Bott-Thom isomorphism, Hopf bundles and Morse theory

Based on Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over the sphere bundle $\mathbb{S}( \mathbb{R} \oplus V)$ in terms of vector bundles and Clifford module bundles over $X$. After passing to topological K-theory these results imply classical Bott-Thom isomorphism theorems.


Introduction
In their seminal paper on Clifford modules [1] Atiyah-Bott-Shapiro describe a far-reaching interrelation between the representation theory of Clifford algebras and topological K-theory. This point of view inspired Milnor's exposition [10] of Bott's proof of the periodicity theorem for the homotopy groups of the orthogonal group. The unique flavor of Milnor's approach is that a very peculiar geometric structure (centrioles in symmetric spaces) which is related to algebra (Clifford representations) leads to basic results in topology, via Morse theory on path spaces.
In the paper at hand we rethink Milnor's approach and investigate how far his methods can be extended. In fact, they allow dependence on arbitrary many extrinsic local parameters. Thus we may replace the spheres in Milnor's computation of homotopy groups by sphere bundles over any finite CW-complex. Among others this leads to a geometric perspective of Thom isomorphism theorems in topological K-theory.
This interplay of algebra, geometry and topology is characteristic for the mathematical thinking of Manfredo do Carmo. We therefore believe that our work may be a worthwhile contribution to his memory.
Recall that a Euclidean vector bundle E of rank p over a sphere S n can be described by its clutching map φ : S n−1 → SO p . In fact, over the upper and lower hemisphere E is the trivial bundle R p , and φ identifies the two fibres R p along the common boundary S n−1 as in the following picture.
Milnor in his book on Morse theory [10] describes a deformation procedure that can be used to simplify these clutching maps φ. The main idea in [10] is viewing the sphere as an iterated suspension and the map φ as an iterated path family in SO p with prescribed end points, and then Morse theory for the energy functional on each path space is applied. However in a strict sense, Morse theory is not applied but avoided: it is shown that the non-minimal critical points (geodesics) have high index, so they do not obstruct the deformation of the path space to the set of minima (shortest geodesics) via the negative gradient flow of the energy. Thus the full path space is deformed into the set of shortest geodesics whose midpoint set can be nicely described in terms of certain totally geodesic submanifolds P j ("centrioles"). In fact there is a chain of iterated centrioles SO p ⊃ P 1 ⊃ P 2 ⊃ · · · such that the natural inclusion of P j into the path space of P j−1 is d-connected for some large d and for all j (that is, it induces an isomorphism in homotopy groups π k for k < d and a surjection on π d ). This is sufficient for Milnor's purpose to understand the topology of the path spaces in order to compute the stable homotopy groups of SO p (Bott periodicity).
In [7] we went one step further and deformed the whole map φ into a special form: the restriction of a certain linear map φ o : R n → R p×p . The latter defines a module structure on R p for the Clifford algebra Cl n−1 , which turns the given bundle E into the Hopf bundle for this Clifford module. As shown in [7] this leads to a conceptual proof of [1,Theorem (11.5)], expressing the coefficients of topological K-theory in terms of Clifford representations, and thus gives a positive response to the remark in [1, page 4]: "It is to be hoped that Theorem (11.5) can be given a more natural and less computational proof".
In the present paper we will put this deformation process into a family context, aiming at a description of vector bundles over sphere bundles in terms of Clifford representations. More specifically, let V → X be a Euclidean vector bundle over a finite CW-complex X and letV = S(R ⊕ V ) → X be the sphere bundle of the direct sum bundle R ⊕ V . It is a sphere bundle with two distinguished antipodal sections (±1, 0). Similar as before a vector bundle E →V can be constructed by a fibrewise clutching function along the "equator spheres" S(V ) in each fibre, and one may try to bring this clutching function into a favourable shape by a fibrewise deformation process similar as the one employed in [7].
We will realise this program if V is oriented and of rank divisible by four in order to derive bundle theoretic versions of classical Bott-Thom isomorphism theorems in topological K-theory. For example let rk V = 8m and assume that V → X is equipped with a spin structure. Let S →V be the spinor Hopf bundle associated to the chosen spin structure on V and the unique (ungraded) irreducible Cl 8mrepresentation, compare Definition 7.8. Then each vector bundle E →V is -after addition of trivial line bundles and copies of S -isomorphic to a bundle of the form E 0 ⊕ (E 1 ⊗ S ), where E 0 , E 1 are vector bundles over X. Moreover the stable isomorphism types of E 0 and E 1 are determined by the stable isomorphism type of E , see Remark 8.3. In K-theoretic language this amounts to the classical Thom isomorphism theorem in orthogonal K-theory, compare part (a) of Theorem 9.7.
Our paper is organized as follows. In Section 2 we recall some notions from the theory of Clifford modules. Section 3 relates the theory of Clifford modules to iterated centrioles in symmetric spaces. This setup, which implicitely underlies the argument in [10], provides a convenient and conceptual frame for our later arguments.
A reminder of the Morse theory of the energy functional on path spaces in symmetric spaces is provided in Section 4 along the lines in [10]. This is accompanied by some explicit index estimates for non-minimal geodesics in Section 5. Different from [10] we avoid curvature computations using totally geodesic spheres instead.
After these preparations Section 6 develops a deformation theory for pointed mapping spaces Map * (S k , SO p ), based on an iterative use of Morse theory on path spaces in symmetric spaces. If R p is equipped with a Cl k -representation then Map * (S k , SO p ) contains the subspace of Hopf maps associated to Clifford subrepresentations on R p (compare Definition 2.1). Our Theorem 6. 13 gives conditions under which this inclusion is highly connected. Section 7 recalls the construction of vector bundle over sphere bundles by clutching data and provides some examples. The central part of our work is Section 8, where we show that if V → X an oriented vector bundle of rank divisible by four, then vector bundles E →V are, at least after stabilisation, sums of bundles which arise from Cl(V )-module bundles over X by the clutching construction and bundles pulled back from X. We remark that up to this point our argument is not using topological K-theory.
The final Section 9 translates the results of Section 8 into a K-theoretic setting and derives the Cl(V )-linear Thom isomorphism theorem 9.3 in this language. This recovers Karoubi's Clifford-Thom isomorphism theorem [8,Theorem IV.5.11] in the special case of oriented vector bundles V → X of rank divisble by four. In this respect we provide a geometric approach to this important result, which is proven in [8] within the theory of Banach categories; see Discussion 9.10 at the end of our paper for more details. Together with the representation theory of Clifford algebras it also implies the classical Thom isomorphism theorem for orthogonal K-theory. Finally, for completeness of the exposition we mention the analogous periodicity theorems for unitary and symplectic K-theory, which are in part difficult to find in the literature.

Recollections on Clifford modules
Let (V, , ) be a Euclidean vector space. Recall that the Clifford algebra Cl(V ) is the R-algebra generated by all elements of V with the relations vw+wv = −2 v, w ·1 for all v, w ∈ V , or equivalently, for any orthonormal basis (e 1 , . . . , e n ) of V , For V = R n with the standard Euclidean structure we write Cl n := Cl(R n ). Let K ∈ {R, C, H}. An (ungraded) Cl(V )-representation is a K-module L together with a map of R-algebras In other words, K is a Cl(V ) ⊗ K-module. We also speak of real, complex, respectively quaternionic Cl(V )-representations.
Let (e 1 , . . . , e n ) be an orthonormal basis of V and put J i = ρ(e i ). Due to (2.1) these are anticommuting K-linear complex structures on L, that is J 2 i = −I and We also speak of a Clifford family (J 1 , . . . , J n ). This implies that all J i are orientation preserving (for K = C, H this already follows from K-linearity).
In the following we restrict to real Cl(V )-modules; for complex or quaternionic Cl(V )-modules similar remarks apply. We may choose an inner product on L such that J i ∈ SO(L); equivalenty all J i are skew adjoint. In this case we also speak of an orthogonal Cl(V )-representation. With the inner product In particular we obtain an isometric linear map R ⊕ V → End(L), By the previous remarks it sends the unit sphere S(R ⊕ V ) ⊂ R ⊕ V into the special orthogonal group SO(L) ⊂ End(L).
Definition 2.1. We call the restriction the Hopf map associated to the orthogonal Clifford representation ρ.
Isometric linear maps R ⊕ V → End(L) are in one-to-one correspondence with isometric embeddings S(R ⊕ V ) → S(End(L)) onto great spheres. This leads to the following geometric characterisation of Clifford representations. Proposition 2.2. Let L be a Euclidean vector space and let be an isometric embedding as a great sphere, which satisfies Then µ is the Hopf map of an orthogonal Clifford representation Cl(V ) → End(L).
Proof. By assumption µ is the restriction of a linear map R ⊕ V → End(L). If A, B ∈ image(µ) ⊂ SO(L), then A + B ∈ R · SO(L) by assumption, hence Taking the trace on both sides we have s = 2 A, B . If A = I and B ⊥ I, this implies B + B T = 0 . For A, B ⊥ id we hence get the Clifford relation The structure of the real representations of Cl n is well known (cf. [9, p. 28]). They are direct sums of irreducible representations ρ n . These are unique and faithful when n ≡ 3 mod 4. Otherwise there are two such ρ n , which are both not faithful and differ by an automorphism of Cl n . The corresponding modules S n and algebras C n := ρ n (Cl n ) are as follows.
Here K(p) denotes the algebra of (p × p)-matrices over K. For n = 3 and n = 7, the two different module structures on S n = K for K = H, O are generated by the left and the right multiplications, respectively, with elements of the "imaginary" subspace R n = Im K = R ⊥ ⊂ K.
The action of (x, ξ) where L(ξ) = L(ξ)⊗id S k denotes the left translation on O, and where x ∈ R k ⊂ Cl k acts on S k = 1 ⊗ S k by ρ k .

