aa r X i v : . [ m a t h . A T ] J un COMMUTATIVE d -TORSION K -THEORY AND ITS APPLICATIONS C˙IHAN OKAY
Abstract.
Commutative d -torsion K -theory is a variant of topological K -theory con-structed from commuting unitary matrices whose order divides d . Such matrices appearas solutions of linear constraint systems that play a role in quantum contextuality and non-local games. Using methods from stable homotopy theory we modify commutative d -torsion K -theory into a cohomology theory which can be used for studying operator solutions oflinear constraint systems. This provides an interesting connection between stable homotopytheory and operator theoretic problems motivated by quantum information theory. Introduction
Commuting unitary matrices can be assembled into a generalized cohomology theory calledcommutative K -theory, a variant of topological K -theory first introduced in [AGLT17]. Thistheory can be further modified by restricting to matrices whose order divides d resulting ina cohomology theory which will be referred to as commutative d -torsion K -theory . Suchmatrices also play a significant role in quantum theory, especially in foundational areasconcerning quantum contextuality [KS67, Bel66] and linear constraint systems in the studyof non-local games [CM14]. The goal of this paper is to make this connection precise. Weintroduce a generalized cohomology theory obtained from commutative d -torsion K -theory,which is tailored for studying operator solutions of linear constraint systems. We expectthat stable homotopical methods introduced in this paper will provide further insight intooperator theoretic problems motivated by quantum information theory.The cohomology theories studied in this paper are based on a classifying space constructionintroduced in [ACTG12]. We write B ( Z /d, G ) to denote the classifying space of a topologicalgroup G constructed from tuples of pairwise commuting group elements where each groupelement has order dividing d , i.e., pairwise commuting d -torsion group elements. When G isthe unitary group U ( m ) this classifying space is constructed from tuples ( A , A , · · · , A n ) ofmatrices satisfying A i A j = A j A i and ( A i ) d = I m . Such matrices also appear as solutions to a linear constraint system specified by an equa-tion
M x = b where M is an r × c matrix over the additive group Z /d of integers modulo d . An operator solution consists of d -torsion m × m unitary matrices A , A , · · · , A c thatsatisfy A M k A M k · · · A M kc c = e πib k /d I m for all 1 ≤ k ≤ r, and A i A j = A j A i whenever M ki and M kj are both non-zero. The data of a linear constraintsystem can be packaged as a pair ( H , τ ), where H is a hypergraph with a vertex set V = Date : June 16, 2020. v , v , · · · , v c } and an edge set E = { e , e , · · · , e r } ; and τ is a function E → Z /d definedby τ ( e k ) = b k . An operator solution can be regarded as a function T : V → U ( m ). Thehomotopical approach initiated in [ORBR17, OR20] associates a 2-dimensional CW complex X , called a topological realization, to the hypergraph H and the function τ represents a2-dimensional cohomology class on this space. In this paper we refine this approach byinterpreting an operator solution as a map of topological spaces. For this a quotient space¯ B ( Z /d, G ) of the classifying space B ( Z /d, G ) is introduced. An operator solution over G = U ( m ) can be turned into a map, defined up to homotopy, f T : X → ¯ B ( Z /d, G ) . Although, in this paper our motivation comes from an urge to understand operator solutionsof linear constraint systems the classifying spaces B ( Z /d, G ) and its variants are of interest toalgebraic topologists as well; see for instance [Oka18, AG15, CS16, ACGV17, OW17, RV18,RS18, OZ20].A generalized cohomology theory is represented by a spectrum. Following [GH19] we showthat B ( Z /d, U ), where U is the stable unitary group, is an infinite loop space and thus spec-ifies a spectrum. This spectrum turns out to be stably equivalent to ku ∧ Bµ d (Proposition2.5), where µ d = { e πik/d | ≤ k ≤ d } and ku is the connective complex K -theory spectrum.Commutative d -torsion K -theory is the generalized cohomology theory associated to thisspectrum. Both the spectrum and the associated cohomology theory will be denoted by kµ d .For applications to linear constraint systems we introduce a stabilized version of the quotientspace ¯ B ( Z /d, U ( m )). The usual stabilization process cannot be carried out in a straight-forward manner. However, by working in the homotopy category of spectra we introduce aspectrum C ( d, m ) such that the associated infinite loop space ¯ B ( d, m ) admits a map¯ ι m : ¯ B ( Z /d, U ( m )) → ¯ B ( d, m ) . This space comes with a canonical cohomology class γ S m in H ( ¯ B ( d, m ) , Z /d ). By construc-tion homotopy groups of C ( d, m ) are concentrated in dimensions ≤ → π C ( d, m ) → Z /d × m −−→ Z /d → π C ( d, m ) → . The kernel consists of the subgroup ( Z /d ) m of m -torsion elements. Using the Atiyah–Hirzebruch spectral sequence we describe C ( d, m )-cohomology of a space. Theorem 3.6.
There is a commutative diagram H ( X, π C ( d, m )) kµ d ( X ) C ( d, m )( X ) H ( X, Z /d ) H ( X, Z /d ) H ( X, π C ( d, m )) cl here cl ( f ) = f ∗ ( γ S m ) and the middle row is an exact sequence. In particular, we have acanonical splitting C ( d, m )( X ) ∼ = H ( X, π C ( d, m )) ⊕ H ( X, π C ( d, m )) . Going back to linear constraint systems we show that the C ( d, m )-cohomology informs usabout the properties of operator solutions over U ( m ). To an operator solution we associatethe class [ f ] of the composite map f : X f T −→ ¯ B ( Z /d, U ( m )) ¯ ι m −→ ¯ B ( d, m )in the C ( d, m )-cohomology of X . It turns out that cl( f ) = 0 if and only if the linearconstraint system has a solution over U (1), i.e., a scalar solution. Corollary 4.9.
Let ( H , τ ) be a linear constraint system and X be a topological realization.(1) If H ( X, ( Z /d ) m ) = 0 then ( H , τ ) has a scalar solution.(2) If d and m are coprime then C ( d, m )( X ) = 0 and ( H , τ ) has a scalar solution.(3) If π ( X ) is trivial and [ τ ] = 0 then ( H , τ ) does not have an operator solution. The most famous example of a linear constraint system, which does not admit a scalarsolution, is the Mermin square construction [Mer93]. This linear constraint system, definedover Z /
2, admits an operator solution in U (2 n ) for n ≥
2. A topological realization ofthe Mermin square linear constraint system can be chosen to be a torus S × S with acertain cell structure. Then an operator solution specifies a class in the C (2 , n )-cohomologygroup M n ∈ C (2 , n )( S × S ) . We refer to this class as the Mermin class. Alternatively, the Mermin class can be identifiedwith the generator of π C (2 , n ) = Z /
2. There is also a real version of these constructionswhich works for the orthogonal group O ( m ). In this case certain generalized cohomologyclasses can be realized as symmetry-protected topological phases (Remark 4.10).The paper is organized as follows. In § B ( Z /d, G ) and thetype of principal bundles classified by this space. Γ-spaces are used to describe the spectrum kµ d and Proposition 2.5 informs us about its stable homotopy type. Low dimensional homo-topy groups are described in § ko sym is studied in § B ( Z /d, G ) and the spectrum C ( d, m ) are introduced in §
3. We prove Theorem 3.6which describes the C ( d, m )-cohomology of a space in this section. Applications of C ( d, m )-cohomology are discussed in §
4. In this section we introduce linear constraint systems and atopological interpretation of operator solutions. Proposition 4.7 provides a computation ofpointed homotopy classes of maps X → ¯ B ( Z /d, G ) when X is a 2-dimensional CW complex.Applications to linear constraint systems are given in Corollary 4.9. The Mermin class isconstructed in this section. Acknowledgement:
The author would like to thank Simon Gritschacher for his commentson an earlier version of this paper; and Daniel Sheinbaum for discussions on symmetry-protected topological phases. This work is supported by NSERC. . Commutative d -torsion K -theory In this section we introduce a new generalized cohomology theory obtained as a variant ofcommutative K -theory introduced in [ACGV17]. Commutative K -theory has nice propertiessuch as the spectrum ku com representing the theory is stably equivalent to ku ∧ C P ∞ asproved in [Gri], where ku is the connective complex K -theory spectrum. For the d -torsioncase the spectrum representing the cohomology theory is denoted by kµ d . It is constructedfrom commuting unitary matrices whose eigenvalues belong to µ d = { e πk/d | ≤ k ≤ d } .To study this spectrum we follow the Γ-space approach of [GH19]. This description allowsus to prove that kµ d is stably equivalent to ku ∧ Bµ d . There is also a real version ko sym constructed from commuting symmetric orthogonal matrices. We describe low dimensionalhomotopy groups of these spectra.2.1. Classifying spaces.
