequence of $P$.
Moreover, the condition $\dim C_{i}\leq\dim P_{i-1}$ for a conic sequence
implies that $h_{k}\geq 1$ for all $k=1,\dots,\dim P$. This observation
establishes the following corollary which provides a necessary condition for a
polytope to be $\Delta$-conic.

######  Corollary 2.6.

If a convex polytope $P$ admits a $\Delta$-conic sequence, then $h_{k}$’s in
(2.7) are positive for all $k=1,\dots,\dim P$.

Using a similar idea of Corollary 2.6, one can also discuss a necessary
condition for a polytope $P$ to have a conic sequence such that the
corresponding sequence $\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ consists of
cubes. Indeed, since the generating function $\Phi_{I^{d-1}}(x)$ for the
$(d-1)$-cube $I^{d-1}$ is

$\Phi_{I^{d-1}}(x)=\sum_{k=1}^{d}{d-1\choose k-1}2^{d-k}x^{k-1},$

we have

$1+x\Phi_{I^{d-1}}(x)=1+x(x+2)^{d-1}.$

Therefore, if $P$ admits a conic sequence such that
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ consists of cubes, Proposition 2.4 gives
us

$\sum_{k=0}^{\dim P}f_{k}(P)x^{k}=\sum_{k=0}^{\dim
P}h_{k}^{\square}\left(1+x(x+2)^{k-1}\right),$

for some positive integers $h_{k}^{\square}$’s with $h_{1}^{\square}=1$.

We refer to Example 2.5, where
$h^{\square}(\mathsf{Q}_{1324,4231})=(h^{\square}_{0},h^{\square}_{1},h^{\square}_{2},h^{\square}_{3})=(1,4,10,1)$.

We remark that the same argument can apply to other conic sequences with
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ consisting of certain polytopes whose
generating functions are written as polynomials depending only on their
dimensions like simplices or cubes.

##  3\. Conic decomposition of a toric variety

We begin this section with a topological model of a projective toric variety
$X_{P}$ associated with a lattice polytope $P$, following [Jor98, Chapter 1]
which is attributed to MacPherson. This gives us a way to understand the
underlying topological space of $X_{P}$. We also investigate several vanishing
theorems for the singular cohomology of the toric variety $X_{P}$, associated
with a conic polytope $P$.

###  3.1. Topological model of a toric variety

Let $M$ be the character lattice of the $n$-dimensional compact torus $T^{n}$.
We consider an $n$-dimensional lattice polytope $P$ in
$M_{\mathbb{R}}\colonequals M\otimes_{\mathbb{Z}}\mathbb{R}$. Let
$\mathcal{F}(P)$ be the set of facets of $P$ and $\lambda(F)$ be the primitive
outward normal vector of $F\in\mathcal{F}(P)$. Regarding $\lambda(F)$ as an
element in the lattice $N$ of one-parameter subgroups of $T^{n}$, we denote by
$T_{F}$ the circle subgroup of $T^{n}$ generated by $\lambda(F)$. In general,
given a face $E$ of $P$, we denote by $N_{E}$ the submodule generated by

(3.1) $\\{\lambda(F)\in N\mid F\in\mathcal{F}(P),~{}F\cap E\neq\emptyset\\}.$

Then $N_{E}$ determines the subtorus $T_{E}\colonequals
N_{E}\otimes_{\mathbb{Z}}\mathbb{R}/\left((N_{E}\otimes_{\mathbb{Z}}\mathbb{R})\cap
N\right)$ of $T^{n}$.

The topological model of the toric variety $X_{P}$ associated with a lattice
polytope $P$ is given

(3.2) $X_{P}\colonequals(P\times T^{n})/_{\sim},$

where we identify $(p,t)\sim(q,s)$ whenever $p=q$ and $t^{-1}s\in T_{E}$ if
$p$ is in the relative interior of $E$. Notice that the action of $T^{n}$ on
$X_{P}$ given by the multiplication on the second factor induces the orbit map
$\pi\colon X_{P}\to P$, that is the projection onto the first factor.

