Bott periodicity for $Z_2$ symmetric ground states of gapped free-fermion systems

Building on the symmetry classification of disordered fermions, we give a proof of the proposal by Kitaev, and others, for a"Bott clock"topological classification of free-fermion ground states of gapped systems with symmetries. Our approach differs from previous ones in that (i) we work in the standard framework of Hermitian quantum mechanics over the complex numbers, (ii) we directly formulate a mathematical model for ground states rather than spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools rather than K-theory. Key to our proof is a natural transformation that squares to the standard Bott map and relates the ground state of a d-dimensional system in symmetry class s to the ground state of a (d+1)-dimensional system in symmetry class s+1. This relation gives a new vantage point on topological insulators and superconductors.


Introduction
In this article we address the following problem of mathematical physics. (We first formulate the mathematical problem as such, and then indicate its origin in physics.) Let there be a Hermitian vector space W ≡ (C 2n , ·, · ) with n a sufficiently large integer, and let W have the additional structure of a non-degenerate symmetric complex bilinear form {·, ·}. Assume that W carries an action by operators J 1 , . . ., J s that satisfy the Clifford algebra relations J l J m + J m J l = −2δ lm Id W (l, m = 1, . . ., s) (1.1) and preserve , as well as { , }. Moreover, let M be a d-dimensional manifold, namely momentum space or phase space, with an involution τ : M → M whose physical meaning is momentum inversion. Our objects of interest then are rank-n complex vector bundles π : A → M whose fibers A k = π −1 (k) ⊂ W are constrained for all k ∈ M by the conditions We refer to them as vector bundles of symmetry class s, or class s for short. The goal is to give a homotopy classification for the classifying maps k → A k of such vector bundles. In the present paper we achieve this goal for the case of the d-sphere, M = S d , and for n large enough relative to d. A companion paper [1] deals with the case of M = T d .
systems with symmetries as sketched above. The investigation of this classification problem was pioneered by Schnyder, Ryu, Furusaki, and Ludwig [2] who observed that there exist, in every space dimension, 5 symmetry classes (among the 10 classes of the "Tenfold Way" of disordered fermions [3,4]) that house topological insulators or superconductors robust to disorder. Building on this observation, Kitaev recognized the mathematical principle behind the emerging pattern, which he named the "Periodic Table of topological insulators and superconductors" [5]. He understood that the constraints due to physical symmetries can be formulated as an extension problem for the Clifford algebra (1.1) of what we propose to call "pseudo-symmetries", and he saw the close connection with a mathematical phenomenon known as Bott periodicity. He also advocated K-theoretic methods as a tool to compute the topological invariants characterizing the different symmetry-protected topological (SPT) phases. A pedagogical discussion of some points outlined by Kitaev was offered by Stone, Chiu, and Roy [6]. Symmetry aspects were elaborated by Abramovici and Kalugin [7]. A remarkable extension of the Bott-type periodicity phenomenon for free-fermion SPT phases was proposed by Teo and Kane [8], who introduced position-like dimensions (associated with defects) in addition to the momentum-like dimensions considered in earlier work. Freedman, Hastings, Nayak, Qi, Walker, and Wang [9] developed this idea further and pointed out that their results lead to a mathematical proof of Kitaev's Periodic Table if one assumes (with referral to unpublished notes by Kitaev) that gapped lattice Hamiltonians are stably equivalent to Dirac Hamiltonians with a spatially varying mass term. Fidkowski and Kitaev [10] gave a complete classification of one-dimensional systems. A recent treatise on the subject at large is by Freed and Moore [11], who set up a comprehensive framework based on the Galilean group and review the relevant notions of twisted equivariant K-theory, an algebraic variant of which is treated in [12].
Let us now highlight the main differences between our work and the current literature. Firstly, to the extent that only the static properties (as opposed to the dynamical response) of the physical system are under investigation, the classification problem at hand is a problem of classifying ground states -that, in any case, is how we view it. Thus we never make any direct reference to a Hamiltonian. (Aside from a locality condition to ensure the continuity of the vector bundle A → M, the only information we need about the Hamiltonian is its symmetry class, as this determines the symmetry class of the ground state.) In particular, there will be no need for any process of "flattening" the Hamiltonian in this paper.
Secondly, all our symmetries are true symmetries in the sense that they commute with the Hamiltonian and leave the ground state invariant. "Symmetries" that anti-commute with the Hamiltonian (such as chirality for the massless Dirac operator) do not appear in our work.
Thirdly, a crucial element of our approach is that we work over the complex number field throughout. As a matter of fact, the vector bundles singled out by the constraints (1.2) are complex, i.e. their fibers are complex vector spaces. While some of them can be regarded as real vector bundles in the sense of Atiyah [13], others cannot be. Moreover, although the operator of Hermitian conjugation does single out a real (or Majorana) subspace R 2n ⊂ C 2n = W , one of our discoveries is that one should keep this real structure flexible in order to attain the best overall perspective. (In fact, the formulation and proof of our results employs two different notions of taking the complex conjugate!) Finally, and most importantly, our work differs from the work of other authors by the principle of topological classification used. Starting with Kitaev [5], most of the past and present literature has relied on the algebraic tools of K-theory to compute the topological invariants given by stable isomorphism classes of vector bundles. An exception is the approach in [14,15], where ordinary (as opposed to stable) isomorphism classes of vector bundles are computed for two of the ten symmetry classes. (These are the classes AI and AII, which are special in that they permit a description of ground states by real and quaternionic vector bundles, respectively. In the present paper we will encounter vector bundles of a more general kind.) In contradistinction, the present work uses homotopy-theoretic methods to establish a homotopy classification for the classifying maps of the vector bundles A → M. It has to be emphasized that the equivalence relation of homotopy is finer than that of (ordinary or stable) isomorphisms of vector bundles: ordinary isomorphism classes are recovered for a large number of valence bands, while stable isomorphism classes are recovered if both the valence and the conduction bands are large in number.
Our third and main achievement is a homotopy-theoretic proof that this map is bijective under favorable conditions. The precise statements are laid down in Theorems 7.1 and 7.2. To give a quick summary, let M be a path-connected Z 2 -CW complex with base point k * fixed by the Z 2 -action, and let A * ∈ C s (n) ⊂ C s−1 (n) ≃ C s+1 (2n) be a target-space base point also fixed by the Z 2 -action. Then if dim M ≪ n there exist two bijections, between homotopy classes of base-point preserving and Z 2 -equivariant maps. The left one, where SM denotes the usual suspension (which is position-type, i.e., the space direction added in going from M to SM is acted upon trivially by the extension of the involution τ), follows rather directly from the Bott periodicity theorems, by employing the G-Whitehead Theorem for G = Z 2 in order to transcribe these classical results to our setting. The right bijection is more difficult to establish. Our strategy of proof is to first consider the special set of symmetry indices s ∈ 2 + 4Z. In these cases, there exists a certain fibration that connects our diagonal map with the standard Bott map; thus, the right bijection in (1.3) follows from the left bijection by an isomorphism due to the projection map p of the fibration. (Applying p amounts to taking a square, which is the squaring operation alluded to earlier.) Unfortunately, for s / ∈ 2 + 4Z the said fibration is not available for finite n, although it does exist in the K-theory limit of infinite n. Therefore, we need to employ an additional argument to complete the proof. Adapting an idea of Teo and Kane [8], we consider generalized momentum spaces (M, τ) with d x position-like and d k momentum-like directions. We then use the left bijection in Eq. (1.3) to dial the symmetry index s to a value where the isomorphism by p applies. The desired result then follows from S rS M =SS r M.
In the results stated above, we gave ourselves the luxury of making the simplifying assumption d = dim M ≪ n. Our fourth and final result are practically useful bounds on the stable regime of d (as a function of n) where Kitaev's Periodic Table holds. The method of derivation used is stable inclusion of symmetric spaces.

Plan of paper.
-This paper is organized as follows. In Section 2 we set up the vector-bundle description of translation-invariant ground states of gapped free-fermion systems for all symmetry classes, starting with class s = 0 (no symmetries; also known as class D) and increasing the number of (pseudo-)symmetries up to s = 7. The passage from vector bundles to an equivalent description by classifying maps is made in Section 3. There we also give a number of examples illustrating the difference between the topological classification by homotopy classes of classifying maps, isomorphism classes of vector bundles, and the stable equivalence of K-theory. In Section 4 we formulate the diagonal map and illustrate it at the special example of making the steps from (s = 0, d = 0) to (1, 1) and further to (2,2).
The role of Section 5 is to collect the results of homotopy theory relevant to our problem. We state the G-Whitehead Theorem and recall the classical Bott periodicity theorem in the complex and real settings. We also exploit the property of Z 2 -equivariance to reformulate homotopy as relative homotopy. By using all this information, we prove in Section 6 that the diagonal map induces a bijection in homotopy for the symmetry classes s = 2 and s = 6. In Section 7 we extend and complete the argument so as to cover all classes s. The final Section 8 presents the precise bounds on stability.

