aa r X i v : . [ m a t h - ph ] F e b CLASSIFICATION OF TOPOLOGICAL INVARIANTS RELATEDTO CORNER STATES
SHIN HAYASHI
Abstract.
We discuss some bulk-surfaces gapped Hamiltonians on a latticewith corners, and propose a periodic table for topological invariants relatedto corner states aimed at studies of higher-order topological insulators. Ourtable is based on four things: (1) the definition of topological invariants, (2)a proof of their relation with corner states (3) computations of K -groups and(4) a construction of explicit examples. Contents
1. Introduction 12. Preliminaries 43. KO -Groups of C ∗ -algebras Associated with Half-Plane and Quarter-Plane Toeplitz Operators 74. Toeplitz Operators Associated with Subsemigroup ( Z ≥ ) n of Z n Z -Spaces of Self-Adjoint/Skew-Adjoint Fredholm Operatorsand Boersema–Loring’s K -theory 35References 441. Introduction
Recent developments in condensed matter physics have greatly generalized thebulk-boundary correspondence for topological insulators to include corner states.Topological insulators have a gapped bulk, which incorporates some topology thatdo not change unless the spectral gap of the bulk Hamiltonian closes under defor-mations. Examples include the TKNN number for quantum Hall systems [65] andthe Kane-Mele Z index for quantum spin Hall systems [36]. It is known that, cor-responding to these bulk invariants, gapless edge states appear, which is called the bulk-boundary correspondence [31]. After Schnyder–Ryu–Furusaki–Ludwig’s classi-fication of topological insulators [60] for ten Altland–Zirnbauer classes [2], Kitaev Mathematics Subject Classification.
Primary 19K56; Secondary 47B35, 81V99.
Key words and phrases.
Topologically protected corner states, higher-order topological insu-lators, K -theory and index theory.AIST-TohokuU Mathematics for Advanced Materials - Open Innovation Laboratory, NationalInstitute of Advanced Industrial Science and Technology, 2-1-1 Katahira, Aoba, Sendai 980-8577,Japan.JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan. noted the role of K -theory and Bott periodicity in the classification problem and ob-tained the famous periodic table [42]. Recently, some (at least bulk) gapped systemspossessing in-gap or gapless states localized around a higher codimensional part ofthe boundary (corners or hinges) are studied [30, 12, 41], which are called higher-order topological insulators (HOTIs) [59]. For example, for second-order topologicalinsulators, not only is the bulk gapped but also the codimension-one boundaries(edges, surfaces), and an in-gap or a gapless state appears around codimension-two corners or hinges. In this framework, conventional topological insulators areregarded as first-order topological insulators. HOTIs are now actively studied andthe classification of HOTIs has also been proposed [26, 40, 52]. Generalizing thebulk-boundary correspondence, relations between some gapped topology and cornerstates are much discussed [66, 5, 63].Initiated by Bellissard, K -theory and index theory are known to provide apowerful tool to study topological insulators. Bellissard–van Elst–Schulz-Baldesstudied quantum Hall effects by means of noncommutative geometry [10, 11], andKellendonk–Richter–Schulz-Baldes went on to prove the bulk-boundary correspon-dence by using index theory for Toeplitz operators [39]. The study of topologicalinsulators, especially regarding its classification and the bulk-boundary correspon-dence for each of the ten Altland–Zirnbauer classes by using K -theory and indextheory has been much developed [39, 24, 64, 16, 29, 48, 57, 64, 17, 38, 44, 1]. In [32],three-dimensional (3-D) class A bulk periodic systems are studied on one piece of alattice cut by two specific hyperplanes, which is a model for systems with corners.Based on the index theory for quarter-plane Toeplitz operators [62, 23, 54], a topo-logical invariant is defined assuming the spectral gap both on the bulk Hamiltonianand two half-space compressions of it. This gapped topological invariant is topolog-ical in the sense that it does not change under continuous deformation of the bulkHamiltonians unless the spectral gap of one of the two surfaces closes. It is provedthat, corresponding to this topology gapless corner states appear. A constructionof nontrivial examples by using two first-order topological insulators (of 2-D classA and 1-D class AIII) is also proposed. Class AIII codimension-two systems arealso studied through this method in [33] and, as an application to HOTIs, the ap-pearance of topological corner states in Benalcazar–Bernevig–Hughes’ 2-D model[12] is explained based on the chiral symmetry. The construction of examples in[33] leads to a proposal of second-order semimetallic phase protected by the chiralsymmetry [51].The purpose of this paper is to expand the results in [32] to all Altland–Zirnbauerclasses and systems with corners of arbitrary codimension. Since class A and classAIII systems (with codimension-two corners) were already discussed in [32, 33] byusing complex K -theory, we focus on the remaining eight cases, for which we usereal K -theory. For our expansion, a basic scheme has already been well developedin the above previous studies, which we mainly follow: some gapped Hamilton-ian defines an element of a KO -group of a real C ∗ -algebra, and its relation withcorner states are clarified by using index theory [39, 24, 16, 29, 44, 64, 17, 38].Although many techniques have already been developed in studies of topologicalinsulators, in our higher-codimensional cases, we still lack some basic results at thelevel of K -theory and index theory; hence, the first half of this paper is devoted tothese K -theoretic preliminaries, that is, the computation of KO -groups for real C ∗ -algebras associated with the quarter-plane Toeplitz extension and the computation LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 3 of boundary maps for the 24-term exact sequence of KO -theory associated with it,which are carried out in Sect. 3. Since the quarter-plane Toeplitz extension [54]is a key tool in our study of codimension-two corners, such a variant for Toeplitzoperators associated with higher-codimensional corners should be clarified, whichare carried out in Sect. 4. These variants of Toeplitz operators were discussed in[23, 22], and the contents in Sect. 4 will be well-known to experts. Since the au-thor could not find an appropriate reference, especially concerning Theorem 4.1which will play a key role in Sect. 5, the results are included for completeness.Note that the idea there to use tensor products of the ordinary Toeplitz extensionfor the study of these variants is based on the work of Douglas–Howe [23], wherethese higher-codimensional generalizations are briefly mentioned. The study ofsome gapped phases for systems with corners in Altland–Zirnbauer’s classificationis carried out in Sect. 5. In the framework of the one-particle approximation, weconsider n -D systems with a codimension k corner and take compressions of thebulk Hamiltonian onto infinite lattices with codimension k − whose in-tersection makes the codimension k corner. We assume that they are gapped. Notethat, under this assumption, bulk, surfaces and corners up to codimension k − k corner are also gapped. For such a system, wedefine two topological invariants as elements of some KO -groups: one is defined forthese gapped Hamiltonians while the other one is related to in-gap or gapless codi-mension k corner states. We then show a relation between these two which statesthat topologically protected corner states appear reflecting some gapped topologyof the system. We first study codimension-two cases (Sect. 5.1 to Sect. 5.4) andthen discuss higher-codimensional cases (Sect. 5.5). This distinction is made be-cause many detailed results have been obtained for codimension-two cases by virtueof previous studies of quarter-plane Toeplitz operators [54, 35] (the shape of thecorner we discuss is more general than in higher-codimensional cases, and a rela-tion between convex and concave corners is also obtained in [33]). Based on theseresults, we propose a classification table for topological invariants related to cornerstates (Table 1). Note that the codimension-one case of Table 1 is Kitaev’s table[42] and Table 1 is also periodic by the Bott periodicity. In order further to clarifya relation between our invariants and corner states, in Sect. 5.6, we introduce Z or Z -valued numerical corner invariants when the dimension of the corner is zero orone. They are defined by (roughly speaking) counting the number of corner states.A construction of examples is discussed in Sect. 5.7. As in [32], this is given byusing pairs of Hamiltonians of two lower-order topological insulators. In the realclasses, there are 64 pairs of them and the results are collected in Table 12. By usingthis method, we can construct nontrivial examples of each entry of Table 1, startingfrom Hamiltonians of first-order topological insulators. The corner invariant for theconstructed Hamiltonian is expressed by corner (or edge) invariants of constituenttwo Hamiltonians. This is given by using an exterior product of some KO -groupsin general, though, as in [32, 33], the formula for numerical invariants introducedin Sect. 5.6 is also included. For computations of KO -groups and classification ofsuch gapped systems, we employ Boersema–Loring’s unitary picture for KO -theory[14] whose definitions are collected in Sect. 2. Basic results for some Toeplitz op-erators are also included there. In Appendix A, we revisit Atiyah–Singer’s study In standard terminologies, they will be called edges , surfaces , hinges or edge of edges dependingon n and k . In this paper, we may simply call them corners but state its codimensions. S. HAYASHI
Table 1.
Classification of (strong) topological invariants relatedto corner state in Altland–Zirnbauer (AZ) classification. In thistable, n is the dimension of the bulk, and k is the codimension ofthe corner.Symmetry n − k mod 8AZ Θ Ξ Π 0 1 2 3 4 5 6 7A 0 0 0 0 Z Z Z Z AIII 0 0 1 Z Z Z Z Z Z Z Z BDI 1 1 1 Z Z Z Z D 0 1 0 Z Z Z Z DIII − Z Z Z Z − Z Z Z Z CII − − Z Z Z Z − Z Z Z Z − Z Z Z Z K -theory. Definitions of some Z -spaces, mapsbetween them, expression of boundary maps of 24-term exact sequences used in thispaper are collected there.Finally, let us point out a relation with our results and the current rapidly devel-oping studies on HOTIs. In [26], the HOTIs are divided into two classes: intrinsic HOTIs, which basically originate from the bulk topology protected by a point groupsymmetry, and others extrinsic
HOTIs. Our study will be for extrinsic HOTIs sinceno point group symmetry is assumed and our classification table (Table 1) is con-sistent with that of Table 1 in [26].2.
Preliminaries
In this section, we collect the necessary results and notations.2.1.
Boersema–Loring’s KO -Groups via Unitary Elements. In this subsec-tion, we collect Boersema–Loring’s definition of KO -groups by using unitaries sat-isfying some symmetries [14]. The basics of real C ∗ -algebras and KO -theory canbe found in [28, 61], for example.A C ∗ ,τ -algebra is a pair ( A , τ ) consisting of (complex) C ∗ -algebra A and an anti-automorphism τ of A satisfying τ = 1. We call τ the transposition and write a τ for τ ( a ). There is a category equivalence between the category of C ∗ ,τ -algebras and thecategory of real C ∗ -algebras: for a C ∗ ,τ -algebra ( A, τ ), the corresponding real C ∗ -algebra is A τ = { a ∈ A | a τ = a ∗ } , and its inverse is given by the complexification.A real structure on a (complex) C ∗ -algebra A is an antilinear ∗ -automorphism r satisfying r = 1. For a real structure r , there is an associated transposition τ givenby τ ( a ) = r ( a ∗ ), which gives a one-to-one correspondence between transpositions i.e., a complex linear automorphism of A that preserves ∗ and satisfies τ ( ab ) = τ ( b ) τ ( a ). LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 5 and real structures on the C ∗ -algebra . We extend the transposition τ on A tothe transposition (for which we simply write τ ) on the matrix algebra M n ( A ) by( a ij ) τ = ( a τji ) where a ij ∈ A and 1 ≤ i, j ≤ n . This induces a transposition τ K on K ⊗ A where K = K ( V ) is the C ∗ -algebra of compact operators on a separablecomplex Hilbert space V . Let ♯ ⊗ τ be a transposition on M ( A ) defined by (cid:18) a a a a (cid:19) ♯ ⊗ τ = (cid:18) a τ − a τ − a τ a τ (cid:19) . If we identify the quaternions H with C by x + yj ( x, y ), the left multiplication by j corresponds to j ( x, y ) = ( − ¯ y, ¯ x ). Then, we have ♯ ⊗ id = Ad j ◦∗ where ∗ denotesthe operation of taking conjugation of matrices and the C ∗ ,τ -algebra ( M ( C ) , ♯ ⊗ id)corresponds to the real C ∗ -algebra H of quaternions. We extend this transpositionto M n ( A ) by ( b ij ) ♯ ⊗ τ = ( b ♯ ⊗ τji ) where 1 ≤ i, j ≤ n and b ij ∈ M ( A ). On M n ( A ),we also consider a transposition e ♯ ⊗ τ defined by (cid:18) c c c c (cid:19) e ♯ ⊗ τ = (cid:18) c τ − c τ − c τ c τ (cid:19) , where c ij ∈ M n ( A ). For an m × m matrix X , we write X n for the mn × mn blockdiagonal matrix diag( X, . . . , X ). For example, we write 1 n for the n × n diagonalmatrix diag(1 , . . . , Definition 2.1 (Boersema–Loring [14]) . Let ( A , τ ) be a unital C ∗ ,τ -algebra. For i = − , , . . . ,
6, let n i be a positive integer, R i be a relation and I ( i ) be a matrix,as indicated in Table 2. Let U ( i ) k ( A , τ ) be the set of all unitaries in M n i · k ( A )satisfying the relation R i . On the set U ( i ) ∞ ( A , τ ) = ∪ ∞ k =1 U ( i ) k ( A , τ ), we consider theequivalence relation ∼ i generated by homotopy and stabilization given by I ( i ) . Wedefine KO i ( A , τ ) = U ( i ) ∞ ( A , τ ) / ∼ i which is a group by the binary operation givenby [ u ] + [ v ] = [diag( u, v )].For a nonunital C ∗ ,τ -algebra ( A, τ ), the i -th KO -group KO i ( A , τ ) is defined asthe kernel of λ ∗ : KO i ( ˜ A , τ ) → KO i ( C , id), where ˜ A is the unitization of A and λ : ˜ A → C is the natural projection. In [14], they also describe the boundary mapsof the 24-term exact sequence for KO -theory associated with a short exact sequenceof C ∗ ,τ -algebras. In Appendix A.3, we discuss an alternative description for someof them through exponentials.2.2. Toeplitz Operators.
In this subsection, we collect the definitions and basicresults for some Toeplitz operators used in this paper [22, 54].Let T be the unit circle in the complex plane C , and let c be the complex conjuga-tion on C , that is, c ( z ) = ¯ z . Let n be a positive integer. On the n -dimensional torus T n , we consider an involution ζ defined as the n -fold product of c . This inducesa transposition τ T on C ( T n ) by ( τ T f )( t ) = f ( ζ ( t )). Let Z ≥ be the set of non-negative integers and P n be the orthogonal projection of l ( Z n ) onto l (( Z ≥ ) n ).For a continuous function f : T n → C , let M f be the multiplication operator on l ( Z n ) generated by f . We consider the operator P n M f P n on l (( Z ≥ ) n ), which is Boersema–Loring called τ the real structure in [14]. In this paper, we distinguish these twosince the antilinear structure naturally appears in our application. We call τ the transpositionfollowing [38]. For notations of the transpositions introduced here, we follow [14].
S. HAYASHI
Table 2.
