1. Introduction
Higher-rank graphs, or $k$
-graphs, are countable categories $\Lambda$
equipped with a degree functor $d : \Lambda \to \mathbb {N}^{k}$
which satisfies a certain factorization property. They were introduced in [Reference Kumjian and Pask11] as a graphical approach to the higher-rank Cuntz–Krieger algebras introduced by Robertson and Steger in [Reference Robertson and Steger21]. Since then $k$
-graphs have been studied by many authors from several points of view (see [Reference Aranda-Pino, Clark, an Huef and Raeburn1–Reference Farsi, Gillaspy, Jorgensen, Kang and Packer5,Reference an Huef, Laca, Raeburn and Sims7,Reference Kimberley and Robertson10,Reference Mutter16,Reference Mutter, Radu and Vdovina17,Reference Raeburn, Sims and Yeend20,Reference Yang23], for example). The motivation for this paper comes from the recent developments of [Reference Kaliszewski, Kumjian, Quigg and Sims9,Reference Kumjian, Pask and Sims12–Reference Kumjian, Pask and Sims14,Reference Pask, Quigg and Raeburn18], where the geometric nature of $k$
-graphs has been investigated and then used to construct new families of twisted $k$
-graph $C^{*}$
-algebras.
In fact, the geometric aspects of a $k$
-graph have been of interest in their own right. In [Reference Pask, Quigg and Raeburn18], the notion of a fundamental group was introduced for a connected $k$
-graph, where connectivity is defined in the categorical sense (see Definition 5.1). In [Reference Kaliszewski, Kumjian, Quigg and Sims9, §4], a connected $k$
-graph is viewed as a connected $k$
-dimensional CW-complex (and so has a $1$
-skeleton). It was then shown that a $k$
-graph may be realized as a topological space in a way which preserves homotopy type. It is therefore of interest to provide a facility to compute the fundamental group of a $k$
-graph in terms of generators and relations. This is the main purpose of this paper.
To clarify the nature of the fundamental group we wish to compute, we briefly describe the approach of [Reference Kaliszewski, Kumjian, Quigg and Sims9,Reference Pask, Quigg and Raeburn18] here. We adapt the description of the fundamental group of a graph as described in [Reference Massey15, Chapter 6] and [Reference Stillwell22, §2] to our situation. The $1$
-skeleton $\operatorname {Sk}_\Lambda$
of a $k$
-graph $\Lambda$
is a directed graph, which we then view as a graph $\operatorname {Sk}_\Lambda ^{+}$
(the augmented graph). We may then quotient out the finite path space $\mathcal {P} ( \operatorname {Sk}_\Lambda ^{+} )$
by trivial paths, and identifications made by the factorization rules in $\Lambda$
to form the fundamental groupoid and fundamental group of $\Lambda$
, see [Reference Pask, Quigg and Raeburn18, §3] and [Reference Kaliszewski, Kumjian, Quigg and Sims9, §4]. To establish notation, we review this construction in slightly more detail in § 2.
Following [Reference Stillwell22], the fundamental group of a graph is easy to compute: Given a maximal spanning tree of the graph, the generators of the fundamental group are indexed by the edges of the graph which are not in the tree. Furthermore, the group generated is a free group (see [Reference Stillwell22, §2.1.8]). We adapt this process to compute the fundamental group of a $k$
-graph: Take a maximal spanning tree of $\operatorname {Sk}_\Lambda ^{+}$
and quotient out the generators of the fundamental group of $\operatorname {Sk}_\Lambda$
by the relations coming from the factorization rules in $\Lambda$
. Hence, the fundamental group of a $k$
-graph is usually not a free group (see Theorem 5.4 below).
Key to our analysis is the frequent use of coloured graphs to visualize and describe the structure of a $k$
-graph (for a complete description of how this works, see [Reference Hazlewood, Raeburn, Sims and Webster8]). Briefly, the $1$
-skeleton $\operatorname {Sk}_\Lambda$
of a $k$
-graph $\Lambda$
is a directed graph, which together with a colouring $c : \operatorname {Sk}_\Lambda ^{1} \to \{ c_1 ,\, \ldots ,\, c_k \}$
of the edges forms a $k$
-coloured graph $( \operatorname {Sk}_\Lambda ,\, c)$
, called the skeleton. As shown in [Reference Hazlewood, Raeburn, Sims and Webster8], a $k$
-coloured graph does not define, or completely determine a $k$
-graph. To make a satisfactory correspondence between $k$
-graphs and $k$
-coloured graphs, we need some additional combinatorial data $\mathcal {C}$
, which encodes the natural quotient structure of $\Lambda$
. To establish notation, we review the relationship between $k$
-graphs and $k$
-coloured graphs in slightly more detail in § 3 and § 4.
In § 5, we implement the method described in the third paragraph above to give a presentation of the fundamental group of a $k$
-graph in terms of the fundamental group of its $1$
-skeleton (see Theorem 5.4). To illustrate the efficacy of our result, we give several computations in Examples 5.5. Finally, in § 6, we show that the abelianization of the fundamental group of a $k$
-graph agrees with its homology as defined in [Reference Kumjian, Pask and Sims12]. Then, in Example 6.3, we compute the fundamental group of a $2$
-graph from the Klein bottle example in [Reference Kaliszewski, Kumjian, Quigg and Sims9, Example 3.13] which reveals a non-standard presentation of this group.