Poles and Centrioles
Clifford modules bear a close relation to the geometry of symmetric spaces. Let P be a Riemannian symmetric space: for any p ∈ P there is an isometry s p of P which is an involution having p as an isolated fixed point. Two points o, p ∈ P will be called poles if s p = s o . The notion was coined for the north and south pole of a round sphere, but there are many other spaces with poles; e.g. P = SO 2n with o = I and p = −I, or the Grassmannian P = G n (R 2n ) with o = R n and p = (R n ) ⊥ . Of course, pairs of poles are mapped onto pairs of poles by isometries of P .
A geodesic γ connecting poles o = γ(0) and p = γ(1) is reflected into itself at o and p and hence it is closed with period 2. 2 : γ shortest geodesic in P with γ(0) = o, γ(1) = p}. For the sphere P = S n with north pole o, this set would be the equator. In general, M need not be connected, but it is still the fixed point set of a reflection (order-two isometry) r on P . 1 Hence the connected components of M , called (minimal) centrioles [5], are totally geodesic subspaces of P -otherwise short geodesic segments σ in the ambient space P with end points in a component of Fix(r) were not unique: γ γ) r( Fix(r) r P Each such midpoint m = γ( 1 2 ) determines its geodesic γ uniquely, and thus the set of minimal geodesics can be replaced with M : if there is another geodesicγ from o to p through m, it can be made shorter by cutting the corner at m, thus it is not minimal: There are chains of minimal centrioles (centrioles in centrioles): Peter Quast [13,14] classified all such chains starting from a compact simple Lie group P = G with at least 3 steps. The result is (3.2) below. The chains 1,2,3 are introduced in Milnor's book [10].
By G p (K n ) we denote the Grassmannian of p-dimensional subspaces in K n for K ∈ {R, C, H}. Further, Q n denotes the complex quadric in CP n+1 , which is isomorphic to the real Grassmannian G + 2 (R n+2 ) of oriented 2-planes in R n+2 , and OP 2 is the octonionic projective plane F 4 / Spin 9 .
A chain is extendible beyond P k if and only if P k contains poles again. E.g. among the Grassmannians P 3 = G p (K n ) only those of half dimensional subspaces (p = n 2 ) enjoy this property: Then (E, E ⊥ ) is a pair of poles for any E ∈ G n/2 (K n ), and the corresponding midpoint set is the group O n/2 , U n/2 , Sp n/2 since its elements are the graphs of orthogonal K-linear maps E → E ⊥ , see figure below.
For compact connected matrix groups P = G containing −I, there is a linear algebra interpretation for the iterated minimal centrioles P j . We only consider classical groups. Theorem 3.1. Let L = K p with K ∈ {R, C, H} and G ∈ {SO p , U p , Sp p } with p even in the real case. Then a chain of minimal centrioles G ⊃ P 1 ⊃ · · · ⊃ P k corresponds to a Cl k -representation J 1 , . . . , J k on L with J j ∈ G, and each P j is the connected component through J j of the set Proof. A geodesic γ in G with γ(0) = I is a one-parameter subgroup, a Lie group homomorphism 2 γ : R → G. When γ(1) = −I, then γ( 1 2 ) = J is a complex structure, J 2 = −I. Thus the midpoint setP 1 is the set of complex structures in G.
By induction hypothesis, we have anticommuting complex structures J i ∈ P i for i ≤ j, and P j is the connected component through J j of the setP j as in (3.3). Suppose that also Consider a shortest geodesic γ from J j to −J j in P j . Put J = γ( 1 2 ). Since P j is totally geodesic, γ is a geodesic also in G, Vice versa, let J ∈P j+1 , that is J is a complex structure anticommuting with J 1 , . . . , J j . Then J o = J −1 j J is a complex structure in G which anticommutes with J j and commutes with J i , i < j. In fact this is shortest in G, being a great circle in the plane spanned by I and J o . Further, the geodesic γ = J j γ o from J j to −J j is contained in P j , due to the subsequent Lemma 3.2 (applied to A = tπJ o ), and it is shortest in P j (even in the ambient space G). Thus J is contained in the midpoint set of (P j , J j ).
Lemma 3.2. J j exp A ∈ P j for some A ∈ g if and only if (3.5) A anticommutes with J j and commutes with J i for i < j.
γ γ s But on the other hand γs = R(gs)γ has the same length as γ since R(gs) is an isometry of G. Thus η = 0.

Deformations of path spaces
Minimal centrioles in P are also important from a topological point of view: Since they represent the set of shortest geodesics from o to p, they form tiny models of the path space of P . This is shown using Morse theory of the energy function on the path space [10].
We may replace (D k , ∂D k ) by an arbitray finite CW-pair (X,    for each path ω ∈ Ω * which is H 1 (almost everywhere differentiable with squareintegable derivative). Applying the gradient flow of −E we may shorten all H 1paths simultaneously to minimal geodesics. 3 Since the energy is not defined on all of Ω * , we will apply this flow only on the subspace of geodesic polygons Ω n ⊂ Ω * for large n ∈ N (to be chosen later), where each such polygon ω ∈ Ω n has its vertices at ω(k/n), k = 0, . . . , n, and the connecting curves are shortest geodesics. For any r ∈ N with n|r we have Ω n ⊂ Ω r . Furthermore Ω 0 * ⊂ Ω n for all n. Lemma 4.3. For all k ≥ 0 there exists an n such that the inclusion Ω n ⊂ Ω * is k-connected.
Let R be the convexity radius on P , which means that for any q ∈ P and any q ′ , q ′′ ∈ B R (q) the shortest geodesic between q ′ and q ′′ is unique and contained in B R (q). By equicontinuity, when n is large enough and x ∈ D k arbitrary, φ(x) maps every intervall [ k−1 n , k+1 n ] into B R (φ(x, k n )) ⊂ P , for k = 1, . . . , n − 1.
Let φ 1 : D k → Ω n such that φ 1 (x) is the geodesic polygon with vertices at φ(x)( k n ) for k = 0, . . . , n. Using the unique shortest geodesic between φ(x)(s) and φ 1 (x)(s) for each s ∈ [0, 1], we define a homotopy φ t between φ and φ 1 with For ω ∈ Ω n let ω k = ω|[ k−1 n , k n ] for k = 1, . . . , n. Its length is L( For c > 0 let Ω c n = {ω ∈ Ω n : E(ω) ≤ c}. We have Ω 0 * = Ω co n where c o is the energy of a shortest geodesic. By continuity, E • φ 1 is bounded, hence φ 1 (X) ⊂ Ω c n for some c > 0. When n is large enough, more precisely n > c/R 2 , any two neighboring vertices of every ω ∈ Ω c n lie in a common convex ball, hence the joining shortest geodesic segments are unique and depend smoothly on the vertices.
Thus we may consider Ω c n as the closure of an open subset of P × · · ·× P ((n− 1)times) with its induced topology.
The space Ω c n is finite dimensional and contains all geodesics of length ≤ √ c from o to p. It is a closed subset of P × · · · × P ((n − 1)-times), and its boundary points are the polygons in Ω n whose energy takes its maximal value c. The gradient of −E on Ω c n is a smooth vector field and its flow is smooth, too.
The index of any geodesic is the same in Ω * and Ω n , see [10,Lemma 15.4]. Hence the index of a non-minimal geodesicγ in Ω n (a "saddle" for E) is at least d + 1, and φ n (D k ) can avoid the domains of attraction for all these geodesics. The energy decreases along the gradient lines starting on φ n (D k ), thus these curves avoid the boundary of Ω c n and they end up on the minimum set of E, the set of minimal geodesics. Hence we can use this flow to deform φ into someφ : ⊂ Ω 0 * . By Lemma 4.3 we may (after a deformation which is constant on ∂D k ) assume that φ(D k ) ⊂ Ω c n for some large n. Now the claim follows from Lemma 4.4.