Boersema–Loring’s unitary picture for KO -theory [14] i KO -group n i R i I ( i ) − KO − ( A , τ ) 1 u τ = u KO ( A , τ ) 2 u = u ∗ , u τ = u ∗ diag(1 , − KO ( A , τ ) 1 u τ = u ∗ KO ( A , τ ) 2 u = u ∗ , u τ = − u (cid:18) i · − i · (cid:19) KO ( A , τ ) 2 u ♯ ⊗ τ = u KO ( A , τ ) 4 u = u ∗ , u ♯ ⊗ τ = u ∗ diag(1 , − )5 KO ( A , τ ) 2 u ♯ ⊗ τ = u ∗ KO ( A , τ ) 2 u = u ∗ , u ♯ ⊗ τ = − u (cid:18) i · − i · (cid:19) the Toeplitz operator associated with the subsemigroup ( Z ≥ ) n of Z n of symbol f .We write T n for the C ∗ -subalgebra of B ( l (( Z ≥ ) n )) generated by these Toeplitzoperators. The algebra T is the ordinary Toeplitz algebra and we simply write T . Note that the algebra T n is isomorphic to the n -fold tensor product of T . Thecomplex conjugation c on C induces an antiunitary operator of order two on theHilbert space l ( Z n ) by the pointwise operation, for which we also write c . Thisinduces a real structure c on B ( l (( Z ≥ ) n )) by c ( a ) = Ad c ( a ) = cac ∗ . We write τ T for the transposition on T n given by its restriction onto T n .We next focus on the case of n = 2. We consider the Hilbert space l ( Z ) and takeits orthonormal basis { δ m,n | ( m, n ) ∈ Z } , where δ m,n is the characteristic functionof the point ( m, n ) on Z . When f ∈ C ( T ) is given by f ( z , z ) = z m z n , we write M m,n for the multiplication operator M f . Let α < β be real numbers, and let H α , H β , ˆ H α,β and ˇ H α,β be closed subspaces of l ( Z ) spanned by { δ m,n |− αm + n ≥ } , { δ m,n |− βm + n ≤ } , { δ m,n |− αm + n ≥ − βm + n ≤ } , and { δ m,n |− αm + n ≥ − βm + n ≤ } , respectively. In the following, we may take α = −∞ or β = ∞ , but not both. Let P α , P β , ˆ P α,β and ˇ P α,β be the orthogonal projectionof l ( Z ) onto H α , H β , ˆ H α,β and ˇ H α,β , respectively. For f ∈ C ( T ), the operators P α M f P α on H α and P β M f P β on H β are called half-plane Toeplitz operators . Theoperator ˆ P α,β M f ˆ P α,β on ˆ H α,β is called the quarter-plane Toeplitz operator , andˇ P α,β M f ˇ P α,β on ˇ H α,β is its concave corner analogue. We write T α and T β for C ∗ -algebras generated by these half-plane Toeplitz operators and ˆ T α,β and ˇ T α,β for C ∗ -algebras generated by the quarter-plane and concave corner Toeplitz operators,respectively. There are ∗ -homomorphisms σ α : T α → C ( T ) and σ β : T β → C ( T ),which map P α M f P α and P β M f P β to the symbol f , respectively. We define the C ∗ -algebra S α,β as a pullback C ∗ -algebra of these two ∗ -homomorphisms. The realstructure c on H = l ( Z ) induces real structures c on T α , T β , ˆ T α,β , ˇ T α,β , and S α,β and thus induces transpositions τ α , τ β , ˆ τ α,β , ˇ τ α,β and τ S on T α , T β , ˆ T α,β ,ˇ T α,β and S α,β , respectively. For transpositions, we may simply write τ when it isclear from the context. The maps σ α and σ β preserve the real structures and we An operator A on a complex Hilbert space V is called the antiunitary operator if A is anantilinear bijection on V satisfying h Av, Aw i = h v, w i for any v and w in V . LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 7 have the following pull-back diagram:(2.1) ( S α,β , τ S ) p β / / p α (cid:15) (cid:15) ( T β , τ β ) σ β (cid:15) (cid:15) ( T α , τ α ) σ α / / ( C ( T ) , τ T )We write σ for the composition σ α ◦ p α = σ β ◦ p β . Let ˆ γ be a ∗ -homomorphism fromˆ T α,β to S α,β which maps ˆ P α,β M f ˆ P α,β to the pair ( P α M f P α , P β M f P β ). This mapˆ γ preserves the real structures, and there is the following short exact sequence of C ∗ ,τ -algebras (Park [54]):(2.2) 0 → ( K ( ˆ H α,β ) , τ K ) → ( ˆ T α,β , ˆ τ α,β ) ˆ γ → ( S α,β , τ S ) → , where the map from ( K ( ˆ H α,β ) , τ K ) to ( ˆ T α,β , ˆ τ α,β ) is the inclusion. Its concavecorner analogue is studied in [33] and the following exact sequence is obtained:(2.3) 0 → ( K ( ˇ H α,β ) , τ K ) → ( ˇ T α,β , ˇ τ α,β ) ˇ γ → ( S α,β , τ S ) → , where ˇ γ is a ∗ -homomorphism mapping ˇ P α,β M f ˇ P α,β to ( P α M f P α , P β M f P β ).3. KO -Groups of C ∗ -algebras Associated with Half-Plane andQuarter-Plane Toeplitz Operators In this section, the KO -theory for half-plane and quarter-plane Toeplitz oper-ators are discussed. In Sect. 3.1, KO -groups of the half-plane Toeplitz algebra iscomputed. Quarter-plane Toeplitz operators are discussed in the following sections,and the KO -groups of the C ∗ ,τ -algebra ( S α,β , τ S ) are computed in Sect. 3.2. InSect 3.3, the boundary maps of the 24-term exact sequence for KO -theory associ-ated with the sequence (2.2) are discussed and the KO -groups of the quarter-planeToeplitz algebra ( ˆ T α,β , ˆ τ α,β ) are computed.3.1. KO -Groups of ( T α , τ α ) . We compute the KO -groups of the C ∗ ,τ -algebra( T α , τ α ). The discussion is divided into two cases whether α is rational (or −∞ )or irrational.We first consider the case when α is a rational number or −∞ . When α ∈ Q ,we write α = pq where p and q are relatively prime integers and q is positive. Let m and n be integers such that − pm + qn = 1 and let(3.1) Γ = (cid:18) n − m − p q (cid:19) ∈ SL (2 , Z ) . Then, the action of Γ on Z induces the Hilbert space isomorphism H α ∼ = H and an isomorphism of C ∗ ,τ -algebras ( T α , τ α ) ∼ = ( T , τ ). Thus, the C ∗ ,τ -algebra( T α , τ α ) is isomorphic to ( T , τ T ) ⊗ ( C ( T ) , τ T ), and its KO -groups are computedas KO i ( T α , τ α ) ∼ = KO i ( C ( T ) , τ T ) ∼ = KO i ( C , id) ⊕ KO i − ( C , id). For the firstisomorphism, see Proposition 1 . . KO i ( C ( T ) , τ T )are obtained in Example 9.2 of [14], and the unital ∗ -homomorphism ι : C → T induces an isomorphism (id ⊗ ι ) ∗ : KO i ( C ( T ) , τ T ) → KO i (( C ( T ) , τ T ) ⊗ ( T , τ T )) ∼ = KO i ( T , τ ). Combined with them, KO -group KO i ( T α , τ α ) and its generators aregiven as follows. • KO ( T α , τ α ) ∼ = Z and its generator is [1 ]. S. HAYASHI • KO ( T α , τ α ) ∼ = Z ⊕ Z . A generator of Z is [ −
1] and that of Z is[ P α M q,p P α ]. • KO ( T α , τ α ) ∼ = ( Z ) . A generator of one Z is [ − I (2) ], and that of another Z is (cid:20)(cid:18) iP α M q,p P α − iP α M − q, − p P α (cid:19)(cid:21) . • KO ( T α , τ α ) ∼ = Z and its generator is [diag( P α M q,p P α , P α M − q, − p P α )]. • KO ( T α , τ α ) ∼ = Z and its generator is [1 ]. • KO ( T α , τ α ) ∼ = Z and its generator is [diag( P α M q,p P α , P α M q,p P α )]. • KO ( T α , τ α ) = KO − ( T α , τ α ) = 0 . The case of α = −∞ is computed similarly, and its generators are given by replacing p and q above with − α . In this case, complex K -groups of T α are computed by Ji–Kaminker and Xia in [34, 69]. Lemma 3.1.
For irrational α and for each i , we have KO i ( T α , τ α ) ∼ = KO i ( C , id) ,where the isomorphism is given by λ α ∗ .Proof. As for complex K -groups, we have K ( T α ) = Z and K ( T α ) = 0 by [34, 69].We consider a split ∗ -homomorphism of C ∗ ,τ -algebras λ α : ( T α , τ α ) → ( C , id) givenby the composition of σ α : ( T α , τ α ) → ( C ( T ) , τ T ) and the pull-back onto a fixedpoint of the involution ζ on T . Let T α = Ker λ α . By the six-term exact sequenceassociated with the extension 0 → T α → T α λ α → C →
0, complex K -groups of T α are trivial. For a C ∗ ,τ -algebra ( A , τ ), it follows from Theorem 1 .
12, Proposition 1 . .
18 of [13] that KO ∗ ( A , τ ) = 0 if and only if K ∗ ( A ) = 0. Therefore, KO ∗ ( T α , τ α ) = 0. The result follows from the 24-term exact sequence of KO -theory for C ∗ ,τ -algebras associated with the short exact sequence 0 → ( T α , τ α ) → ( T α , τ α ) λ α → ( C , id) → (cid:3) KO -Groups of ( S α,β , τ S ) . In this subsection, we compute the KO -groups ofthe C ∗ ,τ -algebra ( S α,β , τ S ). The basic tool is the following Mayer–Vietoris exactsequence associated with the pull-back diagram (2.1) (see Theorem 1 . .
15 of [61],for example):(3.2) · · · / / KO i +1 ( C ( T ) , τ T ) ∂ i +1 r r ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ KO i ( S α,β , τ S ) ( p α ∗ ,p β ∗ ) / / KO i ( T α , τ α ) ⊕ KO i ( T β , τ β ) σ β ∗ − σ α ∗ / / KO i ( C ( T ) , τ T ) ∂ i r r ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ KO i − ( S α,β , τ S ) · · · / / As in [54], the computation of the group KO ∗ ( S α,β , τ S ) is divided into three casescorresponding to whether α and β are rational (or ±∞ ) or irrational. As in Sect. 3,we have a unital ∗ -homomorphism λ α ◦ p α : ( S α,β , τ S ) → ( C , id) which splits. Cor-respondingly, the KO -group KO ∗ ( S α,β , τ S ) have a direct summand correspondingto KO ∗ ( C , id). Noting this, these KO -groups are computed by Lemma 3.1 and thesequence (3.2) when at least one of α and β is irrational. The results are collectedin Tables 4 and 5. In the rest of this subsection, we focus on the cases when both α and β are rational (or ±∞ ). LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 9
When α, β ∈ Q , we write α = pq and β = rs by using mutually prime integerswhere q and s are positive. In the following discussion, the case of α = −∞ corresponds to the case where p = − q = 0, and the case of β = + ∞ corresponds to the case where r = 1 and s = 0. By using the action of Γ ∈ SL (2 , Z )in (3.1) on Z , there are isomorphisms ( T α , τ α ) ∼ = ( T , τ ) and ( T β , τ β ) ∼ = ( T γ , τ γ ),where γ = tu for u = ns − mr and t = − ps + qr . Note that t is positive since α < β .We have the following commutative diagram: KO i ( T α , τ α ) ⊕ KO i ( T β , τ β ) σ β ∗ − σ α ∗ / / ∼ = (cid:15) (cid:15) KO i ( C ( T ) , τ T ) ∼ = (cid:15) (cid:15) KO i ( T , τ ) ⊕ KO i ( T γ , τ γ ) σ γ ∗ − σ ∗ / / KO i ( C ( T ) , τ T )where the vertical isomorphisms are induced by the action of Γ. In the following,we discuss the lower part of the diagram, which is enough for our purpose since theisomorphism KO i ( S α,β , τ S ) ∼ = KO i ( S ,γ , τ S ) is also induced. We write ϕ i for theabove map σ γ ∗ − σ ∗ . By the exact sequence (3.2), we have the following short exactsequence.(3.3) 0 → Coker( ϕ i +1 ) → KO i ( S ,γ , τ S ) → Ker( ϕ i ) → . We first compute kernels and cokernels of ϕ i . Cases for i = − , , , i = 1, groups KO ( T , τ ) and KO ( T γ , τ γ ) are both isomorphic to Z ⊕ Z .The Z direct summand is generated by [ − Z direct summand isgenerated by [ P M , P ] and [ P γ M u,t P γ ], respectively. They map to [ M , ] and[ M u,t ] in KO ( C ( T ) , τ T ) by σ ∗ and σ γ ∗ , respectively. We have KO ( C ( T ) , τ T ) ∼ = Z ⊕ Z , where the Z direct summand is generated by [ − m, n ) ∈ Z , theelement [ M m,n ] ∈ KO ( C ( T ) , τ T ) corresponds to (0 , m, n ) ∈ Z ⊕ Z . Therefore,Ker( ϕ ) ∼ = Z which is generated by ([ − , [ − Coker( ϕ ) ∼ = Z t .We next consider the case of i = 2. We have KO ( C ( T ) , τ T ) ∼ = Z ⊕ ( Z ) ⊕ Z ,where the first Z direct summand is generated by [ − I (2) ]. For ( m, n ) ∈ Z , theelement (cid:20)(cid:18) iM m,n − iM − m, − n (cid:19)(cid:21) in KO ( C ( T ) , τ T ) corresponds to (0 , m mod2 , n mod 2 , ∈ Z ⊕ ( Z ) ⊕ Z (Example 9.2 of [14]). The groups KO ( T , τ ) and KO ( T γ , τ γ ) and their generators are obtained in Sect. 3 and we haveKer( ϕ ) ∼ = ( ( Z ) when t is even, Z when t is odd, Coker( ϕ ) ∼ = (cid:26) Z ⊕ Z when t is even, Z when t is odd.When i = 3, we have KO ( C ( T ) , τ T ) ∼ = ( Z ) . For ( m, n ) ∈ Z , the element[diag( M m,n , M − m, − n )] ∈ KO ( C ( T ) , τ T ) corresponds to ( m mod 2 , n mod 2 , ∈ ( Z ) . By Sect. 3, we haveKer( ϕ ) ∼ = (cid:26) Z when t is even,0 when t is odd, Coker( ϕ ) ∼ = ( ( Z ) when t is even, Z when t is odd. When α = −∞ and β ∈ Q , we have Coker( ϕ ) ∼ = Z s . This is the case when p = − q = 0and t = − ps + qr = s in this case. A similar remark also holds for i = 2 , , When i = 5, we have KO ( C ( T ) , τ T ) ∼ = Z . For ( m, n ) ∈ Z , the element[diag( M m,n , M m,n )] ∈ KO ( C ( T ) , τ T ) corresponds to ( m, n ) ∈ Z . By Sect. 3, wehave Ker( ϕ ) = 0 and Coker( ϕ ) = Z t .Combined with the above computation and the exact sequence (3.3), KO -group KO i ( S α,β , τ S ) are computed, though some complication appears when i = 2 ,
3. Wediscuss these two cases in the following subsections.3.2.1.
The Group KO ( S α,β , τ S ) . We compute the group KO ( S ,γ , τ S ), which isisomorphic to KO ( S α,β , τ S ). The computation is divided into two cases dependingon whether t is even or odd. Note that u is odd when t is even since r and s aremutually prime.When t is odd, Ker( ϕ ) ∼ = Z is generated by ([ − I (2) ] , [ − I (2) ]) and the sequence(3.3) splits. Therefore, KO ( S α,β , τ S ) ∼ = ( Z ) .We next discuss the cases of even t . In this case, both of the kernel and thecokernel of ϕ are isomorphic to ( Z ) . Let g KO ( S ,γ , τ S ) be the kernel of the map λ α ∗ ◦ p α ∗ : KO ( S α,β , τ S ) → KO ( C , id) ∼ = Z which splits. Then, the sequence (3.3)reduces to the following extension:(3.4) 0 → ( Z ) → g KO ( S ,γ , τ S ) → Z → . In the following, we show that this sequence (3.4) splits. We find a lift of thegenerator of Z in g KO ( S ,γ , τ S ) and show this lift has order two. For ( m, n ) ∈ Z and κ = 0 and γ , we write T κm,n for P κ M m,n P κ , and let Q be the projection T γu, T γ − u, . Note that 1 − Q is the projection onto the closed subspace spanned by { δ m,n | ≤ γm − n < t } . For j = 1 , . . . , t , let P j be a projection in T γ , definedinductively as follows: P = (1 − Q ) M , − t +1 (1 − Q ) M ,t − (1 − Q ) ,P j = (1 − Q ) M , − t + j (1 − Q ) M ,t − j (1 − Q ) − j − X k =1 P k . Specifically, P j is the orthogonal projection of H γ onto the closed subspace spannedby { δ n,tn − j +1 } n ∈ Z . Note that P tj =1 P j = 1 − Q . For odd j = 1 , , . . . , t −
1, let s j = P j M , P j +1 − P j +1 M , − P j and s = P t − j =1 , odd s j . The element s satisfies therelations ( i ) s ∗ = − s , (ii) s τ = − s , (iii) s = − Q , (iv) Qs = sQ = 0 and (v) sT γu, = T γ − u, s = 0. Note that σ γ ( s ) = 0 since σ γ (1 − Q ) = 0. We first considerthe following elements: a = (cid:18) i · T − i · T (cid:19) ∈ M ( T ) , b ± = (cid:18) ± is iQ − iQ is (cid:19) ∈ M ( T γ ) , where the double-sign corresponds. Elements a and b ± are self-adjoint unitariessatisfying a τ = − a and b τ ± = − b ± , and pairs ( a, b ± ) are elements of M ( S ,γ );therefore, they define the elements of KO ( S ,γ , τ S ). Lemma 3.2.
As elements of KO ( S ,γ , τ S ) , we have [( a, b + )] = [( a, b − )] = 0 .Proof. We first show that [( a, b + )] = 0. For j = 1 , , . . . , t −
1, let r j = P j M , P j +1 + P j +1 M , − P j and r = P t − j =1 , odd r j . The element r satisfies ( i ) r ∗ = r , (ii) r τ = r ,(iii) r = 1 − Q , (iv) Qr = rQ = 0, (v) rT γu, = T γ − u, r = 0 and (vi) r anticommutes LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 11 ! !" " $ ! % !" $% & $% & $%! Figure 1.
The case of u = 1 and t = 4. 1 − Q is the projectiononto the closed subspace corresponding to lattice points in betweentwo lines (lattice points on the line y = γx are included, while thaton y = γ ( x −
1) are not). P j is the projection onto the closedsubspace spanned by { δ n, n − j +1 | n ∈ Z } . s j interchanges twopoints in a pair up to the sign.with s . For 0 ≤ θ ≤ π , let b θ = (cid:18) is cos θ iQ + ir sin θ − iQ − ir sin θ is cos θ (cid:19) , d = (cid:18) i · T γ − i · T γ (cid:19) . This b θ is a self-adjoint unitary satisfying b τθ = − b θ and b = b + . Therefore, b + and b π are homotopic in U (2)1 ( T γ , τ γ ). We further discuss b π . Let consider lattice points( m, n ) ∈ Z satisfying 0 ≤ γm − n < t , as indicated in Figure 1 for the case where u = 1 and t = 4. As in Figure 1, we divide these points to t pairs of lattice points:for n ∈ Z and odd j = 1 , , . . . , t −
1, a pair consists of { ( n, tn − j ) , ( n, tn − j + 1) } .The action of b π is closed on each pair of lattice points and is expressed by a 4 × C ⊗ C ; one C corresponds to a pair of lattice points, and theother C corresponds to the 2 × V be the followingmatrix. V = 12 −
11 1 − − − . Then V ∈ SO (4) and satisfies V i i − i − i V ∗ = i
00 0 0 i − i − i , where the left matrix inside the conjugation is the restriction of b π onto the closedsubspace spanned by generating functions of these two lattice points tensor C andthe right matrix is that of d (note that Q = 0 on these lattice points). Let W be theunitary on H γ ⊗ C defined by applying V to these pair of lattice points satisfying ≤ γm − n < t and the identity on the lattice points satisfying t ≤ γm − n , then wehave W b π W ∗ = d . Since SO (4) is path-connected, there is a path of self-adjointunitaries from b π to d preserving the relation of the KO -group. Summarizing, wehave a path in U (2)1 ( T γ , τ γ ) from b + to d . By its construction, the pair of the con-stant path at a ∈ M ( T ) and this path gives a path in U (2)1 ( S ,γ , τ S ) from ( a, b + )to ( a, d ). Therefore, we have [( a, b + )] = [( a, d )] = [ I (2) ] = 0 in KO ( S ,γ , τ S ).We next discuss the class [( a, b − )]. For 0 ≤ θ ≤ π , let b ′ θ = (cid:18) − is cos θ iQ + i (1 − Q ) sin θ − iQ − i (1 − Q ) sin θ is cos θ (cid:19) . Then, b ′ θ is a self-adjoint unitary satisfying ( b ′ θ ) τ = − b ′ θ . We have b ′ = b − and b ′ π = I (2) . Therefore, [( a, b − )] = [( a, b ′ π )] = [ I (2) ] = 0 in KO ( S ,γ , τ S ). (cid:3) Let consider the following elements: v = (cid:18) iT u, − iT − u, (cid:19) ∈ M ( T ) , w ± = (cid:18) ± is iT γu, − iT γ − u, (cid:19) ∈ M ( T γ ) , which are self-adjoint unitaries satisfying ( v ) τ = − v and w τ ± = − w ± . Since σ ( v ) = σ γ ( w ± ), pairs ( v , w ± ) are elements of M ( S ,γ ) satisfying ( v , w ± ) τ = − ( v , w ± ) and give elements [( v , w ± )] of the group KO ( S ,γ , τ S ). Lemma 3.3. In KO ( S ,γ , τ S ) , we have [( v , w + )] = [( v , w − )] .Proof. For 0 ≤ θ ≤ π , let consider the following element in M ( T γ ): R θ = is cos θ iT γu, − ir sin θ − iT γ − u, ir sin θ is cos θ iQ − iQ is , Then, we have R = w + ⊕ b + , R π = w − ⊕ b − and R θ is a self-adjoint unitarysatisfying R τθ = − R θ . Since σ γ ( R θ ) = σ ( v ⊕ a ), the pair ( v ⊕ a, R θ ) is containedin U (2)2 ( S ,γ , τ S ) and gives a path from ( v ⊕ a, w + ⊕ b + ) to ( v ⊕ a, w − ⊕ b − ). Byusing Lemma 3.2, we obtain the following equality in KO ( S ,γ , τ S ):[( v , w + )] = [( v ⊕ a, w + ⊕ b + )] = [( v ⊕ a, w − ⊕ b − )] = [( v , w − )] . (cid:3) Lemma 3.4. In KO ( S ,γ , τ S ) , the element [( v , w + )] has order two.Proof. For 0 ≤ θ ≤ π , let A θ = iT u, cos θ i sin θ − iT − u, cos θ − i sin θ − i sin θ iT u, cos θ i sin θ − iT − u, cos θ ∈ M ( T ) ,A γθ = is cos θ iT γu, cos θ i sin θ − iT γ − u, cos θ − i sin θ − i sin θ − is cos θ iT γu, cos θ i sin θ − iT γ − u, cos θ ∈ M ( T γ ) . Then, A θ and A γθ are self-adjoint unitaries satisfying ( A θ ) τ = − A θ and ( A γθ ) τ = − A γθ , and their pair ( A θ , A γθ ) is contained in M ( S ,γ ). Note that ( A , A γ ) = LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 13 ( v ⊕ v , w + ⊕ w − ). Therefore, by Lemma 3.3, the following equality holds in KO ( S ,γ , τ S ):2 · [( v , w + )] = [( v , w + )] + [( v , w − )] = [( A , A γ )] = [( A π , A γ π )] = 0 . (cid:3) Proposition 3.5.
