Units, orientation theory, Thom spectra and the J-invariant

U. Bunke ^{∗}

October 2, 2005

Contents

1 What is this 2

2 Classical groups and their classifying spaces 2 2.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 I-functors and L-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 I^{∗}-spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Delooping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.5 Classicial groups as I^{∗}-functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.6 Spheres and their homotopy equivalences as I^{∗}-functors . . . . . . . . . . . . . . . . . . . 5 2.7 Bar construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.8 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 E_{∞}-ring spectra 7 3.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 I^{∗}-prefunctors and L-spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Thomspectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 More Thom spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.6 E_{∞}-ring spaces and spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Units and orientation theory 14 4.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 E-oriented bordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Orientations of spherical fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4 E-orientations of groups and orientations of Thom spectra . . . . . . . . . . . . . . . . . . 16 4.5 The Atiyah-Bott-Shapiro orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.6 The comparison diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.8 Cannibalistic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 J-theory 20 5.1 Adams operations and related materal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Calculation of cannibalistic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3 The Adams conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.4 The J-theory diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.5 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.6 The decomposition of SF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.7 K-localization of the sphere spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.8 Discrete models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, bunke@unimath.gwdg.de

1 What is this

This material was collected during the process of preparing the talk on Units, Thom spectra and orientation theory at the Arbeitsgemeinschaft in Oberwolfach organized by S. Schwede and J. Rognes. The contents was dictated by the description of the talk, my own path to penetrate the subject, and a lot of small sidesteps to things I have got interested in during the preparation.

2 Classical groups and their classifying spaces

2.1 Remarks

In this subsection we collect material which is presented in detail in [20]. The main point is to put classical groups into a picture which leads smoothly to infinite loop spaces via the bar-construction delooping machine of May. The idea is to go from I^{∗}-functors via I-functors to L-spaces where L is the linear isometries operad. The latter are a natural input of the delooping machine. We obtain the classifying spaces of classical groups as coordinate invariant spectra.

2.2 I-functors and L-spaces

2.2.1 By T we denote the category of based topological spaces (compactly generated, weakly Hausdorff and non-degeneratedly based).

Definition 2.1 Let I denote the topological category of real Hilbert spaces and isometric embeddings. By I^{n} ⊂ I^{fin} ⊂ I we denote the full subcategories of n-dimensional or finite-dimensional spaces, respectively. We set I^{∗} := ⊔_{n≥0}I^{n}.

Observe that in I^{∗} ⊂ I^{fin} is not full since in I^{∗} there are no morphisms between spaces of different dimensions. 2.2.2

Definition 2.2 An I-functor is a pair (T, ω) of a continuous functor T : I → T together with a commutative, associative, continuous natural transformation ω : T × T → T ∘ ⊕ with the following properties.

1. The following composition is the identity:

\xymatrix{T V \ar[r]^{(id,1)} & T V × T 0 \ar[r]^{ω} & T (V ⊕ 0) \ar[r]^{T (V ⊕0→^{∼}V)} & T V}

where 1 ∈ T 0 denotes the basepoint.

2. For each morphism i : V → V′ the induced map T i : T V → T V′ is a homomorphism onto a closed subset.

3. The natural transformation

lim_{I^{fin}} T → T .

is an isomorphism.

Note that by 1. in 2. we have T i = T (V ⊕ (V′ ⊖ V) →^{∼} V′) ∘ ω ∘ (id, 1) with 1 ∈ T (V′ ⊖ V) the base point.

2.2.3

Definition 2.3 An I^{∗}-functor is a pair (T, ω) of a functor I^{∗} → T together with a natural transformation ω : T × T → T ∘ ⊕ such that

1. The following composition is the identity:

\xymatrix{T V \ar[r]^{(id,1)} & T V × T 0 \ar[r]^{ω} & T (V ⊕ 0) \ar[r]^{T (V ⊕0→^{∼}V)} & T V}

where 1 ∈ T 0 denotes the basepoint.

2. For V, V′ ∈ I^{∗} the map \xymatrix{T V \ar[r]^{(id,1)} & T V × T V′ \ar[r]^{ω} & T (V ⊕ V′)} is an homoemorphism onto a closed subset.

There is a natural notion of a morphism between I or I^{∗}-functors.

Definition 2.4 By I[T ] and I^{∗}[T ] we denote the categories of I and I^{∗}-functors.

2.2.4

Lemma 2.5 The forgetful functor I[T ] → I^{∗}[T ] is an isomorphism of categories.

Proof: Let (T, ω) be a I^{∗}-functor. We first use ω in order to extend T to I^{fin} such that T (i) = T (V ⊕ (V′ ⊖ V) →^{∼} V′) ∘ ω ∘ (id, 1) for an inclusion i : V → V′. Then we extend T to I defining T = lim_{I^{fin}} T. □

2.2.5

Definition 2.6 By L we denote the linear isometries operad on ℝ^{∞}. By L[T ] we denote the category of L-spaces.

There is a natural map I[T ] → L[T ]. It maps (T, ω) to an L-space structure on T ℝ^{∞} given by the action the action

θ_{j} : L(j) × (T ℝ^{∞})^{j} →^{(id,ω^{j})} L(j) × T (ℝ^{∞})^{j} →^{eval} T ℝ^{∞}

where eval is the evaluation and ω^{j} is obtained by a j-fold application of ω.

2.3 I^{∗}-spectra

2.3.1 Let I^{∗}(ℝ^{∞}) ⊂ I^{∗} be the full subcategory of subspaces of ℝ^{∞}. Let HT ⊂ T denote the subcategory where the morphisms are homeomorphisms. Furthermore, for any topological category A let hA denote its homotopy category. For V ∈ I^{∗}(ℝ^{∞}) and X ∈ T we let tV denote the functorial one-point compactification of V, Σ^{V} X := W ∧ tV and Ω^{V} X := T (tV, X). 2.3.2

Definition 2.7 A prespectrum (T, σ) is a function T : I^{∗}(ℝ^{∞}) → HT which induces a functor hT : hI^{∗}(ℝ^{∞}) → hT together with maps σ : Σ^{W} T V → T (V ⊕ W) for all W ⊥ V such that the following properties hold true.

1. The adjoint σ̃ : T V → Ω^{V} T (V ⊕ W) is an inclusion with closed image.

2. Let V ⊥ W ⊥ Z ⊥ V Then we have the following commutative diagrams.

\xymatrix{T V \ar[r]^{≅} \ar[d]^{1} & Σ^{0}V \ar[d]^{σ} \\ T V \ar[r]^{≅} & T (V ⊕ 0)} , \xymatrix{Σ^{Z}Σ^{W} T V \ar[r]^{≅} \ar[d]^{Σ^{Z}} & Σ^{W ⊕Z}T V \ar[d]^{σ} \\ Σ^{Z}T (V ⊕ W) \ar[r]^{σ} & T (V ⊕ W ⊕ Z)}

3. The following diagram commutes in hT . Let f ∈ I^{∗}(V, V′), g ∈ I(W, W′), V ⊥ W and V′ ⊥ W′.

\xymatrix{T V ∧ tW \ar[r]^{≅} \ar[d]^{T f∧tg} & Σ^{W} T V \ar[r]^{σ} & T (V ⊕ W) \ar[d]^{T (f⊕g)} \\ T V′ ∧ tW′ \ar[r]^{≅} & Σ^{W′}T V′ \ar[r]^{σ} & T (V′ ⊕ W′)}

A spectrum is a prespectrum where the adjoint maps σ̃ : T V → Ω^{W} T (V ⊕ W) are homeomorphisms. A morphism of prespectra θ : (T, σ) → (T′, σ′) consists of maps θ : T V → T′V such that hθ : hT → hT′ is a morphism of functors, and such that

\xymatrix{Σ^{W} T V \ar[r]^{σ} \ar[d]^{Σ^{W} θ} & T (V ⊕ W) \ar[d]^{θ} \\ Σ^{W} T′V \ar[r]^{σ′} & T′(V ⊕ W)}

commutes. By S ⊂ P we denote the category of prespectra and its full subcategory of spectra.

