# Non-abelian Quantum Statistics on Graphs

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## Abstract

We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space *X*. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of *X* which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.

## 1 Introduction

*n*particles that live in a topological space

*X*this is done by considering some particular tuples of length

*n*that consist of points from

*X*, i.e. elements of \(X^{\times n}\). Namely, these are the unordered tuples of distinct points from

*X*. In other words, we consider space \(C_n(X)\) defined as follows.

*X*correspond to closed loops in \(C_n(X)\) [1, 2, 3]. Under this identification all possible quantum statistics (QS) are classified by unitary representations of the fundamental group \(\pi _1(C_n(X))\). When \(X={\mathbb {R}}^2\) this group is known to be the braid group and when \(X={\mathbb {R}}^k\), where \(k\ge 3\), it is the permutation group \(S_n\). QS corresponding to a one-dimensional unitary representation of \(\pi _1(C_n(X))\) is called abelian whereas QS corresponding to a higher dimensional non-abelian unitary representation is called non-abelian. Quantum statistics can be also viewed as a flat connection on the configuration space \(C_n(X)\) that modifies definition of the momentum operator according to minimal coupling principle. The flatness of the connection ensures that there are no classical forces associated with it and the resulting physical phenomena are purely quantum [4, 5] (cf. Aharonov-Bohm effect [6])

The first part of this paper (Sects. 1–3) contains a meta analysis of literature concerning connections between topology of configuration spaces and the existence of different types of quantum statistics. Because the relevant literature is rather scarce, it was a nontrivial task to make such a meta analysis and we consider it an essential step in describing our results. This is because we see the need of introducing in a systematic and concise way the framework for studying quantum statistics which is designed specifically for graphs. The most challenging part in formulating such a framework is to avoid the language of differential geometry, as graph configuration spaces are not manifolds, whereas the great majority of results in the field concerns quantum statistics on manifolds. As a result, we obtain a universal framework whose many features can be utilised for a very wide class of topological spaces. The framework relies on the following mains steps: (i) defining flat bundles as quotients of the trivial bundle over the universal cover of the configuration space (Theorem 5), (ii) defining Chern characteristic classes solely by pullbacks of the universal bundle (Sect. 3.1), (iii) pointing out the role of the moduli space of flat *U*(*n*)-bundles as an algebraic variety in \(U(n)^{\times r}\), *r* being the rank of the fundamental group of the respective configuration space (Sect. 3.3).

*k*on a topological space

*X*is divided into two steps

- 1.
*Topological classification of wave functions*. Classify isomorphism classes of flat hermitian vector bundles of rank*k*over \(C_n(X)\). Here we also point out that in fact physically meaningful is the classification of vector bundles with respect to the so-called stable equivalence, as nonisomorphic but stable equivalent vector bundles have identical Chern numbers. An important role is played by the reduced*K*-theory and (co)homology groups of \(C_n(X)\). Calculation of those groups for various graph configuration spaces is the main problem we solve in Sect. 5. - 2.
*Classification of statistical properties*. If*X*is a manifold, for each flat hermitian vector bundle, classify the flat connections. The parallel transport around loops in \(C_n(X)\) determines the statistical properties. For general paracompact*X*, this point can be phrased as classification of the*U*(*k*) - representations of the corresponding braid group, i.e. the fundamental group of \(C_n(X)\).

General methods that we describe in the first three sections of this paper, are applied to a special class of configuration spaces of particles on graphs (treated as 1-dimensional CW complexes). Graph configuration spaces serve as simple models for studying quantum statistical phenomena in the context of abelian anyons [7, 8] or multi-particle dynamics of fermions and bosons on networks [9, 10, 11]. Quantum graphs already proved to be useful in other branches of physics such as quantum chaos and scattering theory [12, 13, 14]. Of particular relevance to this paper are explicit physical models of non-abelian anyons on networks. One of the most notable directions of studies in this area is constructing models for Majorana fermions which can be braided thanks to coupling together a number of Kitaev chains [15, 16]. Such models lead to new robust proposals of architectures for topological quantum computers that are based on networks. Another general way of constructing models for anyons is via an effective Chern–Simons interaction [17, 18]. Such models can also be adapted to the setting of graphs as self-adjoint extensions of a certain Chern–Simons hamiltonian which is defined locally on cells of the graph configuration space [19]. All such physical models realise some unitary representations of a graph braid group.

*U*(

*n*) bundles over the graph configuration space. The core result of our paper concerns solving the so-called

*universal presentation*problem of homology groups. This problem relies on constructing

a set of universal generators which generate all homology groups of graph configuration spaces

a set of universal relations which generate all relations between universal generators.

*U*(

*n*) bundles.

While solving the universal presentation problem we used not only the state-of-the-art methods that have been used previously in a different context by us and other authors, but also developed new computational tools. The already existing methods were in particular (i) discrete models of graph configuration spaces by Abrams and Świątkowski [20, 21], (ii) the product-cycle *ansatz* introduced in our previous paper concerning tree graphs [22], (iii) the vertex blow-up method introduced by Knudsen et. al. [23], (iv) discrete Morse theory for graph configuration spaces introduced by Farley and Sabalka [24]. However, these methods have not been used before to tackle the universal presentation problem. New computational tools we used mainly relied on (i) introducing explicit techniques for calculating homology groups appearing in the homological exact sequence stemming from the vertex blow-up, (ii) demonstrating a new strategy of decomposing a given graph by a sequence of vertex blow-ups and using inductive arguments to compute the homology groups, (iii) further formalising and developing the product-cycle *ansatz* so that it can be adapted for more general graphs than just tree graphs (iv) new *ansatz* for non-product universal generators which are homeomorphic to triple tori, (v) implementing discrete Morse theory for graph configuration spaces in a computer code. A non-trivial combination of the above methods that we have applied has proved to be very effective in tackling the universal presentation problem. Nevertheless, while formulating our general framework for studying quantum statistics we already arrive at a number of new very general corollaries. This in particular concerns the structure of abelian statistics on spaces with a finitely-generated fundamental group and pointing out the role of *K*-theory in studying non-abelian statistics of a high rank.

### 1.1 Quantum kinematics on smooth manifolds

*X*is a smooth manifold, known under the name of

*Borel quantisation*, has been formulated by H.D. Doebner et. al. and formalised in a series of papers [25, 26, 27, 28, 29]. Borel quantisation on smooth manifolds can be also viewed as a version of the geometric quantisation [30]. The main point of Borel quantisation is the fact that the possible quantum kinematics on \(C_n(X)\) are in a one-to-one correspondence with conjugacy classes of unitary representations of the fundamental group of the configuration space. We denote this fact bywhere \(QKin_k\) are the quantum kinematics of rank

*k*. i.e. kinematics, where wave functions have values in \({\mathbb {C}}^k\) and \(\pi _1\) is the fundamental group. Let us next briefly describe the main ideas standing behind the Borel quantisation which will be the starting point for building an analogous theory for indistinguishable particles on graphs.

*E*to a vector field

*A*that is tangent to \(C_n(X)\) in the way that respects the Lie algebra structure of tangent vector fields. Namely, we require the standard commutation rule for momenta, i.e.

*A*that is compatible with the hermitian structure. Moreover, commutation rule (1) implies that \(\nabla _A\) is necessarily the covariant derivative stemming from a flat connection. The component proportional to \(\mathrm{div}(A)\) in formula (3) comes from the fact that map \(A\rightarrow {\hat{p}}_A\) must be valid for an arbitrary complete vector field. Usually, one considers momentum operators coming from some specific vector fields that form an orthonormal basis of local sections of \(TC_n(X)\). The divergence of such a basis sections usually vanishes and formula (3) describes the standard minimal coupling principle, see example 1 below. Flat hermitian connections of rank

*k*are classified by conjugacy classes of

*U*(

*k*) representations of \(\pi _1(C_n(X))\) (see [31]). Representatives of these classes can be picked by specifying the holonomy on a fixed set of loops generating the fundamental group. In order to illustrate these concepts, consider the following example of one particle restricted to move on the plane and its scalar wave functions.

### Example 1

*D*the area between the circles. We have \(\partial D=\gamma '-\gamma \). Hence, by the Stokes theorem

*U*(1) representations of \(\pi _1(X)\) are the representations that assign a phase factor \(e^{i\phi }\) to a chosen non-contractible loop. Physically, these representations can be realised as the Aharonov-Bohm effect and phase \(\phi \) is the magnetic flux through point \(*\) that is perpendicular to the plane.

*R*and the relative position

*r*. In terms of the positions of particles, we have

*r*to \(-r\), while

*R*remains unchanged, hence

*U*(1)-representations of the fundamental group that assign an arbitrary phase factor to the wave function when transported around a non-contractible loop. Note that a loop in the configuration space corresponds to an exchange of particles (see Fig. 1).

*U*(1) representations of the fundamental group for each vector bundle. The choice of statistical properties for each vector bundle is a consequence of a general construction of flat vector bundles which we describe in more detail in section 3.3. The fundamental group reads

Bundle \(E_0\) corresponds to the trivial representation of \(\pi _1\), while \(E'\) corresponds to the alternating representation that acts with multiplication by a phase factor \(e^{i\pi }\). Consequently, the holonomy group changes the sign of the wave function from \(E'\) when transported along a non-contractible loop, while the transport of a wave function from the trivial bundle results with the identity transformation. Therefore, bundle \(E_0\) is called bosonic bundle, whereas bundle \(E'\) is called the fermionic bundle.

As we have seen in the above examples, there is a fundamental difference between anyons in \({\mathbb {R}}^2\) and bosons and fermions in \({\mathbb {R}}^3\). Anyons emerge as different flat connections on the trivial line bundle over \(C_2({\mathbb {R}}^2)\), while fermions and bosons emerge as flat connections on non isomorphic line bundles over \(C_2({\mathbb {R}}^3)\). As we explain in section 3, these results generalise to arbitrary numbers of particles.

*B*.

### 1.2 Quantum kinematics on graphs

### Example 2

**Configuration space of two particles on graph***Y*. In \(Y\times Y\) there are 9 two-cells. Six of them are products of distinct (but not disjoint) edges of *Y*. Their intersect with \(\varDelta _2\) is a single point which we denote by (2, 2). The three remaining two-cells are of the form \(e\times e\). They have the form of squares which intersect \(\varDelta _2\) along the diagonal. Graph *Y* and space \(C_2(Y)\) are shown on Fig. 3.

*n*components that are defined as

*n*-cell separately. For each

*n*-cell we require that the connection 1-form \(\varGamma =\sum _{i=1}^n\alpha _i\) is closed, hence locally the connection is flat. In order to impose global flatness of the considered bundle, we require that the parallel transport does not depend on the homotopic deformations of curves that cross different pieces of \(C_n(\varGamma )\). This requirement imposes conditions on the parallel transport operators along certain edges (1-dimensional cells) of \(C_n(\varGamma )\). To see this, we need the following theorem by Abrams [21].