When α and β are rational numbers and t = − ps + qr is even,we have KO ( S α,β , τ S ) ∼ = ( Z ) .Proof. Since u is odd when t is even, the pair ([ v ] , [ w + ]) ∈ KO ( T , τ ) ⊕ KO ( T γ , τ γ )constitutes a nontrivial element of the right Z ⊂ Ker( ϕ ) in the sequence (3.4).The element [( v , w + )] ∈ KO ( S ,γ , τ S ) is a lift of it. Therefore, [( v , w + )] is non-trivial and has order two by Lemma 3.4. This element belongs to g KO ( S ,γ , τ S )and, by mapping 1 ∈ Z to [( v , w + )], we obtain a splitting of the the sequence(3.4). Therefore, g KO ( S ,γ , τ S ) ∼ = ( Z ) and the result follows. (cid:3) The Group KO ( S α,β , τ S ) . We next compute KO ( S α,β , τ S ). Note thatKer( ϕ ) depends on whether t is even or odd. When t is odd, Ker( ϕ ) is zeroand, from the sequence (3.3), we have KO ( S α,β , τ S ) ∼ = Z .We next discuss the cases of even t . In this case, the extension (3.3) is of thefollowing form:(3.5) 0 → Z → KO ( S ,γ , τ S ) → Z → . As in Sect. 3.2.1, we show that this sequence splits by finding a lift of the generatorof the right Z in KO ( S ,γ , τ S ) of order two. Let consider the following elements: v = (cid:18) T u, T − u, (cid:19) ∈ M ( T ) , z ± = (cid:18) T γu, ± s T γ − u, (cid:19) ∈ M ( T γ ) , where the double-sign in the second equality corresponds. Pairs ( v , z ± ) are uni-taries in M ( S ,γ ) satisfying ( v , z ± ) ⊗ τ = ( v , z ± ) and define elements [( v , z ± )]of the KO -group KO ( S ,γ , τ S ). Lemma 3.6. In KO ( S ,γ , τ S ) , we have [( v , z + )] = [( v , z − )] .Proof. For 0 ≤ θ ≤ π , let z θ = (cid:18) T γu, e iθ s T γ − u, (cid:19) ∈ M ( T γ ) which gives a path { z θ } ≤ θ ≤ π of unitaries satisfying ( z θ ) ⊗ τ = z θ . Its endpoints are z = z + and z π = z − . The pair ( v , z θ ) satisfies ( v , z θ ) ⊗ τ = ( v , z θ ) and gives a homotopybetween ( v , z + ) and ( v , z − ) in U (3)1 ( S ,γ , τ S ). (cid:3) Lemma 3.7.
The element [( v , z + )] in KO ( S ,γ , τ S ) has order two.Proof. For 0 ≤ θ ≤ π , let B θ = T u, cos θ θ T − u, cos θ sin θ − sin θ T u, cos θ − sin θ T − u, cos θ ,B γθ = T γu, cos θ s cos θ θ T γ − u, cos θ sin θ − sin θ T γu, cos θ − s cos θ − sin θ T γ − u, cos θ . Table 3. KO ∗ ( S α,β , τ S ) when both α and β are rational (or ±∞ ). i t = − ps + qr — even odd even odd KO i ( S α,β , τ S ) Z ⊕ Z t ( Z ) ⊕ Z Z ⊕ Z ( Z ) ( Z ) Z ) Z Z ⊕ Z t Z Table 4. KO ∗ ( S α,β , τ S ) when one of α and β is rational (or ±∞ )and the other is irrational. i KO i ( S α,β , τ S ) Z ( Z ) ⊕ Z ( Z ) Z Z Z Table 5. KO ∗ ( S α,β , τ S ) when both α and β are irrational. i KO i ( S α,β , τ S ) Z ( Z ) ⊕ Z ( Z ) Z Z Z θ , matrices B θ and B γθ are unitaries satisfying ( B θ ) ⊗ τ = B θ and( B γθ ) ⊗ τ = B γθ . We have B γ = v ⊕ v and B γ = z + ⊕ z − . Note that matri-ces B π and B γ π are contained in M ( C ), where they coincide. Since this unitarysatisfies the symmetry of the KO -group, this is an element of the quaternionicunitary group U (2 , H ). Since U (2 , H ) is path-connected, there is a path of unitariesin U (3)2 ( S ,γ , τ S ) connecting ( B π , B γ π ) to (1 S ) . By using Lemma 3.6, we obtainthe following equality in KO ( S ,γ , τ S ):2 · [( v , z + )] = [( v ⊕ v , z + ⊕ z − )] = [( B π , B γ π )] = [(1 S ) ] = 0 . (cid:3) Proposition 3.8.
When α and β are rational numbers and t = − ps + qr is even,we have KO ( S α,β , τ S ) ∼ = ( Z ) .Proof. The pair ([ v ] , [ z + ]) ∈ KO ( T , τ ) ⊕ KO ( T γ , τ γ ) is contained in Ker( ϕ ) ∼ = Z and is nontrivial. The element [( v , z + )] ∈ KO ( S ,γ , τ S ) is its lift. Therefore,the class [( v , z + )] is nontrivial and has order two by Lemma 3.7. We thus obtaina splitting of the sequence (3.5) and the group KO ( S ,γ , τ S ) is isomorphic to( Z ) . (cid:3) The results in this subsection are summarized in Tables 3, 4 and 5.3.3.
Boundary Maps Associated with Quarter-Plane Toeplitz Extensionsand KO -Groups of ( ˆ T α,β , ˆ τ α,β ) . We next consider the boundary maps of the24-term exact sequence for KO -theory associated with the sequence (2.2):(3.6) ˆ ∂ i : KO i ( S α,β , τ S ) → KO i − ( K ( ˆ H α,β ) , τ K ) . Proposition 3.9.
For each i , the boundary map ˆ ∂ i in (3.6) is surjective.Proof. When i = − , , ,
6, the group KO i − ( K ( ˆ H α,β ) , τ K ) is trivial and the state-ment is obvious. We discuss the other cases. The proof is given by construct-ing explicit elements of the group KO i ( S α,β , τ S ), which maps to a generator ofthe group KO i − ( K ( ˆ H α,β ) , τ K ). As in [35], by using the action of SL (2 , Z ) on We write ˆ ∂ i for boundary maps associated with (2.2) and write ˇ ∂ i for that with (2.3). LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 15 Z , we assume 0 < α ≤ and 1 ≤ β < + ∞ without loss of generality. Letˆ P m,n = ˆ P α,β M m,n ˆ P α,β M − m, − n ˆ P α,β . As in [35], we consider the following elementin ˆ T α,β :(3.7) ˆ A = ˆ P , + M , (1 − ˆ P − , ) + M , ( ˆ P − , − ˆ P , ) . The operator ˆ A is Fredholm whose kernel is trivial and has one dimensional cokernel[35]. We also have the following. • ˆ γ ( ˆ A ) is a unitary in S α,β . • ˆ A is a real operator, that is r ( ˆ A ) = ˆ A , and ˆ A τ = r ( ˆ A ∗ ) = ˆ A ∗ holds.From these preliminaries, the proof of Proposition 3.9 is parallel to the computationin Example 9.4 of [14]. We summarize the results here. • Let u = ˆ γ ( ˆ A ). u is a unitary satisfying u τ = u ∗ and gives an element[ u ] ∈ KO ( S α,β , τ S ). ˆ ∂ ([ u ]) is a generator of KO ( K ( ˆ H α,β ) , τ K ) ∼ = Z . • Let u = (cid:18) i ˆ γ ( ˆ A ) − i ˆ γ ( ˆ A ) ∗ (cid:19) . u is a self-adjoint unitary satisfying u τ = − u and gives [ u ] ∈ KO ( S α,β , τ S ). Its image ˆ ∂ [ u ] is the generator of KO ( K ( ˆ H α,β ) , τ K ) ∼ = Z . • Let u = diag(ˆ γ ( ˆ A ) , ˆ γ ( ˆ A ) ∗ ) . u is a unitary satisfying u ♯ ⊗ τ = u and gives[ u ] ∈ KO ( S α,β , τ S ). Its image ˆ ∂ ([ u ]) is the generator of the group KO ( K ( ˆ H α,β ) , τ K ) ∼ = Z . • Let u = diag(ˆ γ ( ˆ A ) , ˆ γ ( ˆ A )). u is a unitary satisfying u ♯ ⊗ τ = u ∗ andgives [ u ] ∈ KO ( S α,β , τ S ). Its image ˆ ∂ ([ u )] is a generator of the group KO ( K ( ˆ H α,β ) , τ K ) ∼ = Z . (cid:3) Remark . In the case when α , β are both rational (or ±∞ ) and t = − ps + qr iseven, the group KO ( S ,γ , τ S ) ∼ = ( Z ) is generated by [ − I (2) ], [( v , w + )], [ u ] and[( w ′ , I (2)2 )], where w ′ = Y (3)4 diag(1 , − T , T , − , T , T , − − , − Y (3) ∗ . By the map σ ∗ : KO i ( S α,β , τ S ) → KO i ( C ( T ) , τ T ), components generated by [( v , w + )](when i = 2) and [( v , z + )] (when i = 3) maps injectively.The KO -groups of ( ˆ T α,β , ˆ τ α,β ) are computed by the 24-term exact sequence of KO -theory associated with (2.2) and Proposition 3.9. The results are collected inTables 6, 7 and 8. Remark . Similar results in this section also hold for convex corners. Letˇ A ∈ ˇ T α,β be an operator defined by replacing ˆ P α,β in the definition of ˆ A by ˇ P α,β .This ˇ A is a Fredholm operator satisfying ˇ A τ = ˇ A ∗ which have one dimensionalkernel and trivial cokernel [33]. As in Proposition 3.9, by using this example, wesee that the boundary maps ˇ ∂ i of KO -theory associated with the sequence (2.3)is surjective. The KO -groups of ( ˇ T α,β , ˇ τ α,β ) is computed by the 24-term exactsequence associated with (2.3), and the results are the same as in Tables 6, 7 and 8.Through the stabilization isomorphism, we have two boundary maps ˆ ∂ i and ˇ ∂ i from KO i ( S α,β , τ S ) to KO i − ( C , id) associated with (2.2) and (2.3). Since ˆ γ ( ˆ A ) = ˇ γ ( ˇ A ),the relation ˆ ∂ i = − ˇ ∂ i holds, as in Corollary 1 of [33]. The matrix Y (3)4 is introduced in Appendix A.3. Table 6. KO ∗ ( ˆ T α,β , ˆ τ α,β ) when both α , β are rational (or ±∞ ). i t = − ps + qr — even odd even odd even odd KO ∗ ( ˆ T α,β , ˆ τ α,β ) Z ⊕ Z t ( Z ) Z ( Z ) Z Z
04 5 6 7— — — — Z ⊕ Z t Table 7. KO ∗ ( ˆ T α,β , ˆ τ α,β ) when one of α and β is rational (or ±∞ ) and the other is irrational. i KO i ( ˆ T α,β , ˆ τ α,β ) Z ( Z ) ( Z ) Z Table 8. KO ∗ ( ˆ T α,β , ˆ τ α,β ) when both α and β are irrational. i KO i ( ˆ T α,β , ˆ τ α,β ) Z ( Z ) ( Z ) Z Toeplitz Operators Associated with Subsemigroup ( Z ≥ ) n of Z n In this section, Toeplitz operators associated with the subsemigroup ( Z ≥ ) n of Z n for n ≥ n -variable generalization of the ordinaryToeplitz and quarter-plane Toeplitz operators and are briefly discussed in [23, 22],where a necessary and sufficient condition for these Toeplitz operators to be Fred-holm is obtained. We revisit these operators since, in our application to condensedmatter physics, models of higher-codimensional corners are given by using these n -variable generalizations. Since the Toeplitz extension (4.1) and the quarter-planeToeplitz extension (2.3) provide a framework for these applications, we seek thisextension for our n -variable cases (Theorem 4.1). Note that we consider corners ofarbitrary codimension, though of a specific shape compared to the codimension-twocase [54]. In this section, let n be a positive integer bigger than or equals to three.To study such Toeplitz operators, we follow Douglas–Howe’s idea [23] to use thetensor product of the Toeplitz extension,(4.1) 0 → K ι → T γ → C ( T ) → , where K = K ( l ( Z ≥ )). There is a linear splitting of the ∗ -homomorphism γ givenby the compression onto l ( Z ≥ ). For a subset A ⊂ { , . . . , n } , let T n A = A ⊗ · · · ⊗ A n , where A i is C ( T ) when i ∈ A and is T when i / ∈ A . Note that T n ∅ is isomorphicto T n introduced in Sect. 2.2. For subsets D ⊂ R ⊂ { , . . . , n } , let π DR : T n D → T n R be the ∗ -homomorphism induced by γ . Specifically, π DR = a ⊗ · · · ⊗ a n , where a i is id C ( T ) when i ∈ D , is γ when i ∈ R \ D and is id T otherwise. Note that π DR isa surjection and π ∅∅ = id. In the following, we use a subset A of { , . . . , n } as alabel to distinguish C ∗ -algebras and the morphisms between them, which we mayabbreviate brackets {·} in our notation. For example, we write T n , for T n { , } , π i for π ∅{ i } and π , for π { }{ , } . For each A ⊂ { , . . . , n } , the map π A has a linear splitting ρ A : T n A → T n given by the compression onto l (( Z ≥ ) n ). By these preliminaries, LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 17 we consider the following C ∗ -subalgebra of T n ⊕ · · · ⊕ T nn . S n = ( ( T , · · · , T n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) For 1 ≤ i ≤ n, T i ∈ T ni , For 1 ≤ i < j ≤ n, π iij ( T i ) = π jij ( T j ) ) . Let ( T , . . . , T n ) ∈ S n . For a nonempty subset A ⊂ { , . . . , n } , we take i ∈ A andconsider the element π i A ( T i ) ∈ T n A . This element does not depend on the choice of i ∈ A , and we write T A = π i A ( T i ). Let ρ ′ : S n → T n be a linear map defined by ρ ′ ( T , . . . , T n ) = n X k =1 X |A| = k ( − k +1 ρ A ( T A )for ( T , . . . , T n ) ∈ S n , where the second summation is taken over all subsets A ⊂{ , . . . , n } consisting of k elements. Let K n = K ( l (( Z ≥ ) n )), and let γ n : T n → S n be an ∗ -homomorphism given by γ n ( T ) = ( π ( T ) , . . . , π n ( T )). Let ι n be the n -foldtensor product of ι . Theorem 4.1.