2.3.3 We have the suspension spectrum functor Σ^{∞} : T → P and the n-th space functor Ω_{n} : P → T . The functor Ω_{0} is part of an adjoint pair:

Σ^{∞} : T ↔ S : Ω_{0} .

2.3.4 We have further the spectrum functor Ω^{∞} : P → S, Ω^{∞}T (V) := lim_{V ⊥W} Ω^{W} T (V ⊕ W), which is adjoint to the inclusion ν : S → P.

Ω^{∞} : P ↔ S : ν .

The write Ω_{n}^{∞} := Ω_{n} ∘ Ω^{∞}. 2.3.5 We have the composition Q^{∞} : Ω^{∞}Σ^{∞} : T → S such that

Q^{∞} : T ↔ S : Ω_{0} .

2.3.6 We define the composition Q := (…)_{0}∘Q_{∞} : T → T . It extends to a monad (Q, η, μ), μ : Q^{2} → Q and η : id → Q. We let Q[T ] denote the category of Q-algebras. The functor Ω_{0} extends naturally to a functor

QΩ_{0} : S → Q[T ] .

2.4 Delooping

2.4.1 We consider an E_{∞}-operad I′. Let K_{∞} denote the little convex bodies (partial) operad [20, VII.2] and set I = I′ × K_{∞}. We denote the corresponding monads by the same symbols. There is a morphism of monads α_{∞} : K_{∞} → Q. 2.4.2 We consider an I′-algebra or a Q-algebra X. It becomes an I-algebra by pull-back of structure. In this situation there is a functorial construction of a prespectrum (T X, σ) [20, VII.3]. In more detail let K_{V} be the little convex bodies operad of V. Then tV ∧ … is a K_{V} -functor in the sense that there is a map tV ∧ K_{V} (X) → tV ∧ X which is compatible with unit and product of K_{V} (see [18, Def. 9.4]. In fact its adjoint is easier to guess:

\xymatrix{K_{V} (X) \ar[r]^{K_{V} (η)} & K_{V} (Ω^{V} Σ^{V} X) \ar[r]^{action} & Ω^{V} Σ^{V} X} .

Let I_{V} := I′ × K_{V}. Then T XV = B(Σ^{V} , I_{V} , X) with structure map

\xymatrix{σ : Σ^{W} B(Σ^{V} , I_{V} , X) \ar[r] & B(Σ^{V ⊕W} , I_{V} , X) \ar[r]^{I_{V} →I_{V ⊕W}} & B(Σ^{V ⊕W} , I_{V ⊕W} , X)} .

We let ΓX := Ω_{0}^{∞}T X.

2.4.3 Note that X is an H-space. We furthermore have a natural map

i : X → ΓX

which is a weak equivalence if X is group like [20, Thm. 3 1]. 2.4.4 If E ∈ S then there is furthermore a natural map of prespectra ω̃ : T Ω_{0}E → νE which induces an isomorphism on non-negative homotopy groups. Observe that the spectrum Ω^{∞}T X is connective. Therefore we can associate to each spectrum E a connective spectrum E^{≥0} in a functorial way:

E < 0 >:= Ω^{∞}T Ω_{0}E .

We furthermore have natural map

Ω^{∞}ω̃ : E < 0 >→ E ,

which induces an isomorphism on non-negative homotopy groups.

Problem 2.8 Can one exhibit Ω^{∞}T as an adjoint functor and see that it preserves fibration sequences ?

2.4.5 Here we fix some language. An infinite loop space X is by definition zeroth space of a connective spectrum, i.e. X = Ω_{0}E. An infinite loop map X → X′ is by definition Ω_{0} of a map of connective spectra E → E′. In this way we obtain an equivalence of categories of connective spectra and infinite loops spaces. Let U be an E_{∞}-operad. Then delooping gives a functor HU[T ] → HS. It induces an equivalence of the homotopy category of grouplike U-spaces and the homotopy category of connective spectra (each as a full subcategory of HU[T ] and HS, respectively.). Here hC is the homotopy category of a topological category and HC is obtained from hC by inverting weak equivalences (this notion has to be defined in C).

2.5 Classicial groups as I^{∗}-functors

2.5.1 Let K ∈ {ℝ, ℂ, ℍ} and for V ∈ I define V_{K} := V ⊗_{ℝ} K. Then we have the following classical group valued I^{∗}-functors (2.3) which map V ∈ I^{∗} to

\begin{aligned}e & ⊂ & SO(V) & ⊂ & O(V) \\ & & ⊂ & & ⊂ \\ & & SU(V_{ℂ}) & ⊂ & U(V_{ℂ}) \\ & & & & ⊂ \\ & & & & Sp(V_{ℍ}) \end{aligned} .

2.5.2 Let G denote one of these. Then we define ω : G(V) × G(V′) → G(V ⊕ V′) to be usual direct sum embedding. 2.5.3 For V ∈ I^{∗} we define P in^{c}(V) ⊂ Cliff_{ℂ}(V) to be the subgroup generated by the unit vectors of V and U(1). Then we have a natural map ω : P in^{c}(V) × P in^{c}(V′) → P in^{c}(V ⊕ V′) induced by the identification Cliff_{ℂ}(V) ˆ⊗Cliff(V′) ≅ Cliff(V ⊕ V′). Note that ω is neither a group homomorphism nor an embedding. But the restriction of ω to P in^{c}(V) × 1 is an embedding. There is a natural map P in^{c} → O of I^{∗}-functors. The groups Spin^{c}, P in and Spin can be considered as I^{∗}-functors in a similar manner. 2.5.4 We will denote the extensions (see 2.5) of these I^{∗}-functors to I-functors by the same symbols. 2.5.5 If A is an abelian group, then we define the I^{∗}-functor AV := A and ω : AV × AV′ → A(V ⊕ V′) as the sum. We have maps of I^{∗}-functors. O → ℤ/2ℤ and U → U(1) given by the determinant.

2.6 Spheres and their homotopy equivalences as I^{∗}-functors

2.6.1 We consider the continuous functor t : I^{∗} → T such that t ∧ t ≅ t ∘ ⊕. With ω : t × t → t ∘ ⊕ being the projection t × t → t ∧ t we have t ∈ I^{∗}[T ].

2.6.2 We define F̃ ∈ I^{∗}[T ] by F̃V := T (tV, tV) and F̃f = tf∘⋯∘tf^{−1} and ω := ∧. Let F, SF, T op, ST op be the sub-I^{∗}-functors of based homotopy equivalences, degree-one based homotopy equivalences, homeomorphism, degree-one homeomorphisms. 2.6.3 Note that F and SF are grouplike monoid-valued and T op and ST op are group valued. We have natural inclusions

\begin{aligned}SO & ⊂ & O \\ ⊂ & & ⊂ \\ ST op & ⊂ & T op \\ ⊂ & & ⊂ \\ SF & ⊂ & F \end{aligned} .

Again we denote the extensions of these I^{∗}-functors to I-functors by the same symbols.

2.7 Bar construction

2.7.1 The categories I[T ] admits finite products. Our monoid-valued I-functors defined above are monoids in I[T ]. Let G be one of them. Consider X, Y ∈ I[T ] such that X is a right- and Y as a left G-space. Then we can form the simplicial object B(X, G, Y)_{∗} ∈ I[T ]^{Δ^{op}} such that B(X, G, Y)_{n} = X × G^{n} × Y and the simplicial maps are given by the product on G and the actions. The realization B(X, G, Y) is then again an I[T ]-functor. 2.7.2 We define BG := B(∗, G, ∗). If j : H → G is a morphism of grouplike objects in I[T ], then we get a quasifibration sequence (levelwise)

H →^{j} G →^{τ} G/H →^{q} BH →^{Bj} BG ,

where G/H := B(G, H, ∗). The map τ is given by G → B(G, H, H) → B(G, H, ∗), where the first map is an equivalence. 2.7.3 Let G be one of the classical groups introduced above. Then we have two monoid structures on Gℝ^{∞}, one given by the group structure and the other given by the L-algebra structure induced via I[T ] → L[T ].

Lemma 2.9 These two monoid structures are homotopic. In particular, the composition structure is commutative up to homotopy, and the monoid structure given by the action of L is group like.