2. Conventions
For $k \ge 1$
, let $\mathbb {N}^{k}$
denote the monoid of $k$
-tuples of natural numbers under addition and denote the canonical generators by $e_1,\,e_2 ,\, \ldots ,\, e_k$
. For $m \in \mathbb {N}^{k}$
, we write $m = \sum \nolimits _{i=1}^{k} m_i e_i$
then for $m,\,n \in \mathbb {N}^{k}$
we say that $m \le n$
if and only if $m_i \le n_i$
for $i=1,\, \ldots ,\, k$
.
A directed graph $E$
is a quadruple $(E^{0},\,E^{1},\,r_E,\,s_E)$
, where $E^{0}$
is the set of vertices, $E^{1}$
is the set of edges, and $r_E,\,s_E:E^{1} \to E^{0}$
are range and source maps, giving a direction to each edge (if there is no chance of confusion we will drop the subscripts). We follow the conventions of [Reference Raeburn19] which are suited to the categorical setting we wish to pursue: a path of length n is a sequence $\mu =\mu _1\mu _2\cdots \mu _n$
of edges such that $s(\mu _i)=r(\mu _{i+1})$
for $1\le i\le n-1$
. We denote by $E^{n}$
the set of all paths of length $n$
, and define $E^{*}=\bigcup _{n\in \mathbb {N}}E^{n}$
. We extend $r$
and $s$
to $E^{*}$
by setting $r(\mu )=r(\mu _1)$
and $s(\mu )=s(\mu _n)$
.
To align with the established literature on the fundamental groupoid of a $k$
-graph, the following definitions are taken from [Reference Pask, Quigg and Raeburn18, Definition 5.1]: Let $E$
be a directed graph. For each $e \in E^{1}$
, we introduce an inverse edge $e^{-1}$
with $s ( e^{-1} ) = r ( e )$
and $r ( e^{-1} ) = s ( e)$
. Let $E^{-1} = \{ e^{-1} : e \in E^{1} \}$
and $E^{u}=E^{1}\cup E^{-1}$
, then $E^{+} = ( E^{0} ,\, E^{u} ,\, r ,\, s)$
is a directed graph called the augmented graph of $E$
. Let $\mathcal {P}(E^{+})$
be the path category of $E^{+}$
. If we set $(e^{-1})^{-1}=e$
then $e^{-1} \in \mathcal {P} (E^{+})$
for any $e \in E^{u}$
and by extension $\lambda ^{-1} = \lambda _n^{-1} \cdots \lambda _1^{-1} \in \mathcal {P} ( E^{+} )$
for any $\lambda = \lambda _1 \cdots \lambda _n \in \mathcal {P} ( E^{+} )$
. Elements of $\mathcal {P} ( E^{+} )$
are called undirected paths in $E$
, and elements of $\mathcal {P} ( E^{+} )$
which do not contain $e e^{-1}$
for any $e \in E^{u}$
are called reduced undirected paths in $E$
(vertices are reduced paths).
Let $E$
be a directed graph then $E$
is connected if for every $u,\,v \in E^{0}$
there is $\alpha \in \mathcal {P} ( E^{+} )$
with $u = s ( \alpha )$
and $v = r ( \alpha )$
. A tree $T$
is a connected directed graph such that the only reduced $\alpha \in \mathcal {P} ( T^{+} )$
with the same source and range are vertices. Let $E$
be a directed graph with subgraph $T$
which is a tree, then $T$
is a maximal spanning tree if $T^{0} = E^{0}$
. Every connected directed graph has a (not necessarily unique) maximal spanning tree (see [Reference Stillwell22, §2.1.5]).
Remark 2.1 Fix a maximal spanning tree $T$
of a connected directed graph $E$
and $v \in E^{0}$
. For each $w \in E^{0}$
, there is a unique reduced path $\eta _w$
in the augmented graph $T^{+}$
from $v$
to $w$
, which is an element of the fundamental groupoid $\mathcal {G} ( E )$
. For $\lambda \in \mathcal {P} ( E^{+} )$
define $\zeta _\lambda = \eta _{r(\lambda )}^{-1} \lambda \eta _{s(\lambda )} \in v \mathcal {P} (E^{+}) v$
. Note that $\zeta _\lambda ^{-1} = \zeta _{\lambda ^{-1}}$
.
3. Coloured graphs
For $k \ge 1$
, a $k$
-coloured graph $E$
is a directed graph along with a colour map $c_E:E^{1} \to \{c_1,\,\dots ,\,c_k\}$
. By considering $\{c_1,\,\dots ,\, c_k\}$
as generators of the free group $\mathbb {F}_k$
, we may extend $c_E : E^{*} \setminus E^{0} \to \mathbb {F}_k^{+}$
by $c_E (\mu _1 \cdots \mu _n)=c_E (\mu _1)c_E (\mu _2) \cdots c_E (\mu _n)$
. We will drop the subscript from $c_E$
if there is no risk of confusion. For $k$
-coloured directed graphs $E$
and $F$
, a coloured-graph morphism $\phi : F \to E$
is a graph morphism satisfying $c_E \circ \phi ^{1} = c_F$
.
For $2$
-coloured graphs, the convention is to draw edges with colour $c_1$
blue (or solid) and edges with colour $c_2$
red (or dashed).