Let G be a topological group. An element g ∈ G is said to be d -torsion if g d is the identity element 1 G . We are interested in a space constructed frompairwise commuting d -torsion group elements. Definition 2.1.
We define B ( Z /d, G ) to be the geometric realization of the simplicial space[ n ] Hom(( Z /d ) n , G )where Hom(( Z /d ) n , G ) inherits its topology from G n and the simplicial structure is given by d i ( g , g , · · · , g n ) = ( g , · · · , g n ) i = 0( g , · · · , g i g i +1 , · · · , g n ) 0 < i < n ( g , g , · · · , g n − ) i = n, and s j ( g , g , · · · , g n ) = ( g , · · · , g j , G , g j +1 , · · · , g n ) for 0 ≤ j ≤ n .In general, for any cosimplicial group τ • there is a classifying space B ( τ, G ) obtained by asimilar construction, see [OZ20]. When τ is the level-wise free cosimplicial group F • thenthis construction gives the usual classifying space BG . If the level-wise abelianization Z • is used then the resulting space is the classifying space for commutativity B ( Z , G ). Mod- d reduction in each level gives a cosimplicial group ( Z /d ) • and we recover the constructiongiven in Definition 2.1.2.2. TC d -bundles. Let X be a CW complex. The space B ( τ, G ) is a classifying space for theset H τ ( X, G ) of τ -concordance classes of principal G -bundles with τ -structure [OZ20]. When τ • = Z • this notion coincides with transitionally commutative (TC) bundles introduced in[AG15]. For our case of interest, i.e. τ • = ( Z /d ) • , the resulting bundles will be referred to asTC d -bundles. We can carry over this structure to vector bundles. A complex vector bundleof rank n is said to have a τ -structure if the associated principal bundle has a τ -structure.Two complex vector bundles are said to be τ -concordant if the associated principal bundlesare τ -concordant. We write Vect md − tor ( X ) for the set of Z /d -concordance classes of complexvector bundles of rank m over X with a Z /d -structure. More briefly, this set will be referredto as the set of equivalence classes of TC d vector bundles of rank m . .3. Stabilization.
Let C m denote the complex vector space of dimension m with a canoni-cal basis { e , e , · · · , e m } . Inclusion of the canonical basis vectors induces a map C m → C m +1 and the union (colimit) along these inclusions is denoted by C ∞ . Let U ( m ) denote the unitarygroup of m × m matrices. The stable unitary group U is the union along the inclusions U ( m ) → U ( m + 1) , A (cid:18) A
00 1 (cid:19) . (2.1.1)We write B ( Z /d, U ) for the union of B ( Z /d, U ( m )) along the induced stabilization maps.2.4. Γ -spaces. Let
Fin ∗ denote the category whose objects are pointed finite sets k + = { , , · · · , k } ⊔ { + } , k ≥
0, and morphisms are pointed set maps α : k + → l + . Let Top ∗ denote the category of pointed topological spaces. A Γ -space is a functor F : Fin ∗ → Top ∗ .This can be extended to a functor F : Top ∗ → Top ∗ by a coend construction F ( X ) = Z k + F ( k + ) × X k . (2.1.2)There is an assembly map F ( X ) ∧ Y → F ( X ∧ Y ). Associated to the Γ-space there is aspectrum, denoted by F ( S ), consisting of the spaces { F ( S n ) | n ≥ } whose structure mapsare induced by the assembly map.The examples we will encounter are the following. • Let S : Fin ∗ → Top ∗ denote the inclusion functor. This means that we regard k + asa pointed topological space with discrete topology. The associated spectrum is thesphere spectrum and is simply denoted by S . • Let ku denote the Γ-space ku ( k + ) = a d , ··· ,d k ∈ N L ( C d ⊕ · · · ⊕ C d k , C ∞ ) U ( d ) × · · · × U ( d k )where L ( , ) denotes the space of complex linear isometric embeddings between twocomplex inner product spaces. A point in this space is specified by a tuple ( V , · · · , V k )of pairwise orthogonal subspaces. Given α : k + → l + the map ku ( α ) is defined by( V , · · · , V k ) ( ⊕ i ∈ α − (1) V i , · · · , ⊕ i ∈ α − ( l ) V i ) . The spectrum ku ( S ) we obtain is the connective complex K -theory spectrum, whichwill be denoted simply by ku . There is a real version of this construction definedanalogously but using R -vector spaces. The resulting spectrum is the connective real K -theory spectrum ko . • Let M be a commutative discrete monoid. Let M ( k + ) = M k and for α : k + → l + define M ( α ) by sending ( x , · · · , x k ) to ( P j ∈ α − (1) x j , · · · , P j ∈ α − ( l ) x j ). Thenapplying Ω ∞ to the resulting spectrum M ( S ) amounts to group completion M → Ω BM . In particular, we can consider the monoid N and the associated spectrum N ( S ). Since Ω ∞ N ( S ) ≃ Z we see that this spectrum is equivalent to the Eilenberg-Maclane spectrum H Z . There is a map of Γ-spaces dim : ku → N obtained bysending ( V , · · · , V k ) to (dim( V ) , · · · , dim( V k )). .5. The spectrum.
Let µ d ⊂ U (1) denote the subgroup generated by e πi/d . Proposition 2.2.
Sending ( V , · · · , V k ; λ , · · · , λ k ) , where V i are pairwise orthogonal finite-dimensional subspaces of C ∞ and λ ( j ) i ∈ ( µ d ) n , to the n -tuple ( A , · · · , A n ) of pairwise com-muting unitary matrices, where A i acts on V j by multiplication with λ ( j ) i and trivially on thecomplement of V ⊕ · · · ⊕ V k , induces a homeomorphism ku (( µ d ) n ) ∼ = −→ Hom (( Z /d ) n , U ) . Moreover, this homeomorphism is compatible with the simplicial structures and induces ahomeomorphism ku ( Bµ d ) ∼ = −→ B ( Z /d, U ) . Proof.
The statements are proved in [GH19] when λ ( j ) i ∈ U (1) n . These arguments still gothrough when U (1) is replaced by the subgroup µ d . (cid:3) It is instructive to describe the inverse of the first homeomorphism. Let ( A , A , · · · , A n )be a tuple of pairwise commuting matrices in U such that ( A j ) d = I for 1 ≤ j ≤ n . Thesematrices are contained in U ( m ) for some large enough m . We can simultaneously diagonalizethese matrices λ (1)1 I d λ (1)2 I d . . . λ (1) k I d k , · · · , λ ( n )1 I d λ ( n )2 I d . . . λ ( n ) k I d k such that ( λ (1) i , λ (2) i , · · · , λ ( n ) i ) is distinct from ( λ (1) j , λ (2) j , · · · , λ ( n ) j ) whenever i = j . Therefore( A , A , · · · , A n ) amounts to specifying a tuple ( V , V , · · · , V k ) of pairwise orthogonal finitedimensional subspaces V i ⊂ C ∞ , 1 ≤ i ≤ k , together with the eigenvalues λ ( j ) i ∈ µ d . Thenthe inverse map sends ( A , · · · , A n ) to the class of ( V , · · · , V k ; λ , · · · , λ k ) in the coendconstruction 2.1.2.Given a pointed space X and a Γ-space F we write F X for the Γ-space defined by F X ( k + ) = F ( k + ∧ X ). For α : k + → l + the map F X ( α ) is obtained by naturality of the coendconstruction. A Γ-space F is called special if the map F (( k + l ) + ) → F ( k + ) × F ( l + ) inducedby the projections ( k + l ) + → k + and ( k + l ) + → l + is a weak equivalence for all k + , l + . Aspecial Γ-space is called very special if π F (1 + ) is an abelian group. Lemma 2.3.