From symmetries to vector bundles
We begin with some notation and language. A quasi-particle vacuum (or free-fermion ground state, or Hartree-Fock-Bogoliubov mean-field ground state) is a state in Fock space which is annihilated by a set of (quasi-)particle annihilation operators. Two well-known examples are Hartree-Fock ground states, which have a definite particle number, and the paired states of the BCS theory of superconductivity.
To give a precise description of such many-fermion ground states, we set out from the formalism of second quantization. We assume that our translation-invariant physical system (with momentum space M) is built from a unit cell of Hilbert space dimension n. Singleparticle states are then characterized by their momentum k ∈ M and a band index j = 1, . . ., n. The single-particle creation and annihilation operators (denoted by c † k, j resp. c k, j and called Fock operators for short) obey the canonical anti-commutation relations (2.1) Organizing Fock operators by the momentum quantum number, we define W k as the vector space spanned by the Fock operators that lower the momentum by k. Thus where U k is the space of single-particle annihilation operators for momentum k, while V −k is the space of single-particle creation operators for momentum −k.
From now on, we are going to denote the operation of inverting the momentum by This is done in order to prepare the ground for a later modification of the involution τ. (For technical reasons, we will eventually be forced to consider "momentum" spaces M where some of the components of k are position-like instead of momentum-like.) In the present section, we always have τ(k) ≡ −k, and we will take the liberty of frequently writing −k instead of τ(k) for better clarity of the notation. In terms of the basis c k, j , c † k, j , the elements ψ ∈ W k are expressed as with coefficients u j , v j ∈ C. The vector spaces W k are complex, and they all have the same dimension 2n independent of k. In fact, they are canonically isomorphic (by unitary momentum-boost operators taken from the Heisenberg group), and we often omit the momentum quantum number and write simply W k ≡ W ≡ C 2n . The number n is referred to as the (total) number of (valence and conduction) bands. One may think of the collection of vector spaces {W k } k∈M as a complex vector bundle, say W , over the momentum space M. This bundle is trivial in our setting: W ≃ M ×W . It could, however, be non-trivial in a lowenergy effective theory where the bands far from the Fermi surface have been discarded. In any case, W has non-trivial subvector bundles, and these are the objects of our interest.
We now highlight some important structures on the vector spaces W k . First of all, the canonical anti-commutation relations (CAR) for fermion Fock operators induce for all k ∈ M a pairing between W k and W τ(k) , i.e. a non-degenerate bilinear form by dropping the δ -function δ (k − k ′ ) in Eq. (2.1). This pairing has the property of being symmetric for τ-invariant momenta τ(k) = k. We refer to it as the CAR pairing. Expressing ψ ∈ W τ(k) and ψ ′ ∈ W k as in Eq. (2.4) we have Next, Fock space comes equipped with a Hermitian scalar product, which determines an operation of Hermitian conjugation. Since Hermitian conjugation in Fock space takes operators that remove momentum k into operators that create momentum k, it induces a complex anti-linear involution γ : for all k ∈ M. By combining this γ-operation with the CAR pairing between W τ(k) and W k , we get a Hermitian scalar product on each vector space W k : In summary, the set {W k } k∈M is a trivial bundle W of canonically isomorphic Hermitian vector spaces W k ≡ W ≡ C 2n . It comes with the extra structure given by the pairing (2.5).
We are now in a position to formalize the type of free-fermion or Hartree-Fock-Bogoliubov mean-field ground state addressed in the present paper. In the following definition, the abbreviation IQPV stands for a quasi-particle vacuum with the property of being the translation-invariant ground state of an insulator (or gapped system).
Definition 2.1. -By an IQPV we mean a complex subvector bundle A π → M of fibers π −1 (k) ≡ A k ⊂ W k = C 2n of dimension n such that all pairs A k , A τ(k) of τ-opposite fibers annihilate one another with respect to the CAR pairing: (2.10) Remark 2.1. -Physically speaking, the vector space A k ⊂ W k is spanned by the quasiparticle operators of momentum k which annihilate the quasi-particle vacuum. The Fock space description |IQPV of the quasi-particle vacuum is recovered [17] by choosing a basis c 1 (k), . . ., c n (k) of A k for each k and applying their product to a suitable reference state: The condition (2.10) expresses the fact that all annihilation operators must have vanishing anti-commutators with each other. We refer to (2.10) as the Fermi constraint.
We will often use the ⊥-operation to express the Fermi constraint (2.10) as A ⊥ k = A τ(k) (for all k ∈ M). Secondly, given the Hermitian structure , , the orthogonal complement L c of L is defined by (2.12) For present use, we note that the two notions of orthogonality are connected by as a consequence of the relation γψ, ψ ′ = {ψ, ψ ′ }.
In the remainder of this section we will impose various symmetries which centralize the translation group: first time reversal T ; then particle number Q; then particle-hole conjugation C; and so on. The optimal order in which to arrange these symmetries was first understood by Kitaev [5]; we therefore call it the Kitaev sequence.
All of the symmetries T , Q, C, etc., will have the status of true symmetries (i.e., they commute with the Hamiltonians of the appropriate symmetry class; never do they anti-commute). In particular, our operator C of particle-hole conjugation commutes with a particle-hole symmetric Hamiltonian: H = CH C −1 . We emphasize this systematic and rigid feature, as it sets our approach apart from what is usually done in the current literature, with notable exceptions being [7,9].
The resulting free-fermion ground states with symmetries all turn out to fit neatly into the following mathematical framework. To formulate it, recall that the , -orthogonal complement of A ⊂ W is denoted by A c ⊂ W . Please be advised that the process of implementing the framework will convert true physical symmetries into "pseudo-symmetries". Definition 2.2. -By an IQPV of class s (s = 0, 1, 2, . . .) we mean a rank-n complex subvector bundle A π → M as described in Def. 2.1 but with the fibers π −1 (k) ≡ A k ⊂ W ≃ C 2n constrained by the pseudo-symmetry conditions The operator H(k) is commonly referred to as the flattened Hamiltonian, as it may be viewed as a Hamiltonian with energies ±1 independent of k. It is a unitary transformation which is not orthogonal in general, but rather satisfies for all ψ, ψ ′ ∈ W . The notion of flattened Hamiltonian is used in [5,6], along with an orthonormal basis of W consisting of γ-fixed vectors. In this "Majorana" basis, all orthogonal unitary transformations are expressed as real orthogonal matrices.

Remark 2.5.
-Based on the Kitaev sequence, Definition 2.2 arranges for the IQPVs of class s + 1 to be contained in those of class s. The existence of such an inclusion has invited attempts [6] to transcribe the classical result of real Bott periodicity [16,18] so as to derive the desired homotopy classification. In the present paper we pick up on this attempt and show that it can be brought to fruition by invoking additional information.
As a final remark, let us elaborate on a comment made in the introductory section. In our setting and language, a real vector bundle in the sense of Atiyah [13], or quaternionic vector bundle in the sense of [15], would be a complex vector bundle A → M with a C-anti-linear projective involution (T 2 = ±1) mapping the fiber over k to the fiber over τ(k) = −k. Our vector bundles are not of this kind in general. Indeed, for s = 0 we do have the ⊥-operation determining the vector space A τ(k) = A ⊥ k as the annihilator space of A k , yet there exists no canonical map taking the individual vectors in A k to vectors in A τ(k) . Table 1 gives a quick summary of the systematic structure developed in the remainder of this Section (2.1-2.9). Readers who are prepared to take the systematics for granted may want to take a look at Section 2.5 and then proceed directly to Section 3.

2.1.
Class s = 0 (alias D). -The first symmetry class to consider is that of s = 0. This class is realized by gapped superconductors or superfluids with no symmetries (other than translations); it is commonly referred to as class D. By Definition 2.1 an IQPV of class s = 0, or translation-invariant free-fermion ground state of a gapped system in symmetry class D, is a vector bundle A → M with fibers A k ⊂ W ≃ C 2n that are complex n-dimensional vector spaces subject to the Fermi constraint (2.10) or, equivalently, see Remark 2.2. As will be explained in Section 3, there exists an alternative description of such a vector bundle by a so-called classifying map. We seize this opportunity to make two comments. For one, the literature on the subject often construes the relation (2.17) (or rather, its consequences for the Hamiltonian) as a "particle-hole symmetry", although it is actually no more than a fundamental constraint dictated by Fermi statistics -a point forcefully made in [4]. Note especially that no antiunitary or complex anti-linear operations are involved in (2.17).
Our second comment concerns the language used. Borrowing Cartan's notation for symmetric spaces, the terminology for symmetry classes of disordered fermions was introduced in [3]. This was done in the context of mesoscopic metals and superconductors where translation invariance is broken by the presence of disorder. A good fraction of the condensed matter community has adopted the same terminology for the related, but different purpose of classifying translation-invariant ground states (instead of disordered Hamiltonians). This is suboptimal but probably beyond rectification given the developed state of the research field. It is suboptimal because a dictionary is required for the non-expert to translate the terminology into the pertinent mathematics. For example, an IQPV of class D is determined (see Section 3 below) by a Z 2 -equivariant mapping M → Gr n (C 2n ) that maps the τ-fixed points of M to O 2n /U n -a symmetric space not of type D but of type DIII.
where z(k) ∈ C ∪ {∞} and |0 is the Fock vacuum. This state is annihilated for any k by the quasi-particle operator u(k)c k + v(k)c † −k . Thus we may regard it as a vector bundle A → M with fibers

Class s = 1 (alias DIII).
-We now impose the first symmetry (beyond translations), by requiring that our quasi-particle vacua are invariant under the anti-unitary operator T which reverses the time direction. More precisely, we assume time-reversal symmetry for fermions with half-integer spin, so that T 2 = −Id. (Although T is fundamentally defined on the single-particle Hilbert space and then on Fock space, T here denotes the induced action on single-particle creation and annihilation operators.) The resulting symmetry class is commonly called DIII; it is realized, for example, by superfluid 3 He in the B-phase.
Because time reversal inverts the momentum, it gives us a mapping which is actually a pair of maps T : U k → U τ(k) and T : V τ(k) → V k . This pair is compatible with the CAR pairing (2.5) in the sense that Notice that T 2 = −Id requires n to be even. The quasi-particle vacuum encoded in a vector bundle A → M is time-reversal invariant if the quasi-particle annihilation operators at momentum k are transformed by T into quasiparticle annihilation operators at momentum −k = τ(k), i.e., (2.20) Note that even though an anti-unitary operation T is now involved, the fibers A k ⊂ W of the vector bundle A are still complex -and it is not useful to fix any real subspace W R ⊂ W , as W R would have to be (re-)polarized to accommodate the complex vector spaces A k .
To bring (2.20) in line with Eq. (2.14) of Definition 2.2, we observe that the anti-unitary operator T commutes with the operation γ of Hermitian conjugation of Fock operators. Thus by concatenating T with the γ-operation, we get a complex linear operator which has square J 2 1 = −Id since T 2 = −Id and γ 2 = Id. It is easy to check that for all ψ, ψ ′ ∈ W . Thus J 1 is an orthogonal unitary transformation of W . Moreover, the true symmetry condition (2.20) is equivalent to the pseudo-symmetry condition (2.13). Hence the translationinvariant free-fermion ground state of a gapped superconductor or superfluid in symmetry class DIII is precisely modeled by an IQPV of class s = 1 in the sense of Definition 2.2. A one-dimensional example of such a ground state is given in Section 4.3.

Class s = 2 (alias AII).
-Imposing another symmetry (beyond translation and timereversal invariance), we now require that our quasi-particle vacua be compatible with the global U(1) gauge symmetry underlying the law of charge conservation (which is the same as conservation of particle number if all particles carry the same quantum of charge). The resulting symmetry class is commonly called AII. It is realized in band insulators and it hosts, in particular, the so-called quantum spin Hall insulator.
Recall from (2.2) the decomposition W k = U k ⊕V τ(k) by particle annihilation and creation operators. The operator Q for charge (or particle number) acts on U k ⊂ W k as −1 and on V τ(k) ⊂ W k as +1. We say that a quasi-particle vacuum conserves charge (or has fixed particle number) if it is invariant under the action of the U(1) gauge group of operators e iθ Q ; in that case, we prefer to call the quasi-particle vacuum a Hartree-Fock (mean-field) ground state. Noting that invariance of a vector space under a one-parameter group is equivalent to invariance under its generator, we have To bring this symmetry condition in line with Definition 2.2, we observe that the operator iQ is unitary and preserves the CAR pairing { , } since is an automorphism of the canonical anti-commutation relations (2.1). Moreover, J 1 = γ T anti-commutes with Q because T preserves the decomposition W = U ⊕ V while γ swaps the two summands. Therefore, the operator J 2 defined by has the properties of anti-commuting with J 1 and squaring to −Id. Because both J 1 and iQ are orthogonal unitary transformations, so is J 2 . Altogether, we now have two orthogonal unitary generators J 1 , J 2 satisfying the Clifford algebra relations (1.1) for s = 2. Now recall J 1 A k = A c k and use Q A k = A k to do the following computation: Thus the fibers A k of a translation-invariant Hartree-Fock ground state of a band insulator in symmetry class AII are constrained by the pseudo-symmetry conditions k , reflecting the true symmetry conditions TA k = A τ(k) and Q A k = A k . This means that such a ground state is an IQPV of class s = 2 in the sense of Definition 2.2.
2.3.1. Discussion, and class A. -Let us add here some discussion to reveal the physical meaning of the ground-state fibers A k , as this meaning may be somewhat concealed by our comprehensive framework. The condition Q A k = A k of conserved particle number forces A k for all k to be of the form Phrased in physics language, an annihilation operator in the fiber A k ⊂ W k of the Hartree-Fock ground state A is either an operator that annihilates a particle in an unoccupied state of momentum k, or is an operator that annihilates a hole (i.e., creates a particle) in an occupied state of momentum τ(k). For the physical situation at hand (namely, that of a band insulator) the dimension n p ≡ dim A p k is independent of k and is called the number of conduction bands. The dimension n − n p ≡ n h = dim A h k is called the number of valence bands. Now recall that J 1 = T γ and Thus the orthogonality relation A k , J 1 A k = 0 splits into two parts: Since J 1 is unitary and J 2 1 = −Id, these two equations are not independent but imply one another. Moreover, given one of the two spaces, say A h k , they determine the other space A p k as the orthogonal complement of J 1 A h k in U k (and, turning it around, A h k as the orthogonal k is already determined completely by specifying just one of the two components, say A h k . Physically speaking, this means that the number-conserving Hartree-Fock ground states at hand are determined by specifying for each momentum k the space of valence band states. Let us also remark that the vector bundle A → M with (reduced) fibers π −1 (k) = A h k and anti-unitary symmetry T : constitutes a quaternionic vector bundle in the sense of [15]. We take this opportunity to mention one important symmetry class which lies outside the series s = 0, 1, . . ., 7 considered in this paper -namely symmetry class A, where one imposes the symmetry of Q but not that of T . What happens in that case? The answer is that one gets a complex vector bundle without any additional structure. In fact, the process of imposing the symmetry QA k = A k and reducing from A k to A h k simply deletes the Fermi constraint and leaves a rank-n h complex vector bundle with fibers A h k subject to no symmetry conditions at all. Class A plays an important role in the historical development of the subject, as it hosts the class of systems exhibiting the integer quantum Hall effect, where the role of topology was first discovered and understood.