A lower bound for the index
How large is d in Theorem 4.2? This has been computed in [10, §23,24] and [7]. 4 We slightly simplify Milnor's arguments replacing curvature computations by totally geodesic spheres. An easy example is the sphere itself, P = S n . A nonminimal geodesic γ between poles o and p covers a great circle at least one and a half times and can be shortened within any 2-sphere in which it lies (see figure  below). There are n−1 such 2-spheres perpendicular to each other since the tangent vector γ ′ (0) = e 1 is contained in n − 1 perpendicular planes in the tangent space, Span (e 1 , e i ) with i ≥ 2. Thus the index is ≥ n − 1, in fact ≥ 2(n − 1) since any such geodesic contains at least 2 conjugate points where it can be shortened by cutting the corner, see figure. o p o p For the groups P = G = SO p , U p , Sp p (p even) and their iterated centrioles we have similar results. The Riemannian metric on G is induced from the inclusion for any A, B ∈ K p×p where A * :=Ā T . In particular, I, I = 1.
For general P there are two cases which behave quite differently: (a) π 1 (P ) is finite, (b) π 1 (P ) is infinite. We consider first P = SO p , which belongs to Case (a).
Proposition 5.1. The index of any non-minimal geodesic from I to −I in SO p is at least p − 1.
Proof. A shortest geodesic from I to −I in SO p is a product of p/2 half turns, planar rotations by the angle π in p/2 perpendicular 2-planes in R p . A non-minimal geodesic must make an additional full turn and thus a 3π-rotation in at least one of these planes, say in the x 1 x 2 -plane. We project γ onto a geodesic γ 1 in a subgroup SO 4 ⊂ SO p sitting in the coordinates x 1 , x 2 , x k−1 , x k , for any even k ∈ {4, . . . , p}. Then γ 1 consists of 3 half turns in the x 1 x 2 -plane together with (at least) one half turn in the x k−1 x k -plane.
In the torus Lie algebra t of so 4 which is ∈ t (the left and right multiplications by i ∈ H on H = R 4 ). Decomposing v with respect to this basis we obtain v = 2 ( 1 1 ) + 1 −1 . Thus the lift of γ 1 in S 3 × S 3 has two components: one is striding across a full great circle, the other across a half great circle. The first component passes the south pole of S 3 and takes up index 2. Since there are (p − 2)/2 such coordinates x k the index of a non-minimal geodesic in SO p is at least p − 2 (compare [10, Lemma 24.2]).
For U p we have a slightly different situation since we are now in Case (b). To any path ω : I −I we assign the closed curve det ω : [0, 1] → S 1 ⊂ C (assuming p to be even). Its winding number (mapping degree) w(ω) ∈ Z is obviously constant on each connected components of Ω(U p ; I, −I).
Definition 5.2. We will call w(ω) the winding number of ω.
Any geodesic in U p from I to −I is conjugate to γ(t) = exp(tπiD) where D = diag(k 1 , . . . , k p ) with odd integers k i , and w(γ) = 1 2 j k j . Let k j , k h be a pair of entries with k j > k h and consider the projection γ jh of γ onto U 2 acting on This is a geodesic in SU 2 ∼ = S 3 which takes the value ±I when t is a multiple of 1/b. For 0 < t < 1 there are b − 1 such t-values, and at any of these points γ ij takes up index 2. Thus Proof. Let k + be the sum of the positive k j and −k − the sum of the negative k j .
At least one of these inequalities must hold, unless all k i ∈ {1, −1}. In the latter case γ is minimal: it has length π which is the distance between I and −I in U p , being the minimal norm (with respect to the inner product (5.1)) for elements of (exp Next we will show that similar estimates hold for arbitrary iterated centrioles P ℓ of SO p . A geodesic γ : J ℓ −J ℓ in P ℓ has the form for some skew symmetric matrix A, which commutes with J 1 , . . . , J ℓ−1 and anticommutes with J ℓ , cf. (3.5). We split R p as a direct sum of subspaces, such that the subspaces M j are invariant under J 1 , . . . , J ℓ and A and minimal with this property. Then A has only one pair of eigenvalues ±ik j on every M j ; otherwise we could split M j further. Since exp(πA) = −I, the k j are odd integers which can be chosen positive.
. Then s ℓ+1 |p , and all non-minimal geodesics γ : Proof. Choose any pair of submodules M j , M h with j = h in (5.4). We may isometrically identify M j and M h as Cl ℓ+1 -modules, but first we modify the module structure on M h by changing the sign of J ℓ+1 . In this way we view The point γ(t) is fixed under conjugation with the rotation matrix e uB = cI −sI sI cI on M j ⊕ M j with c = cos u, s = sin u if and only if e tπbJ ′ = e −tπbJ ′ , see (5.5). This happens precisely when t is an integer multiple of All γ u are geodesics in P ℓ connecting J ℓ to −J ℓ . By "cutting the corner" it follows that γ can no longer be locally shortest beyond t = 1/b , see figure below. If there is at least one eigenvalue Now we consider the case ℓ = 4m − 2. Then J o := J 1 J 2 . . . J ℓ−1 is a complex structure 6 which commutes with A and J 1 , . . . , J ℓ−1 and anticommutes with J ℓ (since ℓ − 1 is odd). Thus A can be viewed as a complex matrix, using J o as the multiplicaton by i, and the eigenvalues of A have the form ik for odd integers k. As before, we split R p into minimal subspaces M j which are invariant under J 1 , . . . , J ℓ and A. Let E kj ⊂ M j be a complex eigenspace of A| Mj corresponding to an eigenvalue ik j . Then E kj is invariant under J 1 , . . . , J ℓ−1 (which commute with A and i), and also under J ℓ (which anticommutes with both A and i = J o ). By Again we consider two such modules M j , M h . As Cl ℓ -modules they can be identified, M j + M h = M j ⊕ M j , see (2.2). This time, Since e uB commutes with J 1 , . . . , J ℓ , the first equality implies γ u in P ℓ , see (3.5). Now γ(t) is fixed under conjugation with the rotation matrix e uB = cI −sI sI cI with c = cos u, s = sin u if and only if e πtbi = e −πtbi which happens precisely when t is a 5 Another way of saying: J ℓ , J ′ and BJ ′ span a Clifford 2-sphere in SO(M j + M h ) anticommuting with J 1 , . . . , J ℓ−1 and containing γ jh which covers at least three half great circles. 6 Let J 1 , . . . , Jn be a Clifford family and put wn = J 1 · · · Jn. Then w 2 n = J 1 · · · JnJ 1 · · · Jn = (−1) n−1 w 2 n−1 J 2 n = (−1) n w 2 n−1 . thus w 2 n = (−1) s I with s = n+(n−1)+· · ·+1 = 1 2 n(n+1). When n ≡ 2 mod 4, then s is odd, hence w 2 n = −I. Further, when n is even (odd), wn anticommutes (commutes) with J 1 , . . . , Jn. If the Clifford family extends to J 1 , . . . , J n+1 , then wn commutes with J n+1 when n is even and anticommutes when n is odd. integer multiple of 1/b. When b > 1, we obtain an energy-decreasing deformation by cutting b−1 corners, see figure above. Thus the index of a geodesic γ as in (5.3) is similar to (5.2): As before we need a lower bound for this number when γ is non-minimal. Any J ∈ P ℓ defines a C-linear map J −1 ℓ J = −J ℓ J since J ℓ J commutes with all J i and hence with J o . This gives an embedding 7 (5.7) P ℓ ֒→ U p/2 : J → J −1 ℓ J Note that p is divisible by four due to the representation theory of Clifford algebras (2.2), since for ℓ = 4m − 2 the space R p admits at least two anticommuting almost complex structures.
Definition 5.5. For any ω ∈ Ω := Ω(P ℓ ; J ℓ , −J ℓ ) we define its winding number w(ω) as the winding number of J −1 ℓ ω, considered as a path in U p/2 , see Def. 5.2. Proposition 5.6. Let ℓ = 4m − 2. Let γ : J ℓ −J ℓ be a geodesic in P ℓ with winding number w such that Proof. Let r be the number of irreducible Cl ℓ -representations in R p , that is r = p/s ℓ . Let k + be the sum of the positive k j and −k − the sum of negative k j .
One of these inequalities holds unless all k i ∈ {1, −1} which means that γ is minimal, see the subsequent remark.
Remark 5.7. The geodesic γ o (t) = e πtA from I to −I is minimal in U p/2 (with length π) if and only if A has only eigenvalues ±i. In this case, when lies inside P ℓ , the numbers k + of positive signs and k − of negative signs are determined by the above conditions k + + k − = p/s ℓ and k + − k − = 4w/s ℓ . This corresponds to a (J 1 , . . . , J ℓ )-invariant orthogonal splitting (which in particular is complex linear with respect to i · I = J 1 · · · J ℓ−1 ), with A = +i · I on L 0 and A = −i · I on L 1 . Then A 2 = −I, and we obtain another complex structure J ℓ+1 = AJ ℓ = γ( 1 2 ) anticommuting with J 1 , . . . , J ℓ and defining a minimal centriole P ℓ+1 ⊂ P ℓ . Note that In fact, it is the Cartan embedding of P ℓ into the group G ℓ = {g ∈ SOp : gP ℓ g −1 = P ℓ } .
For later use we formulate a necessary and sufficient condition when this Cl ℓ+1representation can be extended to a Cl ℓ+2 -representation.
Then the following assertions are equivalent.
Proof. With the notation from Remark 5.7 recall that J ℓ+1 J −1 When J ℓ+2 exists, then J ℓ+2 J −1 ℓ is another complex matrix which anticommutes with J ℓ+1 J −1 ℓ and thus interchanges the eigenspaces L 0 and L 1 . Therefore Vice versa, when k + = k − =: k, then L 0 and L 1 may be identified as Cl ℓmodules, and by putting J ℓ+2 J −1 we define a further complex structure anticommuting with J 1 , . . . , J ℓ+1 . Remark 5.9. We do not need to consider the group G = Sp p and its iterated centrioles since Sp p = P 4 (SO 8p ), cf. (3.2). For G = U p , the iterated centrioles are U q and G q/2 (C q ) (the complex Grassmannian) for all q = p/2 m . U q has been considered in Prop. 5.3. In G q/2 (C q ) the index of non-minimal geodesics is high when q is large enough as can be seen from the real Grassmannian G q/2 (R q ) = P 3 (Sp q ). In fact, real and complex Grassmannians have the same rank, therefore any geodesic γ in the complex Grassmannian can be conjugated into the real one. Hence the index of γ in the complex Grassmannian is at least as big as in the real Grassmannian, but this is one of the spaces treated in Prop. 5.4. There are analogues of Remark 5.7 and Proposition 5.8 in the complex case (for even ℓ) and for the quaternionic case (for ℓ = 4m − 2).