If $P$ is a simple lattice polytope, then each face $E$ of codimension-$k$ in
$P$ is the transversal intersection of $k$ facets of $P$ by definition. Hence
the elements in (3.1) are linearly independent. In this case, the resulting
toric variety $X_{P}$ has at worst finite quotient singularities and $X_{P}$
is often called a toric orbifold, see [CLS11, Theorem 11.4.8] for instance. We
also refer to [DJ91, Section 7] and [PS10] for more rigorous topological
interpretation of a toric orbifold.

###  3.2. Toric variety with a conic decomposition

In what follows, we mainly consider a conic lattice polytope $P$ in
$M_{\mathbb{R}}$. We may regard each
$C_{i}\in\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ as a rational polytope lying in
an affine subspace of $M_{\mathbb{R}}$. This yields the following diagram,
which we call a _conic decomposition_ of $X_{P}$;

(3.3)
${Y_{2}}$${Y_{i}}$${Y_{i+1}}$${Y_{\ell}}$${X_{1}}$${\cdots}$${X_{i-1}}$${X_{i}}$${\cdots}$${X_{\ell-1}}$${X_{\ell}=X,}$$\scriptstyle{\Phi_{2}}$$\scriptstyle{\Phi_{i}}$$\scriptstyle{\Phi_{i+1}}$$\scriptstyle{\Phi_{\ell}}$

where $X_{i}=\pi^{-1}(P_{i})$ and $Y_{i}=\pi^{-1}(C_{i})$ with respect to the
orbit map $\pi\colon X_{P}\to P$. Moreover, the attaching map $\phi_{i}$
defined in (2.3) induces a map $\Phi_{i}\colon Y_{i}\to X_{i-1}$ by the
following commutative diagram:

${C_{i}\times T^{n}}$${(C_{i}\times T^{n})/_{\sim}=Y_{i}}$${P_{i-1}\times
T^{n}}$${(P_{i-1}\times T^{n})/_{\sim}=X_{i-1}.}$$\scriptstyle{\phi_{i}\times
id}$$\scriptstyle{\Phi_{i}}$

This leads us to understand each $X_{i}$ as the cofiber of $\Phi_{i}\colon
Y_{i}\to X_{i-1}$.

The next two lemmas describe the topology of $Y_{i}$ when (i) $C_{i}$ is a
simplex or (ii) $C_{i}$ is a simple polytope more generally.

######  Lemma 3.1.

If $C_{i}$ is a simplex $\Delta^{d_{i}-1}$ for some $d_{i}\geq 1$, then
$Y_{i}$ is homeomorphic to $S^{2d_{i}-1}/K_{i}$ for some finite group $K_{i}$.

###### Proof.

The proof is essentially the same as the proof of [BNSS21, Propsotion 4.4] and
[SS21, Corollary 3.7]. However, for the reader’s convenience, we shall provide
a sketch of the proof. Consider the conic decomposition in (3.3), which gives
us

$X_{i}-X_{i-1}=\pi^{-1}(P_{i})-\pi^{-1}(P_{i-1})\cong\pi^{-1}(U_{i}),$

where $U_{i}=P_{i}-P_{i-1}$ is homeomorphic to $\mathbb{R}^{d_{i}}_{\geq}$ as
a manifold with corners because $C_{i}=\Delta^{d_{i}-1}$. Then (3.2) implies
that

$\pi^{-1}(U_{i})\cong U_{i}\times T^{d_{i}}/_{\sim}\cong D^{2d_{i}}/K_{i}$

for some finite subgroup $K_{i}$ of $T^{d_{i}+1}$ acting linearly on
$D^{2d_{i}}$. Since $X_{i}$ is a mapping cone of the map $\Phi_{i}\colon
Y_{i}\to X_{i-1}$, we conclude that $Y_{i}\cong S^{2d_{i}-1}/K_{i}$ as
desired. ∎

We notice that the space $S^{2d_{i}-1}/K_{i}$ in Lemma 3.1 above is called an
_orbifold lens space_ in [BSS17]. In Lemma 3.2 below, we consider the case
where $C_{i}$ is a simple polytope. We refer the readers to [Rua02] for the
rigorous definition of an orbifold fiber bundle.