Class s = 3 (alias CII
). -Next, we augment time reversal and particle number by a third symmetry: twisted particle-hole symmetry, which takes us to class CII. The operator, C, of twisted particle-hole conjugation is an anti-unitary transformation exchanging particle creation with particle annihilation operators (or particles with holes, for short); it is a nonrelativistic analog of charge conjugation for Dirac fermions.
In explicit terms, the transformation C : W k → W τ(k) consists of a pair of maps "Twisting" refers to the presence of a linear operator S = S † = S −1 : V k → V k with transpose S * : U k → U k . (Recall that for any linear operator L : X → Y one has a canonically defined adjoint or transpose, L * : Y * → X * . Note also that U k can be regarded as the dual vector space V * k by the CAR pairing.) In the typical examples offered by physics, S exchanges the conduction and valence bands of a system at half filling. We require that S commutes with T . Note the relations (2.25) Now a particle-hole symmetric ground state A → M obeys the symmetry condition (2.26) To bring this in line with the general scheme, consider the linear operator which squares to −Id and is a unitary transformation preserving the CAR pairing of W (because both iQ and γ C are). It anti-commutes with both J 1 and J 2 (because Q does, while γ C commutes), so we now have the Clifford algebra relations (1.1) for s = 3.
Thus a translation-invariant free-fermion ground state of a gapped system in symmetry class CII is an IQPV of class s = 3 in the sense of Definition 2.2.

Class AIII.
-For use in the final Sections 7 and 8, we mention here another "complex" symmetry class, namely AIII, which is like class A in that it lies outside the 8-fold scheme of the "real" symmetry classes (s = 0, . . . , 7). Class AIII differs from CII by the absence of time-reversal symmetry T ; i.e., one has only the Fermi constraint and the symmetries under particle number Q and particle-hole conjugation C. As discussed in Section 2.3.1, the Fermi constraint gets effectively canceled by Q. Nevertheless, in the presence of the true symmetry C there is still the pseudo-symmetry J 3 = iγ CQ. In other words, the situation is formally like that of class DIII (s = 1), but with the Fermi constraint out of force. The pseudo-symmetry J = J 3 is often understood as a so-called sublattice symmetry; the latter, however, is not a true symmetry in our sense, as it anti-commutes with the Hamiltonian.

Going beyond
-To continue the Kitaev sequence beyond s = 3, we need to expand the physical setting by bringing in true symmetries (namely, spin rotations) of a different type than before. We first describe the total algebraic framework that emerges for s ≥ 4 and then explain the physics for each of the symmetry classes s = 4, 5, 6, 7 in sequence.
Thus, let us assume that on W = C 2n we are given two sets of orthogonal unitary operators, { j 1 , j 2 } and { j 5 , . . ., j s }. The former will be recognized as (two of the three) spin-rotation generators and the latter as pseudo-symmetries due to the possible presence of T , Q, and C. Here s ≥ 4 and the second set is understood to be empty when s = 4. The motivation for leaving a gap in the index set will become clear shortly.
We demand that the following algebraic relations be satisfied for our operators: Thus { j 1 , j 2 } and { j 5 , . . . , j s } are two sets of Clifford algebra generators on W , and any two generators belonging to different sets commute with one another. As before, the translation-invariant free-fermion ground state of a gapped system (now of symmetry class s) will be described by a vector bundle over M with n-dimensional fibers a k ⊂ W = C 2n spanned by the quasi-particle annihilation operators at momentum k. (The change of notation from A k to a k is to clear the symbol A k for use with a closely related, but different object.) For reasons that will be explained in detail in the following subsections, the vector spaces a k are required to obey the set of conditions Notice that j 1 , j 2 are true symmetries taking a k to itself, whereas j 5 , . . ., j s are pseudosymmetries taking a k to its orthogonal complement a c k . We will now demonstrate that such a multiplet of (pseudo-)symmetries is equivalent to a set of s pseudo-symmetries J 1 , . . ., J s .
The key step is to double the dimension of W by taking the tensor product with C 2 , and to consider on C 2 ⊗W the set of operators (2.30) By using the algebraic properties laid down in (2.28) one readily verifies that the operators J 1 , . . . , J s so defined satisfy the Clifford algebra relations (1.1).
The strategy now is to transfer all relevant structure of W to C 2 ⊗ W . In the case of the Hermitian scalar product , W we do this by viewing the doubled space as the orthogonal sum W + ⊕W − = C 2 ⊗W of two identical copies W + = W − = W and setting The CAR bracket { , } W is transferred to C 2 ⊗ W by the same principle. The transferred structures define involutions L → L c and L → L ⊥ as before. Note that with these conventions all operators J 1 , . . . , J s are orthogonal unitary transformations of C 2 ⊗W . Now let {a k } k∈M be a vector bundle with n-dimensional fibers a k ⊂ W that satisfy the conditions (2.29). Then we construct a new vector bundle A short computation shows that the relations (2.29) translate into the relations Thus we have assigned to a vector bundle {a k } k∈M constrained by the (pseudo-)symmetry conditions (2.29) an IQPV A → M of class s in the sense of Definition 2.2. This correspondence turns out to be one-to-one. While the proof does have some bearing on the rest of this paper, it is not essential here. We therefore relegate it to the Appendix and proceed with the main message of this section.
2.6. Class s = 4 (alias C). -We are now ready to address class C, which is defined to be the symmetry class of fermions with spin 1/2 and SU 2 spin-rotation symmetry (plus the pervasive translation invariance of the present context). Note that class C does not follow upon CII in the same way that class CII follows upon AII or class AII upon DIII. In fact, the operators T , Q, and C characteristic of the preceding classes cease to be symmetries here; they are superseded by the spin-rotation generators. Examples of quasi-particle vacua of symmetry class C are found among superconductors with spin-singlet pairing.
Let the generators of SU 2 spin rotations be denoted by j 1 , j 2 , and j 3 . As operators on the spinor space C 2 they are represented by 2 × 2 matrices, say One may also think of these matrices j 1 , j 2 and j 3 = j 2 j 1 as a basis (including the unit matrix) for the algebra H of quaternions. For the following, we assume that the quaternion algebra of j 1 , j 2 , j 3 acts reducibly on our vector spaces Here as always, the translation-invariant quasi-particle vacuum of a gapped system (now of class C) is described by a vector bundle over M with n-dimensional fibers a k ⊂ W = C 2n spanned by the quasi-particle annihilation operators at momentum k. These fibers are still subject to the Fermi constraint a ⊥ k = a τ(k) . The property of spin-rotation invariance of the quasi-particle vacuum is expressed by the true symmetry conditions j l a k = a k (l = 1, 2, 3). Altogether, we now have the set of equations Owing to the quaternion relation j 3 = j 2 j 1 we may drop the last condition ( j 3 a k = a k ) as this is already implied by j l a k = a k for l = 1, 2. We then see that the conditions (2.34) coincide with the set of conditions (2.29) for s = 4. Following the blueprint of Section 2.5, we now double up the vector space W to C 2 ⊗W and use the mapping f of (2.32) to transform the vector bundle with fibers a k to an equivalent vector bundle A → M with fibers A k = f (a k ). By the assignments in (4.13), the Clifford algebra H = Cl(R 2 ) generated by j 1 and j 2 becomes the Clifford algebra Cl(R 4 ) generated by J 1 , . . ., J 4 . According to Eq. (2.33) the transformed fibers A k are subject to Since the mapping a k ↔ A k is one-to-one, we see that the translation-invariant free-fermion ground state of a gapped superconductor in symmetry class C is precisely modeled by an IQPV of class s = 4 in the sense of Definition 2.2.

Class s = 5 (alias CI). -
The genesis of the remaining 3 symmetry classes (s = 5, 6, 7) is parallel to that of the classes s = 1, 2, 3: they are obtained by first imposing time-reversal invariance, then charge conservation, and finally particle-hole conjugation symmetry. The difference from the earlier setting is that SU 2 spin rotations now are symmetries throughout. In view of the detailed treatment given in Sections 2.2-2.4, we can be brief here. The first additional symmetry to impose is time-reversal invariance. As before, we assume fermions with spin 1/2, so that T 2 = −Id. The new symmetry condition on the fibers is (2.36) The resulting symmetry class is commonly called CI. By composing T : W → W with γ : W → W we get an orthogonal unitary operator We will now argue on physical grounds that j 5 commutes with the spin-rotation generators j l for l = 1, 2, 3. For this, we first observe that the physical observable of spin, like any component of momentum or angular momentum, is inverted by the operation of time reversal. Since T is complex anti-linear and our generators j l carry an extra factor of i = √ −1 as compared to the physical spin observables, we infer that T j l T −1 = + j l (for l = 1, 2, 3). Secondly, spin rotations g = e ∑ x l j l preserve the CAR pairing { , } and (for x l ∈ R) the Hermitian structure , ; thus they are orthogonal unitary transformations of W . This implies that spin rotations commute with γ and so do their generators j l . Altogether, we obtain We recall the Fermi constraint a ⊥ k = a τ(k) and the symmetry conditions (2.34). By the transcription a k ↔ A k of Section 2.5, it follows that the translation-invariant free-fermion ground state of a gapped superconductor in symmetry class CI is exactly given by an IQPV of class s = 5 in the sense of Definition 2.2.
2.8. Class s = 6 (alias AI). -Next, by including the U(1) symmetry group underlying particle-number conservation, we are led to what is called symmetry class AI. In addition to the previous conditions on fibers we now have ∀k ∈ M : Q a k = a k . (2.39) As before, Q = +1 on creation operators and Q = −1 on annihilation operators. To transcribe this condition to the present framework, we introduce The two operators j 5 and j 6 share the algebraic properties of the pair J 1 , J 2 ; for the detailed reasoning we refer to Section 2.3. Moreover, j 6 like j 5 commutes with the spin-rotation generators j 1 , j 2 , j 3 . Thus we now have the algebraic relations (2.28) for s = 6. The true symmetry conditions a k = Q a k = Ta τ(k) are equivalent to the pseudo-symmetry conditions j 5 a k = j 6 a k = a c k . In conjunction with the Fermi constraint a ⊥ k = a τ(k) and the spin-rotation symmetries (2.34), this means that translation-invariant Hartree-Fock ground states of insulators in symmetry class AI are given by IQPVs of class s = 6.
2.9. Class s = 7 (alias BDI). -Finally, to arrive at class s = 7 (also known as BDI) we augment the symmetry operations of translations, spin rotations, time reversal and U(1) gauge transformations by (twisted) particle-hole conjugation C. Thus we require ∀k ∈ M : Ca k = a τ(k) . (2.41) The properties of the anti-unitary operator C were listed in (2.25). In addition, we demand that the twisting operator γ C commutes with the spin-rotation generators j 1 , j 2 , j 3 .
For reasons that were explained in Section 2.4, the unitary operator preserves the CAR pairing of W . It squares to −Id, anti-commutes with both j 5 and j 6 , and commutes with j 1 , j 2 , and j 3 . Thus we now have the relations (2.28) for s = 7.
The symmetry condition Ca k = a τ(k) is equivalent to the pseudo-symmetry condition j 7 a k = a c k . In view of this and all the other constraints obeyed by a k , the translation-invariant freefermion ground state of a gapped system in symmetry class BDI is an IQPV of class s = 7.
As a final remark, let us mention that there exist simpler ways of realizing class BDI in physics. (A similar remark applies to class AI.) By the (1, 1) periodicity theorem of Section 4.1 and the 8-fold periodicity of real Clifford algebras [19], the effect of 7 "real" pseudosymmetries J 1 , . . . , J 7 is the same (after reducing the number of bands by a factor of 2 4 ) as that of a single "imaginary" pseudo-symmetry K. One may take K = iγ C; thus class BDI is realized by superconductors with particle-hole conjugation symmetry. For another superconducting realization, one may take K = iγT with a time-reversal operator T that squares to +Id.

