(2) Geometric reduction to a problem involving truncated real projective spaces, namely quotients of the formRPn/RPm

TOPOLOGICALK-THEORY AND ITS APPLICATIONS

GEOFFREY POWELL

Notes for the Masterclass, Strasbourg, February 2015

The main aim of these notes is to sketch the proof of Adams’vector fieldTheorem, which identifies the maximum number of linearly independent vector fields which can exist on an odd sphere (there are none on even spheres). This proof is outlined in the final section, Section 5, where two approaches are given. These have the following common structure:

(1) Construction of linearly independent vector fields in the maximal dimen- sion.

(2) Geometric reduction to a problem involving truncated real projective spaces, namely quotients of the formRPⁿ/RP^m.

(3) Calculation of theKO⁰-theory (forRvector bundles) of truncated projec- tive spaces.

(4) Usage of theAdams operationsto prove the non-existence result.

This influences the material presented:

. (The foundational material on topologicalK-theory will be developed in the lectures of Christian Ausoni.)

. A first section, Section 0, covers elementary algebraic preliminaries.

. The Clifford algebras and Clifford modules relevant to the study ofKOare covered in Section 1.

. The relationship with vector bundles is explained in Section 2 which shows both how to give an algebraic model for theKO-theory of truncated projec- tive spaces and how to construct vector fields on odd spheres from Clifford modules.

. Adams operations are introduced in Section 3.

. Section 4 gives first examples and explains the algebraic arguments used in the proof of the non-existence result.

. Finally, Section 5 puts this together to sketch the proof of the vector field problem.

Some comments on the presentation:

(1) Exercises (either in proofs or labelled as such) are highlighted byPin the margin. Slightly harder ones are indicated byPP.

(2) Points where particular attention should be paid are indicated by

 . (3) Sections which can safely be omitted on first reading are indicated by . . .S.

Date: February 28, 2015.

Notes from the MasterclassTopologicalK-theory and its applicationsheld at Strasbourg in February 2015. All corrections will be gratefully received!

1

CONTENTS

0. Preliminaries 3

0.1. Matrix algebras 3

0.2. Gradings 3

0.3. Algebras - graded and otherwise 4

0.4. Filtered algebras 6

0.5. Grothendieck groups of small abelian categories 7 1. Categorification: Clifford modules and algebraic Bott periodicity 8

1.1. Clifford algebras 8

1.2. Clifford modules 12

1.3. Multiplicative structures 14

1.4. References and further reading 14

2. From Clifford modules to vector bundles 15

2.1. The difference construction 15

2.2. Vector bundles from Clifford modules - the ABS construction 16

2.3. Passage toRPⁿ 17

2.4. The bundle case and the Thom isomorphism . . .S 18 2.5. Clifford sections: construction of vector fields on spheres 23

3. Lambda rings and Adams operations 25

3.1. Operations 25

3.2. Preλ-rings and Adams operations 25

3.3. ψ-rings and line elements 28

3.4. ψ-rings associated toλ-rings 29

3.5. ψ-ring structures forKUandKO 30

3.6. Theγ-operations and classification of operations . . .S 31

3.7. The complex orientation ofKU 33

4. Examples and applications ofψ-operations 35

4.1. ψ-rings andψ-modules 35

4.2. Examples from complexK-theory 36

4.3. Non-splitting ofψ-module extensions 40

4.4. Algebraic James periodicity 41

4.5. More extension results forψ-modules 41

4.6. Stabilization 43

4.7. The algebraic Hopf invariant 43

5. Vector fields on spheres 45

5.1. Background 45

5.2. Stiefel varieties 47

5.3. Reduction to truncated projective spaces 48

5.4. Adams’ vector field theorem 49

5.5. KO-theory of truncated projective spaces 51

5.6. Adams’ theorem: the proof of Theorem 5.37 53

5.7. A proof using James periodicity 54

References 57

0. PRELIMINARIES

This section reviews basic material which will be used without further comment in the body of these notes. Most of the exercises proposed (indicated byPin the margin) are straightforward (frequently a case of writing out the details). Those that require slightly more thought are indicatedPP.

Notation0.1. Throughout:

(1) Kdenotes a (commutative) field;

(2) N:={i∈Z|i≥0}; (3) N+ :=N\{0}. 0.1. Matrix algebras.

Notation0.2. ForAan associative algebra andn∈N+, denote byMn(A)the alge- bra ofn×n-matrices with coefficients inAand product matrix multiplication.

Exercise0.3. LetAbe an associativeK-algebra. Show that, fors, t∈N+: P (1) Ms(A)is unital if and only ifAis unital;

(2) Ms(A)∼=Ms(K)⊗AasK-algebras;

(3) Ms(Mt(A))∼=Mst(A)asK-algebras.

0.2. Gradings.

Definition 0.4. AZ-graded (respectivelyZ/2-graded) vector space is a vector space of the formL

n∈ZVn(resp.L

n∈Z/2Vn). A morphism of graded vector spaces is a collection of vector space morphisms{fn :Vn→Wn}.

Remark0.5. With the above structure,Z-graded (resp. Z/2-graded) vector spaces form categoriesV_Z(resp.V_Z/2) and there are forgetful functors

V_Z //V_Z/2 //V

L

n∈ZVn //V⁰⊕V¹ //V, whereV :=L

n∈ZVn,V⁰:=L

n≡0 mod 2Vn,V¹:=L

n≡1 mod 2Vn(considered as ungraded spaces) andVis the category of vector spaces.

Here we are mainly interested inZ/2-graded vector spaces; there are analogues of the following results forZ-gradings.

Definition 0.6. The graded tensor productM⊗Nˆ of twoZ/2-graded vector spaces M, N is defined by

(M⊗Nˆ )⁰ := (M⁰⊗N⁰)⊕(M¹⊗N¹) (M⊗Nˆ )¹ := (M⁰⊗N¹)⊕(M¹⊗N⁰).

Exercise0.7. P

(1) For twoZ/2-graded vector spacesM,N, show that the underlying (un- graded) vector space ofM⊗Nˆ is isomorphic to the ungraded tensor prod- uctM⊗N.

(2) If char(K) 6= 2, show that aZ/2 graded vector space is equivalent to a representation of the groupZ/2(hint: consider the eigenspace decompo- sition for the representation). Show that this extends to an equivalence of categories.

In the Z/2-graded setting, the transposition of tensor factors is always taken with respect to the Koszul sign convention, given by the following definition.

Definition 0.8. ForM, N Z/2-graded vector spaces, the transpositionτ_M,N is the isomorphism ofZ/2-graded vector spaces:

M⊗Nˆ ^τM,N→ N⊗Mˆ

m⊗n 7→ (−1)^|m||n|n⊗m.

The underlying structure onZ/2-graded vector spaces is encoded in the follow- ing statement:

Proposition 0.9. The structure (V_Z/2,⊗,ˆ K, τ)is a symmetric monoidal structure on V_Z/2, whereKis considered as aZ/2-graded space concentrated in degree zero.