######  Lemma 3.2.

If $C_{i}$ is a simple polytope, then $Y_{i}$ is an orbifold $S^{1}$-fiber
bundle over a toric orbifold.

###### Proof.

Let $E_{i}$ be the face of $P$ defined in (2.1). Then it is the minimal face
of $P$ containing $C_{i}$. Since the equivalence relation $\sim$ given in
(3.2) collapses $T_{E_{i}}$ globally on $C_{i}$, we may write

$Y_{i}=(C_{i}\times T^{n}/T_{E_{i}})/_{\sim}.$

Take an affine subspace $\mathcal{A}_{C_{i}}\subset M_{\mathbb{R}}$
intersecting $E_{i}$ transversely such that $C_{i}$ is a rational polytope in
$\mathcal{A}_{C_{i}}$ with respect to the lattice $\mathcal{A}_{C_{i}}\cap M$.
Let $\mu_{C_{i}}\in N\setminus N_{E_{i}}$ be a primitive element which
determines $\mathcal{A}_{C_{i}}$. Then the submodule $N_{C_{i}}\subset N$,
generated by all elements of $N_{E_{i}}$ together with $\mu_{C_{i}}$, gives
rise to the torus

$T_{C_{i}}\colonequals
N_{C_{i}}\otimes_{\mathbb{Z}}\mathbb{R}/\left((N_{C_{i}}\otimes_{\mathbb{Z}}\mathbb{R})\cap
N\right)$

with $\dim T_{C_{i}}=\dim T_{E_{i}}+1$. Accordingly, we have $\dim
T^{n}/T_{E_{i}}=\dim T^{n}/T_{C_{i}}+1$.

Let $S^{1}_{\mu_{C_{i}}}$ be the image of the composition

(3.4) $S^{1}\xrightarrow{\mu_{C_{i}}}T^{n}\twoheadrightarrow T^{n}/T_{E_{i}}$

regarding $\mu_{C_{i}}$ as a one-parameter subgroup of $T^{n}$. We consider
the $S^{1}$-action on $Y_{i}$ given by (3.4). Then we claim that the
corresponding orbit map

$Y_{i}\twoheadrightarrow Y_{i}/S^{1}_{\mu_{C_{i}}}.$

is the desired orbifold $S^{1}$-fiber bundle over a toric orbifold. Since
$\mu_{C_{i}}\in N\setminus N_{E_{i}}$, one can see that

(3.5) $Y_{i}/S^{1}_{\mu_{C_{i}}}\cong(C_{i}\times T^{n}/T_{C_{i}})/_{\sim},$

where the identification $\sim$ is the one induced from (3.2). Since $C_{i}$
is a simple polytope by the assumption and $P_{i}$ is the mapping cone of the
map $\phi_{i}\colon C_{i}\to P_{i-1}$ such that

$P_{i}-P_{i-1}=\bigcup_{F\in[v_{i},E_{i}]}\operatorname{relint}(F),$

any vertex of $C_{i}$, say $w$, is the transversal intersection of $C_{i}$ and
an edge of $E_{i}$, namely

$w=C_{i}\cap\bigcap_{j=1}^{\dim C_{i}}(E_{i}\cap F_{i_{j}})$

for some facets $F_{i_{j}}$’s of $P$. This implies that the image

$\\{\rho_{i}(\lambda(F_{i_{j}}))\mid j=1,\dots,\dim C_{i}\\}$

with respect to the projection $\rho_{i}\colon N\twoheadrightarrow
N/N_{C_{i}}$ is linearly independent, and this is true for each vertex of
$C_{i}$. Therefore, $Y_{i}/S^{1}_{\mu_{C_{i}}}$ in (3.5) is a toric orbifold,
which establishes the claim. ∎

##  4\. Application to singular cohomology

In this section, we prove several vanishing theorems for the singular
cohomology of a toric variety $X_{P}$ with a conic decomposition. In
particular, we prove that if $P$ admits a $\Delta$-conic sequence, then the
(cohomology) Betti numbers of $X_{P}$ are completely determined by the face
numbers of $P$. Throughout this section, we mean by $H^{\ast}(-)$ the singular
cohomology with rational coefficients.