Let X be a pointed space.(1) If F is special then F X is also special.(2) The natural map F ( S ) ∧ X → F X ( S ) is a stable equivalence.Proof. Part (1) is implicitly mentioned in [BF78] and part (2) is therein proved as Lemma4.1. For a more recent exposition of the equivariant version of this statement see [Sch18]when X has finitely many cells and [GH19] for the general case. (cid:3) efinition 2.4. The spectrum ku Bµ d ( S ) will be called the commutative d -torsion K -theoryspectrum and will be denoted by kµ d . The associated generalized cohomology theory will bereferred to as the commutative d -torsion K -theory . Proposition 2.5.
The spectrum kµ d is stably equivalent to ku ∧ Bµ d and the space Ω ∞ kµ d is weakly equivalent to B ( Z /d, U ) .Proof. We modify the argument in [GH19] given for B ( Z , U ). Applying part (1) of the lemmato F = ku and X = Bµ d , and using the well-known fact that ku is special we obtain that ku Bµ d is special. Moreover, ku Bµ d is very special since ku Bµ d (1 + ) = ku (1 + ∧ Bµ d ) = B ( Z /d, U ) (2.5.1)and thus π ( ku Bµ d (1 + )) = π B ( Z /d, U ) = 0. It is a general fact that if F is very special thenΩ ∞ F ( S ) ≃ F (1 + ) [Seg74]. Therefore Ω ∞ kµ d = Ω ∞ ku Bµ d ( S ) ≃ ku Bµ d (1 + ) ∼ = B ( Z /d, U ).The equivalence kµ d ≃ ku ∧ Bµ d follows from part (2) of Lemma 2.3. (cid:3) Remark 2.6.
There is one important difference between ku (( µ d ) n ) and ku ( U (1) n ) worthpointing out. The former is not an infinite loop space whereas the latter is since U (1) n ispath connected. Note that π ku (( µ d ) n ) can be identified with Rep(( Z /d ) n , U ), the union ofthe quotient spaces Hom(( Z /d ) n , U ( m )) /U ( m ) under the conjugation action of U ( m ).Moreover, Rep(( Z /d ) n , U ) ∼ = N (( µ d ) n ) and the quotient mapHom(( Z /d ) n , U ) → Rep(( Z /d ) n , U )can be described using the map of Γ-spaces dim : ku → N ; see [GH19].For example, when d = 2 we have that N (( µ ) n ) = N ∧ ( µ ) n where N has 0 as its base pointand ( µ ) n is based at the identity element. The set of path components is not an abeliangroup.2.6. Low dimensional homotopy groups.
As a consequence of Proposition 2.5 homotopygroups of kµ d coincides with ku -homology of Bµ d . The groups ku ∗ ( Bµ d ) are computed in[BG03, § π r B ( Z /d, U ) ∼ = π r ( kµ d ) ∼ = π r ( ku ∧ Bµ d ) = r = 0 Z /d r = 10 r = 2 . (2.6.1)There is a commutative diagram Bµ d B ( Z /d, U ) Bµ d det (2.6.2)which splits of the Z /d in π B ( Z /d, U ). The determinant map factors through the geometricrealization of the simplicial set of connected components, denoted by | π Hom(( Z /d ) • , U ) | .Proposition 2.2 implies that the connected components of Hom(( Z /d ) n , U ) can be describedas π ku (( µ d ) n ) = N (( µ d ) n ); see also Remark 2.6. Therefore we have π Hom(( Z /d ) • , U ) = N (( µ d ) • ) nd the natural map B ( Z /d, U ) → | π Hom(( Z /d ) • , U ) | is given by the geometric realizationof ku (( µ d ) • ) → N (( µ d ) • )induced by the Γ-space map dim : ku → N which sends a tuple of pairwise orthogonalsubspaces ( V , V , · · · , V k ) to their dimensions ( d , d , · · · , d k ). Since N is a special Γ-spacewe can apply Lemma 2.3 to obtain an equivalence | N (( µ d ) • ) | ≃ Ω ∞ ( N ( S ) ∧ Bµ d ) . Using the equivalence N ( S ) ≃ H Z we obtain the following. Proposition 2.7.
The determinant map factors as B ( Z /d, U ) | π Hom (( Z /d ) • , U ) | Bµ d det where the homotopy groups of the simplicial set of connected components is given by π r | π Hom (( Z /d ) • , U ) | ∼ = ˜ H r ( Bµ d , Z ) . kµ d -cohomology. The determinant map induces a homomorphismdet ∗ : kµ d ( X ) → H ( X, Z /d ) . In general, since the homotopy groups of kµ d are known we can compute kµ d -cohomology us-ing the Atiyah–Hirzebruch spectral sequence [Ada74]. The E -page of the spectral sequenceis given by H p ( X, π − q kµ d ) ⇒ kµ ∗ d ( X ) . (2.7.1)One special case, for which the computation is easy, is when X is a 2-dimensional CWcomplex. In this case the spectral sequence collapses in the E -page and det ∗ becomes anisomorphism kµ d ( X ) ∼ = H ( X, Z /d ) . (2.7.2)Geometrically kµ d ( X ) can be interpreted in terms of vector bundles. We can collect the setof equivalence classes of TC d vector bundles ( § m for various dimensions. LetVect d − tor ( X ) denote the resulting set. This is a monoid under the direct sum operation ofvector bundles. Then we have kµ d ( X ) ∼ = Gr(Vect d − tor ( X ))where Gr is the Grothendieck group of the monoid. When X is a 2-dimensional CW complexthis Grothendieck group becomes isomorphic (by 2.7.2) to the group of line bundles, undertensor product, whose structure group is µ d ⊂ U (1). .8. Real version.
There is a real version of these constructions obtained by replacing U ( m ) with the orthogonal group O ( m ). Every abelian subgroup of O ( m ) can be conjugatedinto SO(2) j × O (1) m − j [HR14, Appendix A]. Thus a homomorphism f : Z m → O ( m ), whenregarded as a representation, is isomorphic to a direct sum f ∼ = η ⊕ η ⊕ · · · ⊕ η j ⊕ ℓ ⊕ ℓ ⊕ · · · ⊕ ℓ m − j where η i : Z m → SO (2) and ℓ i : Z m → O (1). In particular, a matrix is diagonalizable in O ( m ) if and only if it is symmetric , i.e. A T = A . Thus in the real case we will consider 2-torsion orthogonal matrices. The resulting space B ( Z / , O ( m )) is constructed from pairwisecommuting symmetric orthogonal matrices. We can stabilize over m , similar to the complexcase, to obtain B ( Z / , O ). The associated spectrum is given by ko sym = ko Bµ ( S )where ko is the corresponding Γ-space of the connective real K -theory spectrum. There is asimilar stable equivalence ko sym ≃ ko ∧ Bµ and a weak equivalence B ( Z / , O ) ≃ Ω ∞ ko sym by the real versions of Proposition 2.2 and 2.5. Table 1.