Example 3.1 For $k \ge 1$
and $m\in \mathbb {N}^{k}$
, the $k$
-coloured graph $E_{k,m}$
is defined by $E_{k,m}^{0}=\{n\in \mathbb {N}^{k}:0\le n\le m\}$
, $E_{k,m}^{1}=\{\varepsilon _i^{n} : n,\,n+e_i\in E_{k,m}^{0}\}$
, with $r(\varepsilon _i^{n})=n$
, $s(\varepsilon _i^{n})=n+e_i$
and $c_{E}(\varepsilon _i^{n})=c_i$
. The 2-coloured graph $E_{2,e_1+e_2}$
is used often and is depicted below.
Let $E$
be a $k$
-coloured graph and $i \neq j \leq k$
. A coloured graph morphism $\phi : E_{k,e_i+e_j} \to E$
is called a square in $E$
. One may represent a square $\phi$
as a labelled version of $E_{k,e_i+e_j}$
. For instance, the $2$
-coloured graph below on the left has only one square $\phi$
, shown to its right, given by $\phi (n) = v$
for all $n \in E_{2,{e_1+e_2}}^{0}$
, $\phi ( \varepsilon _1^{0}) = \phi ( \varepsilon _1^{e_2}) = e$
and $\phi ( \varepsilon _2^{0} ) = \phi ( \varepsilon _2^{e_1} ) = f$
.
$\mathcal {C}_E = \{ \phi : E_{k,e_i+e_j} \to E : 1 \le i \neq j \le k \}$
denotes the set of squares in a $k$
-coloured graph $E$
.
A collection of squares $\mathcal {C}$
in a $k$
-coloured graph $E$
is called complete if for each $i \neq j \leq k$
and $c_ic_j$
-coloured path $fg \in E^{2}$
, there exists a unique $\phi \in \mathcal {C}$
such that $\phi (\varepsilon _i^{0}) = f$
and $\phi (\varepsilon _j^{e_i}) = g$
. In this case, uniqueness of $\phi$
gives a unique $c_jc_i$
-coloured path $g'f'$
with $g' = \phi (\varepsilon _j^{0})$
and $f' = \phi (\varepsilon _i^{e_j})$
. We will write $fg \sim _\mathcal {C} g'f'$
and refer to elements $(fg,\,g'f')$
of this relation as commuting squares.
Example 3.2 For $n \ge 1$
define $\underline {n} = \{ 1 ,\, \ldots ,\, n \}$
. For $m,\,n \ge 1$
, let $\theta : \underline {m} \times \underline {n} \to \underline {m} \times \underline {n}$
be a bijection. Let $E_\theta$
be the $2$
-coloured graph with $E_\theta ^{0}= \{v\}$
, $E_\theta ^{1} = \{f_1 ,\, \ldots ,\, f_m ,\, g_1 ,\, \ldots ,\, g_n\}$
, and colouring map $c : E^{1}_{\theta } \to \{c_1,\,c_2\}$
by $c ( f_i ) = c_1$
for $i \in \underline {m}$
and $c (g_j ) = c_2$
for $j \in \underline {n}$
. For each $(i,\,j) \in \underline {m} \times \underline {n}$
, define $\phi _{(i,j)} : E_{2,e_1+e_2} \to E_\theta$
by
As $\theta$
is a bijection $\mathcal {C}_{E_\theta } = \{\phi _{(i,j)} : (i,\,j) \in \underline {m} \times \underline {n}\}$
is a complete collection of squares.
4. Higher-rank graphs
A $k$
-graph (or a higher-rank graph) is a countable category $\Lambda$
with a degree functor $d:\Lambda \to \mathbb {N}^{k}$
satisfying the factorization property: if $\lambda \in \Lambda$
and $m,\,n\in \mathbb {N}^{k}$
are such that $d(\lambda )=m+n$
, then there are unique $\mu ,\,\nu \in \Lambda$
with $d(\mu )=m$
, $d(\nu )=n$
and $\lambda =\mu \nu$
.
Given $m \in \mathbb {N}^{k}$
we define $\Lambda ^{m} := d^{-1}(m)$
. Given $v,\,w \in \Lambda ^{0}$
and $F \subseteq \Lambda$
define $vF := r^{-1}(v) \cap F$
, $Fw := s^{-1}(w) \cap F$
, and $vFw := vF \cap Fw$
. The factorization property allows us to identify $\Lambda ^{0}$
with $\operatorname {Obj}(\Lambda )$
, and we call its elements vertices.
By the factorization property, for each $\lambda \in \Lambda$
and $m \leq n \leq d(\lambda )$
, we may write $\lambda =\lambda '\lambda ''\lambda '''$
, where $d(\lambda ')=m,\, d(\lambda ') = n-m$
and $d(\lambda '')=d(\lambda )-n$
; then $\lambda (m,\,n) :=\lambda ''$
. For more information about $k$
-graphs, see [Reference Hazlewood, Raeburn, Sims and Webster8,Reference Kumjian and Pask11,Reference Raeburn, Sims and Yeend20] for example.