######  Theorem 4.1.

Let $X_{P}$ be a projective toric variety and $P$ the associated polytope. If
there exists a conic sequence on $P$ such that $\widetilde{H}^{2k}(Y_{i})=0$
for all $i=2,\dots,\ell$, then $H^{2k+1}(X)=0$.

###### Proof.

Consider the conic decomposition in (3.3) induced from the given conic
sequence on $P$. We prove the claim by the induction on the indices of the
following filtration

(4.1) $X_{1}\hookrightarrow X_{2}\hookrightarrow\cdots\hookrightarrow
X_{\ell-1}\hookrightarrow X_{\ell}=X$

induced from the conic decomposition of $X_{P}$ as in (3.3).

First, since $P_{1}$ is a vertex of $P$, $X_{1}$ is zero-dimensional. Since
$P_{2}$ is an edge of $P$, $X_{2}=\pi^{-1}(P_{2})$ is homeomorphic to $S^{2}$,
which means $H^{odd}(X_{2})$ vanishes. We now claim that $H^{2k+1}(X_{i})=0$,
provided $H^{2k+1}(X_{i-1})=0$ for $i>2$. Since each composition

(4.2) $Y_{i}\xrightarrow{\Phi_{i}}X_{i-1}\to X_{i}$

is a cofibration, we have the following long exact sequence of the cohomology
groups

(4.3)
$\cdots\to\widetilde{H}^{j-1}(Y_{i})\to\widetilde{H}^{j}(X_{i})\to\widetilde{H}^{j}(X_{i-1})\to\widetilde{H}^{j}(Y_{i})\to\cdots.$

Thus, when $j=2k+1$ for some $k>0$, the assumption of $H^{2k}(Y_{i})=0$
together with the induction hypothesis establishes the claim. ∎

We now introduce several applications of Theorem 4.1. Recall from Lemma 3.2
that if $P$ admits a conic sequence such that
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ consists of simple polytopes, then each
$Y_{i}$ is an orbifold $S^{1}$-fiber bundle over a toric orbifold. The
following proposition has been studied in [Luo13], where the author mainly
discusses smooth cases. However, if we restrict our attention to the singular
cohomology with rational coefficients, then the same statement holds for an
orbifold $S^{1}$-fiber bundle over a toric orbifold. The main tool for the
proof is the Gysin sequence. We refer to [PS10, Section 5] for the Gysin
sequence in the orbifold setup.

######  Proposition 4.2.

[Luo13, Remark 2.5.10, Theorem 2.5.48] Let $C_{i}$ be a simple polytope and
$Y_{i}=\pi^{-1}(C_{i})$ as above. Then $H^{k}(Y_{i})=0$ if $k$ is even with
$k>\dim C_{i}$ or if $k$ is odd with $k\leq\dim C_{i}$.

######  Theorem 4.3.

Let $P$ be a lattice polytope admitting a conic sequence as in (2.4) such that
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ consists of simple polytopes, and
$X_{P}$ be the associated toric variety. Then $H^{2k-1}(X)=0$ for all
$2k-1>\frac{1}{2}\dim X_{P}$.

###### Proof.

Similarly to the proof of Theorem 4.1, we prove the claim by the induction on
the filtration in (4.1). Let $2d_{i}\colonequals\dim X_{i}$ for each $1\leq
i\leq\ell$. Then we have $d_{i}\geq d_{i-1}$ and the condition for a conic
sequence on $P$ implies that

(4.4) $\dim Y_{i}\leq 2d_{i}-1.$

We assume that $H^{j}(X_{i-1})=0$ for odd $j$ with $j>d_{i-1}$, and claim that
$H^{j}(X_{i})=0$ for odd $j$ with $j>d_{i}$.