Remark0.10. For current purposes, this means that all standard algebraic notions, such as the definition of aK-algebra, pass formally to theZ/2-graded setting. (The language of symmetric monoidal categories is not required in these lectures.) P Exercise0.11. Show that the embeddingV → V_Z/2that sends a vector space to a

Z/2-vector space concentrated in degree zero is symmetric monoidal with respect to the structures(V,⊗)and(V_Z/2,⊗)ˆ . (This means that⊗ˆ is compatible with⊗on graded spaces concentrated in degree zero.)

Proposition 0.12. There is an equivalence of categories (in fact an involution) V_Z/2 → V_Z/2

(M0, M1) 7→ (M1, M0) (ie. which switches the gradings).

P Proof. Exercise.

P Exercise0.13. Show that the above involution can be identified with the functor

−⊗Σˆ K, whereΣKis theZ/2-graded vector space of total dimension one, concen- trated in degree one.

0.3. Algebras - graded and otherwise.

Definition 0.14. The tensor algebraT(V)of theK-vector spaceV is the free unital, associative algebra onV, given explicitly by

T(V) :=M

n∈N

V^⊗n

which is graded by the natural numbersN(hereV^⊗0=K).

Remark0.15. The tensor algebraT(V)has a universal property: there is a canonical inclusion of vector spacesV ,→T(V), by identifyingV with the factorV^⊗1and, for Aan associative, unital algebra andϕ: V →AaK-linear map, there is a unique extension

V _ ^ϕ //

A

T(V)

∃! ˜ϕ

==

to a morphism ofK-algebras.

P Exercise0.16. Giveϕ˜explicitly in terms ofϕ.

Using the symmetric monoidal structure given in Proposition 0.9, the definition of a unital, associativeZ/2-graded algebra is formal:

Definition 0.17. A unitalZ/2-gradedK-algebra is aZ/2-graded vector spaceA equipped with structure morphisms

η:K → A µ:A⊗Aˆ → A

(the unit and the product) which satisfy the usual associativity and unit axioms.

Exercise0.18. P

(1) Show that the underlying K-vector space of a Z/2-graded algebra is an associativeK-algebra with respect to the ungraded morphisms underlying the product and the unit.

(2) Give an explicit form of the above definition, in terms of the underlying vector spacesA⁰,A¹.

Remark0.19. TheZ/2graded algebra(A, µ, η)is said to be (graded) commutative if the following diagram commutes:

A⊗Aˆ ^τA,A //

µ ""

A⊗Aˆ

|| µ

A.

This isnotin general equivalent to stating that the underlying (ungraded) algebra is commutative.

The following statement is again a formal consequence of working in theZ/2- graded setting.

Proposition 0.20. ForA, BtwoZ/2-graded algebras,A⊗Bˆ has the structure of aZ/2- graded algebra with product

(a⊗b)(a⁰⊗b⁰) = (−1)^|b||a0|aa⁰⊗bb⁰.

Remark0.21. The product of the algebraA⊗Bˆ is given by the composite:

(A⊗B) ˆˆ ⊗(A⊗B)ˆ ^A⊗τˆ−→^B,A⊗Bˆ (A⊗A) ˆˆ ⊗(B⊗B)ˆ ^µA−→^{⊗µˆ B}A⊗Bˆ

(omitting the associativity isomorphisms), whereµ_A,µ_B are the products ofA,B respectively.

Example 0.22. The tensor algebraT(V)is, by construction,N-graded as a vector space. The algebra structure is compatible with the N-grading (this gives an ex- ample of anN-graded algebra). Passing to the associatedZ/2-graded vector space, T(V)has the structure of aZ/2-graded algebra. (Exercise: verify the details.) P Definition 0.23. The Grassmann (or exterior) algebraΛ(V)on theK-vector space V is theZ/2-graded algebra defined as the quotient:

Λ(V) :=T(V)/hx⊗y+y⊗xi, withZ/2-grading inherited fromT(V).

Exercise0.24. P

(1) Describe theZ/2-grading ofΛ(V)explicitly.

(2) Show thatΛ(V)is graded commutative (in factΛ(V)is the freeZ/2-graded commutative algebra on theZ/2-graded vector spaceV[1]consisting ofV concentrated in degree one).

(3) IfV is finite-dimensional, show that the dimension of the underlying vec- tor space ofΛ(V)is2^dimV.

Example 0.25. The complex numbers C have the structure of aZ/2-graded R- algebra, withC0=h1iandC1=hii. (Remark: as aC-algebra,Chas to be concen- trated in degree zero!) Complex conjugationz 7→z(wherex+iy =x−iy) is an automorphism ofZ/2-graded algebras.

Example 0.26. The quaternionsHform a skew field. As anR-vector space:

H=h1, i, j, ki

wherei²=j²=k²=ijk=−1. (Thus,ij=k, for example.) The conjugationh7→his given by:

a+bi+cj+dk=a−bi−cj−dk;

this is an anti-automorphism of algebras (for example:ij=j i=−i j).

There are inclusionsC⇒ Hof algebras given respectively byi7→iandi7→j. These induce aZ/2-grading onH(withi, jof degree1andkof degree zero) and an isomorphism:

C⊗ˆC

∼=

→H ofZ/2-gradedR-algebras. (Exercise: check this!) P

The inclusion given byi7→icorresponds to C∼=C⊗ˆR,→C⊗ˆC∼=H, where the second morphism is induced by the unitR,→C.

Remark0.27.



On a given algebraA, there can exist non-isomorphicZ/2-gradings ofA(as algebras).

PP Exercise0.28. Find non-isomorphicZ/2-gradings on the matrix algebraM2(R).

0.4. Filtered algebras.

Definition 0.29. An increasing filtration of theK-algebraAis a filtration F⁻¹A= 0⊂F⁰A⊂F¹A⊂F²A⊂. . .⊂FⁿA⊂. . . A of the underlying vector space such that

(1) 1∈F⁰A;

(2) for alli, j∈N, the multiplication ofArestricts to:

FⁱA⊗F^jA→F^i+jA.

The associated graded is the graded vector space:

gr(A) :=M

i∈N

FⁱA/Fⁱ⁻¹A.

P Exercise0.30. For Aa filtered K-algebra (as above), show that the product ofA induces a structure of N-graded algebra ongr(A). Explicitly, writinggrⁱ(A) :=

FⁱA/Fⁱ⁻¹A, the product induces

grⁱ(A)⊗gr^j(A)→gr^i+j(A) which satisfy appropriate associativity and unital conditions.

Example 0.31. The tensor algebraT(V)is filtered with respect to:

FⁱT(V) :=

i

M

j=0

V^⊗j.

The associated graded is isomorphic toT(V)as anN-graded algebra.