There is the following short exact sequence of C ∗ -algebras: (4.2) 0 → K n ι n → T n γ n → S n → , where the map γ n has a linear splitting given by ρ ′ .Proof. The map ι n is injective since ι is injective. We first show the exactness at T n .Since γ ◦ ι = 0, we have γ n ◦ ι n = 0, and thus, Im( ι n ) ⊂ Ker( γ n ). Let T ∈ Ker( γ n ).Since π ( T ) = ( γ ⊗ ⊗ · · · ⊗ T ) = 0, there exists some S ∈ K ⊗ T ⊗ · · · ⊗ T such that ( ι ⊗ ⊗ · · · ⊗ S ) = T . Since0 = (1 ⊗ γ ⊗ ⊗ · · · ⊗ T ) = (1 ⊗ γ ⊗ ⊗ · · · ⊗ ι ⊗ ⊗ · · · ⊗ S )= ( ι ⊗ ⊗ · · · ⊗ ⊗ γ ⊗ ⊗ · · · ⊗ S )and ι ⊗ ⊗ · · · ⊗ ⊗ γ ⊗ ⊗ · · · ⊗ S ) = 0. Therefore, thereexists some S ∈ K ⊗ K ⊗ T ⊗ · · · ⊗ T such that S = (1 ⊗ ι ⊗ ⊗ · · · ⊗ S ).By continuing this argument, we see that there exists some S n ∈ K ⊗ · · · ⊗ K ∼ = K n such that ( ι ⊗ · · · ⊗ ι )( S n ) = T . Thus, we have Ker( γ n ) ⊂ Im( ι n ).For the surjectivity of γ n , we see that ρ is a linear splitting of γ n , that is, for( T , . . . , T n ) ∈ S n and 1 ≤ i ≤ n , the relation π i ◦ ρ ′ ( T , . . . , T n ) = T i holds. In thefollowing, we show π ◦ ρ ′ ( T , . . . , T n ) = T and the other case is proved similarly.Note that(4.3) π ◦ ρ ′ ( T , . . . , T n ) = n X k =1 X |A| = k ( − k +1 π ◦ ρ A ( T A )and that π ◦ ρ ( T ) = T . Thus, it is sufficient to show that the sum over A ( = { } )is zero. Note that for 2 ≤ i < · · · < i k − ≤ n , we have π ◦ ρ i ,...,i k − ( T i ,...,i k − ) = π ◦ ρ ,i ,...,i k − ( T ,i ,...,i k − ) . By using this relation, we compute the sum on the right-hand side of (4.3). For k = 1 and k = 2 of the sum, we have the following: X |A| =1 , A6 = { } π ◦ ρ A ( T A ) − X |A| =2 π ◦ ρ A ( T A )= n X i =2 π ◦ ρ i ( T i ) − X ≤ i For each i , we have KO i ( S n , τ S ) ∼ = KO i ( C , id) ⊕ KO i − ( C , id) . The results are collected in Table 9.Proof. Note that KO i ( T n , τ T ) ∼ = KO i ( C , id). The result follows from the 24-termexact sequence of KO -theory associated with the sequence (4.4). (cid:3) A Fredholm Toeplitz operator associated with a codimension- n corner whoseFredholm index is one is constructed as follows. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 19 Example . Let T z be the Toeplitz operatorwhose symbol z : T → C is the inclusion. Its adjoint T ∗ z is a Fredholm operatoron l ( Z ≥ ) of index one. Let p = T z T ∗ z and q = 1 T − p , then p, q ∈ T andare projections onto l ( Z ≥ ) and C δ , respectively, where δ is the characteristicfunction of the point 0 ∈ Z . For a subset A ⊂ { , . . . , n } , let P n A = r ⊗ · · · ⊗ r n ,where r i is p when i ∈ A and is q otherwise. The operator P n A is a projection whichsatisfies P A P n A = 1 T n . Let ˜ T = T ∗ z ⊗ q ⊗ · · · ⊗ q and consider the following elementin T n : G = ˜ T + X A6 = { } P n A , where the sum is taken over all subsets of { , . . . , n } except { } . Then, we can seethat Ker( G ) ∼ = C and Coker( G ) = 0, that is, G is a Fredholm Toeplitz operatorassociated with codimension- n corners whose Fredholm index is one.This example leads to the following result. Proposition 4.6. The boundary maps of the six-term exact sequence for K -theoryassociated with (4 . are surjective. Moreover, the boundary maps of the 24-termexact sequence for KO -theory associated with (4 . are surjective.Proof. The result for complex K -theory is immediate from Example 4.5. For KO -theory, since the operator G in Example 4.5 satisfies G τ = G ∗ , the result follows asin Proposition 3.9. (cid:3) Note that, we have γ n ( G ) = ( π ( G ) , , · · · , ∈ S n by using π ( G ) = M ∗ z ⊗ q ⊗ · · · ⊗ q + X ∅6 = A⊂{ ,...,n − } C ( T ) ⊗ P n − A , The element γ n ( G ) is a unitary and defines an element [ γ n ( G )] of the group K ( S n ).Since the Fredholm index of G is one, this gives a generator of K ( S n ) ∼ = Z . Asin the proof of Proposition 3.9, generators of the KO -groups KO i ( S n , τ S ) are alsoobtained by using G . Remark . Let 1 ≤ j ≤ n . We have the following ∗ -homomorphisms: S n −→ T n − ⊗ C ( T ) γ n − ⊗ −→ S n − ⊗ C ( T ) , where the first map maps ( T , . . . , T n ) to T j . We write σ n,n − for the compositeof the above maps which induces the map σ n,n − ∗ : K i ( S n ) → K i ( S n − ⊗ C ( T )).When i = 0, K ( S n ) ∼ = Z is generated by [1 ] and σ n,n − ∗ [1 ] = [1 ]. When i = 1,the map σ n,n − ∗ is zero since, by Example 4.5, the element [ γ n ( G )] is a generatorof K ( S n ) ∼ = Z and σ n,n − ∗ [ γ n ( G )] = [1] = 0. A similar observation also holdsin real cases. The map σ n,n − ∗ from KO i ( S n , τ S ) to KO i ( S n − ⊗ C ( T ) , τ ) mapsdirect summands corresponding to KO -groups of a point injectively and the othercomponents to zero.5. Topological invariants and corner states in Altland–Zirnbauerclassification In this section, some gapped Hamiltonians on a lattice with corners are discussedin each of the Altland–Zirnbauer classes. Since two of them (class A and AIII) arealready studied in [32, 33], we consider the remaining cases here. The codimen-sion of the corner will be arbitrary, though we mainly discuss codimension-two cases, with many detailed results being obtained by [54, 35, 33] and the results inSect. 3. Higher-codimensional cases are discussed in a similar way, whose resultsare collected in Sect. 5.5.5.1. Setup. Let V be a finite rank Hermitian vector space of complex rank N .Let n be a positive integer. Let Θ and Ξ be antiunitary operators on V whosesquares are +1 or − 1. Let Π be a unitary operator on V whose square is one.These operators Θ, Ξ and Π are naturally extended to the operator on l ( Z n ; V )by the fiberwise operation; we also denote them as Θ, Ξ and Π, respectively. LetHerm( V ) be the space of Hermitian operators on V . We consider a continuous map T n → Herm( V ), t H ( t ), where t = ( t , t , . . . , t n ) in T n . Through the Fouriertransform L ( T n ; V ) ∼ = l ( Z n ; V ), the multiplication operator generated by thiscontinuous map defines a bounded linear self-adjoint operator H on the Hilbertspace l ( Z n ; V ). We consider the lattice Z n as a model of the bulk and call H the bulk Hamiltonian . The Hamiltonian is said to preserve time-reversal symmetry (TRS) if it commutes with Θ (i.e., Θ H Θ ∗ = H ), particle-hole symmetry (PHS) if itanticommutes with Ξ (i.e., Ξ H Ξ ∗ = − H ) and chiral symmetry if it anticommuteswith Π (i.e., Π H Π ∗ = − H ). Furthermore, TRS or PHS is called even (resp. odd )if Θ = 1 or Ξ = 1 (resp. Θ = − = − i ΘΞ such that Π = 1 is satisfied.By taking the partial Fourier transform in the variables t and t , we obtain acontinuous family of bounded linear self-adjoint operators { H ( t ) } t ∈ T n − on H ⊗ V .By taking a compression onto H α ⊗ V , H β ⊗ V and ˆ H α,β ⊗ V , we obtain a familyof operators H α ( t ), H β ( t ) and ˆ H α,β ( t ) parametrized by t = ( t , . . . , t n ) ∈ T n − . H α ( t ) and H β ( t ) are our models for two surfaces (codimension-one boundaries),and ˆ H α,β ( t ) is our model of the corner (codimension-two corner). We assume thefollowing spectral gap condition. Assumption 5.1 (Spectral Gap Condition) . We assume that both H α and H β areinvertible. Under this assumption, the bulk Hamiltonian H is also invertible since, whenwe take a basis of V and identify V with C N , there is a unital ∗ -homomorphism M N ( S α,β ⊗ C ( T n − )) → M N ( C ( T n )) that maps ( H α , H β ) to H . In classes AI andAII, we further assume that the spectrum of H is not contained in either R > or R < .Note that in other classes where Hamiltonians preserve PHS or chiral symmetry,this condition follows from Assumption 5.1. Let h be the pair ( H α , H β ). UnderAssumption 5.1, we set(5.1) sign( h ) = h | h | − . When the bulk Hamiltonian H satisfies TRS, PHS or chiral symmetry, the operators H α , H β , ˆ H α,β and sign( h ) also satisfy the symmetry, that is, commutes with Θ oranticommutes with Ξ or Π.5.2. Gapped Topological Invariants. In the following, starting from a Hamil-tonian satisfying Assumption 5.1 in each class AI, BDI, D, DIII, AII, CII, C andCI, we construct a unitary and see that this unitary satisfies the relation R i inTable 2. By using this unitary, we define a topological invariant as an element ofsome KO -group. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 21 In class AI, the Hamiltonian has even TRS. We take an orthonormal basis of V to identify V with C N and express Θ as C = diag( c, . . . , c ). Under our spectral gapcondition, let(5.2) u = (cid:18) sign( h ) 00 1 N (cid:19) . This u is a self-adjoint unitary satisfying u τ = Ad C⊕C ( u ∗ ) = u ∗ by the TRS.In class BDI, the Hamiltonian has both even TRS and even PHS. Note thatthe chiral symmetry is given by Π = ΘΞ and commutes with Θ and Ξ. For aHamiltonian satisfying chiral symmetry and Assumption 5.1 to exist, the even/odddecomposition V ∼ = V ⊕ V with respect to Π should satisfy rank C V = rank C V ,and we assume that. Then, there is an orthonormal basis of V to identify V with C N such that Π and Θ are expressed as follows:(5.3) Π = (cid:18) − (cid:19) , Θ = (cid:18) C C (cid:19) , where C = diag( c, . . . , c ). Since the Hamiltonian H anticommutes with Π, theoperator sign( h ) in (5.1) is written in the following off-diagonal form:(5.4) sign( h ) = (cid:18) u ∗ u (cid:19) , where u is a unitary. By the TRS, we have u τ = C u ∗ C ∗ = u ∗ .In class D, the Hamiltonian has even PHS. We take an orthonormal basis of V to identify V ∼ = C N and express Ξ as C = diag( c, . . . , c ). Let u = sign( h ), then wehave u τ = Ξ u Ξ ∗ = − u by the PHS.In class DIII, the Hamiltonian has both odd TRS and even PHS. Note that thechiral symmetry is given by Π = i ΘΞ and anticommutes with Θ and Ξ. For such aHamiltonian H satisfying Assumption 5.1 to exist, the complex rank of V must bea multiple of 4 since sign( H ( t )), i Π, i and Θ provides a Cl , ⊗ Cl , ∼ = H (2)-modulestructure on V . We assume rank C V = 4 M for some positive integer M . Lemma 5.2. If a Hamiltonian H satisfying Assumption 5.1 exists, there is anorthonormal basis of V such that Π and Θ are expressed as follows. Π = (cid:18) − (cid:19) , Θ = (cid:18) JJ (cid:19) . We write J = diag( j, . . . , j ) , where j is the quaternionic structure on H .Proof. By using Π, we decompose V ∼ = V ⊕ V . We identify V ∼ = V ∼ = C M ∼ = H M , on which we consider J = diag( j, . . . , j ). Let U = Θ (cid:18) JJ (cid:19) . Since U isa unitary and commutes with Π, we have U = diag( u , u ), where u and u areunitaries on C M . Since Θ = − 1, we have u = Ad J ( u ∗ ). Let W = diag( − u ∗ , W Θ W ∗ = (cid:18) JJ (cid:19) . (cid:3) We take this basis on V and express Π and Θ as above. By the chiral symmetry,we take u in (5.4). By the TRS, we have u ♯ ⊗ τ = J u ∗ J ∗ = u .In class AII, the Hamiltonian has odd TRS. The space V has a quaternionicstructure given by Θ, and the complex rank of V is even, for which we write2 M . There is an orthonormal basis of V for identifying V with C M ∼ = H M and Table 10. i ( ♠ ) and c ( ♠ ) for each of the Altland–Zirnbauer classes ♠ . ♠ AI BDI D DIII AII CII C CI i ( ♠ ) 0 1 2 3 4 5 6 − c ( ♠ ) 1 1 i i J = diag( j, . . . , j ). Let u be a self-adjoint unitary in (5.2). By theTRS, we have u ♯ ⊗ τ = Ad J ⊕J ( u ∗ ) = u ∗ .In class CII, the Hamiltonian has both odd TRS and odd PHS. The chiral sym-metry is given by Π = ΘΞ and commutes with Θ and Ξ. As in the class BDI case,we take an orthonormal basis of V to identify V with C N and express Π and Θ as(5.5) Π = (cid:18) − (cid:19) , Θ = (cid:18) J J (cid:19) , where J = diag( j, . . . , j ). By the chiral symmetry, we take u in (5.4). By the TRS,we have u ♯ ⊗ τ = J u ∗ J ∗ = u ∗ .In class C, the Hamiltonian has odd PHS. Since Ξ provides a quaternionic struc-ture on V , its complex rank is even, for which we write 2 M . We take an orthonormalbasis of V to identify V with C M ∼ = H M and express Ξ as J = diag( j, . . . , j ). Let u = sign( h ), then we have u ♯ ⊗ τ = J u ∗ J ∗ = − u by the PHS.In class CI, the Hamiltonian has both even TRS and odd PHS. The chiral sym-metry is given by Π = i ΘΞ and anticommutes with Θ and Ξ. As in Lemma 5.2, wetake an orthonormal basis of V to express,Π = (cid:18) − (cid:19) , Θ = (cid:18) CC (cid:19) , where C = diag( c, . . . , c ). By the chiral symmetry, we take u in (5.4). By the TRS,we have u τ = C u ∗ C ∗ = u . Definition 5.3. For a Hamiltonian in class ♠ = AI, BDI, D, DIII, AII, CII, C or CIsatisfying Assumption 5.1, let u be the unitary defined as above. As we have seen,this unitary u satisfies the relation R i ( ♠ ) where i ( ♠ ) is as indicated in Table 10. Wedenote its class [ u ] in the KO -group KO i ( ♠ ) ( S α,β ⊗ C ( T n − ) , τ ) by I n, , ♠ Gapped ( H ).The groups KO ∗ ( S α,β ⊗ C ( T n − ) , τ ) are computed by results in Sect. 3.2. Remark . We expressed the symmetry operators in a specific way, though wemay choose another one. In class DIII, for example, the operator Θ can also beexpressed as (cid:18) −CC (cid:19) , where C = diag( c, . . . , c ). Then, we obtain unitariessatisfying u τ = − u , which are treated in [38].5.3. Gapless Topological Invariants. We next define another topological invari-ant by using our model for the corner ˆ H α,β . By Assumption 5.1 and Theorem 2 . { ˆ H α,β ( t ) } t ∈ T n − is a continuous family of self-adjoint Fredholm operators.Corresponding to its Altland–Zirnbauer classes, this family provides a Z -map from( T n − , ζ ) to some Z -spaces of self-adjoint or skew-adjoint Fredholm operators in-troduced in Appendix A as follows. We simply write τ in place of τ S ⊗ τ T . In the following, these abbreviations for tensor productsof transpositions are employed, though the meaning will be clear from the context. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 23 • Class AI, Z -map ˆ H α,β : ( T n − , ζ ) → (Fred (0 , ∗ , r Θ ). • Class BDI, let ǫ = Π. Z -map ˆ H α,β : ( T n − , ζ ) → (Fred (0 , ∗ , r Θ ). • Class D, Z -map i ˆ H α,β : ( T n − , ζ ) → (Fred (1 , ∗ , r Ξ ). • Class DIII, let e = i Π. Z -map ˆ H α,β : ( T n − , ζ ) → (Fred (1 , ∗ , q Θ ). • Class AII, Z -map ˆ H α,β : ( T n − , ζ ) → (Fred (0 , ∗ , q Θ ). • Class CII, let ǫ = Π. Z -map ˆ H α,β : ( T n − , ζ ) → (Fred (0 , ∗ , q Θ ). • Class C, Z -map i ˆ H α,β : ( T n − , ζ ) → (Fred (1 , ∗ , q Ξ ). • Class CI, let e = i Π. Z -map ˆ H α,β : ( T n − , ζ ) → (Fred (1 , ∗ , r Θ ).Here, we write r Θ = Ad Θ when Θ = 1 and q Θ = Ad Θ when Θ = − 1. Involutions r Ξ and q Ξ are defined as Ad Ξ in the same way. By Corollary A.9, the Z -homotopyclasses of Z -maps from ( T n − , ζ ) to the above Z -space of self-adjoint or skew-adjoint Fredholm operators is isomorphic to the KO -group KO i ( C ( T n − ) , τ T ) ofsome degree i . Definition 5.5. For ♠ = AI, BDI, D, DIII, AII, CII, C or CI, let i ( ♠ ), c ( ♠ ) andFred ♠ be numbers and the Z -space as in Tables 10 and 11. For a Hamiltonian H in class ♠ satisfying Assumption 5.1, we denote I n, , ♠ Gapless ( H ) for the class [ c ( ♠ ) ˆ H α,β ]in KO i ( ♠ ) − ( C ( T n − ) , τ T ). We call I n, , ♠ Gapless ( H ) the gapless corner invariant .If the gapless corner invariant is nontrivial, zero is contained in the spectrum ofˆ H α,β . In Sect. 5.6, we discuss more refined relations between the gapless cornerinvariant and corner states when k = n − n . Table 11. In each Altland–Zirnbauer class ♠ , gapped invariantsand gapless invariants are defined as elements of some K - and KO -groups of some degree, as indicated in this table. Classifyingspaces for topological K - and KR -groups through self-adjoint orskew-adjoint Fredholm operators and unitaries are also included.