### Theorem 1

Fix *n*—the number of particles. If \(\varGamma \) has the following properties: (i) each path between distinct vertices of degree not equal to 2 passes through at least \(n-1\) edges, (ii) each nontrivial loop passes through at least \(n+1\) edges, then \(C_n(\varGamma )\) deformation retracts to a *CW*-complex \(D_n(\varGamma )\) which is a subspace of \(C_n(\varGamma )\) and consists of the *n*-fold products of disjoint cells of \(\varGamma \).

*CW*-complex. Therefore, we only need to consider the parallel transport along loops in \(D_n(\varGamma )\). Furthermore, every loop in \(D_n(\varGamma )\) can be deformed homotopically to a loop contained in the one-skeleton of \(D_n(\varGamma )\). The problem of gluing connections between different pieces of \(C_n(\varGamma )\) becomes now discretised. Namely, we require that the unitary operators that describe parallel transport along the edges of \(D_n(\varGamma )\) compose to the identity operator whenever the corresponding edges form a contractible loop. In other words,

More formally, we classify all homomorphisms \(\rho \in {{\mathrm{Hom}}}(\pi _1(C_n(\varGamma )),U(k))\) and consider the vector bundles that are induced by the action of \(\rho \) on the trivial principal *U*(*k*)-bundle over the universal cover of \(C_n(\varGamma )\). For more details, see Sect. 3.

*k*on \(C_n(\varGamma )\) is equivalent to the classification of the

*U*(

*k*) representations of \(\pi _1(D_n(\varGamma ))\). The described method of classification of quantum kinematics in the case of rank 1 becomes equivalent to the classification of discrete gauge potentials on \(C_n(\varGamma )\) that were described in [7].

### Example 3

**Quantum kinematics of rank** 1 **of two particles on graph***Y*. The two-particle discrete configuration space of graph *Y* consists of 6 edges that form a circle (Fig. 4). Therefore, any non-contractible loop in \(C_2(Y)\) is homotopic with \(D_2(Y)\).

*U*(1) operators arising from the parallel transport along the edges in \(D_2(Y)\). These operators are just phase factors

## 2 Methodology

All topological spaces that are considered in this paper have the homotopy type of finite *CW* complexes. This is due to the following two theorems.

### Theorem 2

[20, 21] The configuration space of any graph \(\varGamma \) can be deformation retracted to a finite *CW* complex which is a cube cumplex.

### Theorem 3

[32, 33] The configuration space of *n* particles in \({\mathbb {R}}^k\) has the homotopy type of a finite *CW*-complex.

Using the structure of a *CW*-complex makes some computational problems more tractable. This is especially useful, while computing the homology groups of graph configuration spaces, because the corresponding *CW*-complexes have a simple, explicit form.

*CW*complex is finitely generated [34]. This means that in all scenarios that are relevant in this paper, the fundamental group can be described by choosing a finite set of generators \(a_1,\dots ,a_r\) and considering all combinations of generators and their inverses, subject to certain relations

*n*-particle configuration space of some topological space

*X*will be referred to as the

*n*-strand braid group of

*X*and denoted by \(Br_n(X)\). Notably, there is a wide variety of braid groups when the underlying topological space

*X*is changed. Let us next briefly review some of the flag examples.

- 1.
The

*n*-strand braid group of \({\mathbb {R}}^3\) is the permutation group, \(Br_n({\mathbb {R}}^3)=S_n\). - 2.The
*n*-strand braid group of \({\mathbb {R}}^2\) is often simply called*braid group*and denoted by \(Br_n\). It has \(n-1\) generators denoted by \(\sigma _1,\dots ,\sigma _{n-1}\). One can illustrate the generators by arranging particles on a line. In such a setting, \(\sigma _i\) corresponds to exchanging particles*i*and \(i+1\) in a clockwise manner. By composing such exchanges, one arrives at the following presentation of \(Br_n({\mathbb {R}}^2)\)$$\begin{aligned}&Br_n({\mathbb {R}}^2)=\langle \sigma _1,\dots ,\sigma _{n-1}:\ \sigma _i\sigma _{i+1}\sigma _i=\sigma _{i+1}\sigma _i\sigma _{i+1}\ {{\mathrm{for\ }}}i=1,\dots ,n-2,\\&\quad \sigma _i\sigma _j=\sigma _j\sigma _i{\mathrm{\ for\ }}|i-j|\ge 2\rangle . \end{aligned}$$ - 3.
The

*n*-strand braid group of a sphere \(S^2\) has the same set of generators and relations as \(Br_n({\mathbb {R}}^2)\), but with one additional relation: \(\sigma _1\sigma _2\dots \sigma _{n-1}\sigma _{n-1}\dots \sigma _2\sigma _1=e\). - 4.
The

*n*-strand braid group of a torus \(T^2\). Group \(Br_n(T^2)\) is generated by (i) generators \(\sigma _1,\dots ,\sigma _{n-1}\) where the relations are the same as in the case of \({\mathbb {R}}^2\) and (ii) generators \(\tau _i,\ \rho _i\), \(i=1,\dots ,n\) that transport particle*i*around one of the two fundamental loops on \(T^2\) respectively. As the full set of relations defining \(Br_n(T^2)\) is quite long, we refer the reader to [35]. - 5.
Fundamental groups of

*n*-particle configuration spaces of graphs, also called graph braid groups [24, 36]. The study of integral homology of graph braid groups is a central point of this paper.

*K*(

*G*, 1), i.e. the fundamental group is their only non-trivial homotopy group. Such spaces are also called aspherical. In the following example we aim to provide some intuitive understanding of complications and difficulties that are met while dealing with graph braid groups.

### Example 4

*(Braid groups for two or three particles on*\(\varTheta \)

*-graphs)*Consider graph \(\varTheta \) that consists of two vertices and three parallel edges that connect the vertices. As we show schematically in Fig. 5, group \(Br_2(\varGamma _\varTheta )\) is a free group that has three generators, \(Br_2(\varGamma _\varTheta )=\langle \alpha _D,\alpha _U,\gamma _L\rangle \). Generators \(\alpha _U\) and \(\alpha _D\) correspond to a single particle travelling around a simple cycle in \(\varGamma _\varTheta \) while generator \(\gamma _L\) denotes a pair of particles exchanging on the left junction. Clearly, it is possible to have an analogous exchange on the right junction, \(\gamma _R\). Such an exchange can be expressed by the above generators as

A physical model for a *U*(2) representation of \(Br_2(\varGamma _\varTheta )\) can be constructed using general theory of exchanging Majorana fermions on networks of quantum wires presented in [16]. Here we only briefly sketch the main ideas of this construction. The role of particles is played by two Majorana fermions placed on the spots of black dots from Fig. 5. The two fermions are at the endpoints of the so-called topological region in a network of superconducting quantum wires. Majorana fermions are braided by adiabatically changing physical parameters of the quantum wire.

*i*. Number

*K*is called the rank of \(H_d(\mathfrak {C},{\mathbb {Z}})\), and is equal to the \(d\hbox {th}\) Betti number of complex

*X*.

### Theorem 4

If *X* has the homotopy type of a finite *CW* complex, then ranks of \(H^k(X,{\mathbb {Z}})\) and \(H_k(X,{\mathbb {Z}})\) are equal and the torsion of \(H^k(X,{\mathbb {Z}})\) is equal to the torsion of \(H_{k-1}(X,{\mathbb {Z}})\).

## 3 Vector Bundles and Their Classification

The main motivation for studying (co)homology groups of configuration spaces comes from the fact that they give information about the isomorphism classes of vector bundles over configuration spaces. In the following section, we review the main strategies of classifying vector bundles and make the role of homology groups more precise. Throughout, we do not assume that the configuration space is a differentiable manifold, as the configuration spaces of graphs are not differentiable manifolds. We only assume that \(C_n(X)\) has the homotopy type of a finite *CW*-complex. This means that \(C_n(X)\) can be deformation retracted to a finite *CW*-complex. As we explain in Sect. 4, configuration spaces of graphs are such spaces. The lack of differentiable structure means that the flat vector bundles have to be defined without referring the notion of a connection and all the methods that are used have to be purely algebraic. We provide such an algebraic definition of flat bundles in Sect. 3.3.

In this paper, we consider only complex vector bundles \(\pi :\ E\rightarrow B\), where *E* is a total space and *B* is the base. Two vector bundles are isomorphic iff there exists a homeomorphism between their total spaces which preserves the fibres. If two vector bundles belong to different isomorphism classes, there is no continuous function which transforms the total spaces to each other, while preserving the fibres. Hence, the wave functions stemming from sections of such bundles must describe particles with different topological properties. The classification of vector bundles is the task of classifying isomorphism classes of vector bundles. The set of isomorphism classes of vector bundles of rank *k* will be denoted by \(\mathcal {E}_k^{\mathbb {K}}(B)\).

Before we proceed to the specific methods of classification of vector bundles, we introduce an equivalent way of describing vector bundles which involves *principal bundles* (principal *G*-bundles). A principal *G*-bundle \(\xi :\ P\rightarrow B\) is a generalisation of the concept of vector bundle, where the total space is equipped with a free action of group *G*^{1} and the base space has the structure of the orbit space \(B\cong P/G\). Fibre \(\pi ^{-1}(p)\) is isomorphic to *G* is the sense that map \(\pi :\ P\rightarrow B\) is *G*-invariant, i.e. \(\pi (ge)=\pi (e)\). Moreover, all relevant morphisms are required to be *G*-equivariant. The set of isomorphism classes of principal *G*-bundles over base space *B* will be denoted by \(\mathcal {P}_G(B)\).

*E*. This means that we consider hermitian vector bundles, i.e. bundles with hermitian product \(\langle \cdot ,\cdot \rangle _p\) on fibres \(\pi ^{-1}(p),\ p\in B\) that depends on the base point and varies between the fibres in a continuous way. Choosing sets of unitary frames, we obtain a correspondence between hermitian vector bundles and principal

*U*(

*k*)-bundles. If the base space is paracompact, any complex vector bundle can be given a hermitian metric [39]. Using the fact that principal

*U*(

*k*)-bundles corresponding to different choices of the hermitian structure are isomorphic [39], we have the following bijection

*U*(

*k*)-bundles.