It follows that G/H is a group-like L-algebra. 2.7.4 We can now apply the delooping machine 2.4. We then get the following sequence of spectra:

Ω^{∞}T H →^{j} Ω^{∞}T G →^{τ} Ω^{∞}T G/H →^{q} Ω^{∞}T BH →^{Bj} Ω^{∞}T BG .

Lemma 2.10 This sequence is isomorphic to a fibration sequence in the stable homotopy category.

Proof: The natural map G → ΩBG in I[T ] induces an equivalence (see [20, VII. 3.4] for the second equivalence) Ω^{∞}T G →^{∼} Ω^{∞}T ΩBG →^{∼} ΩΩ^{∞}T BG so that

Ω^{∞}T H →^{∼} Ω^{∞}T ΩBH →^{∼} ΩΩ^{∞}T BH
j ↓  ΩBj ↓
Ω^{∞}T H →^{∼} Ω^{∞}T ΩBG →^{∼} ΩΩ^{∞}T BG

(1)

commutes the stable homotopy category. In this category there is now an up to isomorphism unique fibration sequence (distinguished triangle) extending Bj : Ω^{∞}T BH → Ω^{∞}T BG, the lower horizontal sequence in the following diagram.

Ω^{∞}T H  →^{j}  Ω^{∞}T G →^{τ} Ω^{∞}T G/H →^{q} Ω^{∞}T BH →^{Bj} Ω^{∞}T BG ≅↓  ≅↓  ↓  ≅↓  ≅↓ ΩΩ^{∞}T BH →^{ΩBJ} ΩΩ^{∞}T BG →  C  → Ω^{∞}T BH →^{Bj} Ω^{∞}T BG .

3 E_{∞}-RING SPECTRA 7

We concude that the sequence above is part of a fibration sequence. □

Observe that (1) implies the the two deloops Ω_{1}^{∞}T G and BG of G, one using the L-space structure and the other the monoid structure, coincide as infinite loop spaces (see [20, VII. 3.5]).

2.8 Some examples

2.8.1 The spectrum Ω^{∞}T BO represents connective real K-theory ko. 2.8.2 The spectrum Ω^{∞}T BU represents connective complex K-theory ku. 2.8.3 The spectrum Ω^{∞}T F̃ represents the sphere spectrum S. In fact we have F̃V ≅ F (tV, tV) ≅ Ω^{V} Σ^{V} S^{0} so that F̃ℝ^{∞} → Ω^{∞}Σ^{∞}S^{0} ≅ QS^{0} is the group completion. 2.8.4 Let A be an abelian group. Then it is a L-space such that g ∈ L(j) acts as multiplication A^{j} → A. We get the Eilenberg-MacLane spectrum HU := Ω^{∞}T A. In fact, the L-space structure comes from the I^{∗}-space structure in 2.5.5.

3 E_{∞}-ring spectra

3.1 Remarks

The homotopy category of spectra HS has a smash product so that one speak of rings and modules. The modern point of view is to consider a category of spectra which itself admits a smash product (rather than its homotopy category). This allows for a lot of space-level constructions of algebraic type. In these notes we are in particular interested in the subspace of multiplicative units of SE ⊂ Ω_{0}E of a ring spectrum. Its H-space structure can be lifted to a infinite loop space structure provided the specrum E is nice. In the present notes we adopt the point of view of May [20] (the pre-modern view) of ring spectra over an operad, in particular the linear isomeries operad L. In modern language these are the S-modules. We put some emphasis on machines which construct nice ring spectra.

3.2 Basic definitions

3.2.1 We let T_{e} denote the category of cofibrations S^{0} → X in T . We let 1 ∈ X denote the image of the non-base point of S^{0}. 3.2.2 Similarly we define the categories P_{e} and S_{e} of unital prespectra and spectra by requireing maps e : Σ^{∞}S^{0} → X or e : Q^{∞}S^{0} → X, respectively, which induces cofibrations in the zeroth level. The constructions Σ^{∞}, Ω^{∞}, Ω_{n},Q_{∞} extend to T_{e}, P_{e}, S_{e}. 3.2.3 Let U → L be a map of E_{∞}-operads.

Definition 3.1 A U-prespectrum (T, σ, ξ) is an unital prespectrum (T, σ) together with maps ξ_{j}(g) : U ∧ T V_{1} ∧ ⋯ ∧ T V_{j} → T g(V_{1} ⊕ ⋯ ⊕ V_{j}) for j ≥ 0, g ∈ U(j), V_{i} ∈ I(ℝ^{∞}) (and by convention ξ_{0}(∗) = e) such that

1. The following diagram commutes:

\xymatrix{T V_{1} ∧ ⋯ ∧ T V_{j} \ar[r]^{ξ_{j}(γ(g,h_{1},...,h_{k}))} \ar[d]_{ξ_{j_{1}} (h_{1})∧···∧ξ_{j_{k}} (h_{k})} & T γ(g; h_{1}, …, h_{k})(V_{1} ⊕ ⋯ ⊕ V_{j}) \ar[d] \\ T W_{1} ∧ ⋯ ∧ T W_{k} \ar[r]^{ξ_{k}(g)} & T g(W_{1} ⊕ ⋯ ⊕ W_{k})} ,

where g ∈ U(k), h_{r} ∈ U(j_{r}), 1 ≤ r ≤ k, and j = j_{1}+⋯+j_{k}, W_{r} := h_{r}(V_{j_{1}+···+j_{r−1}} ⊕⋯⊕V_{j_{1}+···+j_{r}}).

2. ξ_{1}(1) = id : T V → T V 3. For g ∈ U(j) and σ ∈ Σ_{j} the following diagram is commutative:

\xymatrix{T V_{1} ∧ ⋯ ∧ T V_{j} \ar[r]^{ξ_{j}(gτ)} \ar[d]_{τ} & T gτ(V_{1} ⊕ ⋯ ⊕ V_{j}) \ar[d]^{=} \\ T V_{τ^{−1}(1)} ∧ ⋯ ∧ T V_{τ^{−1}(j)} \ar[r]^{ξ_{j}(g)} & T g(V_{τ^{−1}(1)} ⊕ ⋯ ⊕ V_{τ^{−1}(j)}))}

4. For fixed V_{i} and W the map ξ_{j} is continuous in g runningin the set of maps such that g(V_{1} ⊕ ⋯ ⊕ V_{j}) ⊂ W.

5. For g ∈ U(j) and V_{i} ⊥ W_{i} the following diagram is commutative:

\xymatrix{T V_{1} ∧ tW_{1} ∧ ⋯ ∧ T V_{j} ∧ tW_{j} \ar[r]^{σ∧···∧σ} \ar[d]_{≅} & T (V_{1} ⊕ W_{1}) ∧ ⋯ ∧ T (V_{j} ⊕ W_{j}) \ar[d]^{ξ_{j}(g)} \\ T V_{1} ∧ ⋯ ∧ T V_{j} ∧ t(W_{1} ⊕ ⋯ ⊕ W_{j}) \ar[d]_{ξ_{j}(g)∧tg} & T g((V_{1} ⊕ W_{1}) ⊕ ⋯ ⊕ (V_{j} ⊕ W_{j})) \ar[d]^{=} \\ T g(V_{1} ⊕ ⋯ ⊕ V_{j}) ∧ tg(W_{1} ⊕ ⋯ ⊕ W_{j}) \ar[r]^{σ} & T (g(V_{1} ⊕ ⋯ ⊕ V_{j}) ⊕ g(W_{1} ⊕ ⋯ ⊕ W_{j}))}

6. If g ∈ U(1), then ξ_{1}(g) : T V → T V is a homemomorphism in the homotopy class T (g_{|V}) and every morphism in I(ℝ^{∞}) is obtainable for some g ∈ U(1).

A U-prespectrum (T, σ, ξ)) is a spectrum, if (T, σ) is so. A morphism of U-prespectra θ : (T, σ, ξ) → (T′, σ′, ξ′) is a morphism of unital prespectra such that the following diagram commutes:

\xymatrix{T V_{1} ∧ ⋯ ∧ T V_{j} \ar[r]^{ξ_{j}(g)} \ar[d]_{θ∧···∧θ} & T g(V_{1} ⊕ ⋯ ⊕ V_{j}) \ar[d]^{θ} \\ T′V_{1} ∧ ⋯ ∧ T′V_{j} \ar[r]^{ξ′_{j}(g)} & T′g(V_{1} ⊕ ∧ ⊕ V_{j})}

By U[P_{e}] and U[S_{e}] we denote the categories of U-prespectra and U-spectra.