( Z -)spaces Fred ♠ and U ♠ are introduced in Appendix A.AZ Gapped Gapless ♠ K -group K -group Fred ♠ U ♠ A K K Fred (0 , ∗ U cpt AIII K K Fred U (0 , AI KO KO − (Fred (0 , ∗ , r Θ ) ( U cpt , r ◦ ∗ )BDI KO KO (Fred (0 , ∗ , r Θ ) ( ∼ = (Fred (1 , ∗ , r Ξ )) ( U (0 , , r )D KO KO (Fred (1 , ∗ , r Ξ ) ( U cpt , r )DIII KO KO (Fred (1 , ∗ , q Θ ) ( ∼ = (Fred (2 , ∗ , r Ξ )) ( U (1 , , r )AII KO KO (Fred (0 , ∗ , q Θ ) ( U cpt , q ◦ ∗ )CII KO KO (Fred (0 , ∗ , q Θ ) ( ∼ = (Fred (1 , ∗ , q Ξ )) ( U (0 , , q )C KO KO (Fred (1 , ∗ , q Ξ ) ( U cpt , q )CI KO − KO (Fred (1 , ∗ , r Θ ) ( ∼ = (Fred (2 , ∗ , q Ξ )) ( U (1 , , q )5.4. Correspondence. By taking a tensor product of the extension (2.2) and( C ( T n − ) , τ T ), we have the following short exact sequence of C ∗ ,τ -algebras,0 → ( K ⊗ C ( T n − ) , τ ) → ( ˆ T α,β ⊗ C ( T n − ) , τ ) → ( S α,β ⊗ C ( T n − ) , τ ) → . Let consider the following diagram containing the boundary map of 24-term exactsequence for KO -theory associated with this sequence: KO i ( ♠ ) ( S α,β ⊗ C ( T n − ) , τ ) ˆ ∂ i ( ♠ ) / / L (cid:15) (cid:15) KO i ( ♠ ) − ( K ( ˆ H α,β ) ⊗ C ( T n − ) , τ )[( T n − , ζ ) , Fred ♠ ] Z ∼ = / / [( T n − , ζ ) , F ♠ ] Z ∼ = exp O O where F ♠ is the Z -subspace of Fred ♠ as in Appendix A, whose inclusion F ♠ ֒ → Fred ♠ is the Z -homotopy equivalence. Maps L and exp are as follows. • When i ( ♠ ) is odd, for [ u ] ∈ KO i ( ♠ ) ( S α,β ⊗ C ( T n − ) , τ ), we take a lift a of u and consider the matrix A as in Definition A.13, and set L ([ u ]) = [ A ].The map exp is defined as in Definition A.13. • When i ( ♠ ) = 0 , 4, for [ u ] ∈ KO i ( ♠ ) ( S α,β ⊗ C ( T n − ) , τ ), we take a self-adjoint lift a of u as in Definition 8.3 of [14] and set L ([ u ]) = [ a ]. The mapexp is defined by exp([ a ′ ]) = [ − exp( πia ′ )]. • When i ( ♠ ) = 2 , 6, for [ u ] ∈ KO i ( ♠ ) ( S α,β ⊗ C ( T n − ) , τ ), we take a self-adjoint lift a of u as in Definition 8.3 of [14] and set L ([ u ]) = [ ia ]. The mapexp is defined by exp([ a ′ ]) = [ − exp( πa ′ )].In each case, the map exp is an isomorphism by Proposition A.7 and Sect. A.2. Forboundary maps ˆ ∂ i ( ♠ ) , we use its expressions through exponentials (see [14] for even i ( ♠ ) and Appendix A.3 for odd i ( ♠ )) and the diagram commutes. Note that, byProposition 3.9, the boundary map ˆ ∂ i ( ♠ ) is surjective. The following is the mainresult of this section. Theorem 5.6. ˆ ∂ i ( ♠ ) ( I n, , ♠ Gapped ( H )) = I n, , ♠ Gapless ( H ) .Proof. The operator ˆ H α,β is a self-adjoint lift of ( H α , H β ) and preserves the sym-metries of the class ♠ . Therefore, we have L ( I n, , ♠ Gapped ( H )) = [ c ( ♠ ) ˆ H α,β ] and theresults follows from the commutativity of the above diagram. (cid:3) Remark . Under Assumption 5.1, the bulkHamiltonian H is also invertible. When we take H in place of h = ( H α , H β )and define the unitary u ′ as in Sect. 5.2, this unitary defines an element [ u ′ ]in KO i ( ♠ ) ( C ( T n ) , τ T ), which classifies bulk invariants in class ♠ . A relation be-tween our gapped invariants I n, , ♠ Gapped ( H ) and these bulk invariants can be discussedthrough the map ( σ ⊗ ∗ : KO i ( ♠ ) ( S α,β ⊗ C ( T n − ) , τ ) → KO i ( ♠ ) ( C ( T n − ) , τ T ),which maps [ u ] to [ u ′ ] and we briefly mention its consequences here. Our gappedinvariant I n, , ♠ Gapped ( H ) has no relation with bulk invariants in the sense that, underAssumption 5.1, bulk invariants are trivial except for a component corresponding to KO -groups of a point and for the cases when α , β are both rational (or ±∞ ) and t = − ps + qr is even. By Remark 3.10, when α and β takes these values, some bulkweak invariants can be nontrivial, though they have no relation with I n, , ♠ Gapless ( H ),which can be seen by comparing the above map ( σ ⊗ ∗ and the boundary mapˆ ∂ i ( ♠ ) . This component maps to zero by the boundary map ˆ ∂ i ( ♠ ) and has no relation with gaplesscorner invariants. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 25 Remark . When we fix α and β , there exist twomodels of corners: convex and concave corners ( ˆ H α,β and ˇ H α,β ). We have discussedconvex corners though, as in [33], similar results also hold for concave corners byusing (2.3) in place of (2.2) in our discussion. By Remark 3.11, the gapless invariantsof these two are related by the factor − Higher-Codimensional Cases. Let n and k be positive integers satisfying3 ≤ k ≤ n . In this subsection, we consider n -D system with a codimension- k corner. Let d = n − k . We consider a continuous map T n → Herm( V ) and thebounded linear self-adjoint operator H on l ( Z n ) generated by this map, whichis our model of the bulk. We next introduce models of corners of codimension k − k corner. For this, we choose d variables t j , . . . , t j d in t , t , . . . , t n and consider the partial Fourier transform inthese d variables to obtain a continuous family of self-adjoint operators { H ( t ) } t ∈ T d on l ( Z k ; V ). On the Hilbert space l ( Z k ; V ) ∼ = ( l ( Z ) ⊗· · ·⊗ l ( Z )) ⊗ V , we considerprojections P k = ( P ≥ ⊗ · · · ⊗ P ≥ ) ⊗ V , and P k,i = ( P ≥ ⊗ · · · ⊗ ⊗ · · · ⊗ P ≥ ) ⊗ V for 1 ≤ i ≤ k where inside the brackets is the tensor products of P ≥ except for the i -th tensor product replaced by the identity. By using these projections, we definethe following operators: H c ( t ) = P k H ( t ) P k , H i ( t ) = P k,i H ( t ) P k,i , for 1 ≤ i ≤ k and for t ∈ T d . These two are our model for a codimension k corner and codimension k − V , wehave ( H ( t ) , . . . , H k ( t )) ∈ M N ( S k ) by the construction. We assume the followingcondition in this subsection. Assumption 5.9 (Spectral Gap Condition) . We assume that our models for codi-mension k − corners H , . . . , H k are invertible. Under this assumption, the model for the bulk, surfaces and corners of codimen-sion less than k , whose intersection makes our codimension- k corner, are invertible.As in Sect. 5.1, let h = ( H , . . . , H k ). Definition 5.10. For a Hamiltonian in class ♠ = AI, BDI, D, DIII, AII, CII, C orCI satisfying Assumption 5.9, let u be a unitary defined by using this h in placeof that in Sect. 5.2. As in Sect. 5.2, this unitary u satisfies the relation R i ( ♠ ) where i ( ♠ ) is as indicated in Table 10. We denote its class [ u ] in the KO -group KO i ( ♠ ) ( S k ⊗ C ( T d ) , τ ) by I n,k, ♠ Gapped ( H ).The KO -groups KO i ( S k ⊗ C ( T d ) , τ ) are computed by using Proposition 4.4. Foreach t ∈ T d , the operator H c ( t ) is Fredholm by Corollary 4.2. Definition 5.11. For ♠ = AI, BDI, D, DIII, AII, CII, C or CI, let i ( ♠ ), c ( ♠ ) andFred ♠ be numbers and the Z -space as in Tables 10 and 11. For a Hamiltonian H in class ♠ satisfying Assumption 5.9, we denote I n,k, ♠ Gapless ( H ) for the class [ c ( ♠ ) H c ]in KO i ( ♠ ) − ( C ( T d ) , τ T ). We call I n,k, ♠ Gapless ( H ) the gapless corner invariant .We next discuss a relation between these two topological invariants. As inSect. 5.4, we consider a tensor product of the extension (4.4) and ( C ( T d ) , τ T ) andlet ∂ i ( ♠ ) : KO i ( ♠ ) ( S k ⊗ C ( T d ) , τ ) → KO i ( ♠ ) − ( K k ⊗ C ( T d ) , τ ) be the boundarymap associated with it expressed through exponentials. Since H c is a self-adjointlift of ( H , . . . , H k ), the following relation holds, as in Theorem 5.6. Theorem 5.12. ∂ i ( ♠ ) ( I n,k, ♠ Gapped ( H )) = I n,k, ♠ Gapless ( H ) . Remark . As in Remark 5.7, under Assumption 5.9, some gapped invariantsrelated to corner states for corners of codimension < k are also defined, though, byRemark 4.7, they are trivial except for a component corresponding to KO -groupsof a point. Remark . Gapless corner invariants for each systems are elements of the group KO i ( C ( T d ) , τ T ) ∼ = L dj =0 (cid:0) dj (cid:1) KO i − j ( C , id). As in the case of (first-order) topologicalinsulators [42], we call the component KO i − d ( C , id) strong and others weak .Complex cases can also be discussed in a similar way . For class A systemswith a codimension ≥ K ( S k ⊗ C ( T d )) and K ( C ( T d )), respectively, andthe boundary map ∂ : K ( S k ⊗ C ( T d )) → K ( K k ⊗ C ( T d )) associated with (4.4)relates these two, which is surjective by Proposition 4.6. In class AIII systems,we use ∂ : K ( S k ⊗ C ( T d )) → K ( K k ⊗ C ( T d )) instead. Gapless corner invariantstakes value in K i ( C ( T d )) ∼ = L dj =0 (cid:0) dj (cid:1) K i − j ( C ), and we call the component K i − d ( C ) strong and others weak .Strong invariants for each system are classified in Table 1.5.6. Numerical Corner Invariants. Our gapless corner invariants are definedas elements of some KO -group. In this subsection, we introduce Z - or Z -valuednumerical corner invariants for our systems in cases where k = n and k = n − k = n and D, DIII, AII and CII when k = n − Case of k = n . In this case, our model of the corner H c is a self-adjoint Fred-holm operator which has some symmetry corresponding to its Altland–Zirnbauerclass . An appropriate definition of numerical topological invariants is introducedin [9] and we put them in our framework.In class BDI, the operator H c is an element of the fixed point set (Fred (0 , ∗ ) r Θ of the involution r Θ , where the Clifford action of Cl , on the Hilbert space is givenby ǫ = Π (see also Lemma A.10 and Remark A.11). We express Π and Θ as in(5.3) and express H c as follows.(5.6) H c = (cid:18) h c ) ∗ h c (cid:19) . The operator h c is a Fredholm operator that commutes with C and thus is a realFredholm operator. Its Fredholm index isind( h c ) = rank C Ker( h c ) − rank C Coker( h c ) = Tr(Π | Ker( H c ) ) , where the right-hand side is the trace of Π restricted to Ker( H c ). The Fredholmindex induces an isomorphism ind BDI : [(pt , id) , (Fred (0 , ∗ , r Θ )] Z → Z . In [32, 33], there is a mistake in the computations of the group K ( S α,β ) in the case where α and β are rational numbers (there is a torsion part in general, as in KO ( S α,β , τ S ) computedin Sect. 4), which is correctly stated in [55]. The author would like to thank Guo Chuan Thiangfor pointing this mistake out. In what follows, we also write H c for ˆ H α,β in k = 2 case. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 27 In class D, iH c commutes with the real structure Ξ and is a real skew-adjointFredholm operator. Its mod 2 index [9] isind ( iH c ) = rank C Ker( H c ) mod 2 , which induces the isomorphism ind D : [(pt , id) , (Fred (1 , ∗ , r Ξ )] Z → Z . In class DIII, H c is an element of (Fred (1 , ∗ ) q Θ , where the action of Cl , is givenby e = i Π. The operator iH c and e commute with the real structure Ξ; thus, iH c is a real skew-adjoint Fredholm operator that anticommutes with e . Its mod2 index [9] is ind ( iH c ) = 12 rank C Ker( H c ) mod 2 , which induces the isomorphism ind DIII : [(pt , id) , (Fred (1 , ∗ , q Θ )] Z → Z . In class CII, the operator H c is an element of (Fred (0 , ∗ ) q Θ , where the Cliffordaction of Cl , is given by ǫ = Π. We express Θ and Π as in (5.5) and express H c as in (5.6). The operator h c commutes with J and is a quaternionic Fredholmoperator. Its Fredholm index ind( h c ) is an even integer that induces an isomorphismind CII : [(pt , id) , (Fred (0 , ∗ , q Θ )] Z → Z . Definition 5.15. For n -D systems with codimension- n corners in classes BDI, D,DIII and CII, we define the numerical corner invariant as follows. • In class BDI, let N n,n, BDIGapless ( H ) = ind( h c ) ∈ Z . • In class D, let N n,n, DGapless ( H ) = ind ( iH c ) ∈ Z . • In class DIII, let N n,n, DIIIGapless ( H ) = ind ( iH c ) ∈ Z . • In class CII, let N n,n, CIIGapless ( H ) = ind( h c ) ∈ Z .Note that by these definitions, they are images of gapless corner invariants I n,n, ♠ Gapless ( H ) for each class ♠ = BDI, D, DIII and CII through the isomorphismind ♠ . In each case, the numerical corner invariant is computed through Ker( H c )and is related to the number of corner states.5.6.2. Case of k = n − . In this case, { H c ( t ) } t ∈ T is a continuous family of self-adjoint Fredholm operators preserving some symmetry. The numerical corner in-variants are given by using ( Z -valued) spectral flow [8] and its Z -valued variants[19, 18]. We first review Z - and Z -valued spectral flow.Spectral flow is, roughly speaking, the net number of crossing points of eigen-values of a continuous family of self-adjoint Fredholm operators with zero [8]. Thefollowing definition of spectral flow is due to Phillips [56]. Definition 5.16 (Spectral flow) . Let A : [ − , → Fred (0 , ∗ be a continuous map.We choose a partition − s < s < · · · < s n = 1 and positive numbers c , c , . . . , c n so that for each i = 1 , , . . . , n , the function t χ [ − c i ,c i ] ( A s ) is con-tinuous and finite rank on [ s i − , s i ], where χ [ a,b ] is the characteristic function of[ a, b ]. We define the spectral flow of A as follows.sf( A ) = n X i =1 (rank C ( χ [0 ,c i ] ( A s i )) − rank C ( χ [0 ,c i ] ( A s i − )) ∈ Z . Spectral flow is independent of the choice made and depends only on the homo-topy class of the path A leaving the endpoints fixed. Thus the spectral flow inducesa map sf : [ T , Fred (0 , ∗ ] → Z which is a group isomorphism. We next discuss Z -valued spectral flow. Let ζ be an involution on the interval[ − , 1] given by ζ ( s ) = − s . Let A be a Z -map from ([ − , , ζ ) to (Fred (0 , ∗ , q ).Then, the spectrum sp( A s ) of A s is symmetric with respect to ζ , and roughlyspeaking, Z -valued spectral flow counts the mod 2 of the net number of pairs ofcrossing points of sp( A s ) with zero. Z -valued spectral flow is studied in [19, 18,21, 15] and we give one definition following [56, 19]. Definition 5.17 ( Z -Valued Spectral Flow) . Let A : ([ − , , ζ ) → (Fred (0 , ∗ , q )be a Z -map. We choose a partition 0 = s < s < · · · < s n = 1 of [0 , 1] and positivenumbers c , c , . . . , c n so that for each i = 1 , , . . . , n , the map t χ [ − c i ,c i ] ( A s ) iscontinuous and finite rank on [ s i − , s i ]. We define the Z -valued spectral flow sf ( A )of A as follows.sf ( A ) = n X i =1 (rank C ( χ [0 ,c k ] ( A s i )) + rank C ( χ [0 ,c i ] ( A s i − ))) mod 2 ∈ Z . Z -valued spectral flow is independent of the choice made and depends only onthe Z -homotopy class of the Z -map A leaving the endpoints fixed or leaving thesepoints in the Z -fixed point set (Fred (0 , b ∗ ) q . Thus Z -valued spectral flow inducesa group homomorphism sf : [( T , ζ ) , (Fred (0 , ∗ , q )] Z → Z . By Appendix A, the Z -homotopy classes [( T , ζ ) , (Fred (0 , ∗ , q )] Z is isomorphic to KO ( C ( T ) , τ T ) ∼ = Z . Example . On C , let consider a family of self-adjoint operators given by B s =diag( s, − s ) for s ∈ [ − , j given by j ( x, y ) = ( − ¯ y, ¯ x ). Then,we have a Z -map B : ([ − , , ζ ) → ( M ( C ) , Ad j ) whose Z -valued spectral flowsf ( B ) is one. We extend this finite-dimensional example to an infinite-dimensionalone to give an example of a family parametrized by the circle of nontrivial Z -valued spectral flow. Let V be a separable infinite-dimensional complex Hilbertspace equipped with a quaternionic structure q . On V ′ = C ⊕ V ⊕ V , we considera quaternionic structure q ′ = j ⊕ q ⊕ q and a family self-adjoint Fredholm operatorsgiven by C s = diag( B s , V , − V ). Let U (0 , ∗ ( V ′ ) the space of unitaries on V ′ whosespectrum is {± } equipped with the norm topology. Then, its endpoints C ± arecontained in U (0 , ∗ ( V ′ ). Through an identification ( V ⊕ V , q ⊕ q ) ∼ = ( V ′ , q ′ ), theoperator diag(1 V , − V ) gives an element v ∈ U (0 , ∗ ( V ′ ) which satisfies Ad ′ q ( v ) = v . The space U (0 , ∗ ( V ′ ) is homeomorphic to the homogeneous space U ( V ′ ) / ( U ( C ⊕V ) × U ( C ⊕V )), which is contractible by Kuiper’s theorem [45]. Thus, there is a path l : [0 , → U (0 , ∗ ( V ′ ) whose endpoints are l (0) = v and l (1) = C . We extend l toa Z -map l ′ : ([ − , , ζ ) → ( U (0 , ∗ ( V ) , Ad q ′ ) by l ′ ( s ) = Ad q ′ ( l ( − s )) for s ∈ [ − , l ′ ( ± 1) = C ± , we connect the endpoints of C and l ′ to construct a Z -map C ′ : ( T , ζ ) → (Fred (0 , b ∗ , q ′ ), where q ′ = Ad q ′ . Then, sf ( C ′ ) = sf ( B ) = 1.In class D, we have a Z -map iH c : ( T , ζ ) → (Fred (1 , ∗ , r Ξ ). The Z -homotopyclasses [( T , ζ ) , (Fred (1 , ∗ , r Ξ )] Z is isomorphic to KO ( C ( T ) , τ T ) ∼ = Z ⊕ Z . By forget-ting the Z -actions and multiplying by − i , there is a map from [( T , ζ ) , (Fred (1 , ∗ , r Ξ )] Z to [ T , Fred (0 , ∗ ]. Combined with this map and the spectral flow sf : [ T , Fred (0 , ∗ ] → Z , we obtain a homomorphismsf D : [( T , ζ ) , (Fred (1 , ∗ , r Ξ )] Z → Z , [ A ] sf( − iA ) . Example . Let B ′ : ([ − , , ζ ) → ( C , Ad c ) be a Z -map defined by B ′ s = is .Then, sf D ( B ′ ) is defined and sf D ( B ′ ) = 1. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 29 In class DIII, H c : ( T , ζ ) → (Fred (1 , ∗ , q Θ ) is a Z -map, where the action of theClifford algebra on the right-hand side is given by e = i Π. The Z -homotopyclasses [( T , ζ ) , (Fred (1 , ∗ , q Θ )] Z is isomorphic to KO ( C ( T ) , τ T ) ∼ = Z ⊕ Z . Since(Fred (1 , ∗ , q Θ ) is a Z -subspace of (Fred (0 , ∗ , q Θ ), the inclusion induces a map from[( T , ζ ) , (Fred (1 , ∗ , q Θ )] Z to [( T , ζ ) , (Fred (0 , ∗ , q Θ )] Z . Combined with the Z -valuedspectral flow, we obtain the following map:sf DIII : [( T , ζ ) , (Fred (1 , ∗ , q Θ )] Z → Z , [ A ] sf ( A ) . For b = 1 or − 1, let i b be the inclusion { b } ֒ → T , and let w b be the composite ofthe following maps:[( T , ζ ) , (Fred (1 , ∗ , q Θ )] Z i ∗ b −→ [( {± } , id) , (Fred (1 , ∗ , q Θ )] Z ind −→ Z . Example . Let j , V , V ′ , q , q ′ , B s and C s be as in Example 5.18. Let e = (cid:18) ii (cid:19) and e ′ = (cid:18) − V V (cid:19) which gives a Cl , -module structure on C and V ⊕ V , respectively. Then, C s = diag( B s , , − 1) gives a Z -map from ([ − , , ζ )to (Fred (1 , ∗ , q ′ ). The operator C is contained in the space of self-adjoint uni-taries on V ′ that anticommutes with e ⊕ e ′ . As in [9], this space of unitariesis contractible by Kuiper’s theorem. We embed [ − , 1] into T by s exp( πis )and, as in Example 5.18, extend C onto T through this contractible space of uni-taries to obtain a Z -map D : ( T , ζ ) → (Fred (1 , ∗ , q ′ ). For this example, we havesf DIII ( D ) = sf DIII ( B ) = 1, w + ( D ) = 1 and w − ( D ) = 0. If we take D ′ as D ′ s = D − s ,then D ′ is also such a Z -map and its invariants are sf DIII ( D ′ ) = 1, w + ( D ′ ) = 0and w − ( D ′ ) = 1.In class AII, H c : ( T , ζ ) → (Fred (0 , ∗ , q Θ ) is a Z -map and its Z -valued spectralflow is defined. We denote sf AII for sf .In class C, we have a Z -map iH c : ( T , ζ ) → (Fred (1 , ∗ , q Ξ ). Note that the Z -homotopy classes [( T , ζ ) , (Fred (1 , ∗ , q Ξ )] Z is isomorphic to KO ( C ( T ) , τ ζ ) ∼ = Z . By forgetting the Z -actions and multiplying by − i , there is a map from[( T , ζ ) , (Fred (1 , ∗ , q Ξ )] Z to [ T , Fred (0 , ∗ ]. Combined with the spectral flow, we ob-tain a homomorphismsf C : [( T , ζ ) , (Fred (1 , ∗ , q Ξ )] Z → Z , [ A ] sf( − iA ) . Note that image the image of sf C are even integers since each eigenspace corre-sponding to the crossing points of the spectrum of − iA t with zero has a quaternionicvector space structure given by Ξ. Example . For s ∈ [ − , B ′′ s = diag( is, is ), and let j be the quaternionicstructure in Example 5.18. Then, B ′′ : ([ − , , ζ ) → ( M ( C ) , Ad j ) is a Z -map,and we have sf C ( B ′′ ) = sf( − iB ′′ ) = 2. Lemma 5.22. (1) sf AII : [( T , ζ ) , (Fred (0 , ∗ , q )] Z → Z is an isomorphism. (2) sf D : [( T , ζ ) , (Fred (1 , ∗ , r )] Z → Z is surjective. (3) sf DIII , w + , w − : [( T , ζ ) , (Fred (1 , ∗ , q )] Z → Z are surjective (4) sf C : [( T , ζ ) , (Fred (1 , ∗ , q )] Z → Z is an isomorphism.Proof. It is sufficient to find examples of Z -maps which maps to generators of Z , Z and 2 Z . Therefore, (1) and (3) follows from Example 5.18 and Example 5.20. For (2) and (4), we can construct such examples from Example 5.19 and Example 5.21,as in Example 5.18. (cid:3) In class DIII cases, we have three surjections sf DIII , w + and w − from Z ⊕ Z to Z . There is the following relation between them. Lemma 5.23. sf DIII = w + + w − .Proof. Let D and D ′ be Z -maps in Example 5.20. Let D ′′ = D ⊕ D ′ , then we havesf DIII ( D ′′ ) = 0, w + ( D ′′ ) = 1 and w − ( D ′′ ) = 1. Invariants sf DIII , w − and w + for D , D ′ and D ′′ tell that non-trivial three elements in the group [( T , ζ ) , (Fred (1 , ∗ , q )] Z consists of classes of D , D ′ and D ′′ . Therefore, we computed three maps sf DIII , w − and w + , from which the result follows. (cid:3) Remark . For our class DIII systems, Z -valued spectral flow counts the stronginvariant. This corresponds to one direct summand of Z ⊕ Z , while the othercorresponds to a weak invariant. When w + = w − , the strong invariant is nonzero.When w + = w − = 1, the strong invariant is zero and the weak invariant is nonzero.When w + = w − = 0, both of them are zero. Definition 5.25. For n -D systems with codimension n − numerical corner invariant as follows. • In class D, let N n,n − , DGapless ( H ) = sf( H c ) ∈ Z . • In class DIII, let N n,n − , DIIIGapless ( H ) = sf ( H c ) ∈ Z . • In class AII, let N n,n − , AIIGapless ( H ) = sf ( H c ) ∈ Z . • In class C, let N n,n − , CGapless ( H ) = sf( H c ) ∈ Z .For each of the above classes ♠ , the numerical invariant N n,n − , ♠ Gapless ( H ) is theimage of the gapless corner invariant I n,n − , ♠ Gapless ( H ) through the map sf ♠ . Thesenumerical invariants account for strong invariants. Remark . In Definition 5.25, the numerical corner invariants for both class DIIIand class AII are defined by using Z -valued spectral flow, though these two Z aredifferent from the viewpoint of index theory in the sense that they sit in differentBott clock. A similar remark holds for, e.g., cases of n = k in classes BDI and CII,where both of these numerical corner invariants are defined as Fredholm indices.5.7. Product Formula. In Sect. 4 of [32], a construction of the second-order topo-logical insulators of 3-D class A systems is proposed, which is given by using theHamiltonians of 2-D class A and 1-D class AIII topological insulators. In thissubsection, we generalize this construction to other pairs in the Altland–Zirnbauerclassification. This provides a way to construct nontrivial examples of each entryin Table 1 from the Hamiltonians of (first-order) topological insulators . For thispurpose, we use an exterior product of topological KR -groups [6].For j = 1 , 2, let H j be a bulk Hamiltonian of an n j -D k j -th order topologicalinsulator in real Altland–Zirnbauer class ♠ j (AI, BDI, D, DIII, AII, CII, C orCI). Let n = n + n , k = k + k and d j = n j − k j for j = 1 , 2, and let d = For the case of k = 2, the construction is restricted to α = 0 and β = ∞ case. that in Sect. 5.1 satisfying Assumption 5.1 when k j = 2 or that in Sect. 5.5 satisfyingAssumption 5.9 when k j ≥ 3. When k j = 1, the bulk Hamiltonian is assumed to be gapped.When k j = 2, we consider the case of α = 0 and β = ∞ . LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 31 d + d . Corresponding to the class in the Altland–Zirnbauer classification (forwhich we write ♠ j ) to which the Hamiltonian belongs, it preserves the symmetriesas (even/odd) TRS, (even/odd) PHS or chiral symmetry. We write Θ j , Ξ j and Π j for the symmetry operator for H j . As in Sect. 5, the models of corners H ci lead toa continuous family of self-adjoint or skew-adjoint Fredholm operators and definesan element of the KO -group KO i ′ ( ♠ j ) ( C ( T d j ) , τ ) where i ′ ( ♠ j ) = i ( ♠ j ) − 1. As inAppendix A.1, we have an exterior product of KO -groups KO i ′ ( ♠ ) ( C ( T d ) , τ T ) × KO i ′ ( ♠ ) ( C ( T d ) , τ T ) → KO i ′ ( ♠ )+ i ′ ( ♠ ) ( C ( T d ) , τ T ) , described through these Fredholm operators. By using this form of the product,we obtain an explicit form of the product of the gapless invariants of H and H . As a result, we can write down a bulk Hamiltonian H of an n -D k -th ordertopological insulator of class ♠ . The lattice on which we consider H c as a model ofthe codimension- k corner is introduced as the product of that of H c and that of H c . By this construction, we have the following relation between gapless invariants. Theorem 5.27. For the Hamiltonian H indicated in Table 12, we have I n ,k , ♠ Gapless ( H ) · I n ,k , ♠ Gapless ( H ) = I n,k, ♠ Gapless ( H ) , where · denotes the exterior product of elements of the KO -groups. Note that Theorem 5.27 is the product formula at the level of KO -group elementsand accounts for both strong and weak invariants. In order to show this theorem,we need to write down the explicit form of H . In the following, we discuss themfor some classes.Let us consider the case where ♠ = BDI and ♠ = BDI. In this case, each H j has even TRS, even PHS and chiral symmetry. We now consider the following n -dimensional Hamiltonian:(5.7) H = H ⊗ ⊗ H , which satisfies even TRS given by Θ = Θ ⊗ Θ , even PHS given by Ξ = Ξ ⊗ Ξ and the chiral symmetry given by Π = Π ⊗ Π . Thus, the Hamiltonian H belongsto the class ♠ = BDI. The model of the codimension- k corner H c of H is writtenby using the model H cj of the codimension- k j corner as follows: H c ( t , t ) = H c ( t ) ⊗ ⊗ H c ( t ) , where t j is an element of the d j -dimensional torus (momentum space) correspondingto a direction parallel to the corner of H cj for j = 1 , 2. Note that ( t , t ) constitutethe parameter of the d -dimensional momentum space in a direction parallel to thecorner of H c . By our assumption, H cj ( t j ) is an element of the space Fred (0 , ∗ andgives a Z -map ( T d j , ζ ) → (Fred (0 , ∗ , r Θ j ). The operator H c ( t , t ) is the image ofthe pair ( H c , H c ) through the map,(Fred (0 , ∗ , r Θ ) × (Fred (0 , ∗ , r Θ ) → (Fred (0 , ∗ , r Θ ) , When k j = 1, the lattice is Z ≥ × Z d j , where H cj is the compression of the bulk Hamiltonianonto this half-space. Topological invariants for them are the one discussed in topological insulators.To clarify our sign choices, we mention that they are obtained by applying the discussion in Sect. 5to the Toeplitz extension (4.1) in place of (2.2) or (4.2). in (A.2), where the action of Cl , to define the left-hand side is given by ǫ j = Π j and that for the right-hand side is given by ǫ = ǫ ⊗ ǫ = Π. Since this map inducesthe exterior product of KO -groups (Appendix A.1), KO ( C ( T d ) , τ T ) × KO ( C ( T d ) , τ T ) → KO ( C ( T d ) , τ T ) , we obtain Theorem 5.27 in this case.We next consider the case where ♠ = DIII and ♠ = D. In this case, H has odd TRS, even PHS and the chiral symmetry, and H has even PHS. As inSect. 5.3, H c ( t ) belongs to (Fred (1 , ∗ , q Θ ), and iH c ( t ) belongs to (Fred (1 , ∗ , r Ξ ).By using Proposition A.4, we identify (Fred (1 , ∗ , q Θ ) with (Fred (2 , ∗ , q Θ ⊕ Θ ) and(Fred (1 , ∗ , r Ξ ) with (Fred (0 , ∗ , r Ξ ⊕ Ξ ). We then use the map (A.2) of the form(Fred (2 , ∗ , q Θ ⊕ Θ ) × (Fred (0 , ∗ , r Ξ ⊕ Ξ ) → (Fred (2 , ∗ , q ′ ) , where q ′ is the conjugation of the fourfold direct sum of Θ ⊗ Ξ . By Proposition A.4,we have the Z -homeomorphism (Fred (2 , ∗ , q ′ ) ∼ = (Fred (0 , ∗ , q Θ ⊗ Ξ ). Thus, weobtain a Z -map H c : ( T d , ζ ) → (Fred (0 , ∗ , q Θ ⊗ Ξ ) from H c and H c which is amodel for the codimension- k corner in class ♠ = AII. Its bulk Hamiltonian H and(odd) TRS operator Θ is expressed as (5.7) and Θ = Θ ⊗ Ξ , respectively. Notethat Π = i Θ Ξ in class DIII and Θ commutes with H . Since the map (A.2)induces the exterior product of KO -groups, Theorem 5.27 holds for this class AIIHamiltonian H .The other cases are computed in a similar way, and the results are summarizedin Table 12, where we write H ⋆ = (cid:18) H ⊗ − i ⊗ H H ⊗ i ⊗ H (cid:19) , Θ ♣ = (cid:18) Θ ⊗ Ξ 00 Θ ⊗ Ξ (cid:19) ,H (cid:3) = (cid:18) − H ⊗ i − ⊗ H H ⊗ i − ⊗ H (cid:19) , Θ △ = (cid:18) Ξ ⊗ Θ 00 Ξ ⊗ Θ (cid:19) , Θ ♦ = (cid:18) − Ξ ⊗ Ξ Ξ ⊗ Ξ (cid:19) and Θ ♥ = (cid:18) ⊗ Θ Θ ⊗ Θ (cid:19) . Table 12: The forms of the Hamiltonians and symmetry opera-tors in class ♠ constructed from two pairs of Hamiltonians andsymmetry operators in classes ♠ and ♠ . Complex cases are alsoincluded [32, 33]. ♠ ♠ ♠ Hamiltonian ( H ) TRS (Θ) PHS (Ξ) Chiral (Π)AI AI CI H ⋆ Θ ♥ i ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H Θ ⊗ Θ — —AI D BDI H ⋆ Θ ♣ Ξ = ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Θ ⊗ Θ Π —AI AII DIII H ⋆ Θ ♥ i ΠΘ diag(1 , − H ⊗ Π + 1 ⊗ H Θ ⊗ Θ — —AI C CII H ⋆ Θ ♣ Ξ = − ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Θ ⊗ Θ Π —BDI AI AI H ⊗ ⊗ H Θ ⊗ Θ — —BDI BDI BDI H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π BDI D D H ⊗ ⊗ H — Ξ ⊗ Ξ — LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 33 BDI DIII DIII H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π BDI AII AII H ⊗ ⊗ H Θ ⊗ Θ — —BDI CII CII H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π BDI C C H ⊗ ⊗ H — Ξ ⊗ Ξ —BDI CI CI H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π D AI BDI H (cid:3) Θ △ Ξ = ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Ξ ⊗ Ξ —D D DIII H ⋆ Θ ♦ i ΠΘ diag(1 , − H ⊗ Π + 1 ⊗ H Ξ ⊗ Θ — —D AII CII H (cid:3) Θ △ Ξ = − ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Ξ ⊗ Ξ —D C CI H ⋆ Θ ♦ i ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H Ξ ⊗ Θ — —DIII AI D H ⊗ ⊗ H — Θ Π ⊗ Θ —DIII BDI DIII H ⊗ Π + 1 ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π DIII D AII H ⊗ ⊗ H Θ ⊗ Ξ — —DIII DIII CII H ⊗ ⊗ H Θ ⊗ Θ Π Π Θ ⊗ Θ Π ⊗ Π DIII AII C H ⊗ ⊗ H — Θ Π ⊗ Θ —DIII CII CI H ⊗ Π + 1 ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π DIII C AI H ⊗ ⊗ H Θ ⊗ Ξ — —DIII CI BDI H ⊗ ⊗ H Θ ⊗ Θ Π Θ Π ⊗ Θ Π ⊗ Π AII AI DIII H ⋆ Θ ♥ i ΠΘ diag(1 , − H ⊗ Π + 1 ⊗ H Θ ⊗ Θ — —AII D CII H ⋆ Θ ♣ Ξ = − ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Θ ⊗ Θ Π —AII AII CI H ⋆ Θ ♥ i ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H Θ ⊗ Θ — —AII C BDI H ⋆ Θ ♣ Ξ = ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Θ ⊗ Θ Π —CII AI AII H ⊗ ⊗ H Θ ⊗ Θ — —CII BDI CII H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π CII D C H ⊗ ⊗ H — Ξ ⊗ Ξ —CII DIII CI H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π CII AII AI H ⊗ ⊗ H Θ ⊗ Θ — —CII CII BDI H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π