Deformations of mapping spaces
We consider a round sphere S = S k , k ≥ 1, with "north pole" N = e 0 ∈ S, and its equator S ′ = S ∩ N ⊥ with "north pole" (or rather "east pole") N ′ = e 1 ∈ S ′ , where the standard basis of R k+1 is denoted e 0 , . . . , e k . Let µ : [0, 1] → S be the meridian from N to −N through N ′ and m = µ([0, 1]) ⊂ S. Further let P be the group G ∈ {SO p , U p , Sp p } or one of its iterated centrioles (see Section 3). We fix some J ∈ P and a shortest geodesic in P , which we consider as a map γ : m → P . Building on the results from Section 4 will will develop a deformation theory for these mapping spaces Map * (S, P ). The main result, Theorem 6.13, says that stably it can be approximated by the subspace of maps which are block sums of constant maps and Hopf maps associated to Clifford representations (see Definition 2.1).
Proof. This can be seen directly by constructing a deformation of arbitrary maps φ ∈ Map * (S, P ) into mapsφ ∈ Map γ (S, P ). However, the proof becomes shorter if we use some elementary homotopy theory, see [4,Ch. VII]. We consider the restriction map is a weak homotopy equivalence, due to the homotopy sequence of this fibration.
Let S ′ = S k−1 ⊂ S be the equator sphere. We may parametrize S by where Ω = Ω(P ; J, −J) is the space of continuous paths ω : [0, 1] → P with ω(0) = J and ω(1) = −J, and Since every minimal geodesic from J to −J in P is determined by its midpoint, which belongs to the midpoint setP ′ ⊂ P whose components are the centrioles P ′ (see Section 3), we furthermore have a canonical homeomorphism

The composition
Definition 6.3. The map Σ J is called the geodesic suspension along J.
We will show that Σ J is highly conected in many cases. At first we will deal with the case k = dim S ≥ 2.
Proposition 6.4. Let k ≥ 2 and let d + dim(S ′ ) be smaller than the index of any non-minimal geodesic in the connected component Ω * ⊂ Ω containing γ. Then Σ J is d-connected.
Proof. By Lemma 6.2, the inclusion Map γ (S, P ) → Map * (S, P ) is a weak homotopy equivalence. Therefore we only have to deal with the inclusion We consider ψ as a mapψ : D j × S ′ → Ω and observe, using k ≥ 2, that the image of this map lies in the connected component Ω * ⊂ Ω determined by γ.
where j+dim S ′ ≤ d+dim S ′ . This results in a deformation ofψ to a map with image contained in Ω 0 * by a deformation which is constant on (∂D j × S ′ ) ∪ (D j × {N ′ }). Hence ψ may be deformed to a map with image contained in Map * (S ′ , Ω 0 ) by a deformation which is constant on ∂D j .
The following picture illustrates the deformation process employed in this proof where ϕ ∈ Map γ (S, P ) lies in the range of ψ.
This process can be iterated. For the sake of exposition we will concentrate on the case G = SO p in the following argument. Let k ≥ 1 and consider an orthogonal Cl k -representation ρ : Cl k → R p×p determined by anticommuting complex structures J 1 , . . . , J k ∈ SO p . Let SO p = P 0 ⊃ P 1 ⊃ · · · ⊃ P k be the associated chain of minimal centrioles (compare Theorem 3.1). Let e 0 , . . . , e k denote the standard basis of R k+1 and let S k−ℓ ⊂ S k be the great sphere in the subspace spanned by e ℓ , . . . , e k . Let Map * (S k−ℓ , P ℓ ), ℓ = 0, . . . , k − 1, is the space of maps S k−ℓ → P ℓ sending e ℓ to J ℓ . Putting Σ ℓ := Σ J ℓ (the suspension along J ℓ ) with J 0 := I we consider the composition Proof. Since the composition of d-connected maps is d-connected it suffices to show that for all ℓ = 0, . . . , k − 2 the suspension in the composition θ is d-connected once p ≥ p 0 . This claim follows from Proposition 6.4, where the assumption on indices of non-minimal geodesics holds for the following reasons: (1) If ℓ + 2 is not divisible by four it holds by Proposition 5.4.
It remains to investigate the space Map * (S 1 , P k−1 ). We restrict to the case k = 4m − 1. Setting ℓ = k − 1 this means ℓ = 4m − 2, and P := P ℓ . From equation (5.7) we recall the canonical embedding In the remainder of this section we will asume that the given Cl krepresentation ρ satisfies J k = +J 1 · · · J ℓ . Orthogonal Cl k -representations of this kind and their Hopf maps will be called positive. Furthermore we denote by In particular w(γ) = −p/4. Definition 6.6. For each loop ω ∈ Map(S 1 , P ℓ ) we define the winding number η(ω) ∈ Z as the winding number of the composition Note that η is constant on path components of Map(S 1 , P ℓ ).
For η ∈ Z let Map * (S 1 , P ℓ ) η ⊂ Map * (S 1 , P ℓ ) denote the subspace of loops with winding number equal to η. This is a union of path components of Map * (S 1 , P ℓ ).
By Proposition 6.5, for sufficiently large p, the map θ in (6.5) induces a bijection of path components. Hence, at least after taking a block sum with a constant map S k → SO p ′ for some large p ′ , the previously defined winding number η : Map * (S 1 , P ℓ ) → Z induces a map η s : Map * (S k , SO p ) → Z , such that η s • θ = η, which we call the stable winding number. This is independent of the particular choice of p ′ , and hence well defined on Map * (S k , SO p ) for the original p. By definition it is constant on path components and therefore may be regarded as a map Since all positive Cl k -representations on R p are isomorphic, this map is independent from the chosen positive Cl k -representation ρ.
For η ∈ Z let Map * (S k , SO p ) η ⊂ Map * (S k , SO p ) denote the subspace with stable winding number equal to η.
Remark 6.7. The stable winding number is additive with respect to block sums of Cl k -representations. More precisely, let ρ i : Cl k → SO pi be positive Cl k -representations, i = 1, 2, with associated chains of minimal centrioles The chain of minimal centrioles associated to the block sum action then takes the form k , and each suspension (6.5) is a product of corresponding suspensions for ρ 1 and ρ 2 . Hence for φ = φ 1 ⊕ φ 2 ∈ Map * (S k , SO p1+p2 ) of block sum form we obtain for the stable winding numbers η s (φ) = η s (φ 1 ) + η s (φ 2 ) .
For w ∈ Z let be the subspaces of maps of winding number w. Notice that we have a canonical identification.
After these preparations we obtain the following analogue of Proposition 6.4. Proposition 6.9. Let d ≥ 0 and let w ∈ Z satisfy p − 4|w| ≥ 4s ℓ · d. Then the suspension Σ ℓ restricts to a d-connected map Proof. Similar as in the proof of Proposition 6.4 we only have to deal with the inclusion where we now have S ′ = S 0 , cf. (6.3). This is equivalent to the inclusion