Homotopy groups of B ( Z / , O ) are isomorphic to the ko -homologyof Bµ [BG10, § ǫ π k + ǫ ( ko sym ) 0 Z / Z / Z / k +3 Z / k +4 Similar to the complex case π ( ko sym ) can be understood by considering the composition of Bµ ⊂ B ( Z / , O ) with the determinant map det : B ( Z / , O ) → Bµ . This composition isthe identity map and splits off the Z / S → ko is 3-connected, i.e. induces an isomorphism on π i for 0 ≤ i ≤ i = 3. From the Atiyah–Hirzebruch spectral sequence we see that S ∧ Bµ → ko ∧ Bµ isalso 3-connected. Therefore the map Q ( Bµ ) → B ( Z / , O ) extending the inclusion Bµ ⊂ B ( Z / , O ) induces an isomorphism on π r for 0 ≤ r ≤
3. Note that π ( QBµ ) = Z / π can be described more concretely. Since O (2 n + 1) ∼ = µ × SO (2 n + 1)there is a fibration sequence B ( Z / , SO ) → B ( Z / , O ) det −→ Bµ which splits due to the splitting of the homomorphism spaces. Since det induces an iso-morphism on π the fiber is simply connected. Looking at the simplicial set of connectedcomponents gives another fiber sequence | Y • | → | N (( µ ) • ) | → Bµ where Y • = π Hom(( Z / • , SO ). This time the fiber is 2-connected. Let O sym ⊂ O denotethe subspace of symmetric orthogonal matrices and SO sym denote the intersection SO ∩ O sym .There is a natural map ΣSO sym → B ( Z / , SO ) (2.7.3) e claim that this map induces a surjection on H , and hence on π . To see this considerthe spectral sequence H p H q ( B ( Z / , SO ) • ) ⇒ H p + q ( B ( Z / , SO ))converging to homology with integer coefficients. Since the degree zero space is just a pointthe only contribution to H can come from H H and H H . First we show that H H term vanishes. The homology groups H ∗ H ( Y • ) are given by the homology H ∗ ( Y • ) of Y • asa simplicial set. We have seen that Y • is 2-connected, in particular, H ( Y • ) = 0. Thus H H term vanishes in the spectral sequence. The only contribution must come from H H . Toput in a different way 2.7.3 is surjective on H , thus, also on π since the target is simplyconnected. Therefore the adjoint mapSO sym → Ω B ( Z / , SO )is π surjective at an arbitrary base point. The space SO sym is a disjoint union of Grassman-nians Gr k ( R ∞ ) where k ≥
0. For positive k each space has π isomorphic to Z /
2. ThereforeGr k ( R ∞ ) → Ω B ( Z / , SO ) induces an isomorphism on π for any k >
0. Alternatively send-ing a line ℓ to the orthogonal matrix which acts by − ℓ and trivially on the complementgives R P ∞ → O sym → Ω B ( Z / , O ) (2.7.4)and the composite induces an isomorphism on π .3. C ( d, m ) -cohomology For each m ≥ C ( d, m ), obtained from the commuta-tive d -torsion K -theory spectrum kµ d . This spectrum serves as a stable version of a quotientof B ( Z /d, U ( m )) in the sense that the infinite loop space associated to the spectrum C ( d, m )will be the corresponding quotient space of the stable version B ( Z /d, U ). In this section wecompute the homotopy groups of C ( d, m ) and describe the C ( d, m )-cohomology of a space.In § C ( d, m )-cohomology informs us about operator solutions of linearconstraint systems. These operator solutions play a significant role in quantum informationtheory.3.1. A quotient space.
Throughout this section let G be a topological group which containsa central subgroup isomorphic to µ d . When G = U ( m ) this will be the subgroup of m × m diagonal matrices with entries in µ d . Definition 3.1.
Let ¯ B ( Z /d, G ) denote the geometric realization of the simplicial space[ n ] Hom(( Z /d ) n , G ) / ∼ where the quotient relation identifies ( A , · · · , A n ) with ( α A , · · · , α n A n ) where α i ∈ µ d .Simplicial structure maps are similar to the ones given in Definition 2.1.There is a fibration sequence Bµ d ∆ G −−→ B ( Z /d, G ) → ¯ B ( Z /d, G )where the fiber inclusion is induced by µ d ⊂ G . By the classification of principal bundlesthis fibration is determined by a cohomology class γ G in H ( ¯ B ( Z /d, G ) , Z /d ). When G is he unitary group U ( m ) we simply write ∆ m for the fiber inclusion and γ m for the cohomol-ogy class. The stabilization maps in 2.1.1 do not descend to ¯ B ( Z /d, U ( m )). However, wewill construct a space which serves as a stabilization using methods from stable homotopytheory.3.2. C ( d, m ) spectrum. We begin with a spectrum level description of ∆ m . For m ≥ δ m : S → ku, (3.1.1)induced by the map 1 + → a m ≥ Gr m ( C ∞ )that sends the element 1 to the subspace C m = h e , e , · · · , e m i and the base point + toGr ( C ∞ ). This assignment determines all the other maps S ( k + ) → ku ( k + ) by equivarianceunder the α maps.Let δ d,m : S Bµ d → ku Bµ d denote the Γ-space map induced by δ m using the functoriality ofthe construction F F X . The associated spectra maps will still be denoted by δ m and δ d,m ,respectively.Consider the cofiber sequence S Bµ d δ d,m −−→ kµ d → C ( δ d,m ) . (3.1.2) Definition 3.2.
We define C ( d, m ) to be the spectrum obtained from C ( δ d,m ) by killing thehomotopy groups of degree greater than 2. We write ¯ B ( d, m ) for the associated infinite loopspace Ω ∞ C ( d, m ).Let ∆ S m denote the map Q ( Bµ d ) → Ω ∞ kµ d obtained by applying Ω ∞ to δ d,m and using theequivalence Q ( Bµ d ) = Ω ∞ ( S ∧ Bµ d ) ≃ Ω ∞ S Bµ d ( S )implied by part (2) of Lemma 2.3. Lemma 3.3.
There is a map of fibrations Bµ d Q ( Bµ d ) B ( Z /d, U ( m )) Ω ∞ kµ d ¯ B ( Z /d, U ( m )) Ω ∞ C ( δ d,m ) ι ∆ m ∆ S m ι m ¯ ι m (3.3.1) Proof.
We can construct a diagram of spaces S Bµ d (1 + ) Ω ∞ S Bµ d ( S ) ku Bµ d (1 + ) Ω ∞ ku Bµ d ( S ) ere S Bµ d (1 + ) = Bµ d , ku Bµ d (1 + ) = ku ( Bµ d ), and Ω ∞ S Bµ d ( S ) = Q ( Bµ d ). Moreover, thebottom horizontal map is a weak equivalence as a consequence of Proposition 2.5. Thus weobtain a commutative diagram Bµ d Bµ d Q ( Bµ d ) B ( Z /d, U ( m )) B ( Z /d, U ) Ω ∞ kµ d ∆ m ι ∆ S m ι m ∼ (3.3.2)where B ( Z /d, U ( m )) → B ( Z /d, U ) is induced by the stabilization map U ( m ) → U . To seethat the middle vertical map turns out to be the composite ι m ∆ m we can replace Bµ d by theset ( µ d ) n of n -simplices. In this case an element ( α , · · · , α n ) of ( µ d ) n is sent to the n -tupleof diagonal matrices ( α I, · · · , α n I ) regarded as a pairwise commuting tuple in U .To construct ¯ ι m , which is defined up to homotopy, we can proceed as follows. Instead ofextending the top square in 3.3.1 downwards we can first extend it upwards by consideringthe fibers to obtain a map of fibrations F F Bµ d Q ( Bµ d ) B ( Z /d, U ( m )) Ω ∞ kµ dfι ∆ m ∆ S m ι m Then ¯ ι m can be obtained from f by delooping. (cid:3) For notational simplicity the composite¯ B ( Z /d, U ( m )) ¯ ι m −→ Ω ∞ C ( δ d,m ) → ¯ B ( d, m ) (3.3.3)will still be denoted by ¯ ι m . Lemma 3.4.