CII C D H ⊗ ⊗ H — Ξ ⊗ Ξ —CII CI DIII H ⊗ ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π C AI CII H (cid:3) Θ △ Ξ = − ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Ξ ⊗ Ξ —C D CI H ⋆ Θ ♦ i ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H Ξ ⊗ Θ — —C AII BDI H (cid:3) Θ △ Ξ = ΘΠ diag(1 , − H ⊗ Π + 1 ⊗ H — Ξ ⊗ Ξ —C C DIII H ⋆ Θ ♦ i ΠΘ diag(1 , − H ⊗ Π + 1 ⊗ H Ξ ⊗ Θ — —CI AI C H ⊗ ⊗ H — Θ Π ⊗ Θ — CI BDI CI H ⊗ Π + 1 ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π CI D AI H ⊗ ⊗ H Θ ⊗ Ξ — —CI DIII BDI H ⊗ ⊗ H Θ ⊗ Θ Π Θ Π ⊗ Θ Π ⊗ Π CI AII D H ⊗ ⊗ H — Θ Π ⊗ Θ —CI CII DIII H ⊗ Π + 1 ⊗ H Θ ⊗ Θ Ξ ⊗ Ξ Π ⊗ Π CI C AII H ⊗ ⊗ H Θ ⊗ Ξ — —CI CI CII H ⊗ ⊗ H Θ ⊗ Θ Π Π Θ ⊗ Θ Π ⊗ Π A A AIII H ⋆ — — diag(1 , − H ⊗ Π + 1 ⊗ H — — —AIII A A H ⊗ ⊗ H — — —AIII AIII AIII H ⊗ ⊗ H — — Π ⊗ Π Our product formula (Theorem 5.27) and the graded ring structure of KO ∗ ( C , id)(Theorem 6.9 of [7]) lead to the following product formula for numerical cornerinvariants. We collect the results here where the form of H is as indicated inTable 12. Corollary 5.28 (Cases of n = k ) . The case of k = n and k = n . • BDI × BDI → BDI , N n ,n , BDIGapless ( H ) · N n ,n , BDIGapless ( H ) = N n,n, BDIGapless ( H ) . • BDI × D → D , ( N n ,n , BDIGapless ( H ) mod 2) · N n ,n , DGapless ( H ) = N n,n, DGapless ( H ) . • BDI × DIII → DIII , ( N n ,n , BDIGapless ( H ) mod 2) · N n ,n , DIIIGapless ( H ) = N n,n, DIIIGapless ( H ) . • BDI × CII → CII , N n ,n , BDIGapless ( H ) · N n ,n , CIIGapless ( H ) = N n,n, CIIGapless ( H ) . • D × D → DIII , N n ,n , DGapless ( H ) · N n ,n , DGapless ( H ) = N n,n, DIIIGapless ( H ) . • CII × CII → BDI , N n ,n , CIIGapless ( H ) · N n ,n , CIIGapless ( H ) = N n,n, BDIGapless ( H ) . Corollary 5.29 (Cases of n = k − . The case of k = n and k = n − . • BDI × D → D , N n ,n , BDIGapless ( H ) · N n ,n − , DGapless ( H ) = N n,n − , DGapless ( H ) . • BDI × DIII → DIII , ( N n ,n , BDIGapless ( H ) mod 2) · N n ,n − , DIIIGapless ( H ) = N n,n − , DIIIGapless ( H ) . • BDI × AII → AII , ( N n ,n , BDIGapless ( H ) mod 2) · N n ,n − , AIIGapless ( H ) = N n,n − , AIIGapless ( H ) . • BDI × C → C , N n ,n , BDIGapless ( H ) · N n ,n − , CGapless ( H ) = N n,n − , CGapless ( H ) . • D × D → DIII , N n ,n , DGapless ( H ) · ( N n ,n − , DGapless ( H ) mod 2) = N n,n − , DIIIGapless ( H ) . • D × DIII → AII , N n ,n , DGapless ( H ) · N n ,n − , DIIIGapless ( H ) = N n,n − , AIIGapless ( H ) . • DIII × D → AII , N n ,n , DIIIGapless ( H ) · ( N n ,n − , DGapless ( H ) mod 2) = N n,n − , AIIGapless ( H ) . • CII × D → C , N n ,n , CIIGapless ( H ) · N n ,n − , DGapless ( H ) = N n,n − , CGapless ( H ) . • CII × C → D , N n ,n , CIIGapless ( H ) · N n ,n − , CGapless ( H ) = N n,n − , DGapless ( H ) . We also have a similar formula by exchanging H and H (e.g. pairs like D × BDI → D). Note that in the case of CII × CII → BDI in Corollary 5.28, we takethe product of two even integers, which is necessarily a multiple of four. A similarremark also holds in the case of CII × C → D in Corollary 5.29. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 35 Appendix A. Z -Spaces of Self-Adjoint/Skew-Adjoint FredholmOperators and Boersema–Loring’s K -theory In this Appendix, we collect necessary results and notations used in this pa-per. The results have been developed in much generality [6, 68, 9, 37, 43, 27, 15],and we contain minimal background for this paper focusing on their relation withBoersema–Loring’s K -theory [14]. In Appendix A.1, we introduce some Z -spacesof self-adjoint and skew-adjoint Fredholm operators following [9]. Some proofs forknown results are contained simply to fix isomorphisms used in this paper (e.g. thederivation of Table 12). In Appendix A.2, we discuss its relation with Boersema–Loring’s K -theory. In Appendix A.3, inspired by exponential maps in [68, 9], wewrite the boundary maps of the 24-term exact sequence of KO -theory in Boersema–Loring’s unitary picture through exponentials. Some of them are already expressedby exponentials in [14]; thus, we consider the remaining cases. This form of bound-ary maps is useful when we discuss a relation between our gapped invariants andgapless invariants through boundary maps [14, 44].A.1. Z -Spaces of Self-Adjoint/Skew-Adjoint Fredholm Operators. Fornon-negative integers k and l , let Cl k,l be the Clifford algebra that is an associativealgebra with unit over R generated by k + l elements e , . . . , e k and ǫ , . . . , ǫ l , whichsatisfy e i = − i = 1 , . . . , k ) and ǫ j = 1 ( j = 1 , . . . , l ) and anticommute with eachother. The following are well-known Clifford algebra isomorphisms [47]. Lemma A.1. (1) Cl k,l +1 ∼ = Cl l,k +1 . (2) Cl k,l ⊗ Cl , ∼ = Cl k +1 ,l +1 . (3) Cl k,l ⊗ Cl , ∼ = Cl k +4 ,l and Cl k,l ⊗ Cl , ∼ = Cl k,l +4 . (4) Cl k,l ⊗ Cl , ∼ = Cl k +8 ,l and Cl k,l ⊗ Cl , ∼ = Cl k,l +8 .Proof. (1) Let e , . . . , e k and ǫ , . . . , ǫ l +1 be generators of the Clifford algebra Cl k,l +1 . Let ˜ e i = ǫ i +1 ǫ ( i = 1 , . . . , l ), ˜ ǫ = ǫ and ˜ ǫ i = e i − ǫ ( i =2 , . . . , k + 1). Then, ˜ e , . . . , ˜ e l and ˜ ǫ , . . . , ˜ ǫ k +1 correspond to generators ofthe Clifford algebra Cl l,k +1 .(2) Let e , . . . , e k and ǫ , . . . , ǫ l be generators of the Clifford algebra Cl k,l , andlet e and ǫ be those of Cl , . We write ω , for e ′ ǫ ′ ∈ Cl , . Then,˜ e i = e i ⊗ ω , ( i = 1 , . . . , k ), ˜ e k +1 = 1 ⊗ e ′ , ˜ ǫ i = ǫ i ⊗ ω , ( i = 1 , . . . , l ) and˜ ǫ l = 1 ⊗ ǫ ′ correspond to generators of the Clifford algebra Cl k +1 ,l +1 .(3) We show that Cl k,l ⊗ Cl , ∼ = Cl k,l +4 ; the other is proved similarly. Let e , . . . , e k and ǫ , . . . , ǫ l be generators of the Clifford algebra Cl k,l , and let ǫ ′ , ǫ ′ , ǫ ′ , and ǫ ′ be those of Cl , . We write ω , for − ǫ ′ ǫ ′ ǫ ′ ǫ ′ ∈ Cl , .Then, ˜ e i = e i ⊗ ω , ( i = 1 , . . . , k ), ˜ ǫ i = ǫ i ⊗ ω , ( i = 1 , . . . , l ) and ˜ ǫ i = 1 ⊗ ǫ ′ i ( i = 1 , . . . , 4) correspond to generators of the algebra Cl k,l +4 .(4) We show that Cl k,l ⊗ Cl , ∼ = Cl k,l +8 ; the other is proved similarly. Let e , . . . , e k and ǫ , . . . , ǫ l be generators of Cl k,l , and let ǫ ′ , . . . , ǫ ′ be those of Cl , . We write ω , for ǫ ′ · · · ǫ ′ ∈ Cl , . Then, ˜ e i = e i ⊗ ω , ( i = 1 , . . . , k ),˜ ǫ i = ǫ i ⊗ ω , ( i = 1 , . . . , l ) and ˜ ǫ i = 1 ⊗ ǫ ′ i ( i = 1 , . . . , 8) correspond togenerators of the algebra Cl k,l +8 . (cid:3) Let W be a (ungraded) complex left Cl k,l -module. We say that W is a (ungraded)real (resp. quaternionic) Cl k,l -module if W is equipped with an antilinear map Note that the “real Z -graded Cliff( R k,l )-module” introduced in [6] is the same as the (un-graded) real Cl l,k +1 -module discussed in this paper. r : W → W (resp. q : W → W ), which commutes with the Cl k,l -action and satisfies r = 1 (resp. q = − r (resp. q ) the real (resp. quaternionic) struc-ture on the Clifford module. Since a real (resp. quaternionic) Cl k,l -module is thesame thing as a module of Cl k,l ⊗ Cl , ∼ = Cl k +1 ,l +1 (resp. Cl k,l ⊗ Cl , ∼ = Cl l +2 ,k )over R , the algebra Cl k,l has one inequivalent irreducible real or quaternionic mod-ule when k − l k − l ≡ Lemma A.2. (1) Let ∆ , be a complex irreducible representation of Cl , . There exists areal structure r , on ∆ , that commutes with the Clifford action. (2) Let ∆ , (resp. ∆ , ) be a complex irreducible representation of Cl , (resp.Cl , ). There exists a quaternionic structure q , (resp. q , ) on ∆ , (resp. ∆ , ) that commutes with the Clifford action. (3) Let ∆ , (resp. ∆ , ) be a complex irreducible representation of Cl , (resp.Cl , ). There exists a real structure r , (resp. r , ) on ∆ , (resp. ∆ , )that commutes with the Clifford action. For the proof of this lemma, see [25], for example. For a Z -space ( X, ζ ) with two Z -fixed points x , x ∈ X ζ , we write P ( X ; x , x ) for the path space starting from x and ending at x , that is, the space of continuous maps f : [0 , → X satisfying f (0) = x and f (1) = x equipped with the compact-open topology. On this space,we consider an involution, for which we also write ζ by abuse of notation, definedas ( ζ ( f ))( t ) = ζ ( f ( t )) for t in [0 , Z -space ( P ( X ; x , x ) , ζ ). When x = x , we write Ω x X for P ( X ; x , x ), which is the based loop space of X withthe base point x . Remark A.3 . Banach Z -spaces and its open Z -subspaces are Z -absolute neigh-borhood retracts [3], and have the homotopy type of Z -CW complexes [46]. Thepath spaces and loop spaces we discuss in the following also have the homotopytype of Z -CW complexes [67]. By the equivariant Whitehead theorem, weak Z -homotopy equivalences between these spaces are Z -homotopy equivalences [50, 4].Let V be a separable infinite-dimensional complex Hilbert space. Let B ( V ) be thespace of bounded complex linear operators on V equipped with the norm topology.Let GL ( V ), U ( V ), Fred( V ) and K ( V ) be subspaces of B ( V ) consisting of invertible,unitary, Fredholm and compact operators on V , respectively. We assume that ourHilbert space V has a real structure r or a quaternionic structure q , that is, an an-tiunitary operator on V satisfying r = 1 or q = − 1, respectively. Correspondingly,the space B ( V ) has an (antilinear) involution r = Ad r or q = Ad q . These involutionsinduce involutions on GL ( V ), U ( V ), Fred( V ) and K ( V ), for which we also write r or q . We write a for r or q and a for r or q . We also assume that there is a complexlinear action of the Clifford algebra Cl k,l on the Hilbert space V that commuteswith the real or the quaternionic structure. For an element v ∈ Cl k,l , we also write v for its action on V , for simplicity. When k − l ≡ Cl k,l has infinite multiplicity. In the following, we discuss the subspaces of B ( V ); we mayabbreviate the Hilbert space V from its notation when it is clear from the context.When the Hilbert space V is such a Cl k,l -module, let B ( k +1 ,l )sk (resp. B ( k,l +1)sa ) bethe subspace of B ( V ) consisting of skew-adjoint (resp. self-adjoint) operators A on V satisfying e i A = − Ae i for i = 1 , . . . , k and ǫ j A = − Aǫ j for j = 1 , . . . , l . Let LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 37 Fred ( k,l )sk = Fred ∩ B ( k,l )sk and Fred ( k,l )sa = Fred ∩ B ( k,l )sa . The involution a on B ( V ) in-duces involutions on Fred ( k,l )sk and Fred ( k,l )sa for which we also write a . Consider thespace Fred ( k,l )sk , and let Υ = e · · · e k − ǫ · · · ǫ l . When k − l is odd, the space Fred ( k,l )sk is decomposed into three components Fred ( k,l )+ , Fred ( k,l ) − and Fred ( k,l ) ∗ correspondingto whether the following element is essentially positive, essentially negative or nei-ther: i − Υ A when k − l ≡ A when k − l ≡ A ∈ Fred ( k,l )sk .As in [9], each of these three components is nonempty. When k − l ≡ a maps Fred ( k,l ) ± to Fred ( k,l ) ∓ (double-sign corresponds), and Fred ( k,l ) ∗ isclosed under the action of a . When k − l ≡ a . The space Fred ( k,l )sa is also decomposed into threecomponents in the same way, except that we take e · · · e k ǫ · · · ǫ l − for Υ in thiscase, and we define the space Fred ( k,l ) ∗ when k − l is odd. When k − l is even, we setFred ( k,l ) ∗ = Fred ( k,l )sk and Fred ( k,l ) ∗ = Fred ( k,l )sa . Summarizing, we have the following Z -spaces:(A.1) (Fred ( k,l ) ∗ , r ) , (Fred ( k,l ) ∗ , r ) , (Fred ( k,l ) ∗ , q ) , (Fred ( k,l ) ∗ , q ) . Proposition A.4. The following Z -homeomorphisms exist. (1) (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k +1 ,l +1) ∗ , a ) and (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k +1 ,l +1) ∗ , a ) , (2) (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k +4 ,l ) ∗ , e a ) and (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k +4 ,l ) ∗ , e a ) , (3) (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k,l +4) ∗ , e a ) and (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k,l +4) ∗ , e a ) , (4) (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k +8 ,l ) ∗ , a ) and (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k +8 ,l ) ∗ , a ) , (5) (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k,l +8) ∗ , a ) and (Fred ( k,l ) ∗ , a ) ∼ = (Fred ( k,l +8) ∗ , a ) , (6) (Fred ( k +1 ,l +1) ∗ , a ) ∼ = (Fred ( l,k +2) ∗ , a ) ,where e a = q when a = r and e a = r when a = q .Proof. Once the Clifford module structure on the left-hand side of these homeo-morphisms is fixed, that on the right-hand side is given following the isomorphismsof Clifford algebras in Lemma A.1. By using Lemma A.2, the Z -homeomorphismsare given as follows.(1) The map (Fred ( k,l ) ∗ ( V ) , Ad a ) → (Fred ( k +1 ,l +1) ∗ ( V ⊗ ∆ , ) , Ad a ⊗ r , ) given by A A ⊗ ω , is a Z -homeomorphism The other one is proved similarly.(3) The map (Fred ( k,l ) ∗ ( V ) , Ad a ) → (Fred ( k,l +4) ∗ ( V ⊗ ∆ , ) , Ad a ⊗ q , ) given by A A ⊗ ω , is a Z -homeomorphism. The other one and (2), (4) and (5)follow in a similar way.(6) The map (Fred ( k +1 ,l +1) ∗ ( V ) , a ) → (Fred ( l,k +2) ∗ ( V ) , a ) given by A Aǫ is a Z -homeomorphism. (cid:3) Proposition A.5. The following Z -homotopy equivalences exist. (1) (Fred ( k +1 ,l ) ∗ , a ) ≃ (Ω e k Fred ( k,l ) ∗ , a ) , for k ≥ and l ≥ . (2) (Fred ( k,l +1) ∗ , a ) ≃ (Ω ǫ l Fred ( k,l ) ∗ , a ) , for k ≥ and l ≥ . (3) (Fred (1 , ∗ , a ) ≃ (Ω Fred , a ) . Proposition A.5 is proved as in [9]. In what follows, we outline its proof sincesome spaces introduced there are of our interest. Proposition A.6. The following maps are Z -homotopy equivalences. (1) α : (Fred ( k +1 ,l ) ∗ , a ) → ( P (Fred ( k,l ) ∗ ; e k , − e k ) , a ) , where α ( A )( t ) = e k cos( πt ) + A sin( πt ) for ≤ t ≤ . (2) α : (Fred ( k,l +1) ∗ , a ) → ( P (Fred ( k,l ) ∗ ; ǫ l , − ǫ l ) , a ) , where α ( A )( t ) = ǫ l cos( πt ) − A sin( πt ) for ≤ t ≤ . (3) α : (Fred (1 , ∗ , a ) → ( P (Fred; 1 , − , a ) , where α ( A )( t ) = cos( πt ) + A sin( πt ) for ≤ t ≤ . Proposition A.5 follows from Proposition A.6 since, in each case, there is a pathconnecting the endpoints of each path space in the unitaries preserving the Cliffordaction and the Z -action. As in [9], the proof of Proposition A.6 reduces to showingthe Z -homotopy equivalences between some spaces of Fredholm operators andsome spaces of unitary operators (Proposition A.7). Let F ( k,l ) ∗ (resp. F ( k,l ) ∗ ) be thesubspace of Fred ( k,l ) ∗ (resp. Fred ( k,l ) ∗ ) consisting of those operators whose essentialspectra are { i, − i } (resp. { , − } ) and whose operator norms are 1. The spaces F ( k,l ) ∗ and F ( k,l ) ∗ are closed under the action of a ; thus, we have Z -spaces ( F ( k,l ) ∗ , a )and ( F ( k,l ) ∗ , a ). Inclusions ( F ( k,l ) ∗ , a ) ֒ → (Fred ( k,l ) ∗ , a ) and ( F ( k,l ) ∗ , a ) ֒ → (Fred ( k,l ) ∗ , a )are Z -homotopy equivalences. Let U cpt be the subspace of U ( V ) consisting ofunitary operators of the form 1 + T , where T ∈ K ( V ). When the Hilbert space V is a Cl k,l -module, let U ( k,l )cpt (resp. U ( k,l )cpt ) be the subspace of U ( V ) ∩ B ( k,l )sk (resp. U ( V ) ∩ B ( k,l )sa ) consisting of a unitary u satisfying u = − u = 1) and u ≡ e k (resp. u ≡ ǫ l ) modulo compact operators. If the Hilbert space has a realor quaternionic structure, these spaces of unitaries are closed under