### 3.1 Universal bundles and Chern classes

*k*over a paracompact topological space can be obtained from a vector bundle which is universal for all base spaces. This is done in the following way. Any continuous map \(f:\ B'\rightarrow B\) between base spaces induces a pullback map of vector bundles over

*B*to vector bundles over \(B'\). The pullback bundle is defined as \(f^*E=\{(p,e)\in B'\times E:\ f(p)=\pi (e)\}\). Similarly, one defines the pullback of principal

*G*-bundles. For a fixed principal

*G*-bundle \(\xi :\ P\rightarrow B\), the pullback map induces a map from [

*A*,

*B*], i.e. from the space of homotopy classes of continuous maps from

*A*to

*B*, to the set of isomorphism classes of principal

*G*-bundles over

*A*by \(f\mapsto f^* \xi \). A space

*B*for which such a map is bijective regardless the choice of space

*A*, is called a

*classifying space*for

*G*and is denoted by

*BG*. If this is the case, bundle \(\xi \) is called a

*universal bundle*. For principal

*U*(

*k*)-bundles, the classifying space is the infinite Grassmannian [39]

*U*(

*k*)-bundle over a paracompact Hausdorff space

*B*can be written as \(f^*(\gamma ^k_{\mathbb {C}})\) for \(f:\ B\rightarrow Gr_k({\mathbb {C}}^\infty )\). The isomorphism class of \(f^*(\gamma ^k_{\mathbb {C}})\) is determined uniquely by the homotopy class of

*f*and vice versa. However, the classification of such homotopy classes of maps, as well as differentiating between different classes are difficult tasks. A more computable criterion for comparing isomorphism classes of vector bundles are invariants called Chern characteristic classes. Let us next briefly introduce this notion. A characteristic class is a map that assigns to each principal

*G*-bundle \(\xi :\ P\rightarrow B\) an element of the cohomology ring of

*B*with some coefficients. Characteristic classes are invariant under isomorphisms of principal bundles, and those that describe principal

*U*(

*k*)-bundles have values in \(H^*(B,{\mathbb {Z}})\). Such characteristic classes are called integral Chern classes. They are evaluated as follows. Let \(a\in H^q(BU(k),{\mathbb {Z}})\). We assign to this element a characteristic class \(c_a\) which is defined defined by its values on an arbitrary principal bundle \(\xi :\ P\rightarrow B\). By the classification theorem, we have \(\xi =f^*_\xi (\gamma _{\mathbb {C}}^k)\) for some continuous map \(f_\xi :\ B\rightarrow BU(k)\). Hence, \(c_a\) is evaluated as \(c_a(\xi ):=f^*_\xi (a)\), where \(f^*_\xi :\ H^q(BU(k),{\mathbb {Z}})\rightarrow H^q(B,{\mathbb {Z}})\) is the pullback of cohomology rings via map \(f_\xi \). Map \(f^*_\xi \) is often called the

*characteristic homomorphism*. It turns out that the only nonzero Chern classes are of even degree.

*B*,

*BU*(1)] is in a bijective correspondence with \(H^2(B,{\mathbb {Z}})\). Hence, we arrive at the first direct application of the knowledge of cohomology ring of space

*B*, namely

*K*-theory and while studying characteristic classes of flat vector bundles.

### 3.2 Reduced *K*-theory

We start with recalling the definition of *stable* equivalence of vector bundles.

### Definition 1

*CW*-complex, group \({{\tilde{K}}}(B)\) fully describes isomorphism classes of vector bundles that have a sufficiently high rank [40]. This concerns vector bundles, whose rank is in the

*stable range*, i.e. is greater than or equal to

*x*. The set of stable equivalence classes of \(Vect^{\mathbb {C}}(B)\) is equal to \(\mathcal {E}_{k_s}^{\mathbb {C}}(B)\). Moreover, \(\mathcal {E}_k^{\mathbb {C}}(B)\) is the same for all \(k\ge k_s\) and equal to \(\mathcal {E}_{k_s}^{\mathbb {C}}(B)\). Therefore,

*K*-theory and cohomology is phrased via the

*Chern character*which induces isomorphism from \({\tilde{K}}(B)\) to \(H^*(B,{\mathbb {Q}})\) when

*B*has the homotopy type of a finite

*CW*-complex.

*B*are torsion-free. In the case when there is non-trivial torsion in \(H^*(B,{\mathbb {Z}})\), torsion of \({{\tilde{K}}}(B)\) is determined by the Atiyah-Hirzebruch spectral sequence [41]. However, the correspondence between torsion of even cohomology and \({{\tilde{K}}}(B)\) is not an isomorphism. In particular, torsion in \({{\tilde{K}}}(B)\) can vanish, despite the existence of nonzero torsion in \(H^{2i}(B,{\mathbb {Z}})\). Finally, we note that stable equivalence of vector bundles is physically important in situations when one has access only to Chern classes or other topological invariants stemming from Chern classes, e.g. the Chern numbers. This is because Chern classes of stably equivalent vector bundles are equal.

### 3.3 Flat bundles and quantum statistics

*G*-bundles over base space

*B*. More precisely, we consider the set of pairs \((\xi ,\mathcal {A})\), where \(\xi \) is a principal

*G*-bundle, and \(\mathcal {A}\) is a connection 1-form on \(\xi \). We divide the set of such pairs into equivalence classes \([(\xi ,\mathcal {A})]\) that consist of vector bundles isomorphic to \(\xi \) and the set of flat connections that are congruent to \(\mathcal {A}\) under the action of the gauge group. The quotient space with respect to this equivalence relation is called the

*moduli space of flat connections*and is denoted by \(\mathcal {M}(B,G)\). The culminating point of this section is to introduce the fundamental relation which says that \(\mathcal {M}(B,G)\) is in a bijective correspondence with the set of conjugacy classes of homomorphisms of the fundamental group of

*B*.

*B*is a smooth manifold. Having fixed a principal connection

*H*on

*P*, we consider parallel transport of elements of

*P*around loops in

*B*. Parallel transport around loop \(\gamma \subset B\) is a morphism of fibres \(\varGamma _\gamma :\ \pi ^{-1}(b)\rightarrow \pi ^{-1}(b)\) which assigns the end point of the horizontal lift of \(\gamma \) (denote it by \({\tilde{\gamma }}\)) to its initial point

*G*, for every choice of the initial point \(p={\tilde{\gamma }}(0)\) there is a unique group element \(g\in G\) such that \({\tilde{\gamma }}(1)=gp\). We denote this element by \(\mathrm{hol}_p(H,\gamma )\) and call the

*holonomy*of connection

*H*around loop \(\gamma \) at point

*p*. Moreover, by the

*G*-equivariance of the connection, we get that

*H*is flat, the parallel transport depends only on the topology of the base space [42], i.e. (i) \(\varGamma _\gamma \) depends only on the homotopy class of \(\gamma \), (ii) parallel transport around a contractible loop is trivial, (iii) parallel transport around two loops that have the same base point is the composition of parallel transports along the two loops \(\varGamma _{\gamma _1\circ \gamma _2}=\varGamma _{\gamma _1}\circ \varGamma _{\gamma _2}\). These facts show that if

*H*is flat, map \(\pi _1(B)\ni [\gamma ]\mapsto \mathrm{hol}_p(H,\gamma )\in G\) is a homomorphism of groups. Because holonomies at different points from the same fibre differ only by conjugation in

*G*, it is not necessary to specify the choice of the initial point. Instead, we consider map

*G*. There is one more symmetry of this map that we have not discussed so far, namely the gauge symmetry. A gauge transformation is a map \(f:\ P\rightarrow G\) which is

*G*-equivariant, i.e. \(f(gp)=g^{-1}f(p)g\). A gauge transformation induces an automorphism of

*P*which acts as \(p\rightarrow f(p) p\). Consequently, transformation

*f*induces a pullback of connection forms. It can be shown that map \(\mathcal {S}_H\) is gauge invariant [42], i.e. depends only on the gauge equivalence class of connection

*H*.

An important conclusion regarding flat bundles on spaces that do not have a differential structure comes from the second part of correspondence (5). This is the reconstruction of a flat principal bundle from a given homomorphism \({{\mathrm{Hom}}}(\pi _1(B),G)\). It turns out that any flat bundle over *B* can be realised as a particular quotient bundle of the trivial bundle over the *universal cover* of *B*. In order to formulate the correspondence, we first introduce the notion of a covering space and a universal cover.^{2} The following theorem is also a definition of a flat principal bundle for spaces that are not differential manifolds.

### Theorem 5

*G*-bundle \(P\rightarrow B\) can be constructed as the following quotient bundle of the trivial bundle over the universal cover of

*B*.

*G*is defined by picking a homomorphism \(\rho : \pi _1(B)\rightarrow G\). Then the action reads \(ag:=\rho (a)g\) for \(a\in \pi _1(B)\), \(g\in G\).

*G*-bundles, one has to classify conjugacy classes of homomorphisms \(\pi _1(B)\rightarrow G\). All spaces that are considered in this paper have finitely generated fundamental group. This fact makes the classification procedure easier. Namely, one can fix a set of generators \(a_1,\dots ,a_r\) of \(\pi _1(B)\) and represent them as group elements \(g_1,\dots ,g_r\). Matrices \(g_1,\dots ,g_r\) realise \(\pi _1(B)\) in

*G*in a homomorphic way iff they satisfy the relations between the generators of \(\pi _1(B)\). This way, the moduli space of flat connections can be given the structure of an algebraic variety. In other words, we consider map

### Fact 3.1

Two points in \(\mathcal {M}(B,G)\) that correspond to two non-isomorphic flat bundles, belong to different path-connected components of \(\mathcal {M}(B,G)\).

Equivalently, if two flat structures, i.e. points in \(\mathcal {M}(B,G)\), belong to the same path-connected component of \(\mathcal {M}(B,G)\), then the corresponding vector bundles are isomorphic. A path connecting the two points in \(\mathcal {M}(B,G)\) gives a homotopy between the corresponding flat structures.

### Example 5

*U*(1)

**bundles over spaces with finitely generated fundamental group.**As conjugation in

*U*(1) is trivial, we have

*U*(1). A standard result from algebraic topology says thatwhere \([\cdot , \cdot ]\) is the group commutator. \(H_1(B,{\mathbb {Z}})\) as any finitely generated abelian group decomposes as the sum of a free component and a cyclic (torsion) part

*p*-torus, whose points correspond to phases \(\phi _i\). In fact, the connected components are in a one-to-one correspondence with isomorphism classes of flat bundles. To see this, recall the fact that set of

*U*(1)-bundles has the structure of a group which is isomorphic to \(H^2(B,{\mathbb {Z}})\). Moreover, as we explain in Remark 3.1, Chern classes of flat bundles are torsion. This means that flat

*U*(1)-bundles form a subgroup of the group of all

*U*(1)-bundles which is isomorphic to the torsion of \(H^2(B,{\mathbb {Z}})\). By the universal coefficient theorem [43], torsion of \(H^2(B,{\mathbb {Z}})\) is the same as torsion of \(H_1(B,{\mathbb {Z}})\). Note that there is exactly the same number of connected components in \({{\mathrm{Hom}}}(H_1(B,{\mathbb {Z}}),U(1))\) as the number of group elements in the torsion component of \(H_1(B,{\mathbb {Z}})\). In this case, fact 3.1 implies that each connected component represents one isomorphism class of flat bundles.

*Characteristic classes of flat bundles*From this point, we can move away from considering connections and use the wider definition of flat

*G*-bundles which makes sense for bundles over spaces that have a universal covering space. As stated in theorem 5, such flat bundles have the form

*U*(

*n*)-bundles over connected

*CW*-complexes we have the following general result about the triviality of rational Chern classes [44].