There is a class γ S m in H ( ¯ B ( d, m ) , Z /d ) such that ¯ ι ∗ m ( γ S m ) = γ m .Proof. The class γ m is the image of the identity homomorphism in H ( Bµ d , Z /d ) ∼ = Hom( Z /d, Z /d )under the differential d : H ( Bµ d , Z /d ) → H ( ¯ B ( Z /d, U ( m )) , Z /d )in the E -page of the Serre spectral sequence associated to the left-hand fibration in 3.3.1.On the other hand, H ( Q ( Bµ d ) , Z /d ) is isomorphic to H ( Bµ d , Z /d ) and γ S m is similarlydescribed as the image of the transgression. The result follows by naturality. (cid:3) Homotopy groups.
To compute the homotopy groups of C ( d, m ) we can use thecofiber sequence S ∧ Bµ d δ m ∧ id −−−→ ku ∧ Bµ d → C ( δ d,m ) (3.4.1) nstead of 3.1.2 since we have a commutative diagram of spectra S ∧ Bµ d S Bµ d ( S ) ku ∧ Bµ d ku Bµ d ( S ) ∼ δ m ∧ id δ d,m ∼ (3.4.2)as a consequence of part (2) of Lemma 2.3. Lemma 3.5.
The homotopy groups of C ( d, m ) fit into an exact sequence → π C ( d, m ) → Z /d φ −→ Z /d → π C ( d, m ) → . (3.5.1) Proof.
The exact sequence comes from the homotopy exact sequence of the cofiber sequence3.4.1 and using the fact that π Σ ∞ Bµ d = Z /d together with the homotopy groups of kµ d given in 2.6.1.We claim that δ m is the m -fold sum δ + · · · + δ . By definition δ is completely determined byits value on S (1 + ) = 1 + , which sends 1 to the subspace h e i . We have an H -space structureon ku (1 + ), which comes from being a special Γ-space, that is induced by ku (1 + ) × ku (1 + ) → ku (1 + ) (3.5.2)that sends ( V, W ) to the direct sum V ⊕ W . This H -space structure is responsible for theabelian group structure on the set of homotopy classes of maps [ S , ku ]. Thus δ + δ iscomputed by using 3.5.2. In effect we obtain a map S (1 + ) → ku (1 + ) that sends 1 to thedirect sum h e i ⊕ h e i ∼ = h e , e i . This is precisely δ . In a similar way we can proceed toshow that δ m is the m -fold sum of δ as claimed.We know that up to homotopy δ m ∧ id induces the map ∆ S m on the level of spaces, by thediagram in 3.4.2. When m = 1 the map φ in 3.5.1 is given by the identity map (from 2.6.2).For m > K, L, M be spectra, X be a space, and f, f ′ : L → M be maps of spectra.(1) ∧ id : [ K, L ] → [ K ∧ X, L ∧ X ], defined by f f ∧ id, is a homomorphism of abeliangroups, i.e, ( f + f ′ ) ∧ id = f ∧ id + f ′ ∧ id.(2) Consider the induced map f ∗ : [ K, L ] → [ K, M ], defined by f ∗ ( g ) = f g , and f ′∗ similarly defined. Then ( f + f ′ ) ∗ = f ∗ + f ′∗ .Both of these results follow from the basic properties of addition of spectrum maps. Weapply (1) to [ S , ku ] → [ S ∧ Bµ d , ku ∧ Bµ d ] and obtain δ m ∧ id = ( δ + · · · + δ ) ∧ id = ( δ ∧ id) + · · · + ( δ ∧ id) . (3.5.3)Note that the map induced on π can be thought of as a map( δ m ∧ id) ∗ : [Σ S , S ∧ Bµ d ] → [Σ S , ku ∧ Bµ d ] (3.5.4)where Σ is the shift operator. Now we apply (2) to the decomposition given in 3.5.3. Weobtain that ( δ m ∧ id) ∗ = ( δ ∧ id) ∗ + · · · + ( δ ∧ id) ∗ and thus φ is given by multiplication with m . (cid:3) .4. C ( d, m ) -cohomology. Let us introduce notation for the abelian groups correspondingto the kernel and the cokernel of the exact sequence in 3.5.10 → ( Z /d ) m i m −→ Z /d × m −−→ Z /d π m −→ Z /dm Z /d → . For a group homomorphism h : A → B we write h ∗ : H n ( X, A ) → H n ( X, B ) for thechange of coefficients map. Note that both the kernel and the cokernel are isomorphic to Z / gcd( d, m ). Theorem 3.6.
There is a commutative diagram H ( X, ( Z /d ) m ) kµ d ( X ) C ( d, m )( X ) H ( X, Z /d ) H ( X, Z /d ) H ( X, Z /dm Z /d ) ( i m ) ∗ cl ( π m ) ∗ where cl ( f ) = f ∗ ( γ S m ) and the middle row is an exact sequence. In particular, we have acanonical splitting C ( d, m )( X ) ∼ = H ( X, Z /dm Z /d ) ⊕ H ( X, ( Z /d ) m ) . Proof.
The first Postnikov section of S ∧ Bµ d is given by a map p : S ∧ Bµ d → Σ Hµ d where Hµ d is the Eilenberg–Maclane spectrum associated to the abelian group µ d . The stablecanonical class can be described using the map p and the shift of the cofiber sequence 3.1.2 kµ d C ( δ d,m ) Σ( S ∧ Bµ d ) C ( d, m ) Σ Hµ d Σ p γ S m The middle row in the statement of the theorem is obtained by evaluating the sequence kµ d → C ( δ d,m ) → Σ Hµ d at the space X . The middle column comes from the Atiyah–Hirzebruch spectral sequence since the spectral sequence collapses in the E -page. The map kµ d ( X ) → H ( X, Z /d ) is induced by the determinant map (see § (cid:3) Remark 3.7.