We have the following results: 3.2.4 We have an extension of the suspension spectrum functor Σ^{∞} : U[T_{e}] → U[P_{e}]. 3.2.5 We have an extension of the spectrum functor Ω^{∞} : U[P_{e}] → U[S_{e}]. 3.2.6 S^{0} is a U-space such that ξ_{j}(g) : S^{0} ∧ ⋯ ∧ S^{0} → S^{0} acts as identity. We have the U-prespectrum Σ^{∞}S^{0}. For each E ∈ U[P_{e}] the map Σ^{∞}S^{0} → E is a map of U-prespectra. 3.2.7 We have Ω^{∞}Σ^{∞}S^{0} ∈ U[S_{e}] and for each E ∈ U[S_{e}] the map e : Ω^{∞}Σ^{∞}S^{0} → E is a map of U-spectra. 3.2.8 We have a functor Q^{∞} = Ω^{∞} ∘ Σ^{∞} : U[T_{e}] → U[S_{e}]. 3.2.9 We have the zeroth space functor U[S_{e}] → U[T_{e}] and the following adjoint pair:

Q^{∞} : U[T_{e}] ↔ U[S_{e}] : Ω_{0} .

3.3 I^{∗}-prefunctors and L-spectra

3.3.1

Definition 3.2 A I^{∗}-prefunctor (T, ω, e) is a continuous functor T : I^{∗} → T together with a commutative, associative and continuous natural transformation ω : T × T → T ∘ ⊕ and a continuous natural transformation e : t → T such that 1. There is a factorization

\xymatrix{T V × T W \ar[rr]^{ω} \ar[dr] &  & T (V ⊕ W) \\  & T V ∧ T W \ar[ur]}

2. The adjoint of

\xymatrix{T V ∧ tW \ar[r]^{1∧e} & T V ∧ T W \ar[r]^{ω} & T (V ⊕ W)}

is an inclusion with closed image and the identity if W = 0.

3. The diagram

\xymatrix{t(V ⊕ W) \ar[r]^{e} \ar[d]^{≅} & T (V ⊕ W) \\ tV ∧ tW \ar[r]^{e∧e} & T V ∧ T W \ar[u]^{ω}}

commutes.

A morphism of I^{∗}-prefunctors (T, ω, e) → (T′, ω′, e′)is a continuous natural transformation θ : T → T′ preserving the addition structure.

3.3.2 Let I_{e}^{∗}[T ] denote the category of I^{∗}-pre-functors. We have a functor Σ_{e}^{∞} : I_{e}^{∗}[T ] → L[P_{e}] which is constructed as follows. We set Σ_{e}^{∞}T V := T V. The structure maps are given by σ := ω(1 ∧ e) : T V ∧ tW → T (V ⊕ W). The unit is

\xymatrix{Σ^{∞}S^{0}V \ar[r]^{≅} & tV \ar[r]^{e} & T V} .

Lemma 3.3 If (T, ω, e) ∈ I_{e}^{∗}[T ], then (Σ_{e}^{∞}T, σ, e) ∈ L[P_{e}] is well-defined.

3.3.3 We have the composition Q_{e}^{∞} = Ω^{∞}Σ_{e}^{∞} : I_{e}^{∗}[T ] → I[S_{e}]. 3.3.4 Here is an example. Let (X, μ, η) be topological abelian monoid. Then we define an I^{∗}-prefunctor (T X, ω, e) such that T XV := X^{+} ∧ tV and T Xf := 1 ∧ tf for V ∈ I^{∗}(ℝ^{∞}) and f : V → V′. We further define

\xymatrix{ω : X^{+} ∧ tV ∧ X^{+} ∧ tW \ar[r]^{≅} & (X × X)^{+} ∧ tV ∧ tW \ar[r]^{μ^{+}∧id} & X^{+} ∧ t(V ⊕ W) \ar[r]^{≅} & T (V ⊕ W)}

and

\xymatrix{e : tV \ar[r]^{≅} & \{1\}^{+} ∧ tV \ar[r]^{η^{+}∧id} & X^{+} ∧ tV \ar[r]^{≅} & T V} .

Applying Σ_{e}^{∞} we get a L-prespectrum Σ_{e}^{∞}T X. 3.3.5 In particular, let X be the trivial monoid {1}. Then Q_{e}^{∞}T {1} ∈ L[S_{e}] is an L-spectrum version of the sphere spectrum S. 3.3.6 This examples is a special case of the construction of Thom spectra 3.4. 3.3.7 Let (T, ω, e) ∈ I_{e}^{∗}[T ]. Then we can form the L-spectrum Q_{e}^{∞}T and take its zeroth space T_{0} := Ω_{0}Q_{e}^{∞}T ∈ L[T ]. We can also define the I^{∗}-functor F̃X by F̃XV := Ω^{V} T V and F̃T f = Ω^{f^{−1}}T f for f : V → V′, and ω : F̃V × F̃W → F̃(V ⊕ W) by

\xymatrix{Ω^{V} T V × Ω^{W} T W \ar[r]^{∧} & Ω^{V ⊕W} (T V ∧ T W)ω \ar[r]^{Ω^{V ⊕W}} & Ω^{V ⊕W} T (V ⊕ W)} .

It determines another L-space F̃T ∈ L[T ].

Lemma 3.4 These two L-spaces T_{0} and F̃T are naturally isomorphic.

3.3.8 Let T ∈ I_{e}^{∗}[T ]. Then F̃T ∈ I^{∗}[T ] has natural I_{∗}-subfunctors SF T ⊂ F T ⊂ F̃T. If fact, SF T V ⊂ F T V ⊂ F̃T V is the subset of those elements which are mapped into the component of the identity or the units of the monoid F̃T ℝ^{∞}. They give rise to natural L-subspaces SF T_{0} ⊂ F T_{0} via the isomorphism above.

3.4 Thomspectra

3.4.1 We consider a morphisms of monoids G → F̃ in I^{∗}[T ]. Then G acts from the left on t and from the right on ∗. Let Y ∈ I^{∗}[T ] by a right G-space. If G is group-like then we have the quasi-fibration

p : B(Y, G, t) → B(Y, G, ∗)

in I^{∗}[T ]. Furthermore, the base point of tV induces a section τ : B(Y, G, ∗) → B(Y, G, t) of p.

Lemma 3.5 The quotient B(Y, G, t)/B(Y, G, ∗) belongs to I_{e}^{∗}[T ] naturally.

Proof: We let

ω : B(Y V, GV, tV)/B(Y V, GV, ∗) ∧ B(Y W, GW, tW)/B(Y W, GW, ∗)
→ B(Y (V ⊕ W), G(V ⊕ W), t(V ⊕ W))/B(Y (V ⊕ W), G(V ⊕ W), ∗)

be induced by the ω-structures of Y, G and t. Furthermore the map e : tV → B(Y V, GV, tV)/B(Y V, GV, ∗) is induced by the base point of Y which determines the inclusion

tV → Y × tV → B(Y V, GV, tV)/B(Y V, GV, ∗) .
□

3.4.2

Definition 3.6 We define T (G; Y) := Σ_{e}^{∞}B(Y, G, t)/B(Y, G, ∗) ∈ L[P_{e}] and let M(G; Y) := Ω^{∞}T (G; Y) ∈ L[S_{e}]. We call these the Thompre- and the Thomspectrum associated to the morphism G → F̃ and Y. In particular we write M(G) := M(G; ∗).