0.5. Grothendieck groups of small abelian categories.

Definition 0.32. ForA a small abelian category, the Grothendieck groupG₀(A) is the free abelian group on the set of isomorphism classes of objects ofA modulo the relation:[E] = [M] + [N]if there exists a short exact sequence

0→M →E→N →0 inA.

Remark 0.33. If the categoryA is semi-simple (that is every short exact sequence splits), then the above short exact sequence givesE ∼=M⊕N, so that the relation becomes the conceptually simpler

[M⊕N] = [M] + [N].

Proposition 0.34. ForA the category of finitely-generated modules over a finite-dimensional K-algebra, the Grothendieck groupG₀(A)is the free abelian group on the set of isomor- phism classes of simple (irreducible) representations ofA.

Proof. Exercise. P

1. CATEGORIFICATION: CLIFFORD MODULES AND ALGEBRAICBOTT PERIODICITY

This section introduces the algebra which provides a model for understanding R-vector bundles on spheres (the relation with vector bundles is explained in Sec- tion 2). The techniques are based upon elementary algebra.

This can be viewed as being a procedure ofCategorification: a description of the coefficient ringKO^∗ofK-theory (that is the algebra associated to vector bundles on spheres, with product given by tensor product of vector bundles) is described in terms of categories of modules over a family of algebras indexed by the natural numbers, the Clifford algebras.

A beautiful aspect of the theory is that the celebrated Bott periodicity from topology has a transparent algebraic counterpart, requiring nothing more than an understanding of the basic case of Morita equivalence, namely that the category of modules over the matrix algebraMn(A)(n ∈ N+) is equivalent (as an abelian category) to the category of modules over the associative algebraA.

1.1. Clifford algebras. Henceforth suppose thatchar(K)6= 2; later we specialize to the caseK=R; there are analogous (but simpler) results forK=C.

Recall that a quadratic form on aK-vector spaceV is a mapq:V →Ksuch that q(ax) =a²q(x),∀a∈K, x∈V, andq(x+y)−q(x)−q(y)is symmetric bilinear in x, y∈V.

Remark 1.1. Sincechar(K) 6= 2, q(x) = b(x, x) for the bilinear formb given by b(x, y) := ¹₂{q(x+y)−q(x)−q(y)}.

A morphism of quadratic spaces(V, q)→(V⁰, q⁰)is aK-linear mapf :V →V⁰ which preserves the quadratic form (ie. q⁰(f(v)) = q(v)). The orthogonal group O(V, q)⊂Aut(V)is the group of automorphisms of the quadratic space(V, q); the groupZ/2 ={±1} ⊂K^×acts diagonally onV and preserves the quadratic form, henceZ/2⊂O(V, q).

Definition 1.2. The Clifford algebraC(V, q)of the quadratic space(V, q)is C(V, q) :=T(V)/hv^⊗2+q(v)i.

Example 1.3. Takingq= 0,C(V,0) =T(V)/hv^⊗2i.This identifies with the exterior algebraΛ(V)(see Definition 0.23). In particular,dim Λ(V) = 2^dimV.

Proposition 1.4. For(V, q)a quadratic space, the filtrationF^kT(V) := Lk

i=0V^⊗i of the tensor algebraT(V)induces a filtration ofC(V, q)

F^qC(V, q) := image{F^qT(V)→C(V, q)}

with associated graded algebragrC(V, q)∼= Λ(V). In particular,dimC(V, q) = 2^dimV. Proof. It is clear thatC(V, q)has the structure of a filtered algebra (exercise: give the details). In the associated graded, the relationv⊗v+q(v) = 0becomesv⊗v= 0, P

hence one recovers the exterior algebra (cf. Example 1.3).

By construction, there is a composite linear map ι_(V,q):V ,→T(V)C(V, q).

Clifford algebras have the following universal property:

Proposition 1.5. For(V, q)a quadratic space andAan associative, unital algebra, a linear mapf :V →Aextends to a morphism of algebras

V ^f //

ι

A

C(V, q)

f˜

;;

if and only iff(v)f(v) =−q(v)1A; in this case, the extension is unique.

In particular, a morphism of quadratic spacesϕ: (V, q)→(V⁰, q⁰)induces a morphism between Clifford algebras fitting into the commutative diagram:

V ^ϕ //

ι

V⁰

ι⁰

C(V, q)

C(ϕ)//C(V⁰, q⁰).

Hence the orthogonal groupO(V, q)acts onC(V, q)by automorphisms of algebras.

Proof. A linear mapf : V → A extends (uniquely) to a morphism of algebras T(V)→A. The extensionf˜is given by factorization across the canonical surjection T(V)C(V, q), hence is unique if it exists. The factorization exists if and only if the ideal generated by the elementsv^⊗2+q(v)is sent to zero.

As an application, one obtains that Clifford algebras have a naturalZ/2-grading.

Corollary 1.6. For(V, q)a quadratic space,C(V, q)is naturallyZ/2-graded. Explicitly C(V, q)∼=C⁰(V, q)⊕C¹(V, q)where

C⁰(V, q) := image{T^even(V)→C(V, q)}

C¹(V, q) := image{T^odd(V)→C(V, q)}.

Proof. The groupZ/2⊂O(V, q)acts onC(V, q);C^ε(V, q)is the(−1)^ε-eigenspace for this action. The groupZ/2acts compatibly onT(V)and the associated eigenspace decomposition isT(V)∼=T^even(V)⊕T^odd(V),which gives the explicit identifica-

tion.

Remark1.7.

(1) With the above structure,T(V)isZ/2-graded and the morphismT(V) C(V, q)is a morphism ofZ/2-graded algebras.

(2) The automorphism ofC(V, q)induced by−1∈Z/2is usually denotedα. Notation1.8. Forn∈N+, write

Cn := C(Rⁿ, q+) C_n⁰ := C(Rⁿ, q₋),

whereq₊(respectivelyq₋) is the standard positive-definite (resp. negative-definite) form.

Example 1.9. Forn= 1, there are isomorphisms of (ungraded)R-algebras:

C1 ∼= C C₁⁰ ∼= R⊕R.

For the second isomorphism of algebras,q₋(e) =−1for the basis elemente∈R, hence a basis corresponding to the decomposition of algebras is ¹₂(1±e), since e²= 1inC₁⁰. (Note that1has degree zero andedegree1.)

Notation 1.10. Denote by (V, q) ⊥ (V⁰, q⁰) the orthogonal sum of the quadratic spaces(V, q),(V⁰, q⁰).