Consider the long exact sequence in (4.3) induced from the cofibration in
(4.2). Since $\dim Y_{i}=2\dim C_{i}+1$, Proposition 4.2 implies that
$\widetilde{H}^{j-1}(Y_{i})=0$ for all even $j-1$ with $j-1>\frac{1}{2}(\dim
Y_{i}-1)$. In particular, since

$d_{i}\geq\dim C_{i}+1=\frac{1}{2}(\dim Y_{i}+1),$

by (4.4), we have $\widetilde{H}^{j-1}(Y_{i})=0$ for all even $j-1$ with
$j-1\geq d_{i}$. Moreover, the induction hypothesis implies that
$\widetilde{H}^{j}(X_{i-1})=0$ for all odd $j$ with $j>d_{i-1}$. Since
$d_{i}\geq d_{i-1}$, we have $\widetilde{H}^{j}(X_{i-1})=0$ for all odd
$j>d_{i}$. Therefore, $H^{j}(X_{i})=0$ if $j$ is odd with $j>\frac{1}{2}\dim
X_{i}$, which completes the induction. ∎

Especially, if a $3$-dimensional polytope $P$ admits a conic sequence, then
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ consists of simple polytopes because
every element in $\mathscr{C}_{\\{\mathcal{P}_{i}\\}}$ is of dimension at most
$2$. Hence Theorem 4.3 implies that $H^{5}(X_{P})=0$ in this case. We refer to
[Jor98, Proposition 3.5.3-(f)].

The next corollary specializes the $\Delta$-conic sequence of $P$, which is
proved in [SS21, Corollary 3.7] as well.

######  Corollary 4.4.

If $P$ is $\Delta$-conic, then $H^{\text{odd}}(X_{P})=0$.

###### Proof.

By Lemma 3.1, each $Y_{i}$ is homeomorphic to $S^{2\dim C_{i}+1}/K_{i}$ for
some finite group $K_{i}$. Now the transfer homomorphism (see for instance
[Bor60, III.2]) implies that

(4.5) $\widetilde{H}^{j}(S^{2\dim
C_{i}+1}/K_{i})=\begin{cases}\mathbb{Q}&\text{ if }j=2\dim C_{i}+1,\\\
0&\text{ otherwise.}\end{cases}$

Hence the proof follows directly from Theorem 4.1. ∎

The converse of Corollary 4.4 is not true in general. Indeed, G. Barthel, et.
al. [BBFK96, Example 3.5] introduced two examples of toric varieties $X$ and
$X^{\prime}$ over the polytope $P$ given in Example 2.3 such that $H^{3}(X)=0$
and $H^{3}(X^{\prime})=\mathbb{Q}$. Note that $H^{1}(X)=H^{1}(X^{\prime})=0$
because a projective toric variety is simply connected [CLS11, Theorem
12.1.10] and $H^{5}(X)=H^{5}(X^{\prime})=0$ by Theorem 4.3.

Our last application of Theorem 4.1 is the Poincaré polynomial

$\operatorname{\mathsf{Poin}}(X_{P},t)=\sum_{i=0}^{\dim X_{P}}\beta_{i}t^{i}$

of a toric variety $X_{P}$, where $\beta_{i}$ denotes the $i$-th cohomology
Betti number. When $P$ is a simple lattice polytope, that is, the resulting
toric variety $X_{P}$ is a toric orbifold, it is well-known that
$H^{\ast}(X_{P})$ vanishes in odd degrees. One can also conclude this using
Corollary 4.4 because every simple polytope admits at least one $\Delta$-conic
sequence. In this case, $(\beta_{0},\dots,\beta_{2i},\dots,\beta_{2n})$ agrees
with the $h$-vector $h(P)=(h_{0},\dots,h_{i},\dots,h_{n})$ of $P$, namely, the
vector obtained by the equation given in (2.7). We refer to [McM71, Sta80].

The next theorem allows us to enjoy Equation (2.7) in a wider class of
polytopes including all simple polytopes, namely, the category of polytopes
admitting $\Delta$-conic sequences.