3.4.3 Let e : 1 → F̃. We have M(e) ≅ Q^{∞}S^{0}. 3.4.4 We have a natural map of I^{∗}-prefunctors q : B(Y, G, t) → B(∗, G, t) which induces the map of Thomspectra q : M(G; Y) → M(G). 3.4.5 Assume the factorization G → F → F̃. Let T ∈ I_{e}^{∗}[T ] be a I^{∗}-pre-functor. Then we form the I^{∗}-functors SF T ⊂ F T. The group G acts on these spaces via G ↦ F from the right (pre-composition of maps). We can form the Thom spectra M(G; SF T), M(G; F T) which classify normally (special) Σ_{e}^{∞}T -oriented bordism. This orientation is given by the natural evaluation map

ε : M(G; F T) → Q_{e}^{∞}T .

3.4.6 Consider the sequence H → G →^{j} F̃ of I^{∗}[T ]-monoids.

Lemma 3.7 There is a map j̄ : H\G → F̃M(H) of I^{∗}-functors such that

\xymatrix{G \ar[r]^{τ} \ar[d]_{j} & H\G \ar[d]^{j̄} \\ F̃ \ar[r]^{e} & F̃M(H)}

commutes.

Proof: Consider

\xymatrix{B(∗, HV, GV) × tV \ar[r]^{≅} & B(∗, HV, GV × tV) \ar[r]^{Bλ} & B(∗, HV, tV) \ar[r] & T HV \\ GV × tV \ar[u]^{τ×1} \ar[rr]^{λ} & & tV \ar[u] \ar[ur]^{e}} ,

where λ is the evaluation. The adjoint of the composite is

HV \GV ≅ B(∗, HV, GV) → Ω^{V} T HV ≅ F̃MHV .
□

If G maps to F or SF, then j̄ maps to F MH or SF MH. 3.4.7 We consider now H → G →^{j} F. Then we have the following diagram (which comes from the corresponding diagram in I_{e}^{∗}[T ]):

\xymatrix{ & M(G; H\G) \ar[dl]^{ϵ} \ar[dd]^{Mj̄} \ar[dr]^{q_{1}} \\ MH &  & MG \\  & M(G; F MH) \ar@{-->}[ul]_{ε} \ar@{-->}[ur]_{q}}

3.4.8 Assume now G = H. Then ϵ and q_{1} are equivalences. We get

\xymatrix{M(G) \ar[r]^{Mj̄∘ϵ^{−1}} \ar[rr]_{id} & M(G, F M(G)) \ar[r]^{q} & MG} ,

i.e. M(G; F M(G)) ≅ M(G) ∨ R for some R ∈ U[S_{e}].

3.5 More Thom spectra

3.5.1 In this Subsection we describe alternative constructions of Thom spectra. We start with the groupvalued I^{∗}-functor BG (we could use F but then we must worry about fibrations versus quasi-fibrations). For V ∈ I^{∗}(ℝ^{∞}) we have the universal spherical bundle ξ(V) → BGV which comes with a natural section. If Y → BG is a map of I^{∗}-functors then we can define the I^{∗}-prefunctor M(f) by the following prescription. We define M(f)V := Y ×_{BO(V)} ξ(V)/Y V, where we consider Y V ⊂ f^{∗}ξ(V) via the section. The base point of Y induces the unit e : t → M(f), and the paring Y (V) × Y (W) → Y (V ⊕ W) (which is preserved by f) together with ξ(V) ∧_{fibrewise} ξ(W) → ξ(V ⊕ W) induces ω : M(f) ∧ M(f) → M(f).

Definition 3.8 We call (M(f), ω, e) and the L-spectrum spectrum derived from it the Thom spectrum associated to f.

3.5.2 We now present a very general construction. Let f : Y → BG be just a map. The we still can construct just a spectrum M(f) as follows. For each V we have BGV ⊂ BGℝ^{∞} and we define Y V := f^{−1}(BGV). Then we define T (f)V := f_{|Y V}^{∗} ξ(V)/Y V. The structure map is induced by the obvious map

\xymatrix{f_{|Y V}^{∗} ξ(V) × tW \ar[r] & ξ(V) × tW \ar[r] & ξ(V ⊕ W)} .

In this way we get a prespectrum (T (f), σ) ∈ P.

Definition 3.9 The associated spectrum (M(f), σ) ∈ S is called the Thom spectrum associated to f.

Actually this construction can be extended to a functor M : T /BG → S. On can show:

Proposition 3.10 ([21], Cor. IX.4.7) The functor M descends to hT /BG → hS.

Proposition 3.11 ([21], Prop. IX.4.10) The equivalence class of M(f) in HS only depends on the homotopy class of Y → G → F.

3.5.3 Using the formalism of parametrized spectra one can redo the construction given in 3.5.2 and shift the complicated verifications into the formlism. We start with the canonical parametrized spectrum M/BF ∈ HS/BF (the stable homotopy category paraterized over B [22]) with fibre Q^{∞}S^{0}. Given f : Y → B we define the spectrum M(f) := c_{!}(f^{∗}M) ∈ HS, where c : Y → ∗ and both operations are taken in the derived sense. It follows from the homotopy invariance properties of f^{∗} and c_{!} verified in [22] that M(f) only depends on the homotopy class of Y/BF. 3.5.4 Orientation theory fits into this formalism as follows. Let E be a ring spectrum and assume that

\xymatrix{Y \ar[r]^{f} & BF \ar[r]^{Be} & BF E}

is homotopic to a constant map. The homotopy induces an isomorphism f^{∗}Be^{∗}E ≅ Y ×E. Let E/BF E be the tautological parametrized spectrum. The unit induces a map e : M → Be^{∗}E. Note that c_{!}(Y × E) ≅ Y^{+} ∧ E. The homotopy induces an isomorphism f^{∗}Be^{∗}E ≅ Y × E. We obtain a map

\xymatrix{M(f) \ar[r]^{c_{!}(f^{∗}Be)} & c_{!}(f^{∗}Be^{∗}E) \ar[r]^{≅} & Y^{+} ∧ E \ar[r]^{^{pr^{2}}} & E} .

At the moment I do not see how to deal nicely with ring structures within this formalism. 3.5.5 We continue to consider f : Y → BG and the associated Thom spectrum M(f). Assume that Y is a grouplike H-space with a homotopy unit, and that f strictly preserves multiplication and unit.

Lemma 3.12 Let Y be a H-space with a homotopy unity and assume that f commutes with unit and multiplication up to homotopy then we can replace it by an homotopy equivalent H-map f′ : Y′ → BG which satifies the strict assumptions above.

Proof: See [21, page 431]. □

We have the diagram

\xymatrix{Y × \{e\} \ar[r]^{1×e} \ar[dr]^{≅} & Y × Y \ar[d]^{+} \ar[r]^{f×f} & BG × BG \ar[d]^{+} \\ & Y \ar[r]^{f} & BG} .

(the triangle commutes up to homotopy, the square commutes). We apply the functor M and get

\xymatrix{M(f) ∧ S \ar[r]^{id∧Me} \ar[dr]^{=} & M(f) ∧ M(f) \ar[d]^{M+} \\  & M(f)}

in HS. Here we use the rule M(f × g) = M(f) ∧ M(g) [21, XI, Prop. 3.6]. This provides a ring spectrum structure on M(f). In order to lift this to an E_{∞}-structure we need more structure on f. 3.5.6 Assume that Y → BG is a morphism in L[T_{e}]. Then M(f) beomes an L-spectrum in a natural way. Then base point of Y induces the unit Σ_{e}^{∞}S^{0} → M(f). This generalizes the construction given in 3.4.