Theorem 3.6 has also a real version where kµ d is replaced by ko sym ≃ ko ∧ Bµ introduced in § C R (2 , m ) denote the cofiber of δ m ∧ id : S ∧ Bµ → ko ∧ Bµ where δ m : S → ko is the real version of 3.1.1 (again we can kill homotopy groups above degree2). From the homotopy groups of ko sym given in Table 1 and of Q ( Bµ ) described in § m is odd then π i C R (2 , m ) = 0 for i = 1 ,
2. Thus the interesting case is C R (2 m ) = C R (2 , m ). The homotopy groups fit into the exact sequence0 → Z / α −→ π C R (2 m ) β −→ Z / −→ Z / → π C R (2 m ) → here is a commutative diagram H ( X, Z / H ( X, Z / H ( X, π C R (2 m )) H ( X, Z / ko sym ( X ) C R (2 m )( X ) H ( X, Z / H ( X, Z / H ( X, Z / δ α ∗ β ∗ cl ∼ = where δ is the connecting homomorphism of the exact sequence associated to 0 → Z / → π C R (2 m ) → Z / →
0. The top and the middle rows are exact.3.5. TC d -bundle interpretation. Let X be a 2-dimensional CW complex. We will providean interpretation for classes in C ( d, m )( X ) as TC d -bundles. Let X γ → X denote the principal Bµ d -bundle corresponding to a cohomology class γ ∈ H ( X, Z /d ). A map X → ¯ B ( d, m )gives rise to a commutative diagram X γ B ( Z /d, U ) X ¯ B ( d, m ) f γ p f (3.7.1)where γ = f ∗ ( γ S m ). The map f γ is constructed as follows. The composition of X γ p −→ X withthe spectrum map X −→ C ( d, m ) corresponding to f lifts to the fiber of C ( d, m ) → Σ Hµ d .This lift factors through kµ d since X is 2-dimensional. Therefore there is a natural map χ : C ( d, m )( X ) → M γ ∈ H ( X, Z /d ) kµ d ( X γ ) (3.7.2)obtained by sending a class represented by f to the class of f γ where γ = f ∗ ( γ S m ). Theorem3.6 implies that χ is injective. Indeed, the H ( X, ( Z /d ) m ) part maps injectively since ( i m ) ∗ is injective. For the H ( X, Z /dm Z /d ) part we consider the fundamental groups. There is acommutative diagram of groups Z /d Z /dπ ( X γ ) Z /dπ ( X ) Z /dm Z /d × m ( f γ ) ∗ f ∗ Under the map given by the composition of χ with the direct sum of the edge homomorphisms kµ d ( X γ ) → H ( X γ , Z /d ) the H ( X, Z /dm Z /d ) part maps injectively into H ( X γ , Z /d ) via theassignment f ∗ ( f γ ) ∗ . s a consequence each class in C ( d, m )( X ) can be represented as a TC d -bundle over X γ foran appropriate cohomology class γ . Moreover, this TC d -bundle is uniquely determined bythe associated determinant line bundle.4. Operator solutions of linear constraint systems
Linear constraint systems arise in quantum information theory in the context of non-localgames. Such games are played among a referee and two players where each player aims towin the game by satisfying a fixed set of rules. For some games if the players use quantumresources, such as entangled quantum states and quantum measurements, then they canincrease their likelihood of winning the game. Other than their significance in quantuminformation theory, linear constraint systems have found applications in resolving problemsin the theory of operator algebras such as Tsirelson problem [Slo19] and Connes embeddingconjecture [JNV + C ( d, m )-cohomology, introduced in §
3. Weshow that operator solutions of linear constraint systems correspond to classes in C ( d, m )-cohomology. The paradigmatic example of a linear constraint system constructed by Mermin[Mer93] gives rise to a non-trivial class in the C (2 , n )-cohomology of a torus for n ≥ Linear constraint systems. A linear constraint system is specified by a system oflinear equations M x = b for some r × c matrix M with entries in Z /d . We say that alinear constraint system has an operator solution if there exist a collection of m × m -unitarymatrices A i , 1 ≤ i ≤ c , such that • ( A i ) d is the identity matrix I m for all 1 ≤ i ≤ c , • A i A j = A j A i whenever M ki and M kj are both non-zero for some 1 ≤ k ≤ r , • A M k A M k · · · A M kc c = ω b k I m , where ω = e πi/d , for all 1 ≤ k ≤ r .When m = 1 we call such a solution a scalar solution . In the physics literature an operatorsolution is usually called a quantum solution and a scalar solution is called a classical solution .A linear constraint system which admits no classical solutions is called contextual ; otherwiseit is called non-contextual . Note that in this paper we restrict our attention to operatorsolutions over finite dimensional Hilbert spaces. The finiteness restriction can be removedfor a more general discussion of the subject. For basic properties of linear constraint systemswe refer to [CM14, CLS17, QW19, OR20].4.2. Topological description.
A linear constraint system can be formulated using hy-pergraphs. The data of a linear constraint system can be turned into a pair ( H , τ ) where H = ( V, E, ǫ ) is a hypergraph with vertex set V , edge set E and an incidence weight ǫ ; and τ isa function E → Z /d . More concretely, let H denote the hypergraph with V = { v , v , · · · , v c } , E = { e , e , · · · , e r } where e k = { v i | M ki = 0 } , and ǫ e k ( v i ) = M ki . The hypergraph is of pecial type, namely, it satisfies the property that every vertex is contained in at least oneedge. The function τ is defined by τ ( e k ) = b k . An operator solution can be regarded as afunction T : V → U ( m ) where T ( v i ) = A i . As before let G be a group which contains acentral subgroup isomorphic to µ d . We can consider solutions over G instead of U ( m ). Wedenote such an operator solution by a function T : V → G .We define a chain complex associated to the hypergraph C ∗ ( H ) : C ∂ → C → C where C = Z /d, C = Z /d [ V ] , C = Z /d [ E ] , ∂ [ e ] = X v ∈ e ǫ e ( v )[ v ] . There is a corresponding cochain complex C ∗ ( H ). The function τ can be regarded as a2-cochain. We write [ τ ] for its cohomology class.For a CW complex X let X n denote the set of n -cells. Definition 4.1 ([OR20]) . A topological realization for the hypergraph H is a connected 2-dimensional CW complex X ( H ) with X = V and X = E together with a homomorphismof chain complexes f ∗ : C ∗ ( X ) → C ∗ ( H ) such that f and f are the identity maps, i.e., C ( X ) C ( X ) C ( X ) Z /d [ E ] Z /d [ V ] Z /d ∂ f ∂ f f ∂ Construction 4.2.
Let X be a topological realization of H . For each 2-cell e ∈ E define asubgroup A ( e ) ⊂ G generated by µ d ∪ { T v | v ∈ e } and let ¯ A ( e ) = A ( e ) /µ d . Associated to anoperator solution T of the linear constraint system ( H , τ ) we construct a map f T : X → ¯ B ( Z /d, G ) , which is defined up to homotopy, as follows(1) send each 0-cell in X to the unique vertex of ¯ B ( Z /d, G ),(2) send the 1-cell labeled by v ∈ X to the 1-cell labeled by [ T ( v )], the equivalence classof T ( v ) under multiplication with elements in µ d ,(3) the boundary of a 2-cell labeled by e ∈ X maps to a contractible loop in the subspace B ¯ A ( e ) ⊂ ¯ B ( Z /d, G ); extend this map to the interior of the disk by choosing acontracting homotopy that lies in B ¯ A ( e ). Remark 4.3.
In part (3) any two choices of a contracting homotopy extending the map onthe boundary of a 2-cell of X are homotopic to each other since the image lies inside thesubspace B ¯ A ( e ), whose homotopy groups above degree 2 vanishes. Therefore the map f T isunique up to homotopy.Let [( X, x ) , ( Y, y )] denote the set of pointed homotopy classes of maps between two basedspaces. We will suppress the base points and simply write [ X, Y ]. This should not result inany confusion since in this paper we will not use the set of unpointed homotopy classes ofmaps. efinition 4.4. Let Sol( X ; d, G ) denote the set obtained from the collection of triples( H , τ, T ), where H admits a topological realization homotopy equivalent to X , by identi-fying ( H , τ , T ) ∼ ( H , τ , T ) if the classes of f T and f T coincide in [ X, ¯ B ( Z /d, G )] forsome choice of base points in the topological realizations. Proposition 4.5.
Let ( H , τ ) be a linear constraint system.(1) ( H , τ ) has a scalar solution if and only if [ τ ] = 0 in H ( C ( H )) , or equivalently, in thesecond cohomology group of any topological realization.(2) If T is an operator solution for ( H , τ ) then f ∗ T ( γ G ) = [ τ ] for any map f T constructedusing the operator solution (Construction 4.2).(3) If ( H , τ ) has an operator solution T and a topological realization X such that f T induces the trivial map between the fundamental groups then ( H , τ ) has a scalar so-lution.Proof. (1) follows from the definition of the chain complex, see also [ORBR17]. Regarding(2) observe that a contractible loop in ¯ B ( Z /d, G ), such as the image of the boundary of a2-cell labeled by an hyperedge e under f T , lifts to a loop in B ( Z /d, G ) that is homotopic toa loop contained in Bµ d . Up to homotopy this loop is specified by an element in µ d . Anoperator solution specifies a lift such that the loop in Bµ d corresponds to τ ( e ). A specialcase of part (3) is proved in [OR20] applicable to hypergraphs with ǫ e ( v ) = ± γ G comes from a class in ¯ G = G/µ d , still denoted by the same symbol. Let H ⊂ G denote the discrete subgroup generated by { T v | v ∈ V } together with µ d . Let ¯ H denote the quotient H/µ d . Since f T induces the trivial map on π we can reduce to the casewhere π ( X ) = 1 by collapsing the non-contractible loops in X . The composite X f T −→ ¯ B ( Z /d, G ) ⊂ B ¯ G factors through a map X → B ¯ H . Since π ( X ) = 1 and the homotopy groups of B ¯ H vanish above dimension 1 this map is null homotopic. Therefore using part (2) we have f ∗ T ( γ G ) = [ τ ] = 0. (cid:3) Example 4.6.