### Theorem 6

*G*be a compact Lie group,

*B*a connected

*CW*-complex and \(\xi : P\rightarrow B\) a flat

*G*-bundle over

*B*. Then, the characteristic homomorphism

### Remark 3.1

Theorem 6 in particular means that if *B* is a finite *CW*-complex, then by the universal coefficient theorem for cohomology (see e.g. [43]), the image of the characteristic map \(f_\xi ^*:\ H^*(BG,{\mathbb {Z}})\rightarrow H^*(B,{\mathbb {Z}})\) consists only of torsion elements of \(H^*(B,{\mathbb {Z}})\).

Specifying the above results for *U*(*n*)-bundles, we get that the lack of nontrivial torsion in \(H^{2i}(B,{\mathbb {Z}})\) has the following implications for the stable equivalence classes of flat vector bundles.

### Proposition 7

Let *B* be a finite *CW* complex. If the integral homology groups of *B* are torsion-free, then every flat complex vector bundle over *B* is stably equivalent to a trivial bundle.

### Proof

If the integral cohomology of *B* is torsion-free, then by the Chern character we get that the reduced Grothendieck group is isomorphic to the direct sum of even cohomology of *B*. Thus, if all Chern classes of a given bundle vanish, this means that this bundle represents the trivial element of the reduced Grothendieck group, i.e. is stably equivalent to a trivial bundle. \(\quad \square \)

- 1.
Configuration space of

*n*particles on a plane. Space \(C_n({\mathbb {R}}^2)\) is aspherical, i.e. is an Eilenberg–Maclane space of type \(K(\pi _1,1)\), where the fundamental group is the braid group on*n*strands \(Br_n\). Cohomology ring \(H^*(C_n({\mathbb {R}}^2),{\mathbb {Z}})=H^*(Br_n,{\mathbb {Z}})\) is known [45, 46]. Its key properties are (i)**finiteness**—\(H^{i}(Br_n,{\mathbb {Z}})\) are cyclic groups, except \(H^{0}(Br_n,{\mathbb {Z}})=H^{1}(Br_n,{\mathbb {Z}})={\mathbb {Z}}\), (ii)**repetition**—\(H^{i}(Br_{2n+1},{\mathbb {Z}})=H^{i}(Br_{2n},{\mathbb {Z}})\), (iii)**stability**—\(H^{i}(Br_{n},{\mathbb {Z}})=H^{i}(Br_{2i-2})\) for \(n\ge 2i-2\). Description of nontrivial flat*U*(*n*) bundles over \(C_n({\mathbb {R}}^2)\) for \(n> 2\) is an open problem. - 2.
Configuration space of

*n*particles in \({\mathbb {R}}^3\). Much less is known about \(H^*(C_n({\mathbb {R}}^3))\). Some computational techniques are presented in [47, 48], but little explicit results are given. Ring \(H^*(C_3({\mathbb {R}}^3)\) is equal to \({\mathbb {Z}},0,{\mathbb {Z}}_2,0,{\mathbb {Z}}_3\) [49] and \(H^q(C_3({\mathbb {R}}^3))=0\) for \(q>4\). However, it has been shown that there are no nontrivial flat*SU*(*n*) bundles over \(C_3({\mathbb {R}}^3)\). - 3.
Configuration space of

*n*particles on a graph (a 1-dimensional*CW*-complex \(\varGamma \)). Spaces \(C_n(\varGamma )\) are Eilenberg–Maclane spaces of type \(K(\pi _1,1)\). The calculation of their homology groups is a subject of this paper. Group \(H_1(C_n(\varGamma ),{\mathbb {Z}})\) is known [8, 37] for an arbitrary graph. We review the structure of \(H_1(C_n(\varGamma ))\) in Sect. 4.1. By the universal coefficient theorem, the torsion of \(H^2(C_n(\varGamma ))\) is equal to the torsion of \(H_1(C_n(\varGamma ))\) which is known to be equal to a number of copies of \({\mathbb {Z}}_2\), depending on the structure of \(\varGamma \). We interpret this result as the existence of different bosonic or fermionic statistics in different parts of \(\varGamma \). The existence of torsion in higher (co)homology groups of \(C_n(\varGamma )\) which is different than \({\mathbb {Z}}_2\), is an open problem. In this paper, we compute homology groups for certain canonical families of graphs. However, the computed homology groups are either torsion-free, or have \({\mathbb {Z}}_2\)-torsion.

## 4 Configuration Spaces of Graphs

*CW*-complex which is a deformation retract of \(C_n(\varGamma )\). The existence of discrete models for graph configuration spaces enables us to use standard tools from algebraic topology to compute homology groups of graph configuration spaces. In particular, we use different kinds of homological exact sequences. There are two discrete models that we use.

- 1.
Abram’s discrete configuration space [21]. The Abram’s deformation retract of \(C_n(\varGamma )\) is denoted by \(D_n(\varGamma )\). We use Abram’s discrete model mainly in the first part of this paper, where we apply discrete Morse theory to the computation of homology groups of some small canonical graphs (Sect. 5.2).

- 2.
The discrete model by Świątkowski [20] that we denote by \(S_n(\varGamma )\). We use this model in Sects. 5.3–5.6 to compute homology groups of configuration spaces of wheel graphs and some families of complete bipartite graphs.

*n*. The dimension of Abram’s model is equal to

*n*for sufficiently large

*n*. Hence, the Świątkowski model is more suitable for rigorous calculations. However, sometimes it is more convenient to use Abram’s model with the help of discrete Morse theory. The computational complexity of numerically calculating the homology groups of \(C_n(\varGamma )\) for a generic graph is comparable in both approaches.

*Abrams discrete model*Let us next describe in detail the discrete configuration spaces \(D_n(\varGamma )\) by Abrams. For the deformation retraction from \(C_n(\varGamma )\) to \(D_n(\varGamma )\) to be valid, the graph must be simple and

*sufficiently subdivided*which means that

each path between distinct vertices of degree not equal to 2 passes through at least \(n-1\) edges,

each nontrivial loop passes through at least \(n+1\) edges.

*n*-dimensional cells in \(D_n(\varGamma )\) are of the following form.

*d*-dimensional cell consists of

*d*edges and \(n-d\) vertices. In other words, cells from \(\varSigma ^{d}(D_n(\varGamma ))\) are of the form

*n*.

*T*by picking a vertex of degree 1 in

*T*. For every \(v\in V(\varGamma )\) there is the unique path in

*T*that joins

*v*and \(*\), called the geodesic \(g_{v,*}\). For every vertex with \(d(v)\ge 2\) we enumerate the edges adjacent to

*v*with numbers \(0,1,\dots ,d(v)-1\). The edge contained in \(g_{v,*}\) has label 0. The remaining edges are labelled increasingly, according to their clockwise order starting from edge 0. The enumeration procedure for vertices goes in an inductive manner. The root has number 1. If vertex

*v*has label

*k*and \(d(v)=2\), the vertex adjacent to

*v*is given label \(k+1\). Otherwise, if \(d(v)\ge 2\), the vertex adjacent to

*v*in the lowest direction with vertices that have not been yet labelled is given label \(k_{max}+1\), where \(k_{max}\) is the maximal label among all of the already labelled vertices. If \(d(v)=1\), we look for essential vertices in \(g_{v,*}\) and go back to the closest essential vertex that contains a direction with unlabelled vertices. In other words, the vertices are labelled in the clockwise direction. This way every edge is given an initial and terminal vertex that we denote by \(\iota (e)\) and \(\tau (e)\) respectively. The terminal vertex is the vertex with the lower index, i.e. \(\tau (e)<\iota (e)\). We can unambiguously specify an edge by calling its initial and terminal vertices, hence we denote the edges by \(e_{\tau }^{\iota }\). Given a cell from \(D_n(\varGamma )\)

*i*th pair of faces from the boundary of \(\sigma \) reads

*Świątkowski discrete model*Świątkowski complex is denoted by \(S_n(\varGamma )\). In order to define it, we regard graph \(\varGamma \) as a set of edges

*E*, vertices

*V*and half-edges

*H*. A half-edge of \(e\in E(\varGamma )\) assigned to vertex

*v*, \(h(v)\subset e\), is the part

*e*which is an open neighbourhood of vertex

*v*. Intuitively, the half-edges are places, where the particles are allowed to ‘slide’. By

*e*(

*h*) we will denote the unique edge, for which \(e\cap h\ne \emptyset \). Similarly, we have vertex

*v*(

*h*) as the vertex, for which

*h*is a neighbourhood. By

*H*(

*v*) we will denote all half edges that are incident to vertex

*v*. Chain complex \(S(\varGamma )=\bigoplus _n S_n(\varGamma )\) reads

*d*-chain \(\chi \). There is a canonical basis for \(S(\varGamma )\), whose elements of degree (

*d*,

*n*) are of the form

### Definition 2

*d*-cell \(c=h_1\dots h_dv_1\dots v_ke_1^{n_1}\dots e_l^{n_l}\in S_n(\varGamma )\) is the set of the corresponding edges and vertices of \(\varGamma \)

*v*to differences of half edges \(h_{ij}:=h_i-h_j,\ h_i,h_j\in H(v)\), \({{\tilde{S}}}_v:={\mathbb {Z}}\langle \emptyset , h_{ij}\rangle \).

### Fact 4.1

*Vertex blowup*In the following, we will explore relations on homology groups that stem from blowing up a vertex of \(\varGamma \): \(\varGamma \rightarrow \varGamma _v\) (Fig. 8).

*v*, \({{\tilde{S}}}^v(\varGamma )\). Any chain \(b\in {\tilde{S}}^v(\varGamma )\) can be decomposed in a unique way by extracting the part that involves generators from \({{\tilde{S}}}_v\). In order to do it, we fix a half-edge \(h_0\in H(v)\) and write

*b*as

*h*-components. It assigns a number of \(n-1\)-particle \(d-1\)-chains to a

*n*-particle

*d*-chain in the following way

*v*(for example, the \(c_Y\) cycles).

### 4.1 *O*-cycles and *Y*-cycles

There are some particular types of cycles that play an important role in this work. These are *O*-cycles and *Y*-cycles. We specify them for the Abram’s model. The construction for \(S_n(\varGamma )\) is fully analogous.

### Definition 3

*O*-cycle in \(D_n(\varGamma )\) is a 1-chain of the form

*O*-cycle in \(S_n(\varGamma )\), note that for all \(v\in V(\varGamma )\cap O\), set \(H(v)\cap O\) contains exactly two half-edges. We denote these half-edges by \(h_v, h_v'\), where the labels are such that \(\partial \sum _{v\in V(\varGamma )\cap O}(h_v'-h_v)=0\). Then,

### Definition 4

*Y*-subgraph of \(\varGamma \) spanned on vertices \(u_0,u_h,u_1,u_2\) such that \(u_0,\ u_1,\ u_2\) are adjacent to \(u_h\) and \(u_0<u_h<u_1<u_2\). The

*Y*-cycle in \(D_2(\varGamma )\) associated to subgraph

*Y*is of the following form

*Y*-cycle in \(D_n(\varGamma )\) is formed by distributing the free particles outside of subgraph

*Y*, i.e.