From vector bundles to classifying maps
In this section we pass from the vector-bundle description to an equivalent description by classifying maps. Recall from Definition 2.2 that an IQPV of class s is a rank-n complex subvector bundle A π → M with the property that its fibers π −1 (k) = A k ⊂ C 2n obey the pseudo-symmetry conditions J 1 A k = . . . = J s A k = A c k and the Fermi constraint A ⊥ k = A τ(k) for all momenta k ∈ M. The equivalent description by a classifying map is as follows.
Let C 0 (n) ≡ ∪ 2n r=0 Gr r (C 2n ) where Gr r (C 2n ) is the Grassmannian of complex r-planes A in W = C 2n . (Although the Fermi constraint A ⊥ = A singles out r = n, we allow r = n here for later convenience.) Given C 0 (n), let C s (n) ⊂ C 0 (n) be the subspace of complex hyperplanes that satisfy the constraints due to s pseudo-symmetries J 1 , . . . , J s : The classifying map Φ for a vector bundle A → M of class s then is simply the map assigning to the momentum k ∈ M the complex hyperplane A k ∈ C s (n).

This reformulation does not yet account for the Fermi constraint
Fixing a class s, we have that the group Z 2 acts on two spaces, M and C s (n), with the nontrivial element acting by τ on the former and τ s on the latter. In view of this, the condition (3.5) can be rephrased as saying that the mapping Φ : An important role is played by the special momenta that satisfy k = τ(k). At these points of M, the condition (3.5) of Z 2 -equivariance constrains Φ to take values in the set of fixed points of τ s . We denote this subspace by The reformulation of the current subsection is summarized by the following statement.
Remark 3.1. -Having recast the Fermi constraint as a condition of Z 2 -equivariance, one may wonder why we could not regard our quasi-particle vacua as Z 2 -equivariant vector bundles. The answer is that although the ⊥-operation gives rise to a well-defined involution τ s on C s (n), it does not determine (not for general values of s) any kind of complex linear or anti-linear mapping from A k to A τ(k) .

Remark 3.2.
-Although the two descriptions by vector bundles and classifying spaces are in principle equivalent, they suggest different notions of topological equivalence. This point is elaborated in the next subsection.
Proposition 3.1 gives a characterization of our vector bundles which is concise and efficient for the purpose of systematic classification by topological equivalence. Yet, the precise nature of the spaces of Z 2 -equivariant classifying maps Φ may not reveal itself immediately to the novice, as the situation seems to get more and more involved and constrained for an increasing number of pseudo-symmetries J 1 , . . . , J s . However, the identification and detailed discussion of the classifying spaces C s (n) and their subspaces R s (n) of τ s -fixed points for all classes s = 0, 1, 2, ..., 7 can be found in the published literature; see [18,6]. (To see that our definition of the "real" spaces R s (n) agrees with that of the literature, one observes that by the relation (2.9) the Hermitian structure , and the CAR bracket { , } reduce to the same Euclidean structure on the real subspace R 2n = W R ⊂ W of γ-fixed points, see Remark 2.4.) The well-known outcome of this exercise is displayed in Table 2, where we substitute n ≡ 8r. One observes that C s+2 (2n) = C s (n). This 2-fold periodicity reflects the fact that doubling the representation space and extending a complex Clifford algebra by 2 generators is the same as tensoring it with the full algebra of complex 2 × 2 matrices. In the same vein, there is an 8-fold periodicity R s+8 (16n) = R s (n), reflecting a similar isomorphism [19] over the real number field.
and the pseudo-symmetry conditions (2.14). On the other hand, we may describe it by a classifying map Φ : Our goal is to establish a topological classification of translation-invariant free-fermion ground states of gapped systems with given symmetries (i.e. of IQPVs in a given symmetry class). To do so, we need to settle on a notion of topological equivalence. In the present paper, we employ the equivalence relation which is given by homotopy: we say that two IQPVs belong to the same topological class if they are connected by a continuous deformation. More precisely, a homotopy between two IQPVs in class s with classifying maps Φ 0 and Φ 1 is given by a continuous family Φ t with τ s • Φ t = Φ t • τ for all t ∈ [0, 1]. We emphasize that all vector bundles in our setting are subbundles of the trivial bundle M ×W = M × C 2n . Understood in this way, the equivalence relation of homotopy leads, in general, to more topological classes than does the equivalence relation given by the notion of isomorphy of vector bundles. This is illustrated by the following example.
Gr q (C n )] as long as 2p ≥ dim M. This bijection breaks down, however, when the inequality of dimensions is violated; it then becomes possible for two IQPVs to be isomorphic without being homotopic. A concrete example is provided by the "Hopf magnetic insulator" [21] for M = S 3 with p = q = 1, where 2p = 2 < 3 = dim S 3 . Indeed, while all complex line bundles over S 3 are isomorphic to the trivial one (Vect C 1 (S 3 ) = 0), such vector bundles, viewed as subbundles of S 3 × C 2 , organize into distinct homotopy classes since [S 3 , Gr 1 (C 2 )] = π 3 (S 2 ) = Z.

These homotopy classes are distinguished by what is called the Hopf invariant.
A standard approach used in the literature is to work with a further reduction of the topological information contained in isomorphism classes, by adopting the equivalence relation of stable equivalence between vector bundles. We will use class A once more in order to illustrate the construction. Two vector bundles A 0 → M and A 1 → M are stably equivalent if they are isomorphic after adding trivial bundles (meaning trivial valence bands in physics language), i.e. if there exist m 1 , m 2 ∈ N such that Under the direct-sum operation, the stable equivalence classes constitute a group called the (reduced) complex K-group of M, which is denoted as K C (M). (Inverses in this group are given by the fact that for compact M, all complex vector bundles A have a partner A ′ such that A ⊕ A ′ ≃ M × C n for some n ∈ N, where the right-hand side represents the neutral element.) In the limit of a large number of valence and conduction bands, namely the stable regime, the elements of the reduced K-group are in bijection with the homotopy classes of maps into the classifying space [20]: Outside the stable regime, stably equivalent vector bundles need not be isomorphic, much less homotopic.

Example 3.2.
-Consider the tangent bundle T S 2 of the two-sphere. By regarding S 2 as the unit sphere in R 3 , we also have the normal bundle NS 2 ≃ S 2 × R. The direct sum of T S 2 and NS 2 is S 2 × R 3 . Thus T S 2 is stably equivalent to the trivial bundle. Yet the isomorphism class of T S 2 differs from that of the trivial bundle.
In the present context, a physical realization of T S 2 is the ground state of a system in symmetry class AI in two spatial dimensions, albeit in the generalized sense that the operation of time reversal is replaced by the combination of time reversal and space inversion, which effectively restricts the fibers A k to be real vector spaces. This realization corresponds to the non-trivial element 1 ∈ N 0 = Vect R 2 (S 2 ) in Table A.1 of [14].

Remark 3.3.
-To compare our approach with that of K-theory, we picked the example of class A. It turns out that only two more of our symmetry classes are accommodated by the standard formulation of K-theory for vector bundles: these are class AI (s = 6), where vector bundles are equipped with a complex anti-linear involution (corresponding to the physical symmetry of time reversal T with T 2 = +1), and class AII (s = 2), where the involution is replaced by a projective involution (time reversal T with T 2 = −1). In the former case, taking stable equivalence classes leads to KR-groups [13,14], while in the latter case it leads to KQ-groups [22,15]. For the other symmetry classes, the corresponding K-theory groups can only be inferred indirectly by an algebraic construction using Clifford modules as in [5,11]. In all cases, the K-theory groups of momentum space M are in bijection with the homotopy classes of Z 2 -equivariant maps M → C s (n) -denoted by [M,C s (n)] Z 2 as a setin the limit of large n (as well as large p and q where applicable, see Table 2).
To sum up, the natural equivalence relation for us to use is that of homotopy. It is a finer tool than stable equivalence (as considered in [5]) and even isomorphy of vector bundles (as considered in [14,15] for s = 6 and s = 2), and is therefore adopted as our topological classification principle. Although we will ultimately work in the stable regime in order to utilize such results as the Bott periodicity theorem, the use of homotopy theory allows us to keep track of the precise conditions under which our equivalences hold. In other words, we are able to say how many bands are required in order for the physical system to be in the stable regime for a given space dimension.

The diagonal map
In this section we introduce the "master diagonal map" -a universal mapping that takes a d-dimensional IQPV of class s and transforms it into a (d + 1)-dimensional IQPV of class s + 1. While there exist in principle many such maps -for some previous efforts in this direction see [6,8] -the one described here stands out in that it can be proven to induce a one-to-one mapping between stable homotopy classes of base-point preserving and Z 2equivariant maps M = S d → C s (n) and S d+1 → C s+1 (2n). For more general choices of M, including the torus M = T d , the mapping to be described is injective. It also bears a close relation to the mapping underlying the phenomenon of real Bott periodicity.
From now on, we will use the model of an IQPV of symmetry class s as a Z 2 -equivariant classifying map φ : M → C s ≡ C s (n). The goal is to construct from φ a new mapping, Φ, which maps M × S 1 (actually the momentum-type suspensionSM of M) into C s+1 . It is not difficult to see that such a map will not induce an injective map of homotopy classes in general, unless the ambient vector space W is enlarged. Therefore our story of constructing Φ begins with a modification of W : we double its dimension by replacing it by C 2 ⊗ W .
The procedure is identical to that of Section 2.5, and we here assume it to be understood. At the same time, we now extend the given Clifford algebra of pseudo-symmetries by two generators, in the process reviewing and exploiting a result known as (1, 1) periodicity.
Let us mention that the physical meaning of the step W → C 2 ⊗W depends on the case. For example, for s = 0 the tensor factor C 2 is to introduce a spin-1/2 degree of freedom. For s = 1 it is to replace a single band by a pair of bands -one valence and one conduction band. 1. (1,1) periodicity. -To offer some perspective on the following, the statement we are driving at is closely related to two standard isomorphisms of complex and real Clifford algebras, namely Cl(C s+2 ) ≃ Cl(C 2 ) ⊗ Cl(C s ) and Cl(R s+1,1 ) ≃ Cl(R 1,1 ) ⊗ Cl(R s ).