Hence the geodesicγ is of block diagonal form
Therefore also ω is in block diagonal form where such that ω 0 is constant with value J ℓ and ω 1 (t) = J ℓ e 2πit , where i = J 1 · · · J ℓ−1 . For 0 ≤ η ≤ p/2 we denote by Map 0 * (S 1 , P ℓ ) η ⊂ Map 0 * (S 1 , P ℓ ) the subspace of loops with winding number η (which is equal to 1 2 dim C L 1 ).

We have a canonical homeomorphism
Map 0 (S 1 , P ℓ ) ≈ Map * (S 0 , Ω 0 ) which replaces the constant map on L 0 by the concatenation ω = γ * γ −1 , restricted to L 0 . Since this is homotopic relative to {0, 1} to the constant loop with value I ∈ SO(L 0 ) by use of the explicit homotopy ω s = γ s * γ −1 s , ω s (t) := ω(st) , 0 ≤ s, t ≤ 1 , we conclude that the composition is homotopic to the canonical inclusion Map 0 * (S 1 , P ℓ ) → Map * (S 1 , P ℓ ) . Since homotopic maps induce the same maps on homotopy groups and a map A → B is d-connected if and only if it induces a bijection on π j for 0 ≤ j ≤ d − 1 and a surjection on π d , we hence obtain the following version of Proposition 6.9. Proposition 6.11. Let d ≥ 1 and p ≥ p 0 (d). Assume s ℓ · d ≤ η ≤ p/4. Then the composition Proof. Notice that for η ≤ p/4 the winding number w = η − p/4 is non-positive. Hence p − 4|w| = p + 4(η − p/4) = 4η ≥ 4s ℓ · d. The assertion now follows from propositions 6.5 and 6.9.
By construction, elements φ ∈ Map * (S k , SO p ) η in the image of θ 0 in (6.6) are described as follows. There is an orthogonal splitting ) is an isometric embedding as a great sphere with image contained in SO(L 1 ) which sends e 0 to I and e i to the restriction of J i to L 1 for 1 ≤ i ≤ k.
Proposition 2.2 then implies that the map φ 1 is in fact equal to the Hopf map associated to the positive Cl k -representation ρ restricted to L 1 . Definition 6.12. We call maps φ of this kind (that is φ = (φ 0 , φ 1 ) on L 0 ⊕ L 1 with φ 0 ≡ I and φ 1 a positive Hopf map on L 1 ) affine Hopf maps of stable winding number η = 1 2 dim C L 1 associated to ρ (a positive Clifford representation). Let be the subspace of affine Hopf maps of stable winding number η.
Summarizing we obtain the following result on deformations of mapping spaces (recall s k = s ℓ from (2.2)). Theorem 6.13. Let d ≥ 1 and k = 4m − 1. Then for all p ≥ p 0 (d) and s k · d ≤ η ≤ p/4 and any positive Clifford representation ρ : Cl k → R p×p the canonical inclusion Proof. The theorem follows from Prop. 6.11 since Hopf ρ (S k , SO p ) η is the image of θ 0 in (6.6).
Remark 6.14. We have finished our homotopy classification of Map * (S k , G) for G = SO p with large p. By a sequence of k steps it can be reduced to the very small subset of affine Hopf maps for a certain Cl k -representation. In the last step we have to consider Map * (S 1 , P ℓ ) with ℓ = k − 1. This is non-contractible only if π 1 (P ℓ ) = 0, but to meet the index condition we need more: the winding number η must be large (cf. 6.11, 6.13 and also Remark 5.9) which implies |π 1 (P ℓ )| = ∞. This restricts us to the cases ℓ = 4m − 2 for G = SO p , Sp p and to ℓ = 2m for G = U p , see table (3.2). The details for the cases G = Sp p and G = U p are left to the reader.

Vector bundles over sphere bundles: Clutching construction
The deformation theory for maps φ : S k → SO p can be applied to the classification of oriented Euclidean vector bundles over n-spheres for n = k + 1 = 4m. Choose N = e n and S = −e n ("north and south poles") in S n ⊂ R n+1 where e 0 , . . . , e n is the standard basis of R n+1 . Let D n ± = {v ∈ S n : ± v, e n ≥ 0}, S n−1 = {v ∈ S n : v, e n = 0} = D n + ∩ D n − ("hemispheres and equator"). Let E → S n an oriented Euclidean vector bundle with fibre R p . Since D n ± is contractible, E | D n ± is a trivial bundle, isomorphic to D n ± × E ± where E ± = E ±en ∼ = R p , pulled back to S n ± along the canonical projections onto the midpoint. The bundle E is obtained from these trivial bundles by identifying {v} × E + to {v} × E − via an oriented orthogonal map, thus by a mapping φ : S n−1 → SO(E + , E − ) ∼ = SO p called clutching function which is well defined by E up to homotopy. In this situation we write Example 7.1. Let ρ : Cl n → End(L) be an orthogonal Cl n -representation. Since n = 4m, the longest Clifford product ω := e 1 · · · e n (the "volume element") commutes with all elements of Cl + n (the subalgebra containing the products of even length) and has order two, ω 2 = 1. 8 The (±1)-eigenspaces 9 L ± of ρ(ω) are invariant under Cl + n , and ρ(v) interchanges the eigenspaces for any v ∈ S n−1 . Using µ := ρ| S n−1 we obtain a bundle L → S n , L = (L + , µ, L − ) This is called the Hopf bundle over S n associated to the Clifford representation ρ.