Examples 4.1
(a) Let $E$
be a directed graph. The collection $E^{*}$ of finite paths in $E$ forms a category, called the path category of $E$, denoted by $\mathcal {P}(E)$. The map $d : \mathcal {P} (E) \to \mathbb {N}$ defined by $d(\mu )=n$ if and only if $\mu \in E^{n}$ is a functor which satisfies the factorization property, hence $( \mathcal {P}(E),\,d)$ is a $1$-graph. It turns out that every $1$-graph arises in this way (see [Reference Kumjian and Pask11, Example 1.3]).(b) For $k \ge 1$
; let $\Delta _k = \{ (m,\,n) : m,\,n \in \mathbb {Z}^{k} : m \le n \}$. With structure maps $r(m,\,n)=m ,\, s (m,\,n)=n$, so that $( \ell ,\,n) = ( \ell ,\,m) (m,\,n)$ then $\Delta _k$ is a category. Set $d ( m ,\, n ) = n - m$, then $d$ is a functor and $( \Delta _k ,\, d )$ is a $k$-graph. The vertices $\Delta _k^{0} = \{ (m,\,m) : m \in \mathbb {Z}^{k} \}$ may be identified with $\mathbb {Z}^{k}$.(c) Resuming the notation of Example 3.2 let $\theta : \underline {m} \times \underline {n} \to \underline {m} \times \underline {n}$
be a bijection. Let $\mathbb {F}^{2}_\theta$ be the semigroup with generators $\{f_1 ,\, \ldots ,\, f_m ,\, g_1 ,\, \ldots g_n \}$ and relations $f_i g_j = g_{j'} f_{i'}$ where $\theta (i,\, j) = (i' ,\, j' )$ for $(i,\, j) \in \underline {m} \times \underline {n}$. Let $d ( f_i ) = e_1$ for $i=1,\,\ldots ,\,m$ and $d ( g_j ) = e_2$ for $j=1,\,\ldots ,\,n$ then $d$ extends to a functor from $\mathbb {F}^{2}_\theta$ to $\mathbb {N}^{2}$ with the factorization property, and so $\mathbb {F}^{2}_\theta$ is a $2$-graph (see [Reference Yang23, §2]).(d) Recall from [Reference Farthing, Pask and Sims6] that if $\Lambda$
is a $k$-graph and $\alpha$ is an automorphism of $\Lambda$, then there is a $(k + 1)$-graph $\Lambda \times _\alpha \mathbb {Z}$ with morphisms $\Lambda \times \mathbb {N}$, range and source maps given by $r(\lambda ,\, n) = (r(\lambda ),\, 0)$, $s(\lambda ,\, n) = (\alpha ^{-n} (s(\lambda )),\, 0)$, degree map given by $d(\lambda ,\, n) = (d(\lambda ),\, n)$ and composition given by $(\lambda ,\, m)(\mu ,\, n) := (\lambda \alpha ^{m} (\mu ),\, m + n)$. In particular $(\Lambda \times _\alpha \mathbb {Z})^{0} = \Lambda ^{0} \times \{0\}$.
We define the skeleton of a $k$
-graph $\Lambda$
to be a $k$
-coloured graph. It consists of the $1$
-skeleton $\operatorname {Sk}_\Lambda$
of $\Lambda$
, which is a directed graph given by $\operatorname {Sk}_\Lambda ^{0} = \operatorname {Obj}(\Lambda )$
, $\operatorname {Sk}_\Lambda ^{1} = \bigcup _{i\leq k}\Lambda ^{e_i}$
, with range and source as in $\Lambda$
. There is a natural colouring map $c : \operatorname {Sk}_\Lambda ^{1} \to \{c_1,\,\dots ,\,c_k\}$
given by $c(f) = c_i$
if and only if $f \in \Lambda ^{e_i}$
. The skeleton $(\operatorname {Sk}_\Lambda ,\,c)$
comes with a canonical set of bi-coloured squares $\mathcal {C}_\Lambda := \{\phi _\lambda : \lambda \in \Lambda ^{e_i+e_j}: i \neq j \leq k \}$
, where the colour-preserving graph morphism $\phi _\lambda : E_{k,e_i+e_j} \to \operatorname {Sk}_\Lambda$
is given by $\phi _\lambda (\varepsilon _\ell ^{n}) = \lambda (n,\,n+e_\ell )$
for each $n \leq e_i+e_j$
and $\ell = i,\,j$
. The collection $\mathcal {C}_\Lambda$
is complete by [Reference Hazlewood, Raeburn, Sims and Webster8, Lemma 4.2].
Conversely, in [Reference Hazlewood, Raeburn, Sims and Webster8, Theorem 4.4, Theorem 4.5], it is shown that for a $k$
-coloured graph $E$
with a complete, associativeFootnote 1 collection of squares $\mathcal {C}_E$
determines a unique $k$
-graph $\Lambda _{E,\mathcal {C}_E}$
.
Examples 4.2
(a) The $2$
-graph $\Lambda _{E_\theta , \mathcal {C}_{E_\theta }}$, determined by the 2-coloured graph $( E_\theta ,\, c)$ with squares $\mathcal {C}_{E_\theta }$ described in Example 3.2 is isomorphic to $\mathbb {F}^{2}_\theta$ defined in Examples 4.1 (c).(b) Recall the $k$
-graph $\Delta _k$ described in Examples 4.1. Part of the skeleton of $\Delta _3$, as seen from the first octant is shown below:
It is straightforward to see that $\operatorname {Sk}_{\Delta _3}^{0} = \mathbb {Z}^{3}$, $\operatorname {Sk}_{\Delta _3}^{1} = \{ ( m ,\, m+e_j ) : m \in \mathbb {Z}^{3} ,\, 1 \le j \le 3 \}$, $r(m,\,m+e_j)=m$ and $s (m,\,m+e_j)=m+e_j$. The commuting squares are
\begin{align*}\mathcal{C} = \{ (m,m+e_i) &(m+e_i,m+e_i+e_j)\\ &= (m,m+e_j)(m+e_i,m+e_i+e_j) : m \in \mathbb{Z}^{3} , 1 \le i \neq j \le 3 \} .\end{align*}One checks that this collection of squares is complete and associative.