3.6 E_{∞}-ring spaces and spectra

3.6.1 In this subsection we present another machine which constructs E_{∞}-ring spectra from categorial data. It is well suited for K-theoretic constructions. 3.6.2 Let C be an operad in T . Let e : S^{0} → X be in T_{e}. A C_{0}-space is a C-space (X, ξ) such that ξ_{j}(g, x_{1}, …, x_{j}) = 0 if x_{i} = 0 (0 is the image of the base point of S^{0}) for one i. Let C_{0}[T_{e}] denote the category of C_{0}-spaces. A C_{0}-space gives rise to a monad C in T_{e}. Let now D be a second operad. There is a notion of an action of D on C such that C will be a monad in D[T_{e}]. We will then call (C, D) an operad pair. The details can be found in [20, VII.2]. Let X be a C-space in D[T_{e}]. If we then apply the delooping theory as in 2.4 we obtain a prespectrum T X ∈ D[P_{e}]. Here is an example. Let M be the monoid operad, M(j) = Σ_{j}. Then (M, M) is an operad pair [20, Lemma 2.3]. Applying the translation space functor we obtain the E_{∞}-operad (Q, Q) with Q(j) = EΣ_{j}. This looks to nice and is actually wrong. It has been corrected in [19] to the price of a new but correctable mistake. The modern approach by [10] corrects the mistakes and superceeds the old work. 3.6.3 A symmetric monoidal category A is permutative if the unit and the associativity is strict. Its nerve NA is then a simplicial M-set. Hence its realization |NA| is a Q-space. There exists a functor Φ from symmetric monoidal categories to permutative categories and a natural equivalence B → ΦB (in symmetric monoidal categories). This allows us to associate to each symmetric monoidal category an infinite loop space. A bipermutative category is a pair of permutative categories (A, ⊕, 0, c) and (A, ⊗, 1, c̃) (c, c̃ implement commutativity) such that

1. 0 is a strict two-sided zero object for ⊗.

2. The right distributive law is satisfied strictly and is compatible with the additive commutativity law.

3. There is some compatibility of c and c̃ (see [20, Def. 3.3 ])

This definition was extended in [10]. In that paper the notion of a multi-category was introduced. In such a category there are morphisms with several inputs and one output. The symmetric group acts on the morphisms and there are composition rules which look like the rules for operad. In fact, an operad is a multicategory with one object. A small bipermutative category gives a ring object in the multicategory P of small permutative categories. Each symmetric monoidal category gives rise to a multicategory. So are the symmetric spectra S. In [10] a multifunctor M : P → S was constructed. Therefore bipermutative categories C ∈ P give rings MC in S. Strict commutativity of rings in P does not make much sense. Commutativity is described using operads. In the E_{∞}-case MC is an E_{∞}-ring spectrum. It is further shown that such a spectrum is weakly equivalent to s strictly commutative ring in S. These constructions replace the ones of [20] whih are based on wrong principles.

There exists a functor Φ from symmetric bi-monoidal categories to bi-permutative categories and a natural equivalence B → ΦB (in symmetric bimonoidal categories). The point now is that the nerve NA of a bipermutative category is a (M, M) simplicial space, and that its realization is a (Q, Q)-space. Combining these construction we can funtorially associate to a symmetric bi-monoidal category A a (Q, Q)-space BA.

3.6.4 Let A be a commutative topological semiring. Then it is a (N , N)-space in a natural way. Geometric realization and delooping produces a Q-ring spectrum. If A is a ring, then the result is the realization of the Eilenberg-MacLane spectrum HA as a Q-spectrum. This construction works because of strict commutativity. Here the mistake of May does not play a role. Further bipermutative categories of K-theoretic relevance will be constructed in 5.8.

4 Units and orientation theory

4.1 Units

4.1.1 Let U be an E_{∞}-operad. We consider a U-spectrum E ∈ U[S_{e}]. The zeroth space E_{0} is an U-space and π_{0}E_{0} is a commutative monoid.

Definition 4.1 We define the U-subspaces SF E ⊂ F E ⊂ E_{0} given by the components of the identity and the set of components which represents invertible elements in π_{0}E_{0}.

4.1.2 By the results stated in 2.4 there exists a connective spectra T SF E, T F E together with a weak equivalences F E ≅ T F E_{0}, SF E ≅ T SF E_{0}. In particular, F E, SF E are infinite loop spaces.

Definition 4.2 The spectrum T F E is called the spectrum of units of E.

4.1.3 We have F E ⊂ E_{0} ≅ Ω^{V} EV. Therefore we obtain a right-action of GV on F E via precomposition for all V ∈ I^{∗}(ℝ^{∞}).

4.2 E-oriented bordism

4.2.1 We consider a map of I^{∗}-monoids G → F̃.

Lemma 4.3 The following definitions define a U-prespectrum T (G; F E):

1. T (G; F E)V := B(F E, GV, tV)/B(F E, GV, ∗), V ∈ I^{∗}(ℝ^{∞}) (Note that F E is a right GV -space by 4.1.3).

2. T (G; F E)f : T (G; F E)V → T (G; F E)V′ is induced by B(id, id, tf), f : V → V′ a morphism in I^{∗}(ℝ^{∞})

3. e : tV → T (G; F E)V is induced by tV → B(F E, G, tV) given by the base point of F E.

4. σ : T (G; F E)V ∧ tV → T (G; F E)(V ⊕ W) is induced by

\xymatrix{B(F E, G, tV) × tW \ar[r]^{≅} & B(F E, G, tV × tW) \ar[r]^{B(id,i,ω)} & B(F E, G(V ⊕ W), t(V ⊕ W))} ,

V ⊥ W

5. ξ_{j}(g) : T (G; F E)V_{1} ∧ … T (G; F E)V_{j} → T (G; F E)(V_{1} ⊕ ⋯ ⊕ V_{j}) is induced by

\xymatrix{B(F E, G, tV_{1}) × ⋯ × B(F E, G, tV_{j}) \ar[d]_{≅} \\ B(F E^{j}, GV_{1} × ⋯ × GV_{j}, tV_{1} × ⋯ × tV_{j}) \ar[d]_{B(ξ_{j}(g),Gg∘ω,tg∘ω)} \\ B(F E, Gg(V_{1} ⊕ ⋯ ⊕ V_{j}), tg(V_{1} ⊕ ⋯ ⊕ V_{j}))} .

The proof is by straight-forward verification. 4.2.2 We write M(G; F E) := Ω^{∞}T (G; F E) ∈ U[S_{e}]. 4.2.3 If E arrises from a I^{∗}-prefunctor M, then the definition given in Lemma 4.3 and Definition 3.6 are equivalent. 4.2.4 We have maps in U[S_{e}].

\xymatrix{M(G) & M(G; F E) \ar[l]_{q} \ar[r]^{ε} & E} .

4.2.5 Definition 4.4 An E-orientation of a Thom spectrum is given by a map γ : M(G) → E in U[S_{e}]. Lemma 4.5 If MG is E-oriented then we can split M(G; F E) ≅ M(G) ⊕ R for some R ∈ U[S_{e}]. Proof: We consider the diagram

\xymatrix{M(G) \ar[dd]_{γ} & M(G, F M(G)) \ar[l]^{ε_{1}} \ar[ddr]^{M(γ)} \ar[r]^{≅} & M(G) ⊕ R \ar[dr]^{q} \ar[dd]^{u} \\ & & & M(G) \\ E & & M(G; F E) \ar[ll]^{ε} \ar[ur]_{q}} .

The composition q ∘ u ∘ (ε_{1})_{|M(G)}^{−1} is the identity. □

4.3 Orientations of spherical fibrations

4.3.1 Consider a morphism of I^{∗}-monoids G → F → F̃. We have the unit Q_{∞}T {1}1 → E which restricts to a map of U-spaces F = F Q_{∞}T {1} → F E (see ??). Let e : G → F E be induced. 4.3.2 We consider E ∈ U[S_{e}]. We have a fibre sequence in U[T_{e}]

\xymatrix{F E \ar[r]^{τ} & B(G; F E) \ar[r]^{q} & BG}

which extends to

\xymatrix{G \ar[r]^{e} & F E \ar[r]^{τ} & B(G; F E) \ar[r]^{q} & BG \ar[r]^{Be} & BF E} .

It gives rise to a sequence in the stable homotopy category

\xymatrix{Ω^{∞}T G \ar[r]^{e} & Ω^{∞}T F E \ar[r]^{τ} & Ω^{∞}T B(G; F E) \ar[r]^{q} & Ω^{∞}T BG \ar[r]^{Be} & Ω^{∞}T BF E} .

As in the proof of Lemma 2.10 one shows: Lemma 4.6 This sequence is a fibration sequence. 4.3.3 The space BG classifies spherical bundles with structure group G. Definition 4.7 An E-orientation of a spherical bundle ξ : W → X classified by ξ : X → BG is given by a lift f̃ in

\xymatrix{ & B(G; F E) \ar[d]^{q} \\ X \ar[ur]^{f̃} \ar[r]^{f} & BG} .