Mermin square [Mer93] is the prominent example of a contextual linearconstraint system, i.e., it admits an operator solution but not a scalar solution. Let P n denote the subgroup in U (2 n ) consisting of matrices of the form i a A ⊗ A ⊗ · · · ⊗ A n where a ∈ Z / A i is one of the Pauli matrices I = (cid:18) (cid:19) X = (cid:18) (cid:19) Y = (cid:18) − ii (cid:19) Z = (cid:18) − (cid:19) . The linear constraint system ( H sq , τ sq ) and an operator solution T sq : V → P is depicted inFigure 1 (left figure). As depicted in the right figure H sq has a topological realization givenby a torus. The class [ τ sq ] is non-zero since the cocycle evaluates to 1 on the torus. Thereforethe linear constraint system does not admit a scalar solution [ORBR17].Another linear constraint system constructed in [Mer93] is the Mermin star linear constraintsystem, which we denote by ( H st , τ st ). An operator solution T st : V → P is displayed in I IX XXIZ ZI ZZXZ ZX Y Y
XIXIZI ZIIX IZY YZX XZXX ZZ
Figure 1. (Left figure) H sq consists of 9 vertices and 6 edges each consistingof 3 vertices in each row and column. The operator solution is given by tensorproduct of two Pauli matrices, where the notation is simplified by omitting ⊗ .The function τ sq takes the value 0 for each hyperedge except the right-mostcolumn. (Right figure) A topological realization given by a torus together witha cell structure consisting of triangles. The operators are placed on the edgesand each triangle corresponds to an hyperedge. The cocycle τ sq assigns 0 toeach triangle except { XX, Y Y, ZZ } , which is assigned 1.Figure 2 (left figure). The corresponding topological realization is again a torus, but with adifferent cell structure (right figure); see [ORBR17]. IY I IXIXXX Y Y XXY Y Y XYXII Y IIIIYIIX
IXIIXIIY I IY IIIY IIXXII Y IIXY Y Y Y XY XYXXX
Figure 2. (Left figure) H st consists of 10 vertices and 5 edges each consistingof 4 vertices in each line. The function τ st takes the value 0 for each hyperedgeexcept the horizontal line. (Right figure) On the torus τ st specifies a 2-cocyclethat assigns 0 to each cell except { XXX, Y Y X, Y XY, XY Y } is assigned 1.4.3. Computing the homotopy classes.
The equivalence classes of operator solutionsmap to the (pointed) homotopy classes of maps θ : Sol( X ; d, G ) ֒ → [ X, ¯ B ( Z /d, G )] . The target can be computed using an algebraic category (the category of crossed modules [Whi49]) which captures the behavior of the homotopy category of 2-dimensional CW com-plexes. et ¯ π i denote i -th homotopy group of ¯ B ( Z /d, G ). Proposition 4.7.
Let X be a -dimensional CW complex. Sending a map to the homomor-phism induced on π gives a surjective map π : [ X, ¯ B ( Z /d, G )] → Hom ( π X, ¯ π ) such that for a fixed homomorphism α the preimage is given by π − ( α ) ∼ = H ( ˜ X, (¯ π ) α ) where (¯ π ) α is the π ( X ) -module determined by the homomorphism α .Proof. The statement holds for [
X, Y ] where Y is an arbitrary CW complex. We will con-struct maps Y r −→ ¯ Y s ←− Y (2) where Y (2) is a 2-dimensional CW complex, and the maps r and s are 3-equivalences, i.e.,each map induces an isomorphism on π i for 0 ≤ i < i = 3. In this case r ∗ : [ X, Y ] → [ X, ¯ Y ], and similarly s ∗ , are bijections [Spa89, Cor 23]. Before the constructionwe first show how to finish the proof of the statement.The set [ X, Y (2) ] can be computed algebraically; for details we refer to [BPHA +
93, Ch. II].Let us write [
X, Y (2) ] α for the set of homotopy classes of maps that induce the homomorphism α between the fundamental groups. The (cellular) chain complex for the universal cover ˜ X consists of π ( X )-modules and we can talk about the cohomology groups H n ( ˜ X, ( π Y (2) ) α )where π Y (2) is regarded as a π ( X )-module via the homomorphism α . The cohomologygroup H ( ˜ X, ( π Y (2) ) α ) acts on [ X, Y (2) ] α in a transitive way. In fact, this action determinesa bijection [ X, Y (2) ] α ∼ = H ( ˜ X, ( π Y (2) ) α ) . We turn to the construction of r and s . The first map is obtained by killing homotopy groupsof Y above dimension 2. Construction of the second map uses the theory of crossed modules.The fundamental property we will use is that any free crossed module over a free base groupis realizable by a 2-dimensional CW complex and maps between such crossed modules comefrom maps between the CW complexes that realize them [BPHA +
93, Ch. II]. Let us applythis to the crossed module given by the connecting homomorphism ∂ : π ( ¯ Y , ¯ Y ) → π ( ¯ Y ) (4.7.1)By the realization result there is a 2-dimensional CW complex Y (2) such that the crossedmodule ∂ : π ( Y (2) , Y ) → π ( Y ) is isomorphic to the one given in 4.7.1. We will showthat this isomorphism is realized by a map s : Y (2) → ¯ Y .
We start the construction of s fromthe 1-st skeleton. We can find a map Y → ¯ Y that induces the desired isomorphism on π . Composing this map with the inclusion ¯ Y ⊂ ¯ Y we obtain Y → ¯ Y . This map liftsto a map Y (2) → ¯ Y since the set of 2-cells is a basis for the free group π ( Y (2) , Y ) andthe isomorphism between the crossed modules implies precisely the lifting condition in thealgebraic language. (cid:3) xample 4.8. We discuss an interesting example. The Pauli group P n defined in Example4.6 has a generalization for all primes p which has a similar description as tensor products of p × p unitary matrices; see for instance [OS19]. As an abstract group P n is the extraspecial2-group of complex type and the extraspecial p -group of exponent p for odd primes. Thereis an irreducible complex representation which allows us to regard it as a subgroup in U ( p n ).Suppose n ≥
2. It is known that π ¯ B ( Z /p, P n ) ∼ = (cid:26) Z / × V p = 2 P n p > , and the higher homotopy groups are given by π i ¯ B ( Z /p, P n ) ∼ = π i ( N p,n _ S n ) , i ≥ , where N p,n has an explicit formula [Oka18]. Therefore according to Proposition 4.7 the map[ X, ¯ B ( Z /p, P n )] → Hom( π X, ¯ π )is an isomorphism when n ≥
3. However, for n = 2 it is only surjective and the kerneldepends on the ¯ π -module structure of ¯ π , which is currently unknown.The canonical class can be described as γ P n = (cid:26) x + P ni =1 x i ∪ z i p = 20 p > , (4.8.1)where { x , x , · · · , x n , z , · · · , z n } is a basis for Z / × V , see [OS19]. Therefore for odd p every linear constraint system has a scalar solution if it has an operator solution over P n .Whereas for p = 2 this depends on the map induced on π , as a result of the cup productdecomposition in 4.8.1 .The operator solution T sq of the Mermin square linear system ( H sq , τ sq ) introduced in Ex-ample 4.6 gives a non-trivial class [ f T sq ] in [ S × S , ¯ B ( Z / , P )]. Let us write T n = T sq ⊗ I n − , (4.8.2)where n ≥
2, for the operator solution obtained by tensoring with the identity matrix: A A ⊗ I n − . Then [ f T n ] gives a non-trivial class in [ S × S , ¯ B ( Z / , P n )] for all n ≥ H st , τ st ) specifies a class in [ S × S , ¯ B ( Z / , P )]. It turnsout that this class coincides with [ f T ] since there is a refined cell structure ([ORBR17]) onthe torus, see Figure 3, on which both the square and the star constructions can be realized.More precisely, there is a commutative diagram X sq X ref X st ¯ B ( Z / , P ) ¯ B ( Z / , P ) ¯ B ( Z / , P ) ⊗ I relating the topological realizations X = S × S with different cell structures as indicatedby the subscripts. XIIXIIY I IY IIIY IIXXII Y IIXY Y Y Y XY XYXXXXY I Y XI XXI Y Y I
Figure 3.