5. Computing the fundamental group of a k-graph
In this section, we define and provide a presentation of the fundamental group of a $k$
-graph. Kaliszewski, Kumjian, Quigg and Sims show in [Reference Kaliszewski, Kumjian, Quigg and Sims9, Corollary 4.2] that the fundamental group of a $k$
-graph may be realized as a quotient of the fundamental group of its skeleton. We provide an alternative proof of this in Theorem 5.4 which yields a natural presentation of the group. We demonstrate the practical use of our result in Examples 5.5.
Definition 5.1 Kumjian et al. [Reference Kumjian, Pask and Sims12, Definition 2.8]
We say that the $k$
-graph $\Lambda$
is connected if the equivalence relation on $\Lambda ^{0}$
generated by the relation $u \sim v$
iff $u \Lambda v \neq \emptyset$
is $\Lambda ^{0} \times \Lambda ^{0}$
.
We review the construction of the fundamental groupoid of a connected $k$
-graph from [Reference Pask, Quigg and Raeburn18]. First, we describe the fundamental groupoid $\mathcal {G} (E)$
of a directed graph $E$
.
Following [Reference Pask, Quigg and Raeburn18, p. 197], let $E$
be a directed graph, then a relation for $E$
is a pair $( \alpha ,\, \beta )$
of paths in $\mathcal {P} (E)$
such that $s ( \alpha ) = s ( \beta )$
and $r ( \alpha ) = r ( \beta )$
. If $K$
is a set of relations for $E$
, then $\mathcal {P} (E) / K$
is the quotient of $\mathcal {P} (E)$
by the equivalence relation generated by $K$
, for more details, see [Reference Pask, Quigg and Raeburn18, §2].
As in [Reference Pask, Quigg and Raeburn18, Definition 5.2], let $C = \{ ( e^{-1}e,\,s(e)) : e \in E^{u} \}$
and call $C$
the set of cancellation relations for $E^{+}$
. The quotient $\mathcal {P} (E^{+}) / C$
is then the fundamental groupoid, $\mathcal {G} (E)$
of $E$
. We denote the quotient functor $\mathcal {P} (E^{+}) \to \mathcal {G} (E)$
by $q_C$
. Elements of $\mathcal {G} (E)$
are reduced undirected paths in $E$
with composition given by concatenation followed by cancellation.
Now we turn to defining the fundamental groupoid of a $k$
-graph $\Lambda$
. First, apply the above construction to form $\mathcal {G} (E)$
where $E=\operatorname {Sk}_\Lambda$
. As in [Reference Hazlewood, Raeburn, Sims and Webster8,Reference Pask, Quigg and Raeburn18], let $S$
be the equivalence relation on $\mathcal {P}(E)$
generated by $\mathcal {C}_\Lambda$
, the commuting squares in $E$
determined by $\Lambda$
. That is, the transitive closure in $\mathcal {P} (E) \times \mathcal {P} (E)$
of
As in [Reference Pask, Quigg and Raeburn18, Observation 5.3], this relation may be extended uniquely to a relation $S^{+}$
on $\mathcal {P} ( E^{+} )$
by adding the relation $( f^{-1} e^{-1} ,\, h^{-1} g^{-1} )$
whenever $(ef,\,gh) \in S$
. This induces a relation, also called $S^{+}$
, on $\mathcal {G} (E)$
.
Definition 5.2 Let $\Lambda$
be a connected $k$
-graph. Then the fundamental groupoid, $\mathcal {G} ( \Lambda )$
is
For $v \in \Lambda ^{0}$
the fundamental group based at $v \in \Lambda ^{0}$
is the isotropy group $\pi _1 ( \Lambda ,\, v ) := v \mathcal {G} ( \Lambda ) v$
.
The above definition of the fundamental groupoid of a $k$
-graph is consistent with the one given in [Reference Pask, Quigg and Raeburn18, Definition 5.6] (see also the accompanying discussion).
Our goal is to obtain a practical way of giving a presentation of $\pi _1(\Lambda ,\,v)$
. First recall that for a connected directed graph $E$
, the quotient functor $q_C : \mathcal {P}(E^{+} ) \to \mathcal {P}(E^{+})/ C = \mathcal {G} ( E )$
restricts to $v \mathcal {P} ( E^{+} ) v$
and the image is the isotropy group $v\mathcal {G}(E)v$
, which is by definition the fundamental group of $E$
at $v$
, denoted $\pi _1 ( E ,\, v )$
.
The following result is well known (see [Reference Stillwell22] for instance).
Lemma 5.3 Let $E$
be a connected directed graph, $v \in E^{0},$
and $T$
be a maximal spanning tree of $E$
. Then $\pi _1 ( E ,\,v ) \cong \langle E^{1} \mid T^{1} \rangle := \langle e \in E^{1} \mid e = 1 \text { if } e \in T^{1} \rangle$
.
Proof. Suppose $e \not \in T^{u} = T^{1} \sqcup T^{-1}$
. With notation as in Remark 2.1, all the edges of $\eta _{r(e)} ,\, \eta _{s(e)}$
are in $T^{u}$
, so $\zeta _e$
is reduced undirected path in $E$
and hence $q_C ( \zeta _e ) = \zeta _e$
. Suppose that $e \in T^{u}$
then $\zeta _e$
is an undirected path in $T$
from $v$
to $v$
and so its reduced form must be $v$
, hence $q_C ( \zeta _e ) = v$
.