Proposition 1.11. For quadratic spaces(V, q),(V⁰, q⁰), there is an isomorphism ofZ/2- graded algebras

C((V, q)⊥(V⁰, q⁰))∼=C(V, q) ˆ⊗C(V⁰, q⁰)

Proof. Exercise (using Proposition 1.5). P

It is also important to understand the underlying algebras by using theungraded tensor product. The key input is provided by the following:

Proposition 1.12. For(V, q)a quadratic space and(R=hei, q+), there are isomorphism of ungraded algebras

(1) C(V, q)→^∼= C⁰((V, q)⊥(R, q+))induced byv7→ve, in particularCk ∼=C_k+1⁰ ; (2) C_k+2→^∼= C_k⁰ ⊗C₂;

(3) C_k+2⁰ →^∼= Ck⊗C₂⁰,

where the last two are induced byv 7→ v⊗e1e2forv ∈ R^k, where{e1, e2} denotes an orthonormal basis of(R², q±).

Proof. The isomorphisms are constructed using the universal property of Clifford algebras (see Proposition 1.5). For the first isomorphism, consider (ve)(ve) in C((V, q) ⊥ (R, q+)); since v,eare orthogonal, ve = −ev in the Clifford algebra and hence(ve)(ve) =−v²e². Butv²=−q(v)ande²=−1, hence(ve)(ve) =−q(v) so that there is an induced morphism of algebras. This is easily checked to be an isomorphism.

The remaining cases are similar.

P Exercise1.13. Fill in the details for the above proof; in particular make the isomor- phisms explicit. For example, the isomorphismCk+2∼=C_k⁰⊗C2is an isomorphism ofC2-algebras, whereC2 acts on the right-hand tensor factor ofC_k⁰ ⊗C2 and on C_k+2via the inclusionR²,→R^k⊕R²∼=R^k+2.

Corollary 1.14. Fork∈N, there are isomorphisms of ungraded algebras:

C4 ∼= C₄⁰ C_k+4 ∼= C_k⊗C₄ C_k+4⁰ ∼= C_k⁰ ⊗C4. In particular,C8∼=C4⊗C4.

Proof. An immediate consequence of Proposition 1.12. For example, this gives iso- morphismsC4 ∼=C₂⁰ ⊗C2andC₄⁰ ∼=C2⊗C₂⁰ and the (ungraded) transposition of tensor factors provides an isomorphism of algebrasC₂⁰ ⊗C2∼=C2⊗C₂⁰. Example 1.15. The quaternions are denotedH(see Example 0.26); by Proposition 1.11 these can bedefinedas the Clifford algebraC2, since the usual presentation of His equivalent to

H∼=C⊗ˆC, andC∼=C₁.

To complete the identification of the underlying algebras of the families of Clif- ford algebrasCk,C_k⁰, one uses the following:

Lemma 1.16. There are isomorphisms of algebras:

(1) C₂⁰ ∼=M2(R); (2) C⊗_RC∼=C⊕C;

(3) C⊗_RH∼=M2(C)∼= End_C(H);

(4) H⊗_RH∼=M4(R)∼= End_R(H).

The latter two fit into a commutative diagram C⊗_RH

∼= //

 _

End_C _(H)

H⊗_RH

∼= //End_R(H)

in which the vertical arrows are induced respectively byC,→HandR,→C.

Proof.

(1) Consider the matrices e₁ =

0 1 1 0

and e₂ =

1 0 0 −1

of M₂(R). These satisfye²₁ = 1 = e²₂ande1e2 =−e2e1, hence establish the isomor- phism withC₂⁰.

(2) The isomorphism C⊗_RC ∼= C⊕C is elementary Galois theory (C ∼= R[x]/(x²+ 1)and, overC,x²+ 1 = (x−i)(x+i)).

(3) Right multiplication by the conjugatehofh ∈ Hinduces a morphism of R-algebras:

H→End_C(H)

viah7→(x7→xh). This extends to a morphism ofC-algebras:

C⊗_RH→End_C(H)

which can be seen to be an isomorphism for dimension reasons.

(4) The identification H⊗_R H ∼= End_R(H) is similar, by considering H → End_R(H).

The compatibility of the latter two isomorphisms is clear from the construction.

Recall (see Exercise 0.3) that, forAan associativeK-algebra ands, t∈N+, there are isomorphisms ofK-algebras:

A⊗Ms(K) ∼= Ms(A) M_s(M_t(A)) ∼= M_st(A).

Proposition 1.17. There are isomorphisms of algebras:

k Ck C_k⁰

0 R R

1 C R⊕R

2 H M2(R)

3 H⊕H M2(C) 4 M2(H) M2(H)

5 M4(C) M2(H)⊕M2(H) 6 M8(R) M4(H)

7 M8(R)⊕M8(R) M8(C) 8 M16(R) M16(R)

and

C_k+8 ∼= C_k⊗M₁₆(R)∼=M₁₆(C_k) C_k+8⁰ ∼= C_k⁰ ⊗M16(R)∼=M16(C_k⁰).

Proof. The result follows by combining Lemma 1.16 with Proposition 1.12, using the standard identifications of matrix algebras stated before the Proposition.

For example, the isomorphismC₃∼=C₁⁰ ⊗C₂givesC₃∼= (R⊕R)⊗H∼=H⊕H.

Similarly,C₅∼=C₃⁰ ⊗C₂gives

C5∼=M2(C)⊗H∼=M2(R)⊗C⊗H∼=M2(R)⊗M2(C)∼=M4(C).

PP Exercise1.18. Establish the analogous (but simpler) results overCand explain their

relationship with the above.

1.2. Clifford modules. For current purposes, we are more interested in modules over the Clifford algebras; these come in two related flavours, depending on whether one takes into account theZ/2-grading.

Definition 1.19. Fork∈N, define the abelian categories:

(1) M_k, the category of finitely-generatedZ/2-gradedC_k-modules;

(2) Nk, the category of finitely-generated (ungraded)C_k⁰-modules.

Remark1.20. By Proposition 1.12, fork >0the underlying algebra ofC_k−1is iso- morphic toC_k⁰. Hence the categoryNk can also be considered as the category of ungradedC_k−1-modules.

Lemma 1.21. Fork >0andM aZ/2-gradedCk-module, (1) by restriction,M⁰andM¹areC_k⁰-modules;

(2) for anyv∈R^k\{0}, multiplication byvinduces an isomorphism ofC_k⁰-modules M⁰→^∼= M¹.

Proof. The first statement is clear. For the second, multiplication byvcommutes with elements ofC_k⁰and multiplication by_q(v)⁻¹vdefines an inverse.

Proposition 1.22. Fork >0, the functors

(−)⁰:Mk → Nk

M 7→ M⁰ and

Ck⊗_C⁰

k−:Nk → Mk

N 7→ Ck⊗_C⁰

kN induce an equivalence of categories betweenM_kandN_k. Proof. It is clear that(Ck⊗_C0

kN)⁰∼=NasC_k⁰-modules. ForM ∈Mk, the inclusion ofC_k⁰-modulesM⁰,→M (see Lemma 1.21) induces a morphism ofCk-modules

C_k⊗_C⁰

kM⁰→M

which is an isomorphism ofC_k⁰-modules in degree zero. Applying Lemma 1.21, it

follows that it is an isomorphism.