4.3.4 Definition 4.8 The map Be : BG → BF E represents the universal E-theory Stifel-Whitney class w(G, E) ∈ T F E^{1}(BG). The E-theory Stiefel-Whitney class of ξ is define by w(ξ, E) := ξ^{∗}w(G, E) ∈ T F E^{1}(X).

4.3.5

Lemma 4.9 The bundle ξ admits an E-orientation if and only if w(ξ, E) = 0. If w(ξ, E) = 0, then the set of orientations is a torsor over T F E^{0}(X)/T G^{0}(X).

4.4 E-orientations of groups and orientations of Thom spectra

4.4.1 Let E ∈ U[S_{e}] and G → F a map of I^{∗}-monoids.

Definition 4.10 An E-orientation of G is a H-map

\xymatrix{BG \ar[r]^{g} & B(G; F E)}

such that q ∘ g ∼ id. If g ≅ g̃_{0} for a map g̃ : Ω^{∞}T BG → Ω^{∞}T B(G, E) and

\xymatrix{Ω^{∞}T BG \ar[rr]_{id} \ar[r]^{g̃} & Ω^{∞}T B(G; F E) \ar[r]^{q} & Ω^{∞}T BG}

holds in HS we call g perfect.

It follows from the sequence 4.3.2 that G admits a perfect E-orientation if and only if e : G → F E is the trivial infinite loop map. 4.4.2 Consider the case E = Hℤ and G = O. Then F Hℤ ≅ {1, −1} and e = det. We see that O does not admit an Hℤ-orientation. On the other hand, SO admits such an orientation. 4.4.3

Lemma 4.11 If g : BG → B(G; E) is a (perfect) E-orientation of G, then

\xymatrix{F E × BG \ar[r]^{τ×g} & B(G; E) × B(G; E) \ar[r]^{+} & B(G; E)}

is an equivalence of (infinite loop) H-spaces.

Proof: Follows from the quasi-fibration sequence

\xymatrix{G \ar[r]^{e} & F E \ar[r]^{τ} & B(G; E) \ar[r]^{q} & BG} .

□

4.4.4

Lemma 4.12 A morphism of (E_{∞}-) ring spectra γ : M(G) → E induces a (perfect) E-orientation g : BG → B(G; F E) in a natural way.

Proof: Let j̄ : GV \GV → F MGV as in Lemma 3.7 with H = G. We consider the following diagram in U[T ].

\xymatrix{B(G\G, G, ∗) \ar[r]^{Bj} \ar[d]^{ϵ} & B(F MG, G, ∗) \ar[r]^{Bγ} & B(F E, G, ∗) \ar[d]^{q} \\ BG \ar@{.>}[urr]^{g} \ar[rr]^{=} &  & BG}

Now G/G ≅ ∗ in U[T ] so that ϵ is a homotopy equivalence and we can fill in g := Bγ ∘ Bj ∘ ϵ^{−1} for some choice of the homotopy inverse such that the diagram homotopy commutes. This yields the perfect E-orientation g. □

4.4.5 We now discuss the converse of Lemma 4.12. Let an E-orientation g : BG → B(G; F E) be given. We are looking for an induced γ : M(G) → E. Assume that g is given as a limit of gV : BGV → B(F E, GV, ∗). Then we consider the diagram

\xymatrix{B(∗, GV, tV) \ar@{.>}[r] \ar[d] & B(F E, GV, tV) \ar[d] \ar[r] & B(∗, GV, tV) \ar[d] \\ B(∗, GV, ∗) \ar[r]^{gV} & B(F E, GV, ∗) \ar[r]^{qV} & B(∗, GV, ∗)}

where the dottet arrow exists because of qV ∘ gV ∼ id. Going to the limit over V this induces the map of Thom spectra T (G) → T (G; E) → E. Since g is a H-map, we in fact obtain a map of ring spectra in the stable homotopy category. In [20] it is an open problem if there is a map γ : M(G) → E of E_{∞}-ring spectra if g is perfect. It looks so that if q ∘ g = id (no homotopy), then things work out.

4.5 The Atiyah-Bott-Shapiro orientation

4.5.1 Here we sketch the construction of E_{∞}-ring spectrum maps M(Spin) → KO and M(Spin^{c}) → KU. This yields by Lemma 4.12 a perfect kO-orientation of Spin and a perfect kU-orientation of Spin^{c}. The construction is based on explicite models. Actually we construct the map on the level of I^{∗}-prefunctors for which we use the same notation as for the associated spectra. We give the details in the Spin-case. 4.5.2 First we construct another model for MSpin. Let V ∈ I^{∗}(ℝ^{∞}). Then we form the Cliff(V)-right module H(V) := L^{2}(V ⊗ Cliff(V)) and let O(V) ⊂ O(H(V)) be the subgroup of Cliff(V)-equivariant orthogonal operators. Observe that O(V) is contractible. We have H(V ⊕ W) ≅ H(V) ⊗ H(W) as Cliff(V ⊕ W) ≅ Cliff(V) ⊗ Cliff(W)-modules. This gives an embedding O(V) × O(W) ⊂ O(V ⊕ W). The group Spin(V) acts on H(V) via its induced action on L^{2}(V) (by Spin(V) → SO(V)) and left multiplication on Cliff(V). We obtain an embedding Spin(V) ⊂ O(V). The following diagram commutes:

\xymatrix{Spin(V) × Spin(W) \ar[d] \ar[r] & Spin(V ⊕ W) \ar[d] \\ O(V) × O(W) \ar[r] & O(V ⊕ W)} .

Observe that O(V) is contractible. We set BSpin(V) := O(V)/Spin(V). It comes with a map BSpin(V)× BSpin(W) → BSpin(V ⊕ W). We thus have a model of BSpin as I^{∗}-functor. We let ξ(V) := O(V) ×_{Spin(V)} tV → BSpin(V) be the universal bundle and define MSpin(V) := ξ(V)/BSpin(V) to be its Thom space. The base point [Spin(V)] ⊂ BSpin(V) determines the unit e : tV → MSpin(V), and we have a natural pairing ω : MSpin(V) ∧ MSpin(W) → MSpin(V ⊕ W). In this way we get the I^{∗}-prefunctor (MSpin, ω, e). 4.5.3 We now describe KO as a I^{∗}-prefunctor. Here we use the unbounded Fredholm picture. It associates to V the one-point compactification of a certain space of unbounded Cliff(V)-equivariant Fredholm operators F (V) on H(V) which we denote by KO(V). The tensor product induces a map F (V) × F (W) → F (V ⊕ W) which extends by continuity to ω : K0(V) ∧ KO(W) → KO(V ⊕ W). The unit is given by the continuous continuation of tV ↦ F (V) which maps v ∈ V to D(v) := D + v, where D is the Dirac operator on L^{2}(V) ⊗ Cliff(V) and v is the left Clifford multiplication by the base point. One checks compatibility with the pairing. Of course, here one has to do a lot of analysis in order to check details. Eventually we get the I^{∗}-prefunctor (KO, ω, e). 4.5.4 We can now describe a map of I^{∗}-prefunctors MSpin → KO. We first define the map O(n)×V → F (V) by (g, v) ↦ gD(v)g^{−1}. On checks Spin(V)-equivariance, i.e. for s ∈ Spin(V) that (gs, s^{−1}v) = gsD(s^{−1}v)s^{−1}g^{−1} = gs(s^{−1}D(v)s)s^{−1}g^{−1} = gD(v)g^{−1}). Therefore we get a map ξ(V) → F (V). This map extends by continuity to MSpin(V) → KO(V). It is easy to see that it preserves the unit and the pairing ω.

4.5.5 For the details we refer to [12] and [13].

4.6 The comparison diagram

4.6.1 For simplicity we will omit the construct Ω^{∞}T in the following. Equivalenty we swith from talking about connective spectra to talking about infinite loop spaces. Consider the map of I^{∗}-monoids

\xymatrix{H \ar[r]^{i} & G \ar[r]^{j} & F} .

We assume that H is perfectly E-oriented by g : BH → B(H; F E). We write also g for the induced map

\xymatrix{BH \ar[r]^{g} & B(H; F E) \ar[r]^{Bi} & B(G; F E)} .