To complete the proof, it suffices to show that $\{ \zeta _e : e \in E^{1} \backslash T^{1} \}$
freely generate $\pi _1 ( E,\,v)$
. This is a standard result, see [Reference Stillwell22, §2.1.7, §2.1.8] for example. □
Since Lemma 5.3 holds for any choice of maximal spanning tree, it follows that $\pi _1(E,\,v)$
does not depend on the choice of basepoint $v$
. We denote the fundamental group of a graph $E$
by $\pi _1(E)$
. Now we turn our attention to computing the fundamental group of a connected $k$
-graph $\Lambda$
. Since $\Lambda \cong \mathcal {P} ( \operatorname {Sk}_\Lambda ) / S$
, we expect the relation $S$
to appear in the description of $\pi _1(\Lambda )$
.
Theorem 5.4 Let $\Lambda$
be a connected $k$
-graph, $v \in \Lambda ^{0}$
and let $T$
be a maximal spanning tree for $\operatorname {Sk}_\Lambda$
. Then $\pi _1 ( \Lambda ,\, v ) \cong \langle \operatorname {Sk}_\Lambda ^{1} \mid t=1 \text { if } t \in T^{1},\, ef=gh \text { if } (ef,\,gh) \in S^{+} \rangle$
.
Proof. Denote by $q_S :\mathcal {G}(\operatorname {Sk}_\Lambda ) \to \mathcal {G}(\operatorname {Sk}_\Lambda ) / S^{+} = \mathcal {G}(\Lambda )$
the quotient map. With notation as in Remark 2.1, observe that $\zeta _e \zeta _f = \zeta _{ef}$
and $\zeta _g \zeta _h = \zeta _{gh}$
in $\mathcal {G} (\operatorname {Sk}_\Lambda )$
. Hence $q_S ( \zeta _e \zeta _f ) = q_S ( \zeta _g \zeta _h )$
if and only if $(ef,\,gh) \in S^{+}$
. Since taking quotients preserves objects, $\pi _1(\Lambda ,\,v) = \pi _1 ( \operatorname {Sk}_\Lambda ) /S^{+}$
. Then Lemma 5.3 implies that $\pi _1(\Lambda ,\,v) \cong \langle \operatorname {Sk}_\Lambda ^{1} \mid t=1 \text { if } t \in T^{1},\, ef=gh \text { if } (ef,\,gh) \in S^{+} \rangle$
. □
Since Theorem 5.4 holds for every choice of $T$
, the group $\pi _1(\Lambda ,\,v)$
does not depend on $T$
. We henceforth denote by $\pi _1(\Lambda )$
the fundamental group of $\Lambda$
. Theorem 5.4 gives us an explicit presentation of $\pi _1(\Lambda )$
, as seen in the following examples.
Examples 5.5
1 Let $\Sigma$
be the 2-graph, which is completely determined by its skeleton, shown below. In [Reference Kaliszewski, Kumjian, Quigg and Sims9, Example 3.10], it was shown that the topological realization of $\Sigma$ is the $2$-sphere $S^{2}$. Let $T$ be the maximal spanning tree for $\operatorname {Sk}_\Sigma$ consisting of edges $T^{1}=\{a,\,b,\,c,\,d,\,e\}$. The commuting squares in $\operatorname {Sk}_\Sigma$ are $(ga,\,ce)$, $(gb,\,cf)$, $(de,\,ha)$ and $(df,\,hb)$, thus Theorem 5.4 gives
\[ \pi_1(\Sigma) \cong \langle \operatorname{Sk}_\Sigma^{1} \mid t = 1 \text{ if } t \in T^{1}, ga=ce,gb=cf,de=ha,df=hb \rangle . \]The first relation forces $g=1$, the second $g=f$, the third $h=1$ and the fourth $f=h$. Hence all the generators of $\pi _1(\Sigma )$ are equal to $1$. Therefore $\pi _1(\Sigma )$ is trivial.
2 Consider the 2-graph $\Pi$
with skeleton $\operatorname {Sk}_\Pi$ shown below, with commuting squares $(ga,\,ce)$, $(gb,\,df)$, $(hb,\,cf)$ and $(ha,\,de)$. In [Reference Kaliszewski, Kumjian, Quigg and Sims9, Example 3.12], it was shown that the topological realization of $\Pi$ is the projective plane. Choose spanning tree $T$ of $\operatorname {Sk}_\Pi$ with $T^{1}=\{a,\,b,\,c,\,f\}$. Then Theorem 5.4 gives
\[ \pi_1(\Pi)=\langle \operatorname{Sk}_\Pi^{1} \mid t=1 \text{ if } t \in T^{1}, ga=ce, gb=df, hb=cf , ha=de \rangle . \]The relations become $g=e$, $g=d$, $h=1$ and $de=1$. So the fundamental group of $\pi _1 ( \Pi ) \cong \langle e \mid e^{2}=1\rangle \cong \mathbb {Z}/2\mathbb {Z}$, the fundamental group of the projective plane.