Matrix algebras provide fundamental examples ofMorita equivalence:

Proposition 1.23. [Wei94, Proposition 9.5.2]ForAan associative algebra andn∈N+, the categories of modules over the ringsAandMn(A)are equivalent as abelian categories.

This establishes algebraic Bott periodicity for the categoriesNk:

Corollary 1.24. Fork >0, the categoriesNk andNk+8 are equivalent as abelian cate- gories.

Proof. Apply Proposition 1.23 to the isomorphism of ungraded algebras C_k+8 ∼= M₁₆(C_k)(using Proposition 1.12 to considerC_k−1instead ofC_k⁰, as in Remark 1.20).

Remark1.25. The categoryMk admits an involution induced by switching grad- ings, (see Proposition 0.12).

Recall the definition of the Grothendieck group of a (small) abelian category from Definition 0.32.

Notation1.26. Denote the Grothendieck groups of the above categories by Mk := G0(Mk)

N_k := G₀(N_k).

Proposition 1.22 and Corollary 1.24 imply:

Corollary 1.27. Fork >0,

(1) (−)⁰:Mk→Nkinduces an isomorphism of abelian groupsMk

∼=

→Nk;

(2) Mk is equivalent to the Grothendieck group of the category of ungradedCk−1- modules;

(3) M_k ∼=M_k+8as abelian groups.

Proposition 1.28. The inclusionR^k ,→ R^k ⊥R∼= R^k+1induces a morphism ofZ/2- graded algebras

C_k ,→C_k+1

and hence compatible (via Proposition 1.22) restriction functors Mk+1 → Mk

Nk+1 → Nk

which induce compatible morphisms of Grothendieck groupsM_k+1→M_k,N_k+1→N_k.

Proof. Exercise. P

Remark1.29. A similar statement holds for the Clifford algebrasC_k⁰. Moreover the (ungraded) isomorphisms of Corollary 1.14 can be used to understand the inclu- sions.

Lemma 1.30. For0≤k≤7, the inclusionsCk ,→Ck+1identify as follows:

C₀ //C₁ //C₂ //C₃ //C₄ //C₅ //C₆ //C₇ //C₈

R //C //H_diag//H⊕H //M2(H) //M4(C) //M8(R)

diag//M8(R)⊕M8(R) //M16(R) where H⊕H ,→ M2(H) is the inclusion as diagonal matrices, M2(H) ,→ M4(C)is induced byH,→M2(C)(viaC⊗_RH∼=M2(C)of Lemma 1.16) andM4(C),→M8(R) is induced byC,→M₂(R).

Proof. The inclusionsCk ,→Ck+1andC_k⁰ ,→C_k+1⁰ identify for0≤k≤1as follows:

C0  //C1  //C2 C₀⁰  //C₁⁰  //C₂⁰

R  //C  //H R  //R⊕R  //M2(R)

whereR,→R⊕Ris the diagonal embedding andR⊕R,→M2(R)the inclusion of diagonal matrices.

The extension to the remaining cases is proved by using Corollary 1.14 together

with the identification of the Clifford algebras. (Exercise: provide the details.) PP Remark 1.31. The matrix algebrasM_n(F), forF ∈ {R,C,H} aresimple algebras, in

particular have a unique isomorphism class of irreducible modules, namely the natural representation Fⁿ. This means that it is straightforward to identify the abelian groupsMkand the morphismsMk →M_k−1, given the identifications pro- vided by Lemma 1.30.



Note that, to apply this remark,M_kis considered (fork >0) as the Grothendieck group of the category ofungradedC_k−1-modules.

Notation1.32. For0< k∈N, letAkdenote the abelian group:

A_k:= Coker{M_k+1→M_k}.

Theorem 1.33. The abelian groupsM_kandA_kare

k 1 2 3 4 5 6 7 8

Ck C H H⊕H M2(H) M4(C) M8(R) M8(R)⊕M8(R) M16(R)

Mk Z Z Z Z⊕Z Z Z Z Z⊕Z

Ak Z/2 Z/2 0 Z 0 0 0 Z

for1≤k≤8.

The groupsMkandAkare determined fork >8by the Bott periodicity isomorphisms M_k ∼=M_k−8andA_k ∼=A_k−8.

Proof. Use Lemma 1.30 to determine the restriction morphismsM_k+1→M_k. PP Exercise1.34. Establish the analogues of the above results overC.

1.3. Multiplicative structures. Proposition 1.11 gives the isomorphism of Z/2- graded algebras

C_k+l→^∼= C_k⊗Cˆ _l.

This leads directly to the following, in whichM₀andA₀are taken asZ.

Lemma 1.35. Fork, l∈N, the graded tensor product induces a functor Mk×Ml→Mk+l.

This induces a morphism of abelian groupsMk⊗Ml→Mk+l.

P Proof. Exercise.

Theorem 1.36. [ABS64]TheN-graded abelian groupL

k∈NM_khas the structure of an N-graded associative algebra with respect to the product given by Lemma 1.35.

This passes to anN-graded associative algebra structure onL

k∈NA_kand this structure is (graded) commutative. Moreover, as a ring:

M

k∈N

Ak∼=Z[η, a, b]/(2η, η³, ηa, a²−4b)

where|η|= 1,|a|= 4and|b|= 8.

Proof. (Sketch.) The algebra structures onLMkandLAk are clear; the fact that the algebraAkis graded commutative requires consideration of the commutation property derived fromC_k+l∼=C_k⊗Cˆ _l. The identification of the algebra structure is straightforward, given the explicit identifications of Theorem 1.33. (Exercise!) PP

1.4. References and further reading. This material is contained in the founda- tional paperClifford Modulesby Atiyah, Bott and Shapiro [ABS64] (note that a dif- ferent sign convention is used in defining their Clifford algebras). It can also be found in the following book references [Hus94, Chapter 12], [HJJS08, Chapter 15], [LM89, Chapter 1].

Remark1.37. Explicit models for the Lie groupsSpin_nare provided by the Clifford algebras; the groupsSpin_nintervene when considering the Thom isomorphism for KO.

Remark1.38. There are two important generalizations:

(1) to theG-equivariant setting, when representations are taken in the category ofG-modules (see [BG10, Chapter 2] and [Hus94, Chapter 12]);

(2) torealrepresentations, where an involution is introduced [Ati66, AS69].

For a unified approach, one should consider the equivariant real setting - or even the generalized equivariant setting introduced by Karoubi, which is reviewed rapidly in [BG10, Chapter 2]. This brings into the picture the real, complex and symplectic representation rings for the groupG.

2. FROMCLIFFORD MODULES TO VECTOR BUNDLES

In this section, we pass from algebra to topology: the algebra of Section 1, in particular Clifford modules, is used to construct vector bundles.

There are two results which are important for the applications considered in Section 5:

(1) Theorem 2.13, which shows how to construct classes inKOg⁰(RPⁿ⁻¹,RP^m−1), hence gives an understanding of vector bundles on truncated projective spacesRPⁿ⁻¹/RP^m−1;

(2) Proposition 2.28, which constructs vector fields on a sphere from a Clifford module.