Remark 7.2. This bundle is isomorpic to
Note that ρ + is positive, 10 that means its Clifford family J i = ρ(e i ), i = 1, . . . , k, on L + satisfies J k = J 1 · · · J k−1 . In particular the stable winding number of µ + (see Definition 6.6) is given by The positivity of this winding number will become important later in the proof of Theorem 8.2.
We now replace the sphere S n by a sphere bundle with two antipodal sections over a finite CW-complex X. More specifically, starting from an n-dimensional Euclidean vector bundle V → X we glue two copies D ± V of its disk bundle DV → X along the common boundary, the unit sphere bundle SV , by the identity map. Thus we obtain an S n -bundleV = S(V ⊕ R) → X, We obtain two sections s ± : X →V of the bundleV → X, which we regard as north and south poles, defined as the zero sections of D ± V .
Let E →V be a Euclidean vector bundle over the total space ofV → X. Let E ± = s * ± E → X which we sometimes tacitly pull back to bundles E ± → D ± V along the canonical fibrewise projection maps D ± V → s ± (X).
Since there are -up to homotopy unique -bundle isometries E | D±V ∼ = E ± restricting to the identity over {s ± }, the bundle E is obtained from E ± by a clutching map σ, that is a section of the bundle which is uniquely determined up to fibrewise homotopy.
Hence, for any x ∈ X we have a map σ x : . Note that we may equivalently consider the clutching map as a bundle isometry σ : E + | SV → E − | SV . In this situation will write Vice versa, if E ± → X are oriented Euclidean bundles and σ ∈ Γ(Map(SV, O(E + , E − ))) a clutching map, then we obtain a vector bundle E = (E + , σ, E − ) →V by gluing the pull back bundles of E ± → D ± V along SV by σ.
We write (F, id, F ) for this triple. Since F v can be identified with F s± for all v ∈ D ± V , this bundle is trivial over every fibreV x , x ∈ X, hence it is isomorphic to a bundle over X, pulled back toV by the projection π :V → X.
Vice versa, for any vector bundle F → X, the pull-back bundle π * F is given by the triple (F, id, F ).
For a generalization of Example 7.1 to sphere bundles we recall the following definition.
Definition 7.4. Let Cl(V ) → X be the Clifford algebra bundle associated to the Euclidean bundle V with fibre Cl(V ) x = Cl(V x ) over x ∈ X.
A real Cl(V )-Clifford module bundle is a vector bundle Λ → X such that each fibre Λ x , x ∈ X, is a real Cl(V x ) module. More precisely, we are given a bundle homomorphism µ : Cl(V ) → End(Λ) which restricts to an algebra homomorphism in each fibre.
We may and will assume that Λ is equipped with a Euclidean structure such that Clifford multiplication with elements in SV is orthogonal.
From now on we will in addition assume that V → X is oriented. Any oriented orthonormal frame 11 of V | U over U ⊂ X induces an orientation preserving orthogonal trivialization b : U × R n ∼ = V | U . Thus L := Λ xo with some fixed x o ∈ U becomes a Cl n -module. Consider the bundle Iso Cln (L, Λ| U ) → U whose fibre over x ∈ U is the space of Cl n -linear isomorphisms φ x : L → Λ x . If U is contractible it has a section φ which intertwines the Cl n -module multiplications (denoted by ·) on L and Λ x , x ∈ U : for all ξ ∈ L and v ∈ R n and x ∈ U we have In other words: We obtain Euclidean trivialisations with a Cl n -module L such that the Cl(V )-module structure on Λ corresponds to the Cl n -multiplication on L. Notice that in particular the Cl n -isomorphism type of L, the typical fibre of Λ → X is uniquely determined over each path component of X.
Let n = 4m. Since V → X is oriented, the volume element ω ∈ Cl n defines a section ("volume section") of Cl(V ) → X, and the ±1-eigenspaces define the positive and negative Clifford algebra bundles Cl ± (V ) → X and subbundles Λ ± → X of Λ → X, which are invariant under Cl + (V ) and such that for every v ∈ V x the endomorphism µ(v) interchanges Λ + x and Λ − x . Correspondingly the typical fibre L decomposes as L = L + ⊕ L − such that the local trivialisations (7.4) preserve positive and negative summands.
Definition 7.5. For n = 4m the bundle L →V defined by the triple is called the Hopf bundle associated to the Cl(V )-Clifford module bundle Λ → X.
Note that each L |V x →V x , x ∈ X, is a Hopf bundle in the sense of Definition (7.1) after passing to local trivialisations. (7.4). Example 7.6. As a particular example we consider the Cl(V )-module bundle is generated as a vector bundle by the Clifford products of even (odd) length.
We define the Clifford-Hopf bundle C →V as the triple where µ : SV → SO(Cl − (V ), Cl + (V )) is the (left) Clifford multiplication.
Example 7.7. Recall that a spin structure on V → X is given by a two fold cover of the SO(n)-principal bundle of oriented orthonormal frames in V , which is equivariant with respect to the double cover Spin n → SO n . Consulting Theorem 2.3 we see that for n = 4m there is exactly one irreducible Cl n -module S = S n , and we obtain the spinor bundle

Since the Clifford algebra bundle can be written in the form
where Spin n ⊂ Cl n acts on Cl n by conjugation, Σ becomes a Cl(V )-module bundle by setting [p, α] · [p, σ] = [p, ασ] for p ∈ P Spin (V ), α ∈ Cl n and σ ∈ S. into Cl(V )-submodule bundles. This induces a triple In the next section we will prove that every vector bundle overV is -after suitable stabilisation -isomorphic to an affine Hopf bundle.