*Y*-cycle in \(S_n(\varGamma )\), denote the half edges of subgraph

*Y*as \(\{h_i\}_{i=0}^2\), where \(h_i\in H(u_h)\) are such that \(e(h_0)=e_{u_0}^{u_h}\), \(e(h_1)=e_{u_h}^{u_1}\), \(e(h_2)=e_{u_h}^{u_2}\). Then,

*Y*-cycles is shown on Figs. 10 and 11.

*Y*-graphs share one cycle \(c_O\) and their free ends are connected by a path \(p_{v_1,v_2}\) which is disjoint with \(c_O\) (Fig. 11). In other words, consider an embedding of a graph which is isomorphic to the \(\varTheta \)- graph.

^{3}

*O*-cycles as differences of

*Y*-cycles, one can compute the first homology group of \(D_n(\varGamma )\). Let us next summarise the results concerning the structure of the first homology group of graph configuration spaces. We formulate the results assuming that the considered graphs are simple. The general form of the first homology group reads

*N*and

*L*are the numbers of copies of \({\mathbb {Z}}\) and \({\mathbb {Z}}_2\) respectively. Numbers

*N*and

*L*depend on the planarity and some combinatorial properties of the given graph [8, 37]. The \({\mathbb {Z}}_2\)-components appear when \(\varGamma \) is non-planar and have the interpretation of different fermionic/bosonic statistics that may appear locally in different parts of a given graph (see [8]).

## 5 Calculation of Homology Groups of Graph Configuration Spaces

This section contains the techniques that we use for computing homology groups of graph configuration spaces. We tackle this problem from the ‘numerical’ and the ‘analytical’ perspective. The numerical approach means using a computer code for creating the boundary matrices and then employing the standard numerical libraries for computing the kernel and the elementary divisors of given matrices. The procedures for calculating the boundary matrices of \(D_n(\varGamma )\), \(S_n(\varGamma )\) and the Morse complex (see Sect. 5.2) were written by the authors of this paper, based on papers [24, 37]. The analytical approach means computing the homology groups for certain families of graphs by suitably decomposing a given graph into simpler components and using various homological exact sequences. Recently in the mathematical community, there has been a growing interest in computing the homology groups of graph configuration spaces. A significant part of the recent work has been devoted to explaining certain regularity properties of the homology groups of \(C_n(\varGamma )\) [50, 51, 52, 53, 54, 55].

### 5.1 Product cycles

*Y*-subgraphs of \(\varGamma \) and the

*O*-type cycles with the remaining particles distributed on the free vertices of \(\varGamma \), one can construct some generators of \(H_*(D_n(\varGamma ))\) or \(H_*(S_n(\varGamma ))\). Such cycles are products of 1-cycles, hence are isomorphic to tori embedded in the discrete configuration space. To construct a product

*d*cycle in \(D_n(\varGamma )\), we choose

*Y*-subgraphs of \(\varGamma \)\(\{Y_i\}_{i=1}^{d_Y}\) and cycles in \(\varGamma \) (

*O*-subgraphs of \(\varGamma \)) \(\{O_i\}_{i=1}^{d_O}\), where \(d_Y+d_O=d\). All the chosen subgraphs must be mutually disjoint.

*i*,

*j*. Product cycle on \(Y_1\times \dots \times Y_{d_Y}\times O_1\times \dots \times O_{d_O}\) with the free particles distributed on \(\{v_1,\dots ,v_{n-2d_Y-d_O}\}\) is the following chain.

### 5.2 Discrete Morse theory for Abrams model

In this subsection, we apply a version of Forman’s discrete Morse theory [58] for Abram’s discrete model that was formulated in [24] (see also [59]). The results are listed in Tables 1 and 2.

*F*which is a linear map mapping

*d*-chains to

*d*-chains. Moreover, map

*F*has the property that for any chain

*c*, we have \(F^{r+1}(c)=F^{r}(c)\) for some

*r*. The Morse complex is the chain complex of chains invariant under

*F*. The basis of such invariant chains consists of

*critical cells*. There are

*a priori*different ways to explicitly realise the discrete gradient flow for graph configuration spaces. We have chosen the realisation introduced in [24]. Here, we do not review the details of this construction, but only present a pseudocode which shows schematically how to compute \(H_d(D_n(\varGamma ))\) using the knowledge of the boundary map in \(D_n(\varGamma )\) and the list of critical cells of

*F*as cells in \(D_n(\varGamma )\). We also direct the reader to public repository [60] where we uploaded a Python implementation of the discrete Morse theory that we used in our work. The results of running the code for different graphs are collected in Tables 1 and 2.

Betti numbers for chosen graphs computed using the discrete Morse theory [24]

\(\varGamma \) | | \(\beta _2(C_n(\varGamma ))\) | \(\beta _3(C_n(\varGamma ))\) | \(\beta _4(C_n(\varGamma ))\) |
---|---|---|---|---|

\(K_4\) | 3 | 3 | 0 | – |

4 | 9 | 0 | 0 | |

5 | 15 | 0 | 0 | |

6 | 21 | 4 | 0 | |

7 | 27 | 16 | 0 | |

8 | 33 | 40 | 1 | |

9 | 39 | 80 | 6 | |

\(K_{3,3}\) | 2 | 0 | – | – |

3 | 8 | 0 | – | |

4 | 19 | 1 | 0 | |

5 | 28 | 10 | 0 | |

6 | 37 | 39 | 0 | |

7 | 46 | 88 | 0 | |

8 | 55 | 157 | 15 | |

\(K_5\) | 2 | 0 | – | – |

3 | 30 | 0 | – | |

4 | 76 | 1 | 0 | |

5 | 116 | 77 | 0 | |

6 | 156 | 381 | 0 | |

7 | 196 | 961 | 0 |

The first regular homology groups of order 2 and 3 for the Petersen family

\(K_6\) | \(P_7\) | \(K_{3,3,1}\) | \(K_{4,4}\) | \(P_8\) | \(P_9\) | \(P_{10}\) | |
---|---|---|---|---|---|---|---|

\(\beta _2(C_4(\varGamma ))\) | 264 | 177 | 172 | 144 | 114 | 70 | 40 |

\(T_2(C_4(\varGamma ))\) | \({\mathbb {Z}}_2\) | \({\mathbb {Z}}_2\) | \({\mathbb {Z}}_2\) | \(\left( {\mathbb {Z}}_2\right) ^2\) | \({\mathbb {Z}}_2\) | \({\mathbb {Z}}_2\) | \({\mathbb {Z}}_2\) |

\(\beta _3(C_6(\varGamma ))\) | 4137 | 2058 | 1919 | 1460 | 986 | 452 | 191 |

\(T_3(C_6(\varGamma ))\) | 0 | 0 | 0 | \(\left( {\mathbb {Z}}_2\right) ^{73}\) | 0 | 0 | 0 |

Table 2 presents the results for the second and third homology groups for graphs from the Petersen family (Fig. 12). These graphs serve as examples, where torsion in higher homology groups appears. Interestingly, the torsion subgroups are always equal to a number of copies of \({\mathbb {Z}}_2\). This phenomenon can be explained by embedding a nonplanar graph in \(\varGamma \) and considering suitable product cycles. The question about the existence of torsion different than \({\mathbb {Z}}_2\) in higher homologies remains open.

### 5.3 Wheel graphs

In this section, we deal with the class of wheel graphs. A wheel graph of order *m* is a simple graph that consists of a cycle on \(m-1\) vertices, whose every vertex is connected by an edge (called a spoke) to one central vertex (called the hub). We provide a complete description of the homology groups of configuration spaces for wheel graphs. In particular, we show that all homology groups are free. Therefore, in addition to tree graphs, wheel graphs provide another family of configuration spaces with a simplified structure of the set of flat complex vector bundles. The general methodology of computing homology groups for configuration spaces of wheel graphs is to consider only the product cycles and describe the relations between them. We justify this approach in Sect. 5.4.

The simplest example of a wheel graph is graph \(K_4\) which is the wheel graph of order 4. Let us next calculate all homology groups of graph \(K_4\) and then present the general method for any wheel graph.

#### 5.3.1 Graph \(K_4\)

*Second homology group*There are three independent cycles in \(K_4\) graph. These are the cycles that contain the hub and two neighbouring vertices from the perimeter. However, any two such cycles always share some vertices. Hence, there are no tori that come from the products of \(c_O\) cycles. Hence, the product 2-cycles are either \(c_Y\otimes c_O\) or \(c_Y\otimes c_{Y'}\). There are four cycles of the first kind: \(c_{Y_1}\otimes c_{O_1}\), \(c_{Y_2}\otimes c_{O_2}\), \(c_{Y_3}\otimes c_{O_3}\) and \(c_{Y_h}\otimes c_{O}\), where \(c_{O}\) is the outermost cycle. However, cycle \(c_{Y_h}\otimes c_{O}\) can be expressed as a linear combination of cycles \(c_{Y_1}\otimes c_{O_1}\), \(c_{Y_2}\otimes c_{O_2}\), \(c_{Y_3}\otimes c_{O_3}\). Therefore, the second homology of the three-particle configuration space is

*AB*-cycles can be expressed as combinations of \(Y\times Y\) cycles. This fact generalises to \(n>4\) in a straightforward way.

*Y*-subgraphs. There are six \(Y\times Y\)-cycles modulo the distribution of free particles. Hence, if there are no free particles, i.e. when \(n=4\), we have

*Y*-subgraphs. The number of all choices of the

*Y*-subgraphs is \({4}\atopwithdelims (){2}\), while the number of possible distributions of \(n-4\) particles on 2 components is \({{n-4+2-1}\atopwithdelims (){2-1}}=n-3\). Hence, the contribution from \(Y\times Y\) cycles reads

*Higher homology groups*The product generators of higher homologies are even simpler than in the case of the second homology. There are only basis cycles of \(Y\times Y\times \dots \times Y\)-type. After removing three and four

*Y*-graphs, \(K_4\) graph always disintegrates into 4 and 6 parts respectively. Taking into account the distributions of free particles, we get the following formulae for the Betti numbers.

*Y*-graphs, group \(H_5(C_n(K_4),{\mathbb {Z}})\) is zero.