3 Recall the skeleton of the $2$
-graph $\mathbb {F}^{2}_\theta$ described in Example 3.2. Since $\mathbb {F}^{2}_\theta$ has a single vertex, $v$, the maximal spanning tree for its $1$-skeleton $\operatorname {Sk}_{\mathbb {F}^{2}_\theta }$ is $v$. The commuting squares of $\operatorname {Sk}_{\mathbb {F}^{2}_\theta }$ are $(f_i g_j ,\, g_{j'} f_{i'} )$ where $\theta (i,\,j) = (i',\,j')$ for $(i,\,j) \in \underline {m} \times \underline {n}$. Hence by Theorem 5.4 the fundamental group of $\mathbb {F}^{2}_\theta$ is
\[ \langle f_i , g_j \mid f_i g_j = g_{j'} f_{i'} \text{ where } \theta (i,j) = (i',j') \rangle . \]For different choices of $\theta$, we get quite different fundamental groups:
(a) If $m=n=2$
and $\theta : \underline {2} \times \underline {2} \to \underline {2} \times \underline {2}$ is the identity map then
\[ \pi_1 ( \mathbb{F}^{2}_\theta ) \cong \langle f_1 , f_2 \rangle \times \langle g_1 ,g_2 \rangle \cong (\mathbb{Z} * \mathbb{Z}) \times (\mathbb{Z} * \mathbb{Z}) = \pi_1 ( S^{1} \vee S^{1} ) \times \pi_1 ( S^{1} \vee S^{1} ) , \]where $*$ is the free product, and $\vee$ is the wedge sum.(b) If $m=n=2$
and $\theta$ is given by $\theta (i,\,j)=(j,\,i)$ then by Theorem 5.4, we have
\[ \pi_1 ( \mathbb{F}^{2}_\theta ) \cong \langle f_1 , f_2 , g_1, g_2 \mid f_1 g_1 = g_1 f_1 , f_1 g_2 = g_1 f_2 , f_2 g_1 = g_2 f_1 , f_2 g_2 = g_2 f_2 \rangle . \]The first and fourth relations give $f_i^{-1}g_i = g_if_i^{-1}$ for $i=1,\,2$, then using the second relation, we have $f_1^{-1}g_1=g_2f_2^{-1}$. Putting these together gives $g_1f_1^{-1} = f_2^{-1}g_2$, and hence $f_2g_1 = g_2f_1$. So the third relation is redundant and
\[ \pi_1(\mathbb{F}^{2}_\theta) \cong (\mathbb{Z}^{2} * \mathbb{Z}^{2})/\langle g_1f_1^{{-}1} = g_2f_2^{{-}1} \rangle = \mathbb{Z}^{2} *_\mathbb{Z} \mathbb{Z}^{2} , \]where $\{f_1,\,g_1\}$ generate the first copy of $\mathbb {Z}^{2}$ and $\{f_2,\,g_2\}$ generate the second copy, and the amalgamation over $\mathbb {Z}$ is with respect to the identifications of $\mathbb {Z}$ in $\mathbb {Z}^{2}$ given by $1 \mapsto g_if_i^{-1}$ for $i=1,\,2$. So $\mathbb {F}^{2}_\theta$ has the same fundamental group as the two-holed torus.
4 It would be nice to include more higher-dimensional examples with a significant geometric content, such as those in [Reference Mutter16,Reference Mutter, Radu and Vdovina17]. However, the computation becomes difficult to work with as the spanning tree only uses relatively few edges of the $1$
-skeleton. The following example takes a $3$-graph whose geometric realization is a sphere and adds two extra edges $x_4',\,y_4'$ to make its fundamental group non-trivial (cf. Example (i) above).
(2)Corresponding to the relations
Choose spanning tree with edges $y_0,\, x_1,\, y_1$
, $x_2,\, y_2,\, x_3,\, y_3$, $x_4,\, y_4$, $u_0,\, v_0$. Then by Theorem 5.4, the fundamental group of the $3$-graph shown above is
\begin{align*} \langle x_i , y_i , u_i , v_i, i=0, \ldots,4 , x_4' , y_4' : \; &x_j = 1 , j=1, \ldots 4,\\ & y_i=1 , i=0, \ldots , 4 , ~ u_0=1, v_0=1 \rangle \end{align*}Applying the relations in the first column, we get $x_0 = u_1 = u_2 = u_3 = u_4 = y_4',\, x_4' u_4 = 1$, and from the second column, we have $x_0 = v_1 = v_2 = v_3 = v_4 = y_4' ,\, x_4' v_4 = 1$. Hence, the fundamental group is $\langle x_0 ,\, x_4' : x_4'x_0 =1 \rangle \cong \mathbb {Z}$ (since the generating set is redundant, $x_4' = x_0^{-1}$).
6. Relationship with first homology group
Full versions of the following definitions may be found in [Reference Kumjian, Pask and Sims12, §3]. Let $X$
be a set. We write $\mathbb {Z} X$
for the free abelian group generated by $X$
. For a $k$
-graph $\Lambda$
, set $C_0 ( \Lambda ) = \mathbb {Z} \Lambda ^{0}$
, $C_1 ( \Lambda ) = \mathbb {Z} \Lambda ^{e_1} \oplus \cdots \oplus \mathbb {Z} \Lambda ^{e_k}$
and $C_2 ( \Lambda ) = \oplus _{1\le i< j\le k} \mathbb {Z} \Lambda ^{e_i+e_j}$
.