2.1. The difference construction. Consider a pair of spaces(X, Y). We will sup- pose that the inclusion ofY in X is ‘nice’ (Y X is a cofibration); for current purposes, one can suppose thatY ⊂X is a subcomplex of a CW-complex.

Form the pushout:

Y //

X

i₁

X

i₂//X∪Y X,

so thatX∪Y X is obtained by gluing two copies ofXalong the subspaceY. The quotient spaceX∪Y X/i1Xis homeomorphic toX/Y.

There are continuous maps X

i₁ //

i₂ //X∪Y X ^ρ //X

whereρis the fold map (induced by the identity on both copies ofX), hence is a retract of bothi1andi2. In particular, ifhis a cohomology theory, there is a split short exact sequence

0 //˜h(X∪Y X/i₁X) //h(X∪Y X)

i^∗₁ //h(X) //

ρ^∗

vv

0

where˜h(X∪Y X/i₁X)∼= ˜h(X/Y)and this can be considered as the relative group h(X, Y)(sinceY X is a cofibration). In particular, ifθis an element ofh(X∪_Y X), thenθ−ρ^∗i^∗₁θis an element ofh(X, Y).

Example 2.1. Let (V₁, V₂;σ)denote the following data: V_i ∈ Vect(X)are vector bundles on X (overRor C) and σ : V₁|Y

∼=

→ V₂|Y is an isomorphism of vector bundles between their restrictions toY. Then, by gluing,(V₁, V₂;σ)defines a vector bundleV_σ ∈ Vect(X∪_Y X)such thati^∗_εV_σ ∼=V_ε ∈Vect(X), forε∈ {1,2}, where the isomorphismσprovides the glue.



This is where the hypothesis onY ⊂X is required.

In particular[V_σ]−[ρ^∗V₁]defines a class inK⁰(X, Y), (whereKis eitherKUor KO.)

Definition 2.2. For(V₁, V₂;σ)as in Example 2.1, let χ(V1, V2;σ) := [Vσ]−[ρ^∗V1] denote the class inK⁰(X, Y).

Lemma 2.3. Let(X, Y)be a pair of spaces as above.

(1) For(V1, V2;σ), ifσextends to an isomorphism of vector bundlesV1

∼=

→ V2 over X, thenχ(V1, V2;σ) = 0.

(2) For triples(V1, V2;σ)and(V₁⁰, V₂⁰;σ⁰), ifϕi:Vi

∼=

→V_i⁰are isomorphisms of vector bundles overX such that

V1|Y σ //

∼= ϕ₁|_Y

V2|Y

∼= ϕ₂|_Y

V₁⁰|Y σ⁰

//V⁰₂|Y,

then(ϕ1, ϕ2)induces an isomorphism of vector bundlesVσ∼=V_σ⁰0andχ(V1, V2;σ) = χ(V₁⁰, V₂⁰;σ⁰)∈K⁰(X, Y).

(3) For any vector bundleW ∈Vect(X),χ(V1⊕W, V2⊕W;σ⊕1W) =χ(V1, V2;σ)∈ K⁰(X, Y).

PP Proof. Exercise.

2.2. Vector bundles from Clifford modules - the ABS construction. Fix0 < n∈ Nand consider the standard inner product onRⁿ, which gives the(n−1)-sphere Sⁿ⁻¹:={x| ||x||= 1}andn-ballDⁿ:={x| ||x|| ≤1}:

Sⁿ⁻¹⊂Dⁿ⊂Rⁿ.

The quotient spaceDⁿ/Sⁿ⁻¹is homeomorphic toSⁿ, hence the relative cohomol- ogyh⁰(Dⁿ, Sⁿ⁻¹)calculates the reduced cohomology˜h⁰(Sⁿ), for any cohomology theoryh. The difference construction of Section 2.1 can therefore be used to con- struct classes inh˜⁰(Sⁿ).

Recall thatCnis the Clifford algebra for the standard positive-definite quadratic formq+onRⁿand there is a linear embeddingRⁿ ,→Cn. This restricts toSⁿ⁻¹,→ C_n^×(the multiplicative units), sincev∈Rⁿis invertible inCnif and only ifq(v)6= 0. Remark 2.4. This is where the relationship between geometry and algebra kicks in. In particular, this is why the Clifford algebrasCnare more important than the algebrasC_n⁰ in topology.

Construction2.5. ForM ∈Mna gradedCn-module, the trivial bundlesM⁰×Dⁿ andM¹×DⁿinVect(Dⁿ)define a triple(M⁰×Dⁿ, M¹×Dⁿ;σ), whereσis defined by Clifford multiplication:

σ:M⁰×Sⁿ⁻¹ → M¹×Sⁿ⁻¹ (m, v) 7→ (vm, v) (compare Lemma 1.35).

Lemma 2.6. The associationM 7→ χ(M⁰×Dⁿ, M¹×Dⁿ;σ)induces a morphism of abelian groups

Mn χn

→KO⁰(Dⁿ, Sⁿ⁻¹)∼=KOg⁰(Sⁿ).

Proof. Lemma 2.3 implies that the image only depends on the isomorphism class ofM ∈ Mn. It is straightforward to verify that the direct sum of graded Clifford modules corresponds to the direct sum of vector bundles via Construction 2.5.

Lemma 2.7. The compositeM_n+1 → M_n ^χ→ⁿ KOg⁰(Sⁿ)is trivial, henceχ_n induces a morphism of abelian groups

χn:An→KOg⁰(Sⁿ).

Proof. Suppose thatM ∈Mnis the restriction of an elementM ∈Mn+1and con- sider the standard inclusions

Sⁿ⁻¹ _  //

Dⁿ  //Rⁿ _

Sⁿ  //Rⁿ⁺¹

where Sⁿ⁻¹ ,→ Sⁿ is the equatorial inclusion and Sⁿ ∼= D₊ⁿ ∪Sⁿ⁻¹ D₋ⁿ, where Dⁿ₊∼=Dⁿ ∼=Dⁿ₋are the upper and lower hemispheres.

Hence the Clifford multiplication by elements ofSⁿ gives an extension of the isomorphismσtoDⁿvia the homeomorphismDⁿ ∼=Dⁿ₊. The result follows from

Lemma 2.3.

Recall from Theorem 1.33 thatL

n∈NAn has the structure of a graded ring in- duces by tensor product of Clifford modules.

Theorem 2.8. [ABS64]Construction 2.5 induces an isomorphism ofN-graded rings:

χ:M

n∈N

An

∼=

→M

n∈N

KOg⁰(Sⁿ),

where the product on the right hand side is induced by the exterior tensor product of vector bundles.

Proof. (Indications.) The compatibility with the product structures is an exercise

with the definitions. The proof thatχis an isomorphism requires the proof of Bott PP periodicity forKOand the calculation of the right hand side.