#### 5.3.2 General wheel graphs

Betti numbers of configuration spaces for chosen wheel graphs computed using the discrete Morse theory

\(\varGamma \) |
| \(\beta _2(D_n(\varGamma ))\) | \(\beta _3(D_n(\varGamma ))\) | \(\beta _4(D_n(\varGamma ))\) |
---|---|---|---|---|

\(W_5\) | 3 | 8 | 0 | – |

4 | 22 | 0 | 0 | |

5 | 34 | 4 | 0 | |

6 | 46 | 30 | 0 | |

7 | 58 | 90 | 0 | |

8 | 70 | 196 | 13 | |

\(W_6\) | 3 | 15 | 0 | – |

4 | 40 | 0 | 0 | |

5 | 60 | 15 | 0 | |

6 | 80 | 90 | 0 | |

7 | 100 | 250 | 5 | |

\(W_7\) | 3 | 24 | 0 | – |

4 | 63 | 0 | 0 | |

5 | 93 | 36 | 0 | |

6 | 123 | 197 | 0 | |

7 | 153 | 527 | 24 |

*Second homology*Since there are no pairs of disjoint

*O*-cycles in wheel graphs, we have

*Y*-subgraphs and \(m-3\) cycles that are disjoint with a fixed

*Y*-subgraph. Hence,

*Y*do not share any edges of the graph (like on Fig. 16b)). Then, as on Fig. 11, cycles \(c_{Y_h}\) and \(c_{Y'}\) are in the same homology class in \(D_2(W_m-Y)\), because they share the same

*O*-cycle and they are connected by a path that is disjoint with

*Y*. Therefore, by multiplying the relation by \(c_{Y}\) we get that

*Y*-subgraphs from the perimeter. For a fixed

*Y*-subgraph, the contribution from \(Y\times Y_h\)-cycles turns out to be equal to the number of independent cycles in the fan graph which is formed by removing subgraph

*Y*from the wheel graph [8]. This number is equal to \(m-3\). Hence,

*Y*-subgraphs from the perimeter may result with the decomposition of the wheel graph into at most two components. This happens iff two neighbouring

*Y*-subgraphs have been removed. The number of nonequivalent ways of distributing the particles is \(n-3\). The number of ways one can choose two neighbouring

*Y*-subgraphs from the perimeter is \(m-1\). This gives us the contribution of \((n-3)(m-1)\). Furthermore, removing a

*Y*-subgraph from the hub and a subgraph from the perimeter always yields two nonequivalent ways of distributing the free particles. The first one being the edge

*e*joining the hub and the central vertex of

*Y*, the second one being the remaining part of the graph, i.e. \(W_m-(Y\sqcup Y_h\sqcup e)\). The contribution is \((n-3)(m-1)(m-3)\). Adding the contribution from \(O\times Y\)-cycles and from non-neighbouring \(Y_p\times Y_p\)-cycles, we get that the final formula for the second Betti number reads

*Higher homologies*In computing the higher homology groups, we proceed in a similar fashion as in the previous section. However, the combinatorics becomes more complicated and in most cases it is difficult to write a single formula that works for all wheel graphs. Let us start with an example of \(H_3(D_n(W_5))\). The possible types of product cycles are \(O\times Y\times Y'\) and \(Y\times Y'\times Y''\). Cycles of the first type arise in \(W_5\) only when graphs

*Y*and \(Y'\) are neighbouring subgraphs from the perimeter. There are four possibilities for such a choice of

*Y*-subgraphs, hence

*Y*-subgraphs or on the connected part of \(W_5\) that is created by removing subgraphs

*Y*and \(Y'\). By arguments analogous to the ones presented in Sect. 5.3.1, the distribution of free particles on the connected component containing cycle

*O*does not play a role. Hence, the contribution to \(\beta _3\) is equal to the number of different distributions of free particles on the edge connecting

*Y*and \(Y'\) and on the connected component. In other words, there are two bins and \(n-5\) free particles. Hence, the total contribution from \(O\times Y\times Y'\)-cycles is \(4(n-4)\). We split the contribution from \(Y\times Y'\times Y''\)-cycles into two groups. The first group consists of cycles only from perimeter (\(Y_p\times Y'_p\times Y''_p\)), for whom the combinatorial description is straightforward. The number of possible choices of

*Y*-subgraphs is \(4\atopwithdelims (){3}\) and it always results with the decomposition of \(W_5\) into 3 components. Hence, with \(n-6\) free particles the number of independent \(Y_p\times Y'_p\times Y''_p\)-cycles is \(4{{n-4}\atopwithdelims (){2}}\). In order to determine the number of independent cycles \(Y_p\times Y'_p\times Y_h\) (two subgraphs from the perimeter and one from the hub), one has to consider different graphs that arise after removing two

*Y*-subgraphs from the perimeter of \(W_5\). The number of independent \(Y_h\)-cycles for a fixed choice of \(Y_p\) and \(Y_p'\) is the same as in a certain fan graph which is determined by the choice of the \(Y_p\)-subgraphs. Choosing \(Y_p\) and \(Y_p'\) to lie on the opposite sides of the diagonal of \(W_5\), the resulting fan graph is the star graph \(S_4\). The free particles outside \(Y_p\) and \(Y_p'\) can always be moved to the \(S_4\)-subgraph. Hence, the contribution from such cycles is given by the number of independent

*Y*-cycles in \(S_4\) for \(n-4\) particles. We denote this number by \(\beta _1^{(n-4)}(S_4)\). The last group of cycles that we have to take into account are \(Y_p\times Y'_p\times Y_h\), where \(Y_p\) and \(Y_p'\) are neighbouring subgraphs. The resulting fan graph is shown on Fig. 17. The \(n-4\) particles that do not exchange on the perimeter subgraphs are distributed between the fan graph and the edge joining \(Y_p\) and \(Y_p'\). There have to be at least 2 particles exchanging on a \(Y_h\)-subgraph of the fan graph. The number of independent \(Y_h\)-cycles for \(k+2\) particles on the fan graph is given in the caption under Fig. 17. After summing all the above contributions, the final formula for the third Betti number reads

*Y*-subgraphs from the perimeter. This always results with the decomposition of \(W_5\) into 5 components. Choosing three

*Y*-subgraphs from perimeter results with the decomposition of \(W_5\) into 3 components: a fan graph and 2 edges. The number of independent \(Y_h\) cycles in the fan graph is the same as in \(S_4\). Taking into account the distribution of \(n-6\) particles between the two edges and the fan graph, we have

*Y*-cycles and one

*O*-cycle. The graph also cannot be too small, i.e. the condition \(m-3\ge d-1\) must be satisfied. Otherwise, there is no cycle that is disjoint with \(d-1\)

*Y*-subgraphs. Hence,

*Y*-subgraphs from the perimeter and what fan graphs are created. We are interested in the number of leaves (\(\mu \)) of the resulting fan graph. The number of cycles in such a fan graph with \(\mu \) leaves is \(m-1-\mu \). It is a difficult task to list all possible fan graphs for any \(W_m\) in a single formula. The results for graphs up to \(W_7\) are shown in Table 4. Using the notation from Table 4, the general formula for \(\beta _d\) reads

The possibilities of choosing a number of *Y*-subgraphs from the perimeter of a wheel graph

\(\varGamma \) | Groups of | Number of possible choices—\(N_\mathbf{n}\) | Number of leaves—\(\mu _\mathbf{n}\) |
---|---|---|---|

\(W_5\) | (1) | 4 | 1 |

(1,1) | 2 | 4 | |

(2) | 4 | 3 | |

(3) | 4 | 4 | |

(4) | 1 | 4 | |

\(W_6\) | (1) | 5 | 2 |

(1,1) | 5 | 4 | |

(2) | 5 | 3 | |

(2,1) | 5 | 5 | |

(3) | 5 | 4 | |

(4) | 5 | 5 | |

(5) | 1 | 5 | |

\(W_7\) | (1) | 6 | 2 |

(1,1) | 9 | 4 | |

(2) | 6 | 3 | |

(1,1,1) | 2 | 6 | |

(2,1) | 12 | 5 | |

(3) | 6 | 4 | |

(2,2) | 3 | 6 | |

(3,1) | 6 | 6 | |

(4) | 6 | 5 | |

(5) | 6 | 6 | |

(6) | 1 | 6 |

*Y*-subgraphs. Group \(n_i\) gives \(n_i-1\) edges. Hence, groups \((n_1,\dots ,n_l)\) give \(|\mathbf{n}|-l\) edges. The final formula reads

*Y*-subgraphs lie on the perimeter—there are \(n-2d\) free particles. The last sum describes the number of independent \(Y_h\times Y_p\times \dots \times Y_p\)-cycles. Here we used the formula for the number of \(Y_h\)-cycles for

*n*particles on a fan graph with \(\mu \) leaves and \(m-1\) spokes [8]

### 5.4 Wheel graphs via Świątkowski discrete model

In this section we show that the homology of configuration spaces of wheel graphs is generated by product cycles. The strategy is to consider two consecutive vertex cuts that bring any wheel graph to the form of a linear tree.

*Y*-subgraph \(Y_i\). Hence, \(H_d(S(T_{m}))\) is freely generated by generators of the form

### Lemma 1

*Y*-cycles and the distributions of free particles on the connected components \(N_m-(v_h(Y_1)\cup \dots \cup v_h(Y_d))\) which we denote by

### Proof

*Y*-subgraphs of \(T_m\) are linearly independent. Hence, any vector from \(\mathrm{im}\delta _{n,d}\) can be uniquely decomposed in this basis and its preimage can be unambiguously determined by subtracting the particles from \(n_1\) and \(n_{d+1}\). By injectivity of \(\delta \),

*d*-subsets of

*Y*-subgraphs of \(T_m\). The result is

*d*connected components of \(N_m-(v_h(Y_1)\cup \dots \cup v_h(Y_d))\). \(\square \)

*d*. The number of the remaining connected components is always equal to

*d*, but their type depends on the distribution of subgraphs \(Y_1,\dots ,Y_d\) in \(N_m\). The situations that are relevant for the description of \(\ker \delta \) are those, where a particle is added by map \(\delta \) to two connected components which contain an edge which before the blow-up was adjacent to the hub of \(W_{m+1}\). There are at most 2

*d*such components, as removing the hub-vertices of two neighbouring

*Y*-subgraphs of \(N_m\) yields a connected component of the edge type which is not adjacent to the hub of \(W_{m+1}\). We label these components by numbers \(1,\dots ,l\) (we always have \(d\le l\le 2d\)) and the occupation numbers of these components are \(n_1,\dots ,n_l\). We choose component 1 to be the component adjacent to edge \(e(h_0)\) and increase the labels in the clockwise direction from the component with label 1. The remaining components are labelled by numbers \(l+1,\dots ,2d\). Map \(\delta \) acts on basis elements of \(\bigoplus _{h\in H(v)-\{h_0\}}H_d\left( S_{n-1}(N_m)\right) \) as follows.

*O*is the cycle in \(W_{m+1}\) which contains edges \(e(h_0)\), \(e(h_i)\) and the hub of \(W_{m+1}\). Similarly, element

*Y*-cycle centred at the hub of \(W_{m+1}\). Such kernel elements correspond to cycles \(c_{Y_1}\dots c_{Y_d}c_{Y_h}\) in \(S_n(W_{m+1})\), where \(Y_h\) is a

*Y*-cycle, whose hub-vertex is the hub-vertex of \(W_{m+1}\). The precise form of such kernel elements is the following.