Let $\partial ^{\Lambda }_1 : C_1 ( \Lambda ) \to C_0 ( \Lambda )$
be the homomorphism determined by $\partial ^{\Lambda }_1 (\lambda ) = s(\lambda )-r(\lambda )$
. Define $\partial ^{\Lambda }_2 : C_2 ( \Lambda ) \to C_1 ( \Lambda )$
as follows. Suppose $\lambda \in \Lambda ^{e_i+e_j}$
where $1\le i< j\le k$
. Factorize $\lambda =f_1 g_1=g_2 f_2$
where $f_r \in \Lambda ^{e_{i}}$
and $g_r \in \Lambda ^{e_{j}}$
for $r=1,\,2$
, then set $\partial _2^{\Lambda } (\lambda )=f_1+g_1-f_2-g_2$
and extend to a homomorphism from $C_2 ( \Lambda )$
to $C_1 ( \Lambda )$
. Then $\partial _2^{\Lambda } \circ \partial _1^{\Lambda } =0$
, and $H_0 (\Lambda ) =\mathbb {Z} \Lambda ^{0} / \operatorname {Im} \partial ^{\Lambda }_1$
, $H_1 (\Lambda ) = \operatorname {ker} \partial ^{\Lambda }_1 /\operatorname {Im} \partial ^{\Lambda }_2$
.
Recall the following definitions from [Reference Kumjian, Pask and Sims12, Definition 2.7, Definition 3.10].
Definition 6.1 Given $h = h_1^{m_1} \cdots h_n^{m_n} \in \mathcal {G} ( \Lambda )$
, where $m_i = \pm 1$
, define $t : \mathcal {G} ( \Lambda ) \to C_1 ( \Lambda )$
by $t(h) = \sum \nolimits _{i=1}^{n} m_i h_i \in C_1 ( \Lambda )$
; then $t(h)$
is called a trail. If $h$
is a circuit (that is $r(h)=s(h)$
) then $t(h)$
is called a closed trail. If $h$
is also simple (that is $s ( h_i^{m_i} ) \neq s ( h_j^{m_j} )$
for $i \neq j$
),then $t(h)$
is called a simple closed trail.
Proposition 6.2 Let $\Lambda$
be a connected $k$
-graph. Then the map $t$
defined in Definition Reference Kumjian, Pask and Sims13 induces an isomorphism $\operatorname {Ab} \pi _1 ( \Lambda ) \cong H_1 ( \Lambda )$
.
Proof. Fix $v \in \Lambda ^{0}$
, then $\pi _1(\Lambda ) \cong \pi _1(\operatorname {Sk}_\Lambda ,\,v)/ S^{+}$
by [Reference Kaliszewski, Kumjian, Quigg and Sims9]. Fix a maximal spanning tree $T \subset \operatorname {Sk}_\Lambda$
, then $\pi _1(\operatorname {Sk}_\Lambda ,\,v) = \langle \zeta _e \mid e \in E^{1} \setminus T^{1} \rangle$
(see [Reference Stillwell22] for example). Then $t: \pi _1(\operatorname {Sk}_\Lambda ,\,v) \to C_1(\Lambda )$
is a homomorphism. Since $t$
sends simple reduced circuits to simple closed trails, [Reference Kumjian, Pask and Sims12, Proposition 3.15] implies that $\ker (\partial _1^{\Lambda }) = t(\pi _1(\operatorname {Sk}_\Lambda ,\,v))$
. As $t ( \zeta _e \zeta _f ) - t ( \zeta _g \zeta _h ) = e+f-g-h \in \operatorname {Im} \partial _2^{\Lambda }$
whenever if $(ef,\,gh) \in S^{+}$
, $t$
descends to a homomorphism $t':\pi _1(\Lambda ) \to H_1(\Lambda )$
which maps $[a]$
to $[t(a)]$
for $a \in \pi _1 ( \operatorname {Sk}_\Lambda ,\,v)$
. Routine calculation then shows that $\ker t'$
is the commutator subgroup of $\pi _1(\Lambda )$
, so $t$
is an isomorphism from $\operatorname {Ab}\pi _1(\Lambda )$
, the abelianization of $\pi _1 ( \Lambda )$
to $H_1(\Lambda )$
.
Example 6.3 Recall the $2$
-graph $\Lambda$
shown below on the right with commuting squares shown on the on the left from [Reference Kumjian, Pask and Sims12, Example 5.7]
which has the same homology as the Klein bottle. However, as we shall see, it does have the same fundamental group, but with a quite different presentation to the one given in [Reference Kaliszewski, Kumjian, Quigg and Sims9, Example 3.13]. To see this, choose spanning tree $T$
with $T^{1}=\{a,\,c,\,g\}$
. By Theorem 5.4, the fundamental group is generated by $\Lambda ^{e_1} \cup \Lambda ^{e_2}$
subject to the relations
which simplify to $b=e ,\, \ 1 = df ,\, \ hb = f ,\, \ h = de$
. Eliminating $b$
and simplifying further, we have
is equal to the fundamental group of the Klein bottle, $\langle a,\,b : aba=b \rangle$
. To see this, set $e=ab$
and $f=b$
, then
A slightly easier calculation shows that in the case $n=2$
, the $2$
-graph in [Reference Kumjian, Pask and Sims12, Example 5.1] has the same fundamental group (6.1) as $\Lambda$
, which is not a surprise as it has the same topological realization as $\Lambda$
(see [Reference Kumjian, Pask and Sims12, Remark 5.9]). The presentation (6.1) in abelian form is $\langle e ,\, f : 2 (f-e) =0 \rangle$
.
One sees that the abelianization of $\pi _1 ( \Lambda )$
is $\mathbb {Z} \oplus \mathbb {Z} / 2 \mathbb {Z}$
, the homology group of the Klein bottle, as stated in [Reference Kumjian, Pask and Sims12, Example 5.7].
Acknowledgements
This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT), No. NRF-2020R1F1A1A01076072 and the Australian Research Council.