######  Theorem 4.5.

Let $X_{P}$ be the toric variety associated with a lattice polytope with a
$\Delta$-conic sequence. Then the Poincaré polynomial
$\operatorname{\mathsf{Poin}}(X_{P},t)$ of $X_{P}$ is given by

(4.6) $\mathsf{Poin}(X_{P},t)=\sum_{k=0}^{n}f_{k}(t^{2}-1)^{k}.$

where $f_{k}$ is the number of $k$-dimensional faces of $P$.

###### Proof.

Let
$\mathscr{C}_{\\{\mathcal{P}_{i}\\}}=\\{C_{2},\dots,C_{\ell}\\}=\\{\Delta^{d_{2}-1},\dots,\Delta^{d_{\ell}-1}\\}$
be the sequence of simplices corresponding to the given $\Delta$-conic
sequence $\\{\mathcal{P}_{i}\\}_{1\leq i\leq\ell}$ of $P$. Notice that
$d_{2}=1$, $d_{\ell}=n$ and $1\leq d_{i}\leq n-1$ for $i=2,\dots,\ell-1$. By
Lemma 3.1 together with (4.5), we have that $\widetilde{H}^{2d_{i}-1}(Y_{i})$
is of rank $1$ and $\widetilde{H}^{j}(Y_{i})$ vanishes for $j<2d_{i}-1$.
Therefore, the long exact sequence in (4.3) implies that the rank of
$\widetilde{H}^{\ast}(X_{i})$ increases by $1$ whenever $\ast$ meets $2d_{i}$.
Repeating this process for each $i=2,\dots,\ell$, we conclude that

(4.7) $\beta_{2j}=\\#\\{i\mid d_{i}=j\\}.$

Since $C_{i}=\Delta^{d_{i}-1}$ for some $d_{i}\geq 1$, we have
$\Phi_{C_{i}}(x)=\sum_{k=1}^{d_{i}}{d_{i}\choose k}x^{k-1}$. Thus we get

(4.8) $1+x\Phi_{C_{i}}(x)=(x+1)^{d_{i}}.$

Therefore, the result of Proposition 2.4 together with (4.7) and (4.8) implies
that

(4.9)
$\sum_{k=0}^{n}f_{k}(P)x^{k}=\sum_{i=2}^{\ell}(x+1)^{d_{i}}=\sum_{k=0}^{n}\beta_{2k}(x+1)^{k}.$

Since $H^{\ast}(X_{P})$ vanishes in odd degrees, the result follows by
plugging $t^{2}-1$ into $x$ in (4.9). ∎

$x_{1}$$x_{2}$$x_{3}$$x_{1}$$x_{2}$$x_{3}$ Figure 6. Two conic sequences of
$GZ_{3}$.

######  Example 4.6.

Let $GZ_{3}$ be the three-dimensional Gelfand–Zetlin polytope, namely the set
of points $(x_{1},x_{2},x_{3})$ in $\mathbb{R}^{3}$ satisfying the following
inequalities:

$\begin{array}[]{ccccccccc}0&&&&1&&&&2\\\
&\mathbin{\rotatebox[origin={c}]{-40.0}{$\leq$}}&&\mathbin{\rotatebox[origin={c}]{50.0}{$\leq$}}&&\mathbin{\rotatebox[origin={c}]{-40.0}{$\leq$}}&&\mathbin{\rotatebox[origin={c}]{50.0}{$\leq$}}&\\\
&&x_{1}&&&&x_{2}&&\\\
&&&\mathbin{\rotatebox[origin={c}]{-40.0}{$\leq$}}&&\mathbin{\rotatebox[origin={c}]{50.0}{$\leq$}}&&&\\\
&&&&x_{3}&&&&\end{array}$

We describe two conic sequences of $GZ_{3}$ in Figure 6, where the first one
is a $\Delta$-conic sequence. Since $f(GZ_{3})=(7,11,6,1)$, Theorem 4.5
implies that the Poincaré polynomial of the toric variety $X_{GZ_{3}}$ is

$\operatorname{\mathsf{Poin}}(X_{GZ_{3}};t)=7+11(t^{2}-1)+6(t^{2}-1)^{2}+(t^{2}-1)^{3}=t^{3}+3t^{4}+2t^{2}+1.$

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