Vector bundles over sphere bundles and affine Hopf bundles
Let V → X be an oriented Euclidean vector bundle of rank n = 4m with associated sphere bundleV = S(R ⊕ V ) → X.
(1) Two (real) vector bundles E,Ẽ over X orV are called stably isomorphic, written for some q.
The following result is central in our paper.
Theorem 8.2. Let E →V be a Euclidean vector bundle. Then possibly after adding copies of C (the Clifford-Hopf bundle, see Example 7.6) and trivial vector bundles R to E we have for a vector bundle E 0 → X (pulled back toV ) and a Hopf bundle L →V associated to some Cl(V )-module bundle Λ → X. Let E 0 ,Ẽ 0 → X be vector bundles and L ,L →V Hopf bundles corresponding to Cl(V )-module bundles Λ,Λ → X. If When V carries a spin structure, we will see in Prop. 9.4 (which is self consistent) that Λ ∼ = E ⊗ Σ when m is even, and Λ ∼ = E ⊗ H Σ when m is odd, where Σ denotes the Spinor bundle associated to P Spin (V ), cf. Example 7.7. Therefore when m is odd where S denotes the spinor-Hopf bundle (cf. Definition 7.8) and E a vector bundle over X, pulled back toV . Further, the stable isomorphism type of Λ determines the stable isomorphism type of E, see Lemma 9.5.
The proof of Theorem 8.2 will cover the remainder of this section. We represent E by a triple (E + , σ, E − ) for Euclidean vector bundles E ± → X, which we consider as bundles over D ± V in the usual way. consists of a pair (φ + , φ − ) of orthogonal bundle homomorphisms φ ± : E ± →Ẽ ± over DV such that the following diagram over SV commutes: , is a homotopy of bundle isomorphisms over DV with φ + = φ 0 + , we may replace φ by φ t = (φ t + , φ − ), replacing the clutching mapσ = φ − σφ −1 + over SV by the homotopic clutching mapσ t =σφ + (φ t + ) −1 which does not changẽ E .
In the proof of Theorem 8.2 we can assume without loss of generality that X is path connected. Let be orthogonal trivialisations over a connected open subset U ⊂ X, where the first trivialisation is assumed to be orientation preserving. (We are not assuming that the the bundle E + → X is orientable). We obtain an induced trivialisation using the isomorphisms σ x (e 1 ) : (E + ) x → (E − ) x for x ∈ U where e 1 ∈ R n is the first standard basis vector. With respect to theses trivialisations σ| U is given by a map (8.4) σ| U : U → Map * (S n−1 , SO p ) .
Here we recall that S n−1 is connected since n ≥ 4 by assumption so that the local clutching function has indeed values in SO p rather than O p . The stable winding number η s (σ x ) ∈ Z (see Definition 6.6) is independent of the chosen trivialisations and constant over U . It is hence an invariant of the triple E = (E + , σ, E − ). Set d = dim X + 1. After adding trivial bundles and Clifford-Hopf bundles to E we can assume that the stable mapping degree η = η s (σ) satisfies the conditions s k · d < η ≤ p/4 from Theorem 6.13 (with k := n − 1, recall s k = s ℓ for ℓ = k − 1), compare Remarks 6.7 and 7.2. After adding trivial bundles to E , if necessary, we can furthermore assume that p is divisible by s k . In particular, by Theorem 6.13, with these choices of p and η the canonical inclusion is d + 1-connected for any positive representation ρ : Cl k → End(R p ) (which is equivalent to the (p/s k )-fold direct sum of the positive S k ). We will work under these assumptions from now on.
Next we note that we may add on both sides of (8.1) a bundle F → X, pulled back via π :V → X (which further inreases p, but not the winding number η, so that the assumptions of Theorem 6.13 are preserved), that is we add π * F = (F, id, F ).
Then the original statement is obtained by embedding F into a trivial bundle over X and adding a complement F ⊥ on both sides of (8.1).
We use this freedom to put E + into a special form. Since E + embeds into a trivial bundle R N → X and each bundle over X of rank larger than dim(X) splits off a trivial line bundle (its Euler class vanishes), E + embeds into any sufficiently large vector bundle over X, in particular into Λ + for some Cl(V )-module bundle Λ → X (e.g. Λ = R q ⊗ Cl(V ) = Cl(V ) q ). Then E + is a direct summand of Λ + .
After adding (F, id, F ) to (E + , σ, E − ) with F = (E + ) ⊥ ⊂ Λ + we may hence assume that Note that this bundle is oriented. After these preparations Theorem 8.2 is proven by induction over a cell decomposition of X. The decomposition (8.1) results from the following fact.
Proposition 8.5. There is an orthogonal decomposition of Λ ± into Cl(V )-invariant subbundles with the following property: the triple E = (E + , σ, E − ) is isomorphic to the triple -module multiplication on Λ 1 . Furthermore this isomorphism can be chosen as the identity on the first bundle E + = Λ + . Hence, by (8.3), it is given by an isomorphism of vector bundles over the total space DV , Proof. In the induction step let X = X ′ ∪ D be obtained by attaching a cell D to X ′ and assume that the assertion holds for the restriction to X ′ of the bundle We denote by V ′ → X ′ the restriction of V . Hence we have the decomposition (8.5) over X ′ and an isomorphism . We need to extend f to a similar isomorphism over X = X ′ ∪ D. In particular we need to extend the bundle decomposition Λ = Λ 0 ⊕ Λ 1 from X ′ to X. This will ultimately be achieved by applying Theorem 6.13 to the restriction of E →V tô V | D ∼ = D × S n . In fact, trivializing E ± over D ⊂ X, the clutching map σ of E will become a mapσ : D × S n−1 → SO(L + ) where L + = R p is the standard fibre of E ± . However, in order to apply Theorem 6.13, this mapσ : D → Map(S n−1 , SO(L + )) needs (i) taking values in Map * (S n−1 , SO(L + )) (the maps φ : S n−1 → SO(L + ) with φ(e 1 ) = I, see Def. 6.1), (ii)σ(∂D) ⊂ Hopf ρ+ (S n−1 , SO(L + )) η with ρ + : Cl k → End(L + ) as in (7.1). Requirement (ii) will be met (using the clutching maps σ, ω as trivializations) by transforming f tof , see (8.12), which will change (8.8) to (8.14). From (8.14) we will see using Remark 7.2 that the clutching mapfσ : ∂D → Map * (S n−1 , SO(L + )) takes values in Hopf ρ+ (S n−1 , SO(L + )). In order to meet the requirement (i) we need an extensionF off from ∂D to all of D such thatF (x, e 1 ) = id L+ for all x ∈ D, see the "Assertion" below. Then we can apply our deformation theorem 6.13 to the clutching map τ =Fσ, thus obtaining a new clutching map τ 1 with values in Hopf ρ (S n−1 , SO(L + )) η . It fits together with the given clutching map along X ′ since there was no change along ∂D = X ′ ∩ D.
Now we explain these steps in detail. Choose a trivialization of V | D , that is an oriented orthonormal frame over D. It induces trivialisations of DV and Cl(V ). Also we choose a compatible trivialization of Λ, compare Example 7.4: Identifying SV | D with D × S n−1 , we choose e 1 ∈ S n as a base point and put We use these bundle isomorphisms as trivializations η and ϑ of the bundles E − | D and (Λ + 0 ⊕ Λ − 1 )| ∂D (which below will be pulled back to DV | D and DV | ∂D , respectively): (8.11) η : Thus we can express the isomorphism f in (8.7), restricted to DV | ∂D , as a map At the base point e 1 ∈ S n−1 we obtain for all x ∈ ∂D: f (x, e 1 ) = ϑ(f (σ 1 (x))) (8.13) are evaluated as follows: such thatF (x, e 1 ) = id L + for all x ∈ D.
Proof of assertion. By (8.13),f is equal to the identity on the fibre over ∂D × {e 1 }, where as usual e 1 ∈ S n−1 ⊂ D n . Notice that {e 1 } ⊂ D n is a strong deformation retract. Let r t : D n → D n , t ∈ [0, 1], be a deformation retraction with r 1 = id D n and r 0 ≡ e 1 : This gives a homotopyf t : id L + ≃f , withf t (x, e 1 ) = id L + for allx ∈ ∂D and t ∈ [0, 1]. Considering D as the cone over ∂D, this can be viewed as a map Since p ≥ p 0 (d) where d = dim X and s ℓ · d ≤ η ≤ p/4, we can apply Theorem 6.13 to obtain a deformation (τ t ) t∈[0,1] of τ = τ 0 such that τ t = τ on ∂D and τ 1 (D) ∈ Hopf ρ+ (S n−1 , SO(L + )) η .
Thus the trivial bundle L + decomposes over D into orthogonal Cl n−1 -invariant subbundles, (8.17) (L + ) D = L + 0 ⊕ L + 1 and τ 1 = id ⊕μ , whereμ (8.15) = µ(e 1 ) −1 µ = ρ + | S n−1 is the Hopf map induced by the Cl n−1 -module structure on L + 1 (compare Remark 7.2). We have to connect these clutching data over D to the given ones over X ′ . In (8.9) we have chosen a trivialization Λ + | D ∼ = → (L + ) D which over ∂D transforms the decomposition Λ + = Λ + 0 ⊕Λ + 1 into a bundle decomposition (L + ) ∂D = (L + 0 ⊕L + 1 )| ∂D . Now we extend the decomposition (8.5) of Λ + | ∂D to all of D by applying the inverse of this trivialization to (8.17) obtaining a decomposition Λ + = Λ + 0 ⊕ Λ + 1 on all of X. Similarly, we obtain subbundles Now we can define a bundle isomorphims . In fact, over X ′ this is true by assumption, and over D we use (8.18) for the commutativity of the diagram Hence the isomorphism f from Equation (8.7) extends to an isomorphism φ over X between the given bundle E and a bundle of the form (8.6). This finishes the proof of Proposition 8.5 and the existence part of Theorem 8.2.
After this preparation we may hence assume that →Ẽ is a vector bundle isomorphism, which we may assume to be orthogonal.
Hence we obtain stable isomorphisms E 0 ∼ =sẼ0 and Λ 1 ∼ =sΛ1, finishing the proof of the uniqueness statement in Theorem 8.2.