Refined topological realization4.4.
Application of C ( d, m ) -cohomology. Now we focus on operator solutions in unitarygroups. For notational simplicity let us write Sol( X ; d, m ) for the operator solutions over U ( m ). Recall the map ¯ ι m : ¯ B ( Z /d, U ( m )) → ¯ B ( d, m )introduced in 3.3.3. Composing with ¯ ι m gives a mapˆ θ : Sol( X ; d, m ) ֒ → [ X, ¯ B ( Z /d, U ( m ))] (¯ ι m ) ∗ −−−→ C ( d, m )( X )where we have identified [ X, ¯ B ( d, m )] with the 0-th C ( d, m )-cohomology of X since the targetspace is the infinite loop space associated to the spectrum representing the cohomologytheory. Given an operator solution T the image of f T under (¯ ι m ) ∗ will be denoted by ˆ f T .By Lemma 3.4 the pull-back ˆ f ∗ T ( γ S m ) coincides with f ∗ T ( γ m ). Therefore for a linear constraintsystem existence of a scalar solution is determined in a stable manner, i.e., ˆ f ∗ T ( γ S m ) = 0 ifand only if a scalar solution exists.Theorem 3.6 gives us the decomposition C ( d, m )( X ) ∼ = H ( X, Z /dm Z /d ) ⊕ H ( X, ( Z /d ) m )and we will denote classes in this group by pairs ( ϕ ; ϕ ). Corollary 4.9.
Let ( H , τ ) be a linear constraint system and X be a topological realization.(1) If H ( X, ( Z /d ) m ) = 0 then ( H , τ ) has a scalar solution.(2) If d and m are coprime then C ( d, m )( X ) = 0 and ( H , τ ) has a scalar solution.(3) If π ( X ) is trivial and [ τ ] = 0 then ( H , τ ) does not have an operator solution.Proof. (1) follows from Lemma 3.4, Proposition 4.5 part (1) and (2) since ϕ = ( ˆ f T ) ∗ ( γ S m ) = f ∗ T ( γ m ) = [ τ ] for any operator solution T . (2) follows from part (1) since if ( d, m ) = 1 then H ( X, ( Z /d ) m ) = 0. Part (3) follows from Proposition 4.5 part (3). Existence of an operatorsolution implies that [ τ ] = 0 since X is simply connected. (cid:3) .5. The Mermin class.
In the physics literature a quantum system with Hilbert space( C ) ⊗ n is called an n -qubit . Such systems play a significant role in quantum informationtheory. Operator solutions in U (2 n ) of linear constraint systems over Z / C ( d, m )-cohomology, where d = 2 and m = 2 n . Theorem 3.6 gives an isomorphism C (2 , n )( X ) ∼ = H ( X, Z / ⊕ H ( X, Z / . We will construct non-trivial classes that come from operator solutions to the Mermin squarelinear constraint system in Example 4.6. Our topological realization is a torus X = S × S .An operator solution for n = 2 is given in Figure 1. Let T denote this solution. We definean operator solution in U (2 n ) by tensoring with the identity as in 4.8.2, i.e., by constructingan operator solution T n defined by T n ( v ) = T ( v ) ⊗ I n − for v ∈ V . Let [ T n ] denote the classof this solution in Sol( S × S ; 2 , n ). Let M n denote the class ˆ θ ( T n ) in C (2 , n )( S × S ).This class will be called the Mermin class . We want to determine M n in terms of therepresentation given by the pair ( ϕ ; ϕ ). For each n ≥ τ ] = 0since, as we have seen in Example 4.6, the Mermin square linear constraint system does notadmit a scalar solution. Therefore ϕ is the non-trivial class in H ( S × S , Z /
2) = Z /
2. Todetermine ϕ consider the diagram (see 3.7.1) X ϕ B ( Z / , U ) X ¯ B (2 , n ) (4.9.1)where X ϕ is the Bµ -bundle determined by the non-trivial class ϕ . The class ϕ is deter-mined by the map induced on π X → π ¯ B (2 , n ) and this can be computed using diagram4.9.1. Applying π to the diagram we obtain π X ϕ π B ( Z / , U ) π X π ¯ B (2 , n ) ∼ = Let ˜ x and ˜ z denote the elements lifting the generators x = (1 ,
0) and z = (0 ,
1) of thequotient group π X = Z . It suffices to determine the images of ˜ x and ˜ z under the tophorizontal map. Figure 1 tells us that ˜ x maps to the loop determined by X ⊗ I n − and˜ z maps to Z ⊗ I n − . We can understand the induced map on π by composing with thedeterminant map det : B ( Z / , U ) → Bµ . This amounts to taking the determinant of thematrices representing the loops, which gives 1 in both cases. Thus both of the loops map tothe trivial loop in Bµ d . Therefore ϕ = (0 , ∈ ( Z / . In summary, the Mermin class M n is represented by (0 ,
0; 1). Since f T n induces the trivial map on π it factors as S × S ¯ B (2 , n ) S f Tn ¯ f (4.9.2) here the vertical map collapses the non-contractible loops corresponding to x and z . Thehomotopy class of ¯ f is the generator of π C (2 , n ) = Z /
2. By slight abuse of notation wewill also write M n for this class and refer to it as the Mermin class as well.Let χ ( M n ) denote the class in kµ d ( X ϕ ) where χ is defined in 3.7.2. This class correspondsto a TC d -bundle over X ϕ . The associated determinant line bundle, which can be describedby the class given by the image of the edge homomorphism kµ d ( X ϕ ) → H ( X ϕ , Z / ϕ is trivial. Thus the TC d -bundle is also trivial by injectivity of χ . A similarstatement holds for X = S and the corresponding Mermin class in π C (2 , n ).Let us compare to the unstable situation. The operator solution T n is over the Pauli group P n . The diagram 4.9.1 factors as X ϕ B ( Z / , P n ) B ( Z / , U ) X ¯ B ( Z / , P n ) ¯ B (2 , n ) ˜ gf g where g ∗ ( γ S n ) = γ P n is given as in 4.8.1. The homotopy class [ f ] is non-trivial in [ X, ¯ B ( Z / , P n )],which surjects onto Hom( π X, ¯ π ) as we have seen in Example 4.8. However, the composite gf induces the trivial map on π . This is not in conflict with Proposition 4.5 part (3) if wetake G = U . This is because the subgroup µ ֒ → U is not a central, or even not a normal,subgroup. Proposition 4.5 part (3) also implies that the diagonal map ¯ f in 4.9.2 does notfactor through ¯ B ( Z / , U (2 n )). Remark 4.10.
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Department of Physics and Astronomy, University of British Columbia, Vancouver BC V6T1Z4, Canada
E-mail address : [email protected]/[email protected]@math.ubc.ca/[email protected]