2.3. Passage toRPⁿ. Recall thatRPⁿ ∼=Sⁿ/_Z/2, whereZ/2∼={±1}acts freely by multiplication on Sⁿ ⊂ Rⁿ⁺¹. The inclusionRⁿ ,→ Rⁿ⁺¹ is equivariant for the action of{±1}hence, for any0< m < n∈N, induces a diagram

S^m−1  //

Sⁿ⁻¹

RP^m−1  //RPⁿ⁻¹ where the surjections are theZ/2-Galois coverings.

Remark2.9. ForE×Sⁿ⁻¹the trivial vector bundle associated to the vector spaceE, Z/2acts by bundle isomorphisms via(e, v)7→(−e,−v)and passage to the quotient defines a vector bundle:

E×_Z/2Sⁿ⁻¹

RPⁿ⁻¹.

Writingπn−1:RPⁿ⁻¹→ ∗for the projection andλn−1for the canonical line bundle onRPⁿ⁻¹, this bundle can be identified as:

E×_Z/2Sⁿ⁻¹∼=λ_n−1⊗π^∗_n−1E,

namely the trivial bundleπ_n−1^∗ Etwisted by the line bundleλn−1.

Remark2.10. The above is a case of a general construction for discrete groups (and, more generally, topological groups). For a discrete groupG, consider a GaloisG- coveringE→B ∼=E/Gand letMbe a finite rankG-module defined overR. Then the groupGacts diagonally onE×M and the associated projection

(E×M)/G→B

is a vector bundle. For instance, this can be applied to the universalG-bundle EG→BG

to construct a vector bundle on the classifying spaceBGofG.

In the above,G=Z/2; the classifying space has the homotopy type ofRP^∞and the construction forRPⁿis simply given by restriction. In the general case, it is not possible to simply twist a trivial bundle by a line bundle to describe the associated bundle.

Construction2.11. ForM ∈M_mandn > m, let(λ_n−1⊗π^∗_n−1M⁰, π_n−1^∗ M¹;σ)denote the triple where

σ:λ_m−1⊗π^∗_m−1M⁰ → π^∗_m−1M¹ [(m, v)] 7→ [(vm, v)]

under the identifications of the restrictions toRP^m−1:

(λ_n−1⊗π_n−1^∗ M⁰)|_RPm−1 ∼= λ_m−1⊗π^∗_m−1M⁰∼=M⁰×_Z/2S^m−1 π_n−1^∗ M¹|_RPm−1 ∼= π^∗_m−1M¹∼=M¹×RP^m−1.



The twisting is necessary so that the Clifford multiplication passes to the quo- tient.

Proposition 2.12. For natural numbers0< m < n, the construction M 7→χ(λ_n−1⊗π_n−1^∗ M⁰, π^∗_n−1M¹;σ) induces a morphism of abelian groups:

χm,n:Mm→KOg⁰(RPⁿ⁻¹,RP^m−1).

Moreover, the compositeM_n→M_m→KOg⁰(RPⁿ⁻¹,RP^m−1)is trivial, hence induces χm,n:Mm/Mn →KOg⁰(RPⁿ⁻¹,RP^m−1).

Proof. (Indications.) The proof follows those of Lemmas 2.6 and 2.7.

The following result gives aconceptualdescription of the groupsKOg⁰(RPⁿ⁻¹,RP^m−1) in terms of Clifford modules.

Theorem 2.13. [ABS64]For0< m < n,

χm,n:Mm/Mn→KOg⁰(RPⁿ⁻¹,RP^m−1).

is an isomorphism.

Proof. The best approach to this result is in the equivariant setting (cf. [Kar02] for

example).

Remark2.14. The paper [ABS64] gives the result as aconsequenceof the calculation of the right hand side by Adams [Ada66].

2.4. The bundle case and the Thom isomorphism . . .S.

Remark 2.15. This subsection is included for the reader who wishes to have a glimpse of the extension of the above methods to the bundle setting and how these results are applied to derive the Thom isomorphism forKO.

The ABS construction of Section 2.2 generalizes to the bundle setting. Namely, replaceRⁿby a Euclidean vector bundle:

V

X,

so that each fibre is equipped with a positive-definite quadratic form. The Clifford algebra construction globalizes to define the bundle of Clifford algebras:

C(V)

X,

so thatC(V)is a bundle ofZ/2-graded algebras, in particular the multiplication is a morphism of vector bundlesC(V)⊗C(V)→C(V).More precisely, the multipli- cation isZ/2-graded:

C(V) ˆ⊗C(V)→C(V).

A (graded) Clifford module forC(V)is aZ/2-graded vector bundleM → X which is aZ/2-graded module overC(V), namely equipped with a structure map ofZ/2-graded vector bundles

C(V) ˆ⊗M →M

which satisfies the usual unital and associativity conditions.

Definition 2.16. ForV → Xa (finite rank) Euclidean vector bundle, so that each fibreV_xis equipped with a norm|| − ||,

(1) the sphere bundle S(V) → X is the bundle with fibre S(V)_x := {v ∈ V_x| ||v||= 1};

(2) the disc bundleD(V)→Xis the bundle with fibreD(V)_x:={v∈V_x| ||v|| ≤ 1},

so that there are bundle maps (overX):

S(V)  // $$

D(V)

 //V

||X.

The Thom spaceX^V is the quotient spaceD(V)/S(V)of the total space of the disc bundle by the total space of the sphere bundle. (IfXis compact, this is homeomor- phic to the one-point compactification of the total space ofV.)

Construction 2.17. For V → X and M → X as above, let π : D(V) → X and S(V) → X denote the associated disc and sphere bundles, defined by the Eu- clidean metric of V, so that there is an inclusion of total spaces S(V) ⊂ D(V). There is a triple for(D(V), S(V)):

(π^∗M⁰, π^∗M1;σ)

whereσ:π^∗M⁰|_S(V)→^∼= π^∗M1|_S(V)is defined by the Clifford multiplication (glob- alizing Construction 2.5).

The difference construction provides the associated class:

χ(π^∗M⁰, π^∗M₁;σ)∈KO⁰(D(V), S(V)).

Remark2.18.

(1) Recall that the Thom spaceX^V isD(V)/S(V), henceKO⁰(D(V), S(V))∼= KOg⁰(X^V).

(2) The above construction is the basis for the construction of the Atiyah-Bott- Shapiro orientation ofKOand is related to the Thom isomorphism theo- rem. To complete their construction, one requires to understand how to associate a bundle of Clifford modules to a Clifford module (over a point);

the approach is sketched below.

Recall thatO(V, q)denotes the orthogonal group of the quadratic space(V, q); the determinant defines a group homomorphismO(V, q) → Z/2andSO(V, q)is the kernel.

The groupPin(V, q)is the subgroup ofC^×(V, q)of elements withq(v) = 1and the groupSpin(V, q)is defined by

Spin(V, q) := Pin(V, q)∩C⁰(V, q).