the action of a , and we obtain Z -spaces. Proposition A.7. The following maps are Z -homotopy equivalences: (1) p : ( F ( k +1 ,l ) ∗ , a ) → ( − U ( k,l )cpt , a ) , p ( A ) = e k exp( πAe k ) , for k ≥ , l ≥ . (2) p : ( F ( k,l +1) ∗ , a ) → ( − U ( k,l )cpt , a ) , p ( A ) = ǫ l exp( πAǫ l ) , for k ≥ , l ≥ . (3) p : ( F (1 , ∗ , a ) → ( − U cpt , a ) , p ( A ) = exp( πA ) . (4) p : ( F (0 , ∗ , a ) → ( − U cpt , a ◦ ∗ ) , p ( A ) = exp( πiA ) .Proof. By Remark A.3, it is sufficient to show that these maps are weak Z -homotopy equivalences. Equivalently, to show that p i and its restriction to the Z -fixed point sets (the map p Z : ( F ( k +1 ,l ) ∗ ) a → ( − U ( k,l )cpt ) a in the case of (1)) areweak homotopy equivalences. They are proved by using quasifibrations on somedense subspaces of contractible fibers as in [9]. (cid:3) Lemma A.8. There is a Z -homeomorphism (Fred , a ) ∼ = (Fred (0 , ∗ , a ) .Proof. This is given by a Z -map (Fred( V ) , a ) → (Fred (0 , ∗ ( V ⊕ V ) , a ⊕ a ), A (cid:18) A ∗ A (cid:19) , where the action of Cl , on V ⊕ V is given by ǫ = diag(1 , − (cid:3) Proposition A.4, Proposition A.5 and Lemma A.8 lead to the following. Corollary A.9. The following Z -homotopy equivalences exist. (1) (Fred ( k,l ) ∗ , r ) ≃ (Ω k − l Fred , r ) . (2) (Fred ( k,l ) ∗ , q ) ≃ (Ω k − l +4 Fred , r ) . (3) (Fred ( k,l ) ∗ , r ) ≃ (Ω l − k +6 Fred , r ) . LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 39 (4) (Fred ( k,l ) ∗ , q ) ≃ (Ω l − k +2 Fred , r ) .When the subscript m on Ω m is negative, this should be replaced by m +8 n by takinga sufficiently large integer n to make the subscript non-negative. Note that when k and l are relatively small, we further have the following Z -homeomorphisms. Lemma A.10. Multiplication by the imaginary unit i = √− induces the following Z -homeomorphisms: (1) (Fred (0 , ∗ , Ad r ) → (Fred (1 , ∗ , Ad e r ) , where ˜ r = rǫ . (2) (Fred (0 , ∗ , Ad q ) → (Fred (1 , ∗ , Ad e q ) , where ˜ q = − qǫ . (3) (Fred (1 , ∗ , Ad q ) → (Fred (2 , ∗ , Ad e r ) , where ˜ r = qe . (4) (Fred (1 , ∗ , Ad r ) → (Fred (2 , ∗ , Ad e q ) , where ˜ q = − re .Remark A.11 . The Z -spaces in Lemma A.10 appear in the study of topologicalinsulators. Specifically, Table 11 is obtained by taking the quantum symmetries asreal or quaternionic Clifford module structures as follows. • In class BDI, we put r = Θ and ǫ = Π in (1); then, ˜ r = Ξ. • In class CII, we put q = Θ and ǫ = Π in (2); then, ˜ q = Ξ. • In class DIII, we put q = Θ and e = i Π in (3); then, ˜ r = Ξ. • In class CI, we put r = Θ and e = i Π in (4); then, ˜ q = Ξ.For l ≥ 1, let us consider the map(A.2) (Fred ( k,l +1) ∗ , a ) × (Fred ( k ′ ,l ′ +1) ∗ , a ′ ) → (Fred ( k + k ′ ,l + l ′ ) ∗ , a ⊗ a ′ )defined by ( A, B ) A ⊗ ǫ l ⊗ B , where the Clifford action to define Fred ( k + k ′ ,l + l ′ ) ∗ is generated by ˜ e i = e i ⊗ i = 1 , . . . , k ), ˜ e k + i = ǫ l ⊗ e i ( i = 1 , . . . , k ′ ), ˜ ǫ i = ǫ i ⊗ i = 1 , . . . , l − 1) and ˜ ǫ l + i − = ǫ l ⊗ ǫ i ( i = 1 , . . . , l ′ ). This map induces the exteriorproduct of topological KR -groups as in [9].A.2. Relation with Boersema–Loring’s Unitary Picture. In this subsection,we discuss a relation between these Z -spaces of self-adjoint/skew-adjoint Fredholmoperators and Boersema–Loring’s K -theory.Let { W i } i ∈ I be the set of mutually inequivalent irreducible real (resp. quater-nionic) representations of Cl k,l with hermitian inner-products which { W i } i ∈ I con-sists of one or two elements corresponding to k and l . Let W = ⊕ i ∈ I W i and V = l ( Z ≥ ) ⊗ W which has a real (resp. quaternionic) Cl k,l -module structureinduced by that of { W i } i ∈ I . We take a complete orthonormal basis { δ j } j ∈ Z ≥ of l ( Z ≥ ) given by generating functions of each points in Z ≥ . Let V n be thesubspace of V spanned by { δ j ⊗ w | ≤ j ≤ n, w ∈ W } , which is a real (resp.quaternionic) Cl k,l -module. Let GL cpt be the space of invertible operators on V of the form e k + T for some compact operator T . Let GL ( k,l )cpt = GL cpt ∩ B ( k,l )sk ( V )and GL ( k,l )cpt = GL cpt ∩ B ( k,l )sa ( V ). Let GL ( k,l ) n (resp. GL ( k,l ) n ) be the subspace of B ( k,l )sk ( V n ) (resp. B ( k,l )sa ( V n )) consisting of invertible operators, and let U ( k,l ) n (resp. U ( k,l ) n ) be its subspace of unitaries. We have an injection GL ( k,l ) n ֒ → GL ( k,l ) n +1 (resp. GL ( k,l ) n ֒ → GL ( k,l ) n +1 ) given by mapping A to A ⊕ e k (resp. A ⊕ ǫ l ), and let GL ( k,l ) ∞ (resp. GL ( k,l ) ∞ ) be its inductive limit lim −→ GL ( k,l ) n (resp. lim −→ GL ( k,l ) n ). We also define U ( k,l ) ∞ and U ( k,l ) ∞ for unitaries in the same way. The space GL ( k,l ) n (resp. GL ( k,l ) n ) is identified with the subspace of GL ( k,l )cpt (resp. GL ( k,l )cpt ) consisting of operatorsof the form e k + T (resp. ǫ l + T ), where T ∈ B ( V n ), and we have an injective Z -map ( GL ( k,l ) ∞ , a ) → ( GL ( k,l )cpt , a ) (resp. ( GL ( k,l ) ∞ , a ) → ( GL ( k,l )cpt , a )). As in [53],the following holds . Proposition A.12. The map ( GL ( k,l ) ∞ , a ) → ( GL ( k,l )cpt , a ) and the map ( GL ( k,l ) ∞ , a ) → ( GL ( k,l )cpt , a ) are Z -homotopy equivalences. By using a deformation of invertibles to unitaries, ( U ( k,l ) ∞ , a ) and ( U ( k,l ) ∞ , a ) are Z -homotopy equivalent to ( U ( k,l )cpt , a ) and ( U ( k,l )cpt , a ), respectively. We denote U ♠∞ for these subspaces of U ♠ as indicated in Table 11.These Z -spaces of unitaries appears in Boersema–Loring’s KO -theory [14]. Let( X, ζ ) be a compact Hausdorff Z -space, and consider a C ∗ ,τ -algebra ( C ( X ) , τ ζ ) ofcontinuous functions on X , whose transposition τ ζ is given by f τ ζ ( x ) = f ( ζ ( x )).Then, the Z -homotopy classes [( X, ζ ) , U ♠∞ ] Z can be identified with the group KO i ( ♠ ) − ( C ( X ) , τ ζ ) where i ( ♠ ) is as indicated in Table 10. In the following, wediscuss two of eight KO -groups and the others are discussed in a similar way.As for the KO − -group, an element of the set [( X, ζ ) , ( U ∞ , r ◦∗ )] Z is representedby a Z -map f : ( X, ζ ) → ( U n , r ◦ ∗ ). This f is a unitary element of M n ( C ( X ))satisfying f ( ζ ( x )) = r ( f ( x )) ∗ which is the same as the relation f τ ζ = f to define KO − -groups. Thus, the set [( X, ζ ) , ( U ∞ , r ◦ ∗ )] Z is the same as KO − ( C ( X ) , τ ζ )by the definition of Boersema–Loring’s KO − -group.Finally, we discuss the KO -group. By the multiplication of − i , we have a Z -homeomorphism ( U (1 , ∞ , q ) → ( U (0 , ∞ , − q ). A Z -continuous map f : ( X, ζ ) → ( U (0 , n , − q ) is a self-adjoint unitary in M n ( C ( X )) satisfying f ♯ ⊗ τ ζ = − f ∗ = − f .The Clifford algebra Cl , has just one irreducible quaternionic representation upto equivalence, which is constructed as follows. On W = C , we consider the action ρ of Cl , ⊗ Cl , defined as follows: ρ (1 ⊗ e ) = (cid:18) i i (cid:19) , ρ (1 ⊗ e ) = (cid:18) − cc (cid:19) , ρ ( e ⊗ 1) = (cid:18) − 11 0 (cid:19) , where c is the complex conjugation on C . The space U (1 , ∞ is defined as the inductivelimit of maps U (1 , n → U (1 , n +1 , A A ⊕ I where I = ρ ( e ⊗ 1) and the space U (0 , ∞ is defined as that of maps A A ⊕ − iI where − iI = I (6) .A.3. Boersema–Loring’s K -Theory and Exponential Maps. We describeboundary maps of the 24-term exact sequence of KO -theory (which we denoteas ∂ BL i in this section) in Boersema–Loring’s unitary picture through exponentialmaps. The map ∂ BL i for even i has already been expressed as an exponential mapin [14]; thus, we focus on ∂ BL i for odd i . A clue is the exponential maps given inProposition A.7. For a short exact sequence of C ∗ ,τ -algebras,(A.3) 0 → ( I , τ ) → ( A , τ ) ϕ → ( B , τ ) → , and for each odd i , we construct a group homomorphism(A.4) ∂ exp i : KO i ( B , τ ) → KO i − ( I , τ ) In [53], an upper semicontinuous function is introduced to show that an injection GL ∞ → GL cpt is a homotopy equivalence. In our setup, this function is Z -invariant, and the result followsas in [53]. LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 41 and show they coincides with ∂ BL i up to a factor of − 1. Let W n ∈ M n ( C ) and Q n ∈ M n ( C ) be the following matrices: W n = 1 √ (cid:18) i · n n n i · n (cid:19) , Q n = 1 √ n − I (2) n I (2) n n ! , and let V n ∈ M n ( R ) and X n ∈ M n ( R ) be the permutation matrices satisfying V n diag( x , . . . , x n ) V ∗ n = diag( x , x n +1 , x , x n +2 , . . . , x n , x n ) ,X n diag( x , . . . , x n ) X ∗ n = diag( x , x , x n +1 , x n +2 , x , x , x n +3 , x n +4 , . . . , x n ) . As in [14], let Y ( − n = V n W n , Y (1)2 n = V n , Y (3)4 n = V n Q n W n and Y (5)4 n = X n . Definition A.13. Suppose we have a short exact sequence of C ∗ ,τ -algebras asin (A.3). We assume I = Ker( ϕ ) and identify the unit in e I with that of e A .For i ∈ {− , , , } , suppose [ u ] ∈ KO i ( B , τ ), where u ∈ M n i · n ( e B ) is a unitarysatisfying the relation R i and λ ( u ) = I ( i ) n , for which n i , R i and I ( i ) are as inTable 2. Let a in M n i · n ( e A ) be a lift of u satisfying the relation R i and k a k ≤ ∂ exp i ([ u ]) = h − Y ( i )2 n i · n (cid:0) ǫ exp( πAǫ ) (cid:1) Y ( i ) ∗ n i · n i ∈ KO i ( I , τ ) , where A = (cid:18) a ∗ a (cid:19) and ǫ = diag(1 n i · n , − n i · n ). Lemma A.14. ∂ exp i for odd i are well-defined group homomorphisms.Proof. We need to show that (a) the unitaries constructed all satisfy the correctrelation, (b) the choice of lift is not important, (c) some lift is always available,(d) homotopy is respected, (e) compatible with respect to the stabilization by I ( i ) and (f) the addition is respected. (c), (b) and (d) are proved in the same wayas in Lemma 8 . C ( a ) = − ǫ exp( πAǫ ) and C ′ ( a ) = Y ( i )2 n i · n C ( a ) Y ( i ) ∗ n i · n .(1) We first consider the case of i = 1. Let u ∈ M n ( e B ) be a unitary satisfying u τ = u ∗ and λ n ( u ) = I (1) n . We take a lift a ∈ M n ( e A ) of u such that k a k ≤ a τ = a ∗ . Since ϕ ( C ′ ( a )) = V n ǫ V ∗ n = I (0) n , we have C ′ ( a ) ∈ M n ( e I )and λ ( C ′ ( a )) = I (0) n . Since A τ = A , we have, C ( a ) τ = − exp( πAǫ ) τ ǫ τ = − ǫ exp( πǫ τ A τ ) ǫ ∗ = − ǫ exp( πǫ Aǫ ∗ ) = C ( a ) . Since Y (1)2 n = V n is the orthogonal matrix, ( Y (1)2 n ) τ = Y (1) ∗ n , and thus, C ′ ( a ) τ = C ′ ( a ) holds. When u = 1, we can take a = 1 and C ′ (1) = I (1) in this case. Combined with this, the proof is completed once we havechecked that ∂ exp1 preserves the addition. Let u ∈ M m ( e B ) and v ∈ M n ( e B ).We take their lift a and b such that a τ = a ∗ and b τ = b ∗ . Then, we have C ′ (diag( a, b )) = diag( C ′ ( a ) , C ′ ( b )) since C (cid:18)(cid:18) a b (cid:19)(cid:19) = − (cid:18) m + n − m + n (cid:19) exp π (cid:18) − a ∗ , − b ∗ )diag( a, b ) 0 (cid:19)! , Matrices W n , Q n , V n and X n are what we borrowed from Sect. 8 of [14]. Some of thebasic formulas that they satisfy can be found there. Ad V m +2 n exp π (cid:18) − a ∗ , − b ∗ )diag( a, b ) 0 (cid:19) ! = exp (cid:18) π · diag (cid:18) Ad V m (cid:18) − a ∗ a (cid:19) , Ad V n (cid:18) − b ∗ b (cid:19)(cid:19)(cid:19) , which shows that ∂ exp1 ([ u ] + [ v ]) = ∂ exp1 ([ u ]) + ∂ exp1 ([ v ]).(2) We next consider the case of i = − 1. Let u ∈ M n ( e B ) be a unitary satisfying u τ = u and λ n ( u ) = I ( − n . We take a lift a ∈ M n ( e A ) of u such that k a k ≤ a τ = a . Since ϕ ( C ′ ( a )) = I (6) n , we have C ′ ( a ) ∈ M n ( e I ) and λ ( C ′ ( a )) = I (6) n . Since A e ♯ ⊗ τ = − A , we have C ( a ) e ♯ ⊗ τ = − exp( πAǫ ) e ♯ ⊗ τ ǫ e ♯ ⊗ τ = exp( πǫ e ♯ ⊗ τ A e ♯ ⊗ τ ) ǫ = exp( πǫ A ) ǫ = ǫ exp( πAǫ ) = − C ( a ) . Since ( V n xV ∗ n ) ♯ ⊗ τ = V n x e ♯ ⊗ τ V ∗ n and W e ♯ ⊗ τ n = − W ∗ n , we have C ′ ( a ) ♯ ⊗ τ = − C ′ ( a ). For u = 1, we take a = 1 and C ′ (1) = I (6) holds. Therefore, asin (1), all we have to show is the additivity of ∂ exp − . Let a ∈ M m ( e A )and b ∈ M n ( e A ) be lifts of the unitaries u and v . Then, C ′ (diag( a, b )) =diag( C ′ ( a ) , C ′ ( b )) follows from V m +2 n W m +2 n (cid:18) − a ∗ , − b ∗ )diag( a, b ) 0 (cid:19) W ∗ m +2 n V ∗ m +2 n = diag (cid:18) V m W m (cid:18) − a ∗ a (cid:19) W ∗ m V ∗ m , V n W n (cid:18) − b ∗ b (cid:19) W ∗ n V ∗ n (cid:19) . (3) Let us consider the case of i = 5. Let u ∈ M n ( e B ) be a unitary satisfying u ♯ ⊗ τ = u ∗ and λ n ( u ) = I (5) n = 1 n . We take a lift a ∈ M n ( e A ) of u such that k a k ≤ a ♯ ⊗ τ = a ∗ . Since ( X n xX n ) ♯ ⊗ τ = X n x ♯ ⊗ τ X n , A ♯ ⊗ τ = A and ǫ ♯ ⊗ τ = ǫ , the relation C ′ ( a ) ♯ ⊗ τ = − X n exp( πǫ A ♯ ⊗ τ ) ǫ X ∗ n = − X n exp( πǫ A ) ǫ X ∗ n = C ′ ( a )holds. We have C ′ (1 ) = I (4) , and for the additivity of ∂ exp5 , note that X m +4 n (cid:18) − a ∗ , − b ∗ )diag( a, b ) 0 (cid:19) X ∗ m +4 n = diag (cid:18) X m (cid:18) − a ∗ a (cid:19) X ∗ m , X n (cid:18) − b ∗ b (cid:19) X ∗ n (cid:19) . (4) Consider the case of i = 3. Let u ∈ M n ( e B ) be a unitary satisfying u ♯ ⊗ τ = u and λ n ( u ) = I (3) n = 1 n . We take a lift a ∈ M n ( e A ) of u such that k a k ≤ a ♯ ⊗ τ = a . Since A e ♯ ⊗ ♯ ⊗ τ = − A and ǫ e ♯ ⊗ ♯ ⊗ τ = − ǫ , therelation C ( a ) e ♯ ⊗ ♯ ⊗ τ = − C ( a ) holds. Since ( Q n xQ ∗ n ) τ = Q n x e ♯ ⊗ ♯ ⊗ τ Q ∗ n and W e ♯ ⊗ ♯ n = − W ∗ n , we have C ′ ( a ) τ = − C ′ ( a ). For the remaining part, we LASSIFICATION OF TOPOLOGICAL INVARIANTS RELATED TO CORNER STATES 43 note that C ′ (1 ) = I (2)2 and Y (3)4 m +4 n (cid:18) − a ∗ , − b ∗ )diag( a, b ) 0 (cid:19) Y (3) ∗ m +4 n = diag (cid:18) Y (3)4 m (cid:18) − a ∗ a (cid:19) Y (3) ∗ m , Y (3)4 n (cid:18) − b ∗ b (cid:19) Y (3) ∗ n (cid:19) . (cid:3) Lemma A.15. Each ∂ exp i is natural with respect to the morphisms of short ex-act sequences of C ∗ -algebras. That is, suppose we have the following commutativediagram of exact lows: / / ( I , τ ) / / ι (cid:15) (cid:15) ( A , τ ) ϕ / / α (cid:15) (cid:15) ( B , τ ) / / β (cid:15) (cid:15) / / ( I , τ ) / / ( A , τ ) ϕ / / ( B , τ ) / / Then, we have ι ∗ ◦ ∂ exp i = ∂ exp i ◦ β ∗ .Proof. As in Lemma 8 . ϕ j ) = I j ( j = 1 , 2) for simplicity. Let [ u ] ∈ KO ui ( B , τ )be an element represented by a unitary u ∈ M n i · n ( e B ) satisfying the symmetryrelation R i . Let a ∈ M n i · n ( e A ) be a lift of u such that || a || ≤ 1, and satisfy therelation R i . Then, a = α ( a ) is a lift of u satisfying the symmetry, and thus, ∂ exp i ([ u ]) = [ C ′ ( a )] holds. Since α ( C ′ ( a )) = C ′ ( α ( a )) = C ′ ( α ), we have ι ∗ ◦ ∂ exp i ([ u ]) = ι ∗ [ C ′ ( a )] = [ α ( C ′ ( a )] = ∂ exp i ([ u ]) = ∂ exp i ◦ β ∗ [ u ] . (cid:3) Proposition A.16. ∂ BL i = − ∂ exp i for odd i . As in the proof of Theorem 8 . ∂ exp1 : K ( B ) → K ( I ) is defined by forgetting the real structure inthe case of i = 1 of Definition A.13. Lemma A.17. The boundary maps ∂ BL1 and ∂ exp1 from K ( B ) to K ( I ) satisfy therelation ∂ BL1 = − ∂ exp1 .Proof. Suppose that [ u ] ∈ K ( B ) where u ∈ M n ( ˜ B ) and λ ( u ) = 1 n . We takea lift a of u in M n ( ˜ A ) satisfying || a || ≤ 1. Consider the partial isometry v = (cid:18) a √ − a ∗ a (cid:19) and let V = (cid:18) v ∗ v (cid:19) . ∂ exp1 ([ u ]) is computed as ∂ exp1 ([ u ]) = h − Y (1)4 n (cid:0) ǫ exp( πV ǫ ) (cid:1) Y (1) ∗ n i = [ Y (1)4 n (1 − vv ∗ ) Y (1) ∗ n ] − [ Y (1)4 n (1 − v ∗ v ) Y (1) ∗ n ] . 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