Thom isomorphism theorems
Theorem 8.2 can be reformulated concisely in the language of topological Ktheory. Let X be a finite CW-complex and V → X an oriented Euclidean vector bundle of rank n = 4m ≥ 0 with associated Clifford algebra bundle Cl(V ) → X. Each Cl(V )-module bundle Λ → X is isomorphic to a Cl(V )-submodule bundle of Cl(V ) q → X for some q. This holds in the special case X = {point}, since each Cl n -module is a direct sum of irreducible Cl n -modules, each of which also occurs as a summand in the Cl n -module Cl n . 12 For more general X we use a partition of unity subordinate to a finite cover of X by open trivializing subsets for the given Cl(V )-module bundle, similar as for ordinary vector bundles.
In particular, two Cl(V )-module bundles Λ,Λ represent the same element in K Cl(V) (X), if and only if they are stably isomorphic as Cl(V )-module bundles in the sense of Definition 8.1 (2).
Let K O (X) denote the real topological K-theory of X. If X is equipped with a base point x 0 recall the definition of the reduced real K-theorỹ Let X V =V /s + be the Thom space associated to V → X with base point [s + ] (recall that s + ⊂V denotes the zero section of D + V ). The fibrewise projection π :V → s + induces a group homomorphism Definition 9.2. The group homomorphism 12 Recall that by Theorem 2.2 there is, up to isomorphism, just one irreducible Cln-module for n = 4m.
In this language Theorem 8.2 translates to the following result.
Theorem 9.3. The Clifford-Thom homomorphism Ψ V is an isomorphism.
We will now point out that Theorem 9.3 implies classical Thom isomorphism theorems for orthogonal, unitary and symplectic K-theory (the orthogonal and unitary cases are also treated in [8,Theorem IV.5.14]).
Assume that V → X is equipped with a spin structure and let Σ → X be the spinor bundle associated to V → X, compare Example 7. In this case Cl(V )-module bundles are of a particular form.
Recall the notion of the tensor product over H (e.g. see [6, p. 16]): Let P a H-right vector space and Q a H-left vector space. Then This is a vector space over R, not over H. If m is odd then the right H-multiplication on S commutes with the left Cl n -action (see Table (2.2)), and hence Σ is a bundle of right H-modules in a canonical way. We equivalently regard S and Σ as left H-modules by setting λ · s := sλ −1 .
Proposition 9.4. Let Λ be a Cl(V )-module bundle. Then Λ is a twisted spinor bundle, that is there exists a vector bundle E → X over R or H when m is even or odd, respectively, such that The isomorphism in (9.2) is given by This is a Cl(V )-linear bundle homomorphism: For all α ∈ Cl(V ), To check bijectivity we look at the fibres L of Λ and S of Σ. Set C := Cl n . The fibre of E = Hom Cl(V ) (Σ, Λ) is Hom C (S, L) and the homomorphism j is fibrewise the linear map Since S is the unique irreducible C-representation we have L ∼ = S p = R p ⊗ S for some p, and Hom C (S, L) = Hom C (S, S) p = End C (S) p . Applying j o to φ = (0, . . . , id S , . . . , 0) with the identity at the k-th slot for k = 1, . . . , p, we see that this map is onto. When m is even we have C = End R (S), and when m is odd, C = End H (S) (cf. Table (2.2)). Thus End C (S) = R · id S when m is even and End C (S) = H · id S when m is odd. Note that in the second case an element λ ∈ H corresponds toλ := R λ −1 ∈ End C (S) (right multiplication with λ −1 on S, that is left multiplication with λ with respect to the left H-module structure defined before) in order to make the identification a ring map. 13 In the first case, Hom C (S, L) = End C (S) p ⊗ S = R p ⊗ S = S p . Thus j o is an isomorphism whence j is an isomorphism.
In the second case we consider Hom C (S, L) as a H-right vector space using precomposition φ → φ •λ, whereλ = R λ −1 ∈ End C (S) = H. We hence compute Therefore φ = f ⊗ id Σ for some R-linear isomorphism f : E →Ẽ in the first case, and φ = f ⊗ H id Σ for some H-linear isomorphism f : E →Ẽ in the second case.
(ii) There is some q ≥ 0, such that Λ ⊕ Σ q ∼ =Λ ⊕ Σ q are isomorphic Cl(V )-module bundles Proof. This follows since Σ → X is a Cl(V )-submodule bundle of Cl(V ) q → X for some q ≥ 0, and vice versa Cl(V ) → X is a Cl(V )-submodule bundle of Σ q → X for some q. Again this is obvious if X is equal to a point and follows for general X by a partition of unity argument.
Together with Proposition 9.4 this implies Proposition 9.6. Let V → X be of rank 4m and equipped with a spin structure, and let Σ → X denote the associated spinor bundle.
(a) For even m the map is an isomorphism. 13 End C (S) is an (associative) division algebra, thus isomorphic to H or C or R. In fact, kernel and image of any A ∈ End C (S) are C-invariant subspaces of the irreducible C-module S, hence A is invertible or 0. When m is odd, C = End H (S) and therefore End C (S) ⊃ H, hence End C (S) = H. When m is even, C = End R (S). In fact, any A ∈ End C (S) has a real or complex eigenvalue and hence an invariant line or plane in S. Since it commutes with all endomorphisms on S, every line or plane is A-invariant since GL(S) ⊂ End(S) acts transitively on the Grassmannians. Thus A is a real multiple of the identity. (See also Wedderburn's theorem.) (b) For odd m the map is an isomorphism, where K Sp denotes quaternionic K-theory based on H-right vector bundles.
Hence we obtain the classical Thom isomorphism by composing the isomorphisms of 9.6 and 9.3, using that the spinor Hopf bundle S (cf. 7.8) is the Hopf bundle (cf. 7.1) associated to the spinor bundle Σ: Theorem 9.7. Let V → X be equipped with a spin structure and let S →V denote the spinor Hopf bundle associated to V → X (see Definition 7.8).
(a) For even m the map Based on Remark 6.14 there are analogues of the theorems 8.2 and 9.3 for complex and quaternionic vector bundles overV . Detailed proofs of the following statements are left to the reader. Theorem 9.8. Let X be a finite CW-complex and V → X an oriented Euclidean vector bundle with associated Clifford algebra bundle Cl(V ) → X. (a) Let rk V = 4m. Then there is a Clifford-Thom isomorphism K Cl(V)⊗H (X) ∼ =K Sp (X V ).
(b) Let rk V = 2m. Then there is a Clifford-Thom isomorpism The analogues of Proposition 9.4 for complex and quaternionic Cl(V )-module bundles are as follows.
Assume that V → X is equipped with a spin structure. Recall from (2.2) that the spinor bundle Σ associated to P Spin (V ) is quaternionic when rk V = n = 4m with m odd and it is real when n = 4m with m even. If Λ is a (Cl(V ) ⊗ H)module bundle (with left Cl(V )-multiplication and right H-multiplication), we put E = Hom Cl(V )⊗H (Σ, Λ) for m odd and E = Hom Cl(V ) (Σ, Λ) if m is even; in both cases precisely one of the bundles E and Σ is (right) quaternionic, and we obtain an isomorphism of (Cl(V ) ⊗ H)-module bundles analogue to (9.3), For (Cl(V ) ⊗ C)-module bundles Λ with dim V = n = 2m we can apply [2,Prop. 3.34]: Let Σ c be the complex spinor bundle associated to P Spin c (V ) by means of a Spin c -structure on V (cf. [9,Appendix D]). Putting E = Hom Cl(V )⊗C (Σ c , Λ) we obtain that j : is an isomorphism of (Cl(V ) ⊗ C)-module bundles. Together with Theorem 9.8 we hence arrive at the following classical Thom isomorphism theorems, which complement Theorem 9.7. Theorem 9.9. (a) Let V → X be a spin bundle of rank n = 4m. Then multiplication with the spinor Hopf bundle S →V induces Thom isomorphisms K Sp (X) →K Sp (X V ) for even m , (b) Let V → X be a spin c -bundle of rank n = 2m. Then multiplication with the complex spinor Hopf bundle S c →V induces a Thom isomorphism K U (X) →K U (X V ) .
Discussion 9.10. We will point out some connections of the argument at the beginning of this section to the classical monograph [8]. Since V is oriented and of rank divisible by four we have C(V, Q) ∼ = C(V, −Q) as algebra bundles, induced by the map V → C(V, Q), v → ω · v, where ω ∈ Cl(V ) denotes the volume section of C(V, −Q) (note that v 2 = − v 2 and 4 | rk V imply (ω · v) 2 = + v 2 ). Hence (9.4) K(E (V,Q) (X)) ∼ = K(E (V,−Q) (X)) = K Cl(V) (X) for the given V → X.
In the following we denote by C(V ) → X the algebra bundle C(V, Q) → X and C = E V (X) the Banach category of module bundles over C(V ) → X.
Hence E V ⊕R (X) ∼ = C × C and the forgetful functor φ V can be identified with the functor ψ : C × C → C , sending a pair of C(V )-module bundles (E, F ) to their direct sum E ⊕ F , compare [8,III.4.9] for the cases q = 0 and q = 4. Since this functor has a right inverse E → (E, 0) the first and last maps in the exact sequence   Put differently: For the given V → X Karoubi's theory K V (X) is the topological K-theory based on Cl(V )-module bundles over X. We remark that this is not true in general for bundles V → X of rank not divisible by four.