The groupSpin(Rⁿ, q₊)is written simplySpin_n, so thatSpin_n,→C_n^×is a subgroup (consisting of elements of degree zero).

The adjoint action induces a short exact sequence:

0→Z/2→Spin_n ^Ad→SO(n)→0.

Remark2.19. Forn≥3,Spin_nis the universal cover ofSO(n). The adjoint action fits into a commutative diagram

Spin_n

Ad

%%

SO(n)

Ad //Aut(C_n).

IfPG → X is a principalG-bundle (forGa topological group) andM is aG- representation (overR), there is an associated vector bundle:

P_G×GM

X∼=PG/G.

This construction is compatible with the respective tensor products. Namely, if M, N areG-representations, equipM⊗Nwith the diagonalG-module structure, then

P_G

×_G(M ⊗N)∼= (P_G×_GM)⊗(P_G×_GN) as vector bundles overX.

Remark 2.20. The existence of a universal example for a rank n vector bundle, which is defined over the spaceBO(n)leads to the Stiefel-Whitney characteris- tic classes. Namely, there is an isomorphism of algebras

H^∗(BO(n);F2)∼=F2[w₁, . . . , w_n], where the universal Stiefel-Whitney classwihas degreei.

The classifying map of a ranknbundleV over a (paracompact) spaceX, X →^V BO(n)

induces a morphism of algebras:

H^∗(BO(n);F2)∼=F2[w1, . . . , wn]^V

∗

→H^∗(X;F2).

By definition, theith Stiefel-Whitney classw_i(V)ofV is the image ofw_iunder the above morphism.

IfV is oriented, the structure group O(n)can be replaced bySO(n). In coho- mology, the induced morphism of algebras is:

H^∗(BO(n);F2)∼=F2[w1, . . . , wn]H^∗(BSO(n);F2)∼=F2[w2, . . . , wn], corresponding to the quotient byw1. In particular, a vector bundle is oriented if and only ifw1(V) = 0.

Definition 2.21. A vector bundleV → X of ranknhas aSpin_n-structure if there exists a principalSpin_n-bundlePSpin_n(V)→X such thatV →Xis isomorphic to

P_Spin

n(V)×_Spin

nRⁿ

X

whereRⁿ is considered as aSpin_n-module viaSpin_n ,→ SO(n) ,→ O(n)and the canonical action.

Remark2.22. In algebraic topology, the existence of aSpin_n-structure onV is usu- ally interpreted in terms of the vanishing of the first two Stiefel-Whitney classes w1(V)∈H¹(X;F2),w2(V)∈H²(X;F2).

This corresponds to the vanishing of the obstructions to lifting in the following diagram:

BSpin_n

BSO(n) ^w2 //

K(Z/2,2)

X V // ;; DD

BO(n) _w

1 //K(Z/2,1),

where the morphism indicatedV classifies the vector bundle. The vertical mor- phisms are fibrations and the maps to the Eilenberg-MacLane spaces represent the universal Stiefel-Whitney classesw1andw2respectively.

IfV →Xadmits aSpin_n-structure, then the associated bundle of Clifford alge- brasC(V)is isomorphic to

PSpin_n(V)×AdCn



Note that the adjoint representation is not the same as the representation by left multiplication associated toSpin_n ⊂C_n^×.

IfM is aZ/2-gradedCn-module, then it is aSpin_n-module via left multiplica- tionµ(observe that the left multiplication preserves theZ/2-grading, sinceSpin_n⊂ C_n⁰), hence one may form the associatedZ/2-graded vector bundle:

P_Spin

n(V)×µM

X

Proposition 2.23. ForM,V as above,P_Spin

n(V)×µM →X is aZ/2-gradedC(V)-

module. PP

Proof. Exercise - the fact thatC(V)is defined by using the adjoint representation plays an essential r ˆole. (See [LM89, Proposition II.3.8], for example.) Forn= 8k, choose a generator ofM8k(this corresponds to an element generated by the periodicity class). Then the above construction provides, for any Spin_8k- vector bundleV →Xof rank8k, a Thom class inKOg⁰(X^V).

Remark 2.24. Under the above conditions onV, the Thom isomorphism theorem forKOstates that there is an isomorphism ofKO⁰(X)-modules:

KO⁰(X)→^∼= KOg⁰(X^V) induced by the Thom class construction outlined above.

The module structure comes from the Thom diagonal. Namely, the diagonal morphismX →X×Xfits into a morphism of vector bundles:

V //

p^∗₂V

X

∆//X×X,

wherep₂:X×X →Xis the projection onto the second factor.

Passage to Thom spaces induces the Thom diagonal:

X^V →(X×X)^p∗2V ∼=X₊∧X^V, whereX+isX equipped with a disjoint basepoint.

In a multiplicative cohomology theoryh, the composite:

h^∗(X)⊗˜h^∗(X^V)→h˜^∗(X₊∧X^V)→˜h^∗(X^V),

where the first morphism is the exterior product, givesh˜^∗(X^V)the structure of an h^∗(X)-module.

Remark2.25. The Thom isomorphism theorem is used in defining theBott canni- balistic characteristic classesusing the Adams operations (which are introduced in Section 3), which are natural transformations:ψⁿ:KO⁰(−)→KO⁰(−), forn∈N.

Using the above notation, one obtains a morphismρⁿ_V via the following commuta- tive diagram:

KO⁰(X)

ρⁿ_V

∼= //KOg⁰(X^V)

ψⁿ

KO⁰(X) _∼

= //KOg⁰(X^V),

in which the horizontal isomorphisms are the Thom isomorphisms. By the multi- plicative structure, the key elements are

ρⁿ_V :=ρⁿ_V(1)∈KO⁰(X)

where 1 ∈ KO⁰(X) is the unit. These are the Bott cannibalistic characteristic classes; their construction is analogous to the construction of the Stiefel-Whitney classes inH^∗(X;F2)using the Thom isomorphism forH^∗(−;F2)(this requires no hypothesis onV) and the Steenrod squaring operationsSqⁿ,n∈N.

Remark 2.26. In stable homotopy theory, the existence of the Thom classes and their compatibility lead to the Atiyah-Bott-Shapiro orientation of the spectrumKO which represents orthogonalK-theory:

MSpin→KO.

This is a morphism ofring spectra, whereMSpinis the Thom spectrum associated to spin cobordism. There is a rigidification of this result: the map can be shown to be a morphism ofhighly structured ring spectra.

For unitaryK-theory there is an analogous orientation (which is easier to con- struct, since the Thom isomorphism is more elementary forKU):

M U →KU,

whereM U is the complex cobordism spectrum (the existence of such a morphism of ring spectra is equivalent to having acomplex orientation). Moreover, these fit into a commutative diagram:

MSpin //

M U

KO //KU whereKO→KUrepresents complexification.