*O*-cycles and

*Y*-cycles) in a configuration space of the disconnected graph \(W_{m+1}-(v_h(Y_1)\cup \dots \cup v_h(Y_d))\). More specifically, the disconnected graph \(W_{m+1}-(v_h(Y_1)\cup \dots \cup v_h(Y_d))\)

^{4}is a disjoint sum of a number of edges and of one fan graph. We regard the 1-cycles (

*O*-cycles or

*Y*-cycles) at the hub as generators of the first homology group of the configuration space of the fan graph multiplied by different distributions of particles on the disjoint edge-components of \(W_{m+1}-(v_h(Y_1)\cup \dots \cup v_h(Y_d))\). Fan graphs are planar, hence by equation (13) there is no torsion in \(\ker \delta _{n,d}\). Hence, \(H_d(S_n(W_{m+1}))\) is torsion-free and short exact sequence for \(H_d(S_n(W_{m+1}))\) gives in this case

### 5.5 Graph \(K_{3,3}\)

Graph \(K_{3,3}\) has the property that all its vertices are of degree three. High homology groups of graphs with such a property have been studied in [23]. In particular, we have the following result.

### Theorem 8

*N*the number of vertices of graph \(\varGamma \) and label the vertices by labels \(1,\dots , N\). Moreover, denote by \(\mathcal {Y}=\{Y_1,\dots Y_N\}\) the set of

*Y*-subgraphs of \(\varGamma \) such that the hub of \(Y_k\) is vertex

*k*. Group \(H_N(S_n(\varGamma ))\) is freely generated by product cycles

As we show in Sect. 5.7, the second homology group of configuration spaces of such graphs is also generated by product cycles. Later in this section, by comparing the ranks of homology groups computed via the discrete Morse theory, we argue that \(H_4(C_n(K_{3,3}))\) is also generated by product cycles. Interestingly, in \(H_3(C_n(K_{3,3}))\) there is a new non-product generator. Using this knowledge, we explain the relations between the product and non-product cycles that give the correct rank of \(H_3(C_n(K_{3,3}))\).

*Second homology group*There are no pairs of disjoint cycles in \(K_{3,3}\), hence the product part for \(n=2\) is empty. When \(n=3\), there are 12 \(O\times Y\)-cycles. This can be seen by choosing the

*Y*-graph centered at vertex 1 on Fig. 20b)—there are 2 cycles disjoint with such a

*Y*-subgraph. There are 6

*Y*-subgraphs in \(K_{3,3}\), hence we get the number of \(O\times Y\)-cycles. One checks by a straightforward calculation that 8 of them are independent. Hence,

*Y*-cycle. This happens only when we have a situation as on Fig. 11. Therefore, cycles of the \(Y\times Y\)-type, where the hubs of the

*Y*-subgraphs, are connected by an edge, are all independent (Fig. 21a)). The number of such cycles is 9. The relations occur between \(Y\times Y\)-cycles, where the hubs of the subgraphs are not connected by an edge (Fig. 21b)). There are 6 such cycles. The number of relations is 4. To see this, consider

*Y*-subgraph, whose hub is vertex 1 (Fig. 20). Denote this subgraph by \(Y_1\). It is straightforward to see that in graph \(K_{3,3}-Y_1\) we have \(c_{Y_3}\sim c_{Y_6}\). Hence,

*Y*-subgrphs that lie on the same side of the \(K_{3,3}\) graph as \(Y_1\) (see Fig. 20a)) read

*Y*-subgraphs are considered, all distributions of free particles are equivalent (Fig. 21b)). When the subgraphs are adjacent, there are two different parts of \(K_{3,3}\), where the particles can be distributed, see Fig. 21a). This gives the formula

*Higher homology groups*Let us first look at the third homology group. The are no product cycles for \(n=4\) however, from the Morse theory for the subdivided graph from Fig. 22 we have

*Y*-subgraphs are adjacent. For every pair of adjacent

*Y*subgraphs there is an unique

*O*-cycle. An example of such a cycle is

*Y*-subgraphs which is 9. Adding the properly embedded generator of \(H_3(D_4(K_{3,3}))\), we get

*Y*-subgraphs. The first way is to remove two

*Y*-graphs from the same side and one from the opposite side. This results with the partition of \(K_{3,3}\) into three components (Fig. 23a)). Removing three

*Y*-graphs from the same side splits \(K_{3,3}\) into three parts (Fig. 23b)). Therefore,

*Y*-subraphs. There are no \(Y\times Y\times \dots \times Y\times O\)-cycles in \(H_p(D_n(K_{3,3}))\) for \(p\ge 4\). As direct computations using discrete Morse theory show, there are also no non-product generators (see Table 1). Therefore, only \(Y\times Y\times \dots \times Y\)-cycles contribute to \(H_p(D_n(K_{3,3}))\) for \(p\ge 4\). Removing four

*Y*-graphs from \(K_{3,3}\) always results with the splitting into 5 parts, removing five

*Y*-graphs gives 7 parts and removing all six

*Y*-graphs gives 9 parts. Summing up,

### 5.6 Triple tori in \(C_n(K_{2,p})\)

*n*-particle configuration space are not product. This is the family of complete bipartite graphs \(K_{2,p}\) (see Fig. 24a). The first interesting graph from this family is \(K_{2,4}\). As we show below, its 3-particle configuration space gives rise to a 2-cycle which is a triple torus. It turns out that such triple tori together with products of

*Y*cycles generate the homology groups of \(C_n(K_{2,p})\). The most convenient discrete model for studying \(C_n(K_{2,p})\) is the Świątkowski model. In fact, we study the Świątkowski configuration space of graph \(\varTheta _p\) (see 24b) which is topologically equivalent to \(K_{2,p}\), but it has the advantage that its discrete configuration space is of the optimal dimension. Because there are no 3-cells in \(S_n(\varTheta _p)\), hence automatically we get that

### Lemma 2

The first homology group of \(C_n(K_{2,p})\) is equal to \({\mathbb {Z}}^{p(p-1)}\) for \(n\ge 2\) and \(p-1\) for \(n=1\).

By counting the number of 0-, 1- and 2-cells in \(S_n(K_{2,p})\), we compute the Euler characteristic (see also [61]).

### Lemma 3

### Example 6

**Generators of**\(H_2(S_n(\varTheta _3))\). Group \(H_2(S_n(\varTheta _3))\) is generated by products of

*Y*-cycles at vertices

*v*and \(v'\). More precisely, consider the following two

*Y*-cycles

*Y*-cycles as

*Y*-cycle of the

*Y*-subgraph, whose hub vertex is

*v*and which is spanned on edges \(e_i,e_j,e_k\). Cycle \(c'_{ijk}\) corresponds to an analogous

*Y*-subgraph, whose hub is \(v'\).

### Example 7

**The generator of**\(H_2(S_3(\varTheta _4))\). Formula (18) tells us that \(\beta _2(C_3(K_{2,4}))=1\). The corresponding generator in \(S_3(\varTheta _4)\) has the following form.

*Y*-cycles, one can see that the above chain is a combination of all 2-cells of \(S_3(\varTheta _4)\), hence, \(C_n(K_{2,4})\) has the homotopy type of a closed 2-dimensional surface. Its Euler characteristic is equal to \(-4\), hence this is a surface of genus 3. By the classification theorem of surfaces [62], we identify \(C_n(K_{2,4})\) to have the homotopy type of a triple torus (fig. 25).

### Proposition 9

*Y*-cycles and new relations. First of all, by proposition 10 different distributions of additional particles in the \(\varTheta \)-cycle can be realised are combinations of different products of

*Y*-cycles.

### Proposition 10

Hence, all \(\varTheta \)-cycles generate a subgroup of \(H_2(S_n(\varTheta _p))\) which is isomorphic to \({\mathbb {Z}}^{{p-1}\atopwithdelims (){3}}\). The last type of relations we have to account for^{5} are the new relations between products of *Y*-cycles.

### Proposition 11

*Y*-cycles satisfy

### 5.7 When is \(H_2(C_n(\varGamma ))\) generated only by product cycles?

In this section we prove the following theorem.

### Theorem 12

Let \(\varGamma \) be a simple graph, for which \(|\{v\in V(\varGamma ):\ d(v)>3\}|=1\). Then group \(H_2(C_n(\varGamma ))\) is generated by product cycles.

In the proof we use the Świątkowski discrete model. The strategy of the proof is to first consider the blowup of the vertex of degree greater than 3 and prove theorem 12 for graphs, whose all vertices have degree at most 3. For such a graph, we choose a spanning tree \(T\subset \varGamma \). Next, we subdivide once each edge from \(E(\varGamma )-E(T)\). We prove the theorem inductively by showing in lemma 4 that the blowup at an extra vertex of degree 2 does not create any non-product generators. The base case of induction is obtained by doing the blowup at every vertex of degree 2 in \(\varGamma -T\). This way, we obtain graph which is isomorphic to tree *T* and we use the fact that for tree graphs the homology groups of \(S_n(T)\) are generated by products of *Y*-cycles.

### Lemma 4

Let \(\varGamma \) be a simple graph, whose all vertices have degree at most 3. Let *T* be a spanning tree of \(\varGamma \). Let \(v\in V(\varGamma )\) be a vertex of degree 2 and \(\varGamma _v\) the graph obtained from \(\varGamma \) by the vertex blowup at *v*. If \(H_2(S_n(\varGamma _v))\) is generated by product cycles, then \(H_2(S_n(\varGamma ))\) is also generated by product cycles.

### Proof

*Y*-cycles and

*O*-cycles, subject to the \(\varTheta \)-relations (equations (12) and (11)) and the distribution of free particles which say that \([ce]=[cv]\) whenever

*v*is a vertex of

*e*. Recall that cycle

*c*represents element of \(\ker \left( \delta _{n,1}\right) \) whenever \([ce]=[ce']\), where

*e*and \(e'\) are the edges incident to vertex

*v*. This happens if and only if cycles

*ce*and \(ce'\) are related by a \(\varTheta \)-relation or a particle-distribution relation. However, because all vertices of \(\varGamma \) have degree at most 3, it is not possible to write the \(\varTheta \) relations in the form \(ce-ce'=\partial (b)\) for any

*c*. Hence, cycles

*ce*and \(ce'\) must be related by the particle distribution, i.e. there exists a path in \(\varGamma _v\) which is disjoint with \({\mathrm{Supp}}(c)\) and which joins edges

*e*and \(e'\). The desired homomorphism

*f*is constructed as follows. For a generator

*c*of \(H_{1}\left( S_{n-1}(\varGamma _v)\right) \), find path

*p*(

*c*) which joins

*e*and \(e'\) and is disjoint with \({\mathrm{Supp}}(c)\). Having found such a path, we complete it to a cycle \(O_{p(c)}\) in a unique way by adding to

*p*vertex

*v*and edges \(e,e'\). From cycle \(O_{p(c)}\) we form the

*O*-cycle \(c_{O_{p(c)}}\) (see definition 3). Homomorphism

*f*is established after choosing the set of independent generating cycles and paths that are disjoint with them. It acts as \(f:\ [c]\mapsto [c\otimes c_{O_{p(c)}}]\). Clearly, we have \(\varPsi _{n,2}([c\otimes c_{O_{p(c)}}])=[c]\) by extracting from \(c_{O_{p(c)}}\) the part which contains half-edges incident to

*v*.