4.
Let there be Clifford algebra generators j 1 , . . ., j s that satisfy the relations (1.1) and are orthogonal unitary transformations of W = C 2n , which means that they preserve , W and { , } W . Then we take the tensor product of W with C 2 and pass to a Clifford algebra with s + 2 generators J 1 , . . ., J s+2 defined on C 2 ⊗W by J l = 0 1 1 0 ⊗ j l (l = 1, . . ., s), The Hermitian scalar product and the CAR bracket of W are transferred to the doubled space in the natural way explained in Section 2.5. Note that on C 2 ⊗W we have the two involutions A → A c and A → A ⊥ as before.
We now observe that all Clifford algebra generators J 1 , . . . , J s+2 are orthogonal unitary transformations of C 2 ⊗ W but for the distinguished generator K = J s+2 , which is unitary but sign-reverses the extended CAR bracket: We call K "imaginary" while using the adjective "real" for the generators J 1 , . . . , J s+1 . Let us note the alternative option of working with the modified generator iK instead of K. The former would be a bona fide orthogonal transformation of C 2 ⊗W , but it has square plus one, and one would call it "positive" (in contradistinction with the "negative" generators J 1 , . . . , J s+1 ) as is done in [5,6]. We prefer the present convention of a negative but imaginary generator K, as it will render our later discussion of the diagonal map more concise.
We are now ready to get to the point. Let us recall from Section 3 the spaces (s ≥ 0) By C 0 (n) we simply mean the space of all complex r-planes (0 ≤ r ≤ 2n) in W = C 2n . The subspace R 0 (n) consists of all complex planes a ⊂ W with the "Lagrangian" property a = a ⊥ ; such planes are necessarily of dimension n. The spaces C s and R s will be called the "complex" and "real" classifying spaces for our IQPVs (or vector bundles) of class s. As major players of our classification work they were tabulated in Table 2 of Section 3. Next, we decree the corresponding definitions at the level of the doubled space C 2 ⊗W : Here the more elaborate notation R s+1,1 reflects the fact that the generators J 1 , . . . , J s+1 are real, whereas the last generator J s+2 = K is imaginary. Now consider the mapping f : a → A defined by Eq. (2.32) of Section 2.5. It is clear that f is a map from C s (n) to C s+2 (2n). Indeed, one easily checks that for a ∈ C s (n) the image plane A = f (a) satisfies the relations J l A = A c (l = 1, . . . , s + 2) of C s+2 (2n). Moreover, from a = a ⊥ one deduces that A = A ⊥ . Thus f restricts to a map f ′ : R s (n) → R s+1,1 (2n). The statement of (1, 1) periodicity is now as follows. The proof of the proposition will consume the rest of this subsection. It remains to show that, given the Clifford algebra generators J 1 , . . . , J s+2 on the doubled space, W , one can reconstruct the original framework built on the generators j 1 , . . . , j s on W so as to invert the mapping f : a → A. For the inverse direction, we may not assume the decomposition W = C 2 ⊗W and the connecting relations (4.1) but must construct them. This is done in the following. We begin with some preparation and state a useful lemma along the way.
Like the other generators, the distinguished operator K = J s+2 is unitary and anti-Hermitian and has eigenvalues ±i. Let the corresponding eigenspaces be denoted by Note that all operators J 1 , . . . , J s+1 exchange these spaces: J l W ± = W ∓ (l = 1, . . ., s + 1) and that dimW + = dimW − . The idea of the sequel is to carry out a reduction from W to First of all, the non-degenerate symmetric bilinear form { , } (the CAR pairing) given on W descends by restriction to a non-degenerate symmetric bilinear form (4.9) Indeed, if w + ∈ W + and w − ∈ W − , then since K sign-reverses the CAR pairing. By similar reasoning, the Hermitian scalar product , : W × W → C descends to a Hermitian scalar product , W : W ×W → C. Writing L ≡ iIK, observe that L 2 = Id W and L A = A. It follows that A has an orthogonal decomposition by L-eigenspaces: As we shall see, A is already determined by one of the two summands, say A ∩ E +1 (L). To show that, consider the operator Π = 1 2 (Id − iK) of orthogonal projection from W to W , and let A (±) ⊂ W be the image of A ∩ E ±1 (L) under the projector Π.

Lemma 4.1. -The linear maps
because L anti-commutes with K and hence exchanges W + with W − . The map Π : v → v + is surjective by the definition of A (+) . It is also injective since v + = 0 implies v = v + +Lv + = 0 and therefore w ∈ E −1 (L). Thus the map Π : A∩E +1 (L) → A (+) is an isomorphism of vector spaces. The argument for Π : To prove the second statement, let w ∈ A (+) and w ′ ∈ A (−) . Then w + Lw ∈ A and and from A, A c = 0 we infer that 0 = w + Lw , w ′ + Lw ′ = 2 w, w ′ . Thus A (+) and A (−) are orthogonal to each other. Because of A (+) ⊂ W and A (−) ⊂ W are in fact orthogonal complements of each other.
As an immediate consequence, we have: -The vector space W has an orthogonal decomposition by the following four subspaces: We now carry out a reduction based on the isomorphism Π : A ∩ E +1 (L) → A (+) . For this we observe that the relations J l A = A c have the following refinement: due to the fact that J l commutes with L = iIK. We recall that the operators J 1 , . . ., J s exchange the subspaces W ± = E ±i (K). The operators L J 1 , . . . , L J s then preserve these subspaces and hence commute with the projector Π. By applying Π to Eq. (4.13) and using Corollary 4.1 it follows that L J l A (±) = A (∓) (l = 1, . . . , s). (4.14) In view of the above, we introduce the restricted operators j l := L J l W (l = 1, . . ., s). .
because L = iIK preserves the CAR bracket. Therefore, A = A ⊥ implies {a, a} W = 0. By the same reasoning, we have {a c , a c } W = 0, since A = A ⊥ entails A c = (A c ) ⊥ . Now the combination of {a, a} W = 0 with {a c , a c } W = 0 implies that a is half-dimensional: dim a = dim a c = 1 2 dimW . Hence a = a ⊥ . Thus the mapping C s+2 (2n) → C s (n) by A → a restricts to a mapping from R s+1,1 (2n) to R s (n). This inverts the maps f : C s (n) → C s+2 (2n) and f ′ : R s (n) → R s+1,1 (2n) and completes the proof of the proposition.

Z 2 -equivariant Bott map. -
We now turn to the construction of the Z 2 -equivariant Bott map, or diagonal map for short. Fixing any symmetry index s ≥ 0, we are given a pair of classifying spaces C s (n) and R s (n). We then apply to them the (1, 1) periodicity theorem in the expanding direction. That is, starting from s real generators j 1 , . . ., j s on W , we follow Section 4.1 to pass to an extended system of s + 2 generators J 1 , . . ., J s , I, K on C 2 ⊗W . The ensuing construction begins with the space By imposing on it the two additional pseudo-symmetries, first I and subsequently K, we get From Proposition 4.1 we recall the bijection C s (n) ≃ C s+2 (2n).
As before, we denote the subspaces of fixed points of the Fermi involution A → A ⊥ by R j (2n) ⊂ C j (2n) (for j = s, s + 1). Now the last one of the s + 2 Clifford algebra generators, namely K, is imaginary, i.e., it sign-reverses the CAR pairing. While this is of no relevance for the spaces above, it does matter for the subspace of fixed points of the Fermi involution in C s+2 (2n). Recall that this subspace is denoted by (4.18) By construction, we have a bijective correspondence R s+1,1 (2n) ≃ R s (n); cf. Prop. 4.1.
Next, for any C-linear operator X on C 2 ⊗W , let X → X T denote the operation of taking the transpose w.r.t. the CAR pairing, i.e. {X T w, w ′ } = {w, X w ′ } for all w, w ′ ∈ C 2 ⊗W .

Lemma 4.2.
-If an automorphism τ car of GL(C 2 ⊗ W ) is defined by τ car (g) = (g −1 ) T , then for any subvector space A ⊂ C 2 ⊗W one has the relation Proof. -By definition, the vectors of (g · A) ⊥ have zero CAR pairing with those of g · A.
It follows that g T · (g · A) ⊥ = A ⊥ and hence (g To prepare the next formula, let each complex hyperplane A in C 2 ⊗W be associated with an anti-Hermitian operator where Π A and Π A c project on A and its orthogonal complement A c , respectively.  Recall that E +i (K) ⊂ C 2 ⊗ W denotes the eigenspace of K with eigenvalue +i. Since the Clifford generators J 1 , . . . , J s , and I anti-commute with K, they swap the two eigenspaces E +i (K) and E −i (K) = E +i (K) c . This means that the two 2n-planes E ±i (K) lie in C s+1 (2n). Applying this to any w ∈ E +i (K) we get Turning to the last stated property, we note that the automorphism τ car of Lemma 4.2 sends J(A) to J(A ⊥ ). Since K is imaginary, we have τ car (K) = −K and E +i (K) ⊥ = E −i (K). Therefore, To summarize, our map t → β t (A) ∈ C s+1 (2n) is a Z 2 -equivariant curve (actually, a minimal geodesic in the natural Riemannian geometry of C s+1 (2n)) which joins the invariable point E +i (K) with its antipode E −i (K) by passing through the variable point A at t = 1/2.
Let the space of all paths in C s+1 (2n) from E +i (K) to E −i (K) be denoted by Ω K C s+1 (2n). Then as an immediate consequence of Lemma 4.3 we have: from the classifying space C s (n) to the path space Ω K C s+1 (2n). By its property of Z 2 -equivariance, β induces a mapping between the sets of Z 2 -fixed points: There is an induced action of Z 2 on the space of paths Ω K C s+1 (2n). The symbol (Ω K C s+1 (2n)) Z 2 denotes the subspace of paths that are fixed by this Z 2 -action.
which correspond to the empty and the fully occupied state, |0 and |1 , respectively. The procedure of doubling by (1, 1) periodicity here amounts to forming the tensor product with the two-dimensional spinor space, (C 2 ) spin . As input A ∈ R 0 (1) ≃ R 1,1 (2) we take the complex line of the state with both spin states occupied: The operator I is to be identified with the first pseudo-symmetry J 1 of the Kitaev sequence, where the left tensor factor (denoted by "BdG" for Bogoliubov-deGennes) acts in the twodimensional quasi-spin space with basis c and c † . The simplest choice of imaginary generator K is We then apply the one-parameter group of β to produce an IQPV of class s = 1. By using β 1/2 (A) = A and switching from the path parameter t ∈ [0, 1] to the momentum parameter k = π(t − 1/2), we write the fibers A k(t) = e (t π/2)KJ(A) · E +i (K) as (For a more informed perspective on this construction, please consult Remark 5.1 below.) To the physics reader this may look more familiar when written as a BCS-type ground state: |g.s. = e ∑ k cot(k/2)P k |vac , P k = c † k,↑ c † −k,↓ . If the imaginary generator is chosen as K = K(α) = i(σ 1 ) BdG ⊗ (σ 1 cos α + σ 3 sin α) spin , the Cooper pair operator P k takes the more general form which clearly displays the spin-triplet pairing of the superconductor at hand. The physical system is in a symmetry-protected topological phase, since the winding in its ground state cannot be undone without breaking the time-reversal invariance. 1) to (2,2). -To give a second example, we start from the outcome of the previous one and progress to a two-dimensional band insulator with conserved charge in class AII. This time, the effect of (1, 1) doubling for the already spinful system is to introduce two bands, which we label by p and h. To implement charge conservation directly and avoid working through all the details of (1, 1) doubling, we first perform a change of basis (by a particle-hole transformation) on our class-DIII superconductor to turn it into a particle-number conserving reference IQPV of class s = 1:

Example 2: from
We see that the change of basis has turned each quasi-particle operator into either an annihilation operator a k 1 ,• or a creation operator b † −k 1 ,• . We stress that {A k 1 } is still the ground state in disguise of our one-dimensional class-DIII superconductor -it has only been jacked up by (1, 1) periodicity. The pseudo-symmetry operators now are In this representation, the operator J(A k ) is expressed by Now we again apply the one-parameter group of β , still with parameter k 0 = π(t − 1/2) for t ∈ [0, 1]. In this way we get a two-dimensional IQPV of class s = 2: where k = (k 0 , k 1 ). Notice that there is a redundancy here: the four operators spanning A k are not independent; rather, the subspace of conduction bands ( a) is already determined by the subspace of valence bands ( b) and vice versa; cf. the discussion in Section 2.3.1. This is the price to be paid for our comprehensive formalism handling all classes at once.
At k 0 = ±π/2 -the two poles of a two-sphere with polar coordinate k 0 + π/2 -, the k 1 -dependence goes away by construction. These two points are easily seen to be the only points where the Kane-Melé Pfaffian vanishes, which implies that our IQPV of class s = 2 has non-trivial Kane-Melé invariant [23] and lies in the quantum spin Hall phase.

Homotopy theory for the diagonal map
In this section, we collect and develop a number of homotopy-theoretic results related to the diagonal map β . This is done en route to our goal of proving that β induces the desired bijection in homotopy: a one-to-one mapping, β Z 2 * , between stable homotopy classes of base-point preserving and Z 2 -equivariant maps f : S d → C s (n) and F : S d+1 → C s+1 (2n).
We have introduced β somewhat informally, but now state precisely how it is to be used. Let f : M → C s (n), k → A k , be the classifying map of an IQPV of class s. By composing f with β we get a new map Recall that Ω K C s+1 (2n) denotes the space of paths in C s+1 (2n) from E +i (K) to E −i (K). Next, we choose to view the path coordinate t as a coordinate for the second factor in the direct product M × [0, 1]. We then re-interpret β • f as a mapping Since β t degenerates at the two points t = 0 and t = 1, the mapping F descends to a map (still We let the non-trivial element of Z 2 act onSM by Thus,SM is the "momentum-type" suspension of M. (The symbol SM is reserved for the position-type suspension invoked later on.) Then, since f : . Thus, starting from an IQPV of class s over the d-dimensional space M, the composition with β produces an IQPV of class s + 1 over the (d + 1)-dimensional spaceSM.
In the sequel, we restrict all discussion to the case of topological spaces with base points, say (X , x * ) and (Y, y * ), and to base-point preserving maps f : X → Y , f (x * ) = y * . Borrowing the language from physics, this means that there is (at least) one distinguished momentum k * ∈ M whose fiber A k * is not free to vary but is kept fixed: A k * ≡ A * . This condition is physically well motivated in many (if not all) cases. For example, for a superconductor one takes k * to be a momentum far outside the Fermi surface of the underlying metal, and A * = U is then the "vacuum" space spanned by the bare annihilation operators.
We adopt the convention of denoting by [X ,Y ] * the set of homotopy classes of base-point preserving maps f from a topological space (X , x * ) to another topological space (Y, y * ). If X and Y are G-spaces (with base points that are fixed by G), the symbol [X ,Y ] G * denotes the set of homotopy classes of G-equivariant and base-point preserving maps f : X → Y .
In our concrete setting, we choose a base point k * = τ(k * ) for M and the corresponding τS M -fixed point (k * , 1/2) as the base point ofSM. Our classifying maps f : M → C s (n) are required to assign to k * an invariable fiber f (k * ) = A * = A ⊥ * ∈ R s (n). Proof.
Moreover, if f u for u ∈ [0, 1] is a homotopy of Z 2 -equivariant maps from M to C s (n), then F u given by F u (k,t) = (β t • f u )(k) is a homotopy of Z 2 -equivariant maps fromSM to C s+1 (2n). Thus β * descends to β Z 2 * as claimed.
Remark 5.1. -It is perhaps instructive to highlight the workings of β and β Z 2 * for a zerodimensional momentum space consisting of two τ-fixed points, M = {k ∈ R | k 2 = 1} ≡ S 0 . In this case the suspensionS(S 0 ) can be regarded as the circle S 1 ⊂ C of unitary numbers with involution τS M given by complex conjugation. This viewpoint is realized by the map Now, starting from f : S 0 → R s (n) ≃ R s+1,1 (2n) with f (k * ) = A * and f (−k * ) = A (for some choice of base point k * = ±1), we apply β to obtain F : S 1 → C s+1 (2n) as It is easy to verify that F is continuous and satisfies F(e iθ ) ⊥ = F(e −iθ ). Thus F is a Z 2equivariant loop F : S 1 → C s+1 (2n). Half of the loop is determined by the choice of base point A * ; the other half is variable and parameterized by A ∈ R s (n). By the reasoning given above, this construction induces a mapping of homotopy classes,

Connection with complex Bott periodicity. -
In the previous subsection we introduced a mapping in homotopy, β Z 2 * , which makes sense for any momentum space M with an involution τ. Our goal now is to show that, under favorable conditions, this map is bijective.
Let us recapitulate the situation at hand: we have a Z 2 -equivariant mapping β : C s (n) → Ω K C s+1 (2n), cf. Eq. (4.22), doubling the dimension of W and increasing the symmetry index and the momentum-space dimension by one. The first step of the following analysis is to investigate β as an unconstrained map; which is to say that we forget the Z 2 -actions on C s (n) and Ω K C s+1 (2n) for the moment. Note that π d (X ) ≡ π d (X , x * ) denotes the homotopy group of base-point preserving maps from S d into the topological space (X , x * ).

Remark 5.2.
-We reiterate that all our maps are understood to be base-point preserving.
The base point of C s (n) lies in R s (n) ⊂ C s (n).

Remark 5.3.
-From the original paper by Bott [16] one has quantitative bounds on d in order for the Bott map β * to be an isomorphism; these are 2 ≤ d + 1 ≤ 2 (3−s)/2 n for odd s, and 1 ≤ d ≤ 2 (2−s)/2 n for even s. We will not need these stability bounds in the following.

G-Whitehead
Theorem. -The mapping β under consideration is Z 2 -equivariant, and the question to be addressed now is whether it is a homotopy equivalence between topological spaces carrying Z 2 -actions. The main tool to simplify (if not answer) this question is the so-called G-Whitehead Theorem, a standard homotopy-theoretic result that we now quote for the reader's convenience. Although we will be concerned only with the case of G = Z 2 , we will state the theorem for any group G. To do so in a concise way, we need to introduce some terminology first.
The statement of the G-Whitehead Theorem makes use of the notion of a G-CW complex, which we assume to be understood; see [24] for an introduction. (This reference deals with the case of the trivial group G = {e}. For the case of a general group G, see [25].) A fact of importance for us is that all products of spheres with factor-wise Z 2 -action are Z 2 -CW complexes, as this covers all cases considered later.
Suppose, then, that we are given a G-equivariant mapping f : Y → Z between G-spaces. If X is another G-space, consider the mapping induced by f , We are now in a position to write down the desired statement; for a reference, see [25].

Reformulation by relative homotopy.
-We return to our task of investigating the mapping β Z 2 * of Proposition 5.1. The link with the material above is made by the identifications Y = C s (n), Z = Ω K C s+1 (2n), and G = Z 2 . To apply the G-Whitehead theorem, we need to look at our map β : Y → Z and determine how connected (in the sense of Def. 5.2) are its restrictions Y H → Z H to the fixed-point sets of all subgroups H ⊂ Z 2 . There are only two subgroups to consider: H = {e} (trivial group), and H = G = Z 2 . In the former case, the required result has been laid down in Proposition 5.2. What remains to be dealt with is the latter case, namely β H : Y H → Z H for H = Z 2 .
By the first property, such a map F is already determined by its values on one of the two hemispheres ofS(S d ) = S d+1 . Such a hemisphere is a disk D d+1 parameterized by t for, say 0 ≤ t ≤ 1/2, with boundary S d at the equator t = 1/2. The values of F at the equator are constrained by F(x, 1/2) = F(x, 1/2) ⊥ ∈ R s+1 (2n). Thus the restriction of F to 0 ≤ t ≤ 1/2 is a mapping that takes D d+1 to C s+1 (2n), the boundary S d to R s+1 (2n), and the base point x * to A * . It is clear that this correspondence is bijective. Indeed, from the restricted data for 0 ≤ t ≤ 1/2 the full function F is reconstructed by the relation F(x, It is also clear that this bijection of maps descends to a bijection of homotopy classes. Now, using the identification offered by Lemma 5.1 we reformulate the maps of (5.6) as The G-Whitehead Theorem then prompts us to ask under which conditions these maps are isomorphisms. A partial answer is given in the next section.

Bijection in homotopy for s ∈ {2 , 6}
In this section we are going to show that for two symmetry classes, namely for s = 2 and s = 6, the issue in question can be settled rather directly. What distinguishes these two cases is the existence of a fiber bundle projection that allows us to reduce the task at hand to the standard scenario of real Bott periodicity. (The other cases, s / ∈ {2 , 6}, will have to be handled by a less direct argument.) To anticipate the strategy in somewhat more detail, the main idea is as follows. When s = 2 or s = 6, we are able to construct a fibration (actually, a fiber bundle) for a certain base space R s,1 (2n) ≃ R s,1 (2n). The projection p sends the base point A * ∈ R s+1 (2n) to the base point E −i (K) ∈ R s,1 (2n) and induces an isomorphism by basic principles. This isomorphism p * will be shown to compose with β ′ * to give the isomorphism underlying real Bott periodicity: Thus the desired statement will be reduced to a known result in topology. Let us make the historical remark that, in order to discover the space R s,1 (2n) which is central to our argument, it was necessary for us to abandon the usual (Majorana) convention of realizing the involution τ car by complex conjugation. In fact, we find it optimal to work with two such involutions at once. In the next subsection, which is preparatory, we introduce the second involution, τ car . By using the fact that I is real and K imaginary, which is to say that I preserves the bracket { , } while K reverses its sign, one computes Thus the roles of I and K get exchanged: while K was imaginary with respect to τ car it is real with respect to τ car , and vice versa for I. The remaining generators J l = τ car (J l ) = τ car (J l ) for l = 1, . . ., s are real with respect to both structures, CAR and CAR. Guided by the above, we employ τ s+1 to define a space R s,1 (2n) by While this is a curve in C s+1 (2n) when A ∈ C s+2 (2n) is in general position (and our true goal is to characterize the mapping β ′ to Z 2 -equivariant curves; see (4.23)), we now observe that t → β t (A) for A = A ⊥ ∈ R s+1,1 (2n) has the following alternative interpretation.
Proof. -By inspecting the definitions (6.8) and (4.18) one sees that R s+1,1 (2n) = R s,1 (2n) ∩ R s+1 (2n). (6.9) Indeed, the two spaces on the right-hand side have the same pseudo-symmetries including IA = A c , but the points of the second space are fixed with respect to τ s+1 while the first space is the fixed-point set of τ s+1 . In view of Eq. (6.5) this implies that A ∈ R s,1 (2n) ∩ R s+1 (2n) is invariant under multiplication by IK. Since I is a pseudo-symmetry, it follows that so is K, i.e., KA = A c . Therefore the intersection on the right-hand side of Eq. (6.9) does give the space on the left-hand side.
Hence β t (A) ∈ R s,1 (2n) as claimed. To prepare the next statement, note that the continuous map induces a mapping β * in homotopy; more precisely, by concatenating f : , and this construction induces a mapping between homotopy groups as usual; cf. the beginning of Section 5.
is an isomorphism for 1 ≤ d ≪ n .