We recall the fibration sequences in HS from 2.10 and 4.3.2 which we put in the following homotopy commutative diagram

\xymatrix{G \ar@{.>}[d]^{χ} \ar[r]^{τ} & G/H \ar@{.>}[d]^{f} \ar[r]^{q} & BH \ar[d]^{g} \ar[r]^{Bj} & BG \ar[d]^{=} \ar[r]^{Bτ} & BG/H \ar@{.>}[d]^{Bf} \\ G \ar[r]^{e} & F E \ar[r]^{τ} & B(G; F E) \ar[r]^{q} & BG \ar[r]^{Be} & BF E} .

We have two maps (see Lemma 4.11)

\xymatrix{G/H \ar[rr]_{Be} \ar[r]^{q} & BH \ar[r]^{g} & B(H; F E) \ar[rr]_{q} \ar[r] & BH × F E \ar[r]^{^{pr^{1}}} & BH} .

We have q ∘ g ∘ q = q = q ∘ Be as infinite loop maps. Therefore

\xymatrix{G/H \ar[r]^{gq/Be} & B(H; F E) \ar[r]^{Bi} & B(G; F E) \ar[r]^{q} & BG}

is the trivial infinite loop map and we have a factorization f as an infinite loop map in view of Lemma 4.11. We can now fill in BF. Finally we fill in χ. □

If G → SF the we have an analogous diagram

\xymatrix{G \ar@{.>}[d]^{χ} \ar[r]^{τ} & G/H \ar@{.>}[d]^{f} \ar[r]^{q} & BH \ar[d]^{g} \ar[r]^{Bj} & BG \ar[d]^{=} \ar[r]^{Bτ} & BG/H \ar@{.>}[d]^{Bf} \\ G \ar[r]^{e} & SF E \ar[r]^{τ} & B(G; SF E) \ar[r]^{q} & BG \ar[r]^{Be} & BSF E} .  (2)

4.7 Examples

4.7.1 We consider the case G = O and E = Hℤ. Then E_{0} ≅ ℤ and F E ≅ {1, −1} has two points. We have e = det : O → {1, −1}. The corresponding spectrum is the Eilenberg-MacLane spectrum T F E = Hℤ/2ℤ. The universal Stiefel-Whitney class is the class w(O, Hℤ) ∈ H^{1}(BO, ℤ/2ℤ) represented by B det : BO → Hℤ/2ℤ. This is the classical result. 4.7.2 Since kO is connective we have SF kO^{0} = kO = BO_{⊗}. On BO we have two infinite loop space structures (one from the spectrum ko and one from the ring spectrum structure). We write BO_{⊗} for BO with the latter multiplicative structure.

Lemma 4.13 We have a splitting as infinite loop spaces BO_{⊗} ≅ BO(1) × BSO_{⊗}.

Proof: By 2.5.5 and 2.8.4 w_{1} = B det : BO → BO(1) ≅ Hℤ/2ℤ is an infinite loop map. It is claimed in [20, V.3.1] that this is also an infinite loop map BO_{⊗} → BO(1)^{1}. We have another map η : BO(1) → BO ⊂ BO × ℤ which classifies the element ξ − \underline{ℝ} ∈ K(BO(1)), where ξ is the universal one-dimensional linebundle. It is claimed that this is again an infinite loop map^{2}. It is then further claimed that the infinite loop space structure on Hℤ/2ℤ is unique. It follows that

\xymatrix{BO(1) \ar[r]^{η} & BO_{⊗} \ar[r]^{w_{1}} & BO(1)}

splits BO_{⊗} as an infinite loop space. □

4.7.3 In asimilar manner one shows:

Lemma 4.14 We have a splitting as infinite loop spaces BU_{⊗} ≅ BU(1) × BSU_{⊗}.

4.7.4 Let us consider the case G = Spin and E = kO. We have the Atiyah-Bott-Shapiro orientations

g : BSpin → B(BSpin; SF kO) ≅ BSpin × SF kO

and

g : BSpin^{c} → B(BSpin^{c}; SF kU) ≅ BSpin^{c} × SF kU .

It is claimed that these are perfect^{3}. 4.7.5 The Atiyah-Bott-Shapiro orientations induce the diagrams

\xymatrix{BSpin \ar[r]^{g} \ar[rr]_{Bi} & B(SO, SF kO) \ar[r]^{q} & BSO}

\xymatrix{BSpin^{c} \ar[r]^{g} \ar[rr]_{Bi} & B(SO, SF kU) \ar[r]^{q} & BSO}

Lemma 4.15 The following compositions are equivalences of infinite loop spaces.

\xymatrix{BSO_{⊗} × BSpin \ar[r]^{τ×g} & B(SO, SF kO) × B(SO, SF kO) \ar[r]^{+} & B(SO, SF kO)}

\xymatrix{BSU_{⊗} × BSpin^{c} \ar[r]^{τ×g} & B(SO, SF kU) × B(SO, SF kU) \ar[r]^{+} & B(SO, SF kU)}

Proof: We set up the following diagram (note that SO/Spin ≅ BO(1))

\xymatrix{SO/Spin \ar[r]^{=} \ar[d] & SO/Spin \ar[d]^{(q,∗,−f)} \\ B(Spin, SF kO) \ar[d] & BSpin × BSO_{⊗} × BO(1) \ar[l]^{≅g+τ} \ar[d]^{a} \\ B(SO, SF kO) \ar[d] & BSpin × BSO_{⊗} \ar[l]^{g+τ} \ar[d] \\ B(SO/Spin) \ar[r]^{=} & B(SO/Spin)} .

1I do not see this easily!

2I only see a H-map at the moment. I do not have a good picture of the infinite loop space structure on BO_{⊗}

3I do not see this

The calculation of the cofibre a follows from the fact that f : SO/Spin → BO(1) is an isomorphism (look at the π_{1}-level of

\xymatrix{SO \ar[r]^{τ} \ar[rr]_{e∘χ} & SO/Spin \ar[r]^{f} & BO_{⊗} \ar[r]^{≅} & BSO_{⊗} × BO(1)}

given by the first square in (2) and use that e_{∗} : π_{1}(SO) → π_{1}(BO_{⊗}) is not trivial ). The existence of the upper square follows from the second square in the comparison diagram (2). For the complex case the argument is similar. □

4.7.6 The splitting B(Spin, SF kO) ≅ BSpin × BSO_{⊗} has the follwing consequence (the only if part).

Corollary 4.16 An oriented spherical bundle admits a kO-orientation if and only if it admits a Spinstructure. A Spin-structure gives a distinguished kO-orientation. A similar statement holds in the complex case for Spin^{c} and kU.

4.8 Cannibalistic classes

4.8.1 We consider a map G → F of monoid valued I^{∗}-functors. Let E ∈ U[S_{e}] be an E_{∞}-ring spectrum and consider a map ψ : E → E in of rings in HS. It induces the map ψ : F E → F E of H-spaces. Then we have the following homotoy commutative diagram (we write the H-space structure of F E and related operations multiplicatively).

\xymatrix{F E \ar[r]^{ψ/id} \ar[d]^{τ} & F E \ar[d]^{τ} \\ B(G, F E) \ar[r]^{Bψ/id} \ar[d]^{q} \ar@{.>}[ur]^{c(ψ)} & B(G, F E) \ar[d]^{q} \\ BG \ar[r]^{=} & BG} .

Definition 4.17 The class c(ψ) : B(G; F E) → F E is called the universal cannibalistic class associated to ψ.

4.8.2 If ψ : F E → F E is a map of U-spaces (e.g. if ψ refines to a map in U[S_{e}]) then this is a diagram of infinite loop spaces. 4.8.3 We assume that we have a perfect E orientation g : BG → B(G; F E) of E (somtimes we consider G → SF and BG → B(G; SF E)). Then c(ψ) = 0.

5 J-theory

5.1 Adams operations and related materal

5.1.1 The Adams operations on K^{0}(X) and KO^{0}(X) are derived from the λ-ring structure. Note that by the splitting principle every operation on K^{0} is determined by its action on one-dimensional (and two-dimensional in the real case) bundles. Let V ect(X) denote the semiring of vector bundles over X, a finite CW -complex. We