This way, we obtained that \(H_2\left( {\tilde{S}}^v_n(\varGamma )\right) \cong \ker \left( \delta _{n,1}\right) \oplus \mathrm{coker}\left( \delta _{n,2}\right) \) and that elements of \(\ker \left( \delta _{n,1}\right) \) are represented by product \(c_O\otimes c_Y\) cycles. By the inductive hypopaper, elements of \(\mathrm{coker}\left( \delta _{n,2}\right) \) are the product cycles that generate \(H_2\left( S_{n-1}(\varGamma _v)\right) \) subject to relations \(ce\sim ce'\). \(\square \)

The last step needed for the proof of theorem 12 is showing that the blowup of \(\varGamma \) at the unique vertex of degree greater than 3 does not create any non-product cycles. Here we only sketch the proof of this fact which is analogous to the proof of lemma 4. Namely, using the knowledge of relations between the generators of \(H_{1}\left( S_{n-1}(\varGamma _v)\right) \), one can show that the elements of \(\ker \left( \delta _{n,1}\right) \) are of two types: i) the ones that are of the form \(\partial (c\otimes b_{p(c)})\), where \([c]\in H_{1}\left( S_{n-1}(\varGamma _v)\right) \) and \(b_{p(c)}\) is the 1-cycle corresponding to path \(p(c)\subset \varGamma _v\) which is disjoint with \({\mathrm{Supp}}(c)\) and whose boundary are edges incident to *v*, ii) pairs of cycles of the form \((c(e_j-e_0),c(e_0-e_j))\), where \(e_0, e_i, e_j\) are edges incident to *v* and \([c]\in H_{1}\left( S_{n-2}(\varGamma _v)\right) \). Such pairs are mapped by \(\delta _{n,1}\) to \(c\otimes \left( (e_j-e_0)(e_0-e_i)+(e_0-e_i)(e_0-e_j)\right) \) which is equal to \(\partial (c\otimes c_{0ij})\), where \(c_{0ij}\) is the *Y*-cycle corresponding to the *Y*-graph in \(\varGamma \) centred at *v* and spanned by edges \(e_0,e_i,e_j\). Next, in order to show splitting of the homological short exact sequences, we consider a homomorphism \(f:\ \ker \left( \delta _{n,1}\right) \rightarrow H_2\left( {\tilde{S}}^v_n(\varGamma )\right) \), for which \(\varPsi _{n,2}\circ f=id_{\ker (\delta _{n,1})}\). Such a homomorphism maps [*c*] to \([c\otimes c_{O_{p(c)}}]\), where \(O_{p(c)}\) is the cycle which contains path *p*(*c*) and vertex *v*. Pairs \(([c(e_j-e_0)],[c(e_0-e_j)])\) are mapped by *f* to cycles \(c\otimes c_{0ij}\). We obtain that \(H_2\left( {{\tilde{S}}}^v_n(\varGamma )\right) \cong \ker \left( \delta _{n,1}\right) \oplus \mathrm{coker}\left( \delta _{n,2}\right) \), where the generators of \(\ker \left( \delta _{n,1}\right) \) are in a one-to-one correspondence with the product homology classes of \(H_2\left( {{\tilde{S}}}^v_n(\varGamma )\right) \) described above. Elements of \(\mathrm{coker}\left( \delta _{n,2}\right) \) are also represented by product cycles. These cycles are the generators of \(H_2\left( S_{n}(\varGamma _v)\right) \) subject to relations \(ce_0\sim ce_i\), \(i=1,\dots ,d(v)\), where \(e_0,e_1,\dots ,e_{d(v)}\) are edges incident to *v*.

The task of characterising all graphs, for which \(H_2(S_n(\varGamma ))\) is generated by product cycles requires taking into account the existence of non-product generators from Sect. 5.6. As we show in Sect. 5.6 the existence of pairs of vertices of degree greater than 3 in the graph implies that there may appear some multiple tori in the generating set of \(H_2(C_n(\varGamma ))\) stemming from subgraphs isomorphic to graph \(K_{2,4}\). Furthermore, the class of graphs, for which higher homologies of \(C_n(\varGamma )\) are generated by product cycles is even smaller. Recall graph \(K_{3,3}\) whose all vertices have degree 3, but \(H_3(C_n(K_{3,3}))\) has one generator which is not a product of 1-cycles (see Sect. 5.5).

## 6 Summary

*X*are classified by conjugacy classes of unitary representations of the fundamental group of the configuration space \(C_n(X)\). Conversely, every unitary representation of the graph braid group gives rise to a flat complex vector bundle over space \(C_n(X)\). We interpret different isomorphism classes of flat complex vector bundles over \(C_n(X)\) as fundamentally different families of particles. Among these families we find for example bosons, corresponding to the trivial flat bundle, and fermions that may correspond to a non-trivial flat bundle. Interestingly, there also exist intermediate possibilities called anyons who can live on a trivial as well as on a non-trivial bundle. The existence of more than two isomorphism classes is

*a priori*possible. However interesting and desirable, an explicit construction of non trivial flat bundles for configuration spaces of \(X={\mathbb {R}}^2\) or \(X={\mathbb {R}}^3\) is difficult, hence some simplified mathematical models are needed. This motivates the study of configuration spaces of particles on graphs which are computationally more tractable. Topological invariants that give a coarse grained picture of the structure of the set of isomorphism classes of flat complex vector bundles over \(C_n(X)\) are the homology groups of configuration spaces. In particular, we point out the important role of Chern characteristic classes that map the flat vector bundles to torsion components of the homology groups of \(C_n(X)\) with coefficients in \({\mathbb {Z}}\). In the second part of this paper, we compute homology groups of configuration spaces of certain families of graphs. We summarise the computational results as follows.

Configuration spaces of tree graphs, wheel graphs and complete bipartite graphs \(K_{2,p}\) have no torsion in their homology. This means that the set of flat bundles over configuration spaces of such graphs has a simplified structure, namely every flat vector bundle is stably equivalent to a trivial vector bundle. Hence, these families of graphs are good first candidates for a class of simplified models for studying the properties of non-abelian statistics.

Computation of the homology groups of configuration spaces of some small canonical graphs via the discrete Morse theory shows that in some cases there is a \({\mathbb {Z}}_2\)-torsion in the homology. However, we were not able to provide an example of a graph which has a torsion component different than \({\mathbb {Z}}_2\) in the homology of its configuration space.

It is a difficult task to accomplish a full description of the homology groups of graph configuration spaces using methods presented in this work. One fundamental obstacle is that such a task requires the knowledge of possible embeddings of

*d*-dimensional surfaces in \(C_n(\varGamma )\) which generate the homology. However, cycles generating the homology in dimension 2 of graph configuration spaces have the homotopy type of tori or multiple tori. This fact allowed us to find all generators of the second homology group of configuration spaces of a large family of graphs in Sect. 5.7.

**Trivial bundles**. They are relevant in the context of quantum computing where one is interested mainly in universality of unitary representations of braid groups and the dimension of the representations grow exponentially with the number of particles. In that context, the fact that one can have different isomorphism classes of vector bundles does not seem to play a significant role. In fact, it is even better not to have many isomorphism classes. If we know that there is just one isomorphism class (all bundles are isomorphic to the trivial bundle) then all representations of the braid group are related to each other via the isomorphism of the corresponding bundles and the problem of classifying them should become more tractable. As we show in this paper, this happens when one considers high-dimensional representations (stable range) of graph braid groups where there is no torsion in the homology of \(C_n(\varGamma )\).**Non-trivial bundles**. They become relevant in situations where the rank of the bundle is not too high, i.e. if the considered bundles are not in the stable range. The corresponding representations of braid groups appear in the effective models of non-abelian Chern-Simons particles which are point-like sources mutually interacting via a topological non-Abelian Aharonov–Bohm effect. A model of such particles constrained to move on graphs would be constructed by defining a separate Chern–Simons hamiltonian for each cell of the closure of \(C_n(\varGamma )\) viewed as a subset of \(\varGamma ^{\times n}\). The non-abelian braiding would show up as proper gluing conditions for the wave-functions on the boundaries of cells from \(C_n(\varGamma )\) while studying self-adjoint extensions of such a hamiltonian. The moduli space of flat*U*(*n*) bundles over \(C_n(\varGamma )\) is the space of possible parameters that appear as the gluing conditions (see e.g. [19]). This area is still quite unexplored and some more progress has to be made to see how this theory works explicitly for concrete graphs.

*a priori*large number of possible unitary representations of braid groups can be cut down by taking into account the anyonic fusion rules [63], i.e. by applying the framework of modular tensor categories. Unitary representations of the braid group that are described by modular tensor categories, are defined by specifying the fusion rules, the so-called

*F*-matrix that ensures associativity of fusion and the

*R*-matrix that describes braiding of pairs of particles [63]. Ocneanu rigidity [64] asserts that for anyons on the plane, there is only a finite number of representations of the braid group that arise from the above construction. However, graphs in principle provide more freedom, as pairs of particles may braid differently in different parts of the graph.There are two main differences between braiding particles on the plane and braiding particles on a graph: i) the generating braids are more complicated than just the ones that correspond to braiding of pairs of neighbouring particles, ii) the variety of relations is much bigger than in the case of particles on the plane. In other words, one can have more than just one

*R*-matrix for particles on a graph. This property of graph braid groups can be seen already on the level of its abelian representations [8]. Namely, consider a 2-connected graph which consists of a number of 3-connected components that are connected to each other (Fig. 26). The theory of abelian representations of graph braid groups [8] tells us that a pair of particles can exchange as bosons or fermions, depending in which component of the graph the exchange takes place. An analogous situation will take place in the case of non-abelian representations—one can assign different

*R*-matrices to different components of the graph.

One expects that the fusion rules will nevertheless significantly restrict the number of admissible representations of graph braid groups.

## Footnotes

- 1.
The action of

*G*on*P*can be left or right. In this work we pick up the convention of right action. This means that \(g(h(p))=(gh)(p)\) for \(g,h\in G\), \(p\in P\). Group action is free iff for all \(g\in G\) and \(p\in P\), \(gp\ne p\). - 2.
Universal covers of graph configuration spaces have a particularly nice structure, as they have the homotopy type of a

*CAT*(0) cube complex [21] which is contractible. - 3.
The \(\varTheta \) graph consists of two vertices which are connected by three edges. It can be also viewed as complete bipartite graph \(K_{2,3}\).

- 4.
\(W_{m+1}-(v_h(Y_1)\cup \dots \cup v_h(Y_d))\) is a disconnected topological space. We give this space the structure of a graph by adding a vertex to the open end of each open edge.

- 5.
We do not mention here the typical relations between different

*Y*-cycles on*Y*-subgraphs of the \(S_p\) graphs which are met while computing the first homology group of the configuration spaces of star graphs (see [8]). Such relations are also inherited by the products of*Y*-cycles.

## Notes

### Acknowledgements

TM acknowledges the financial support of the National Science Centre of Poland – grants *Etiuda* no. 2017 / 24 / *T* / *ST*1 / 00489 and *Preludium* no. 2016 / 23 / *N* / *ST*1 / 03209. AS was supported by the National Science Centre of Poland grant *Sonata Bis* no. 2015 / 18 / *E* / *ST*1 / 00200.

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