Squaring by the CAR involution. -
We now adopt the simplified notation where the index 0 means the connected component containing the base point A * . Also, and from the above we record the relations Then we recall that the eigenspaces E ±i (K) of the generator K are exchanged by each of the linear operators I, J 1 , . . ., J s . Thus I, J 1 , . . ., J s are pseudo-symmetries for E ±i (K) and we have E ±i (K) ∈ C. This allows us to regard the connected space C as the orbit of, say E +i (K), under the action of its symmetry group, U : From this perspective, we may also think of C as a coset space U /U K where U K is the isotropy group of E +i (K): This subgroup U K can be viewed as the group of fixed points of a Cartan involution θ : (One may compute θ more simply by θ (u) = KuK −1 as I commutes with all u ∈ U .) On basic grounds, the fact that the elements of U K are fixed by a Cartan involution implies that U K is a symmetric subgroup and C ≃ U /U K is a symmetric space. (In fact, C in all cases is either a unitary group or a complex Grassmannian; see Table 2 in Section 3.) Note the relation τ car = θ • τ car . (6.16) Beyond U K ⊂ U , two more groups of relevance for the following discussion are the subgroups G and L of elements fixed by the CAR and CAR involutions respectively: As subgroups of U , both G and L act on C ≃ U /U K . These Lie group actions of G and L have nice properties due to the fact that both involutions, τ car and τ car , commute with θ , as is immediate from τ car (IK) = −IK = τ car (IK).
To get ready for Lemma 6.2 below, we need to accumulate a few more facts. First of all, the space R s,1 can be seen as the L-orbit in C through E +i (K). Alternatively, we may think of R s,1 = L · E +i (K) as R s,1 ≃ L/H for H = L ∩U K . Here we note that θ : U → U restricts to an involution θ : L → L and that H = Fix L (θ ) is a symmetric subgroup. It is sometimes useful to identify the symmetric space L/H with its Cartan embedding into L ⊂ U . This is defined to be the space A similar discussion can be given for the G-orbit in C through A * , but the only fact we need in this case is the identification G · A * = R s+1 . has the following properties: Proof. -We first show that p is into R s,1 . For this we write p(A) = Σ(A) · E +i (K) with Σ(A) = τ car (σ (A)) −1 σ (A) and send p(A) to its image under the Cartan embedding: Let Σ(A) ≡ Σ for short, and notice that τ car (Σ) = Σ −1 . Applying τ car to ℓ one gets τ car (ℓ) = τ car Σθ Now a short calculation using the assumption (i) shows that Σ commutes with θ (Σ) −1 . We therefore have τ car (ℓ) = ℓ ∈ L . This means that ℓ = θ (ℓ) −1 lies in the Cartan embedding U(L/H), which in turn implies that p(A) ∈ L · E +i (K). Thus p is into R s, 1 . To see that p : C → R s,1 is surjective, let A = σ (A) · E +i (K) ∈ R s,1 . By assumption (ii), the expression for p(A) in this case takes the form Thus p : R s,1 → R s,1 is the operation of squaring (or doubling the geodesic distance) from the point E +i (K): in normal coordinates by the exponential mapping w.r.t. E +i (K) it is the map p(A) = p(exp(X ) · E +i (K)) = exp(2X ) · E +i (K). (6.21) Since the squaring map is surjective, it follows that p : C → R s,1 is onto. Now recall R s+1,1 ⊂ R s,1 and β t (A) = e (t π/2)KJ(A) · E +i (K). The second stated property is then an immediate consequence of the relation (6.21): Turning to the third property, we observe that σ as a section of U → U /U K satisfies for some h(u, A) taking values in the isotropy group U K of E +i (K). By specializing this to A = g · A * ∈ R s+1 for u = g ∈ G and using g = τ car (g) we obtain From the second property of p we know that p(A * ) = p(β 1/2 (A * )) = β 1 (A * ) = E −i (K). Now the subgroup U K of θ -fixed points is stable under τ car , since θ and τ car commute. Therefore, τ car (h) −1 ∈ U K and we conclude that The principal bundle U → U /U K = C is trivial, and we may take σ to be of the form σ (A) = (u, Id), with the second factor being the neutral element. The involution τ car does not mix the two factors; therefore, the second factor of τ car (σ (A)) is still neutral. Because the Cartan involution θ swaps factors and thus moves the neutral element to the first factor, θ (σ (A)) commutes with σ (A) and τ car (σ (A)), as is required in order for the first condition of Lemma 6.2 to be met. Moreover, an element A ∈ R 2,1 lifts to σ (A) = (uτ Sp (u) −1 , Id) for some u ∈ U n . In this case one has τ car (σ (A)) = (τ Sp (u) u −1 , Id) = σ (A) −1 , which means that also the second condition of Lemma 6.2 is satisfied. The case of s = 6 is the same but for the substitutions n → n/4 and Sp → O.
Thus Lemma 6.2 applies, and from the properties stated there it follows that for s ∈ {2 , 6} we have a short exact sequence of spaces where the first map is simply the inclusion of R s+1 = p −1 (E −i (K)) into C.  [24]) that the mapping p * induced by the projection p of a fibration -in the concrete setting at hand, that's the map is an isomorphism of homotopy groups for all d. By composing p * with the mapping β ′ * of Eq. (5.7), we arrive at the map By the second property of p stated in Lemma 6.2, this composition is identical to the standard Bott map of Proposition 6.1. Since the latter is an isomorphism for 1 ≤ d ≪ n and p * is an isomorphism for all d, it follows that β ′ * is an isomorphism for 1 ≤ d ≪ n . Remark 6.3. -To draw the same conclusion for all classes s, one would need eight fibrations of the following type: The third (s = 2) and seventh (s = 6) of these are the fibrations discussed in the proof of Proposition 6.2. While the others are available [26] in the K-theory limit of infinitely many bands (n → ∞), they do not seem to exist at finite n .
Anticipating the further developments of the next section, the fruit of all our labors in this paper will be Theorem 7.2, which applies to all symmetry classes s. Here we state and prove that result in a preliminary version restricted to s ∈ {2, 6}. Proof. -After the identification [SM,C s+1 (2n)] Z 2 * = [M, Ω K C s+1 (2n)] Z 2 * , our statement is an immediate consequence of the G-Whitehead Theorem as explained in Section 5. Recall that in order for that theorem to apply in the case of a Z 2 -equivariant mapping β : Y → Z, one has to show that β H : Y H → Z H is highly connected for all subgroups H of Z 2 . We have done so (with the identifications Y = C s (n) and Z = Ω K C s+1 (2n), and for s ∈ {2, 6}) for H = {e} (by Proposition 5.2) and H = Z 2 (by Prop. 6.2). In both cases, the fact that (for s = 2, 6) there is no bijection between π 0 (C s (n)) and π 0 (Ω K C s+1 (2n)) (resp. between π 0 (R s (n)) and π 0 (Ω K C s+1 (2n)) Z 2 ) is remedied by the assumption that M is path-connected. Indeed, under that condition the image of the base-point preserving map β (resp. β Z 2 ) lies entirely in the connected component of Ω K C s+1 (2n) (resp. (Ω K C s+1 (2n)) Z 2 ) containing the base point and we may simply restrict to that single connected component. With this detail in mind, the G-Whitehead Theorem indeed applies to give the stated result.

Bijection in homotopy for all s
In this section we extend the statement of Proposition 6.3 to all symmetry classes s. In order to do so, we find it necessary to generalize the momentum sphere M = S d to include position-like coordinates. Recall that the momentum sphere is defined as the compactification of R d with involution τ(k) = −k. We now introduce d x position-like coordinates and denote by The generalization to S d x , d k was previously used in [8] for the purpose of classifying topological phases in the presence of a defect. In fact, if the defect has codimension d x + 1, it can be enclosed by a large sphere S d x , and at every point of this sphere, the classification by Kitaev's Periodic Table without defect applies (whenever valid). Thus, the domain is enhanced to S d x × T d k if the system without defect has discrete translational symmetry, or S d x × S d k for continuous translation invariance. In [1], it is proved that one may replace S d x × T d k (resp. S d x × S d k ) by S d x , d k at the expense of losing "weak" invariants. For d x = d k = 0, the three entries of Z change to Z 2n+1 (class A), Z n/2+1 (class AII) and Z n/4+1 (class AI), corresponding to the connected components of C 0 (n) (class A), R 2 (n) (class AII) and R 6 (n) (class A); see Table 2.
The resulting sets [S d x , d k ,C s (n)] Z 2 * of equivariant homotopy classes are listed in Table 3. This was derived previously in [8] and will follow from the results of the present section.
In the following, we use the same definition, albeit with A ∈ C s (n) (rather than the previous A ∈ C s+2 (2n)) and with τ car (K) = K (rather than τ car (K) = −K). The latter change, i.e., replacing the imaginary generator K by a real one, has an important consequence: the second property listed in Lemma 4.3 changes from β t (A) ⊥ = β 1−t (A ⊥ ) to β t (A) ⊥ = β t (A ⊥ ). Hence, the additional coordinate t is now position-like rather than momentum-like. This means that the modified curve t → β t (A) agrees with the original Bott map [18]: all Z 2 -fixed points A ∈ R s (n) ⊂ C s (n) are now mapped to Z 2 -fixed points β t (A) ∈ R s−1 (n) ⊂ C s−1 (n) for all t.  [25]. For the trivial subgroup {e} ⊂ Z 2 , the map β : C s (n) → Ω K C s−1 (n) is the complex Bott map and therefore highly connected. Similarly, for the full group Z 2 , the map β restricts to the real Bott map R s (n) → Ω K R s−1 (n), which is also highly connected. The obstruction that there may be a mismatch between π 0 for C s (n) resp. R s (n) and Ω K C s−1 (n) resp. Ω K R s−1 (n), is avoided by the reasoning described in the proof of Proposition 6.3.
By specializing the result above to the case of M = S d x , d k (which is path-connected unless d x = d k = 0) and using SM = S(S d x , d k ) = S d x +1, d k we immediately get the following: Proof. -The idea of the proof is to first apply Theorem 7.1 repeatedly in order to adjust the symmetry index s to be either 2 or 6 (for concreteness, we settle on the arbitrary choice of 2 here), then use the statement of Proposition 6.3 to increase d k by one unit, and finally go to the symmetry index s + 1 by retracing the initial steps.
To spell out the details, let s = 2 + r. Then Theorem 7.1 implies that there is a bijection [M,C s (n)] Z 2 * ≃ [S r M,C 2 (n)] Z 2 * , where S r M is the r-fold suspension of M. Here we made use of the fact that if M is pathconnected, then so is its suspension. We next apply Proposition 6.3 to obtain a bijection [S r M,C 2 (n)] Z 2 * ≃ [SS r M,C 3 (2n)] Z 2 * . Finally, we observe thatSS r M = S rS M and do r applications of Theorem 7.1 in reverse: Specializing once more to M = S d x , d k we have -Although this result follows directly from the more general one in Theorem 7.2, it may be instructive to repeat the proof: as it clearly shows our chain of reasoning for a special case of importance in physics.
From the combination of the Corollaries 7.1 and 7.2, the entries of Table 3 are determined by just specifying one column of entries for variable symmetry index s but fixed values for the dimensions d x and d k , subject to d x +d k ≥ 1. For example, one may take (d x , d k ) = (1, 0), in which case [S 1,0 ,C s (n)] Z 2 * is none other than the well-known fundamental group π 1 (R s (n)) (or π 1 (C s (n)) for the classes A and AIII).

Stability bounds
In stating our theorems, 7.1 and 7.2, we simply posed the qualitative condition d = dim M ≪ n, leaving their range of validity unspecified. To fill this quantitative void, we are now going to formulate precise conditions on d (as a function of n) in order for the theorems to apply. where C s (n + m s ) is defined as stated above, but now with Clifford generators J l ⊕ J ′ l (for l = 1, . . ., s). The map i s has the property of being equivariant with respect to the (induced) Z 2 -action on its image and domain: In particular, its restriction i Z 2 s to the subspace R s (n) has image in R s (n + m s ). The goal of this section is to prove the following theorem: Proof. -Since M is path-connected and all maps are base-point preserving, we may replace C s (n) = C s (m s r) by its connected component (denoted by C s (m s r) 0 in the table) containing the base point A * . Then, by applying the Z 2 -Whitehead Theorem, we obtain the desired statements provided that i s is d 1 -connected and i Z 2 s is d 2 -connected, with numbers d 1 and d 2 that are yet to be determined. The latter is done in the remainder of the proof, where we distinguish between four cases.
Case (i). -We start with the three rows attributed to case (i) in the tables. These enjoy the property of having Lie groups for their target spaces and we can make use of the following three fiber bundles: Sp 2r ֒→ Sp 2r+2 → Sp 2r+2 /Sp 2r = S 4r+3 , each of which gives rise to a long exact sequence in homotopy. By using π m (S d ) = 0 for m < d, we infer from these sequences the following values of d 1 and d 2 : for U r ֒→ U r+1 , d 2 = 4r + 2 for Sp 2r ֒→ Sp 2r+2 .
For the next two cases, (ii) and (iii), the target spaces are quotients G r /H r with G r and H r being either an orthogonal, a unitary or a symplectic group. The strategy in the following will be to apply the result of case (i) to the exact sequence associated to the fiber bundle H r ֒→ G r → G r /H r .
We distinguish between case (ii) where the inclusion G r ֒→ G r+1 is less connected than the inclusion H r ֒→ H r+1 , and case (iii) where it is the other way around.
Case (iv). -In the remaining three rows of the table, the target space has the form of a quotient G p+q /G p × G q . For the product of any two topological spaces X and Y , one has a natural isomorphism [24] π m (X ×Y ) ≃ π m (X ) × π m (Y ) for all m ≥ 0. In the case of X = G p and Y = G q , it is compatible with the inclusions G p ֒→ G p+1 and G q ֒→ G q+1 . In other words, these maps form a commutative diagram Hence, if G p is m-connected and G q m ′ -connected, then G p × G q is min(m, m ′ )-connected.
In particular, G p × G q is always less connected than G p+q and we can follow the steps of case (iii) with H r replaced by G p ×G q . As a result, d 1 = min(m, m ′ )+1 = min(m+1, m ′ +1) (and the same for d 2 ). This completes the determination of d 1 and d 2 and, hence, the proof of the theorem.
Specializing to the physically most relevant case of M = S d x , d k , we obtain Once the conditions for (i s ) * to be bijective are met, we are in what is called the stable regime. In that case, Corollary 8.1 can be applied repeatedly to give a bijection (i s ) * : [S d x , d k ,C s (n)] Z 2 * → [S d x , d k ,C s (∞)] Z 2 * , where C s (∞) is the direct limit under i s . This is the limit where K-theory applies.
As discussed in Section 3.1, there is a difference between homotopy classes, K-theory classes, and isomorphism classes. This distinction is relevant for s = 2 (alias class AII), s = 6 (alias class AI) and class A (see Section 2.3.1), corresponding to case (iv) in the proof of Theorem 8.1. In these classes, there is a U 1 -symmetry leading to a decomposition of the fibers A k ∈ C s (n) as A k = A p k ⊕ A h k , where p stands for particles or conduction bands and h for holes or valence bands. Recall from Section 2.3.1 that A k is already determined by A h k . The bundle with fiber A h k over k ∈ M is a Q-vector bundle (class AII, see [15]), an R-vector bundle (class AI, see [14]) or an ordinary complex vector bundle (class A) over M. In [14] and [15], isomorphism classes of these vector bundles have been classified for M = S d x , d k with d k ≤ 4 and d x ≤ 1. However, as was emphasized in Section 3.1, in the situation at hand, where we have subvector bundles, isomorphism classes agree with homotopy classes only when dim A h is large compared to dim M and dim M Z 2 . It is the goal of the following to specify precisely what is meant by "large" in this context.
0 r = 1 r = 1 r = 1 r ≤ 2 1 r = 1 r = 1 2 3 r = 1 r = 1 4 r = 1 r = 1 r = 1 r ≤ 2 5 r = 1 r = 1 r = 1 r ≤ 2 r ≤ 2 6 q = 1 q = 1 q = 1 q ≤ 2 7 r = 1 r = 1 r = 1 r ≤ 2 r ≤ 2 r ≤ 3 For M = S d x , d k , the table in the Corollary simplifies to the following: bijective surjective class A d x + d k < 2q + 1 d x + d k < 2q + 1 class AI d x + d k < 2q + 1 and d x < q d x + d k ≤ 2q + 1 and d x ≤ q class AII d x + d k < 4q + 3 d x + d k ≤ 4q + 3 If the configuration space M meets the conditions for bijectivity as listed above, the set of (equivariant) homotopy classes is in bijection with the set of isomorphism classes of complex (class A), R-(class AI) and Q-(class AII) vector bundles (with fixed fibers over the base point of M). The rank of these bundles is determined by q (for class A and AI) or 2q (for class AII). Thus, we have derived the exact boundary to the part of the unstable regime which is described by isomorphism classes of certain vector bundles. We are now in a position to list all potentially unstable cases. There are infinitely many possibilities in general if d x and d k are unrestricted. However, the physically most relevant cases are those with d k ≤ 3 and d x < d k . The latter inequality is needed on physical grounds since the dimension of the defect is d k − d x − 1 ≥ 0. Table 4 lists all cases which are not in the stable regime and may therefore differ from the stable classification.
In Table 4, the cases in which isomorphism classes of vector bundles give the same classification as homotopy classes are included. In order to leave this intermediate regime, the conditions for q need to additionally be met by p. For instance, neither the stable classification nor the classification of complex vector bundles give any non-trivial topological  Table 3 which are neither captured by K-theory nor by isomorphism classes of vector bundles. Entries here correspond to the case of r = q = 1 in Table 4.
phases for d k + d x = 3 in class A, but the Hopf insulator with q = p = 1 has a homotopy classification by Z. It may also happen that non-trivial phases disappear in the unstable regime: in class AIII with d k + d x = 3, the stable Z classification is lost for r = 1 since [S d x , d k , U 1 ] * = π 3 (U 1 ) = 0.
For d x = 0, there is at most one exception for all entries which is neither in the stable regime nor in the regime of vector bundle isomorphism classes (since for the latter p = q = 1). The resulting change of the classification is shown in Table 5. The changes in the first two rows for d k = 3 are the ones described before. There are only two additional changes in the remainder of the table: For s = 5 (class CI) all non-trivial topological phases vanish in dimension d k = 3 for similar reasons as in class AIII. However, there is an important change for s = 4 (class C) from trivial (0) to non-trivial (Z 2 ) by a superconducting analog of the class-A Hopf insulator.

Appendix: proof of Proposition 2.1
Recall the mathematical setting of s ≥ 4 pseudo-symmetries J 1 , . . . , J s constraining the vector spaces A k by Eqs. (2.33). We must show that the solutions A k of (2.33) are in bijection with the solutions a k of Eqs. (2.29) for the reduced system of generators j 1 , j 2 , j 5 , . . ., j s .
Thus, let there be on W ≡ C 2 ⊗ W a set of s ≥ 4 orthogonal unitary operators J 1 , . . ., J s subject to the relations (1.1). Forming the two operators where K is seen to be imaginary, let the shortened system J 5 , . . . , J s , I, K define complex and real classifying spaces C s−2 (2n) and R s−3,1 (2n) by the exact analog of Eqs. (4.5) and (4.6) with s replaced by s − 4. We then know from Proposition 4.1 that there exist bijections C s−2 (2n) → C s−4 (n), R s−3,1 (2n) → R s−4 (n), A k → a k , which are given by intersecting A k with E +1 (L) for L ≡ J 1 J 2 J 3 J 4 and applying the projector Π = 1 2 (Id − iK) to obtain a k . The spaces on the right-hand side are determined by Eqs. (4.3) and (4.4) via the system j l = L J l W (l = 4, . . . , s) defined as in (4.15). Note that the restricted generators j l (5 ≤ l ≤ s) satisfy the third set of relations in (2.28).
It remains to take into account the presence of the additional generators J 1 , J 2 , and J 3 . These commute with K = iJ 1 J 2 J 3 and thus preserve the decomposition W = W + ⊕ W − = E +i (K) ⊕ E −i (K). Simply restricting them to the subspace W = W + as we obtain the relations stated in the first and second line of Eqs. (2.28). We also observe that the process of reduction to W makes j 3 and j 4 redundant as j 3 = j 2 j 1 and j 4 = −Id W .
To prove Proposition 2.1, we have to show that the conditions on A k due to the pseudosymmetries J 1 , J 2 are equivalent to the conditions on a k due to the symmetries j 1 , j 2 . The key observation here is that the pseudo-symmetry relations J l A k = A c k for l = 1, 2, 3 have the following refinement: , because J 1 , J 2 , J 3 anti-commute with L = J 1 J 2 J 3 J 4 and hence exchange the two eigenspaces E +1 (L) and E −1 (L). By applying the projector Π = 1 2 (Id − iK) to this equation in order to distill a k = Π(A k ∩ E +1 (L)), it follows from Corollary 4.1 that j l a k = (Π • J l )(A k ∩ E +1 (L)) = Π(A c k ∩ E −1 (L)) = a k (1 ≤ l ≤ 3), owing to the fact that the operators J l (l = 1, 2, 3) preserve the decomposition W = W + ⊕W − .
Conversely, the conditions j l a k = a k transform into the conditions J l A k = A c k (l = 1, 2, 3) by the inverse map a k → A k given in (2.32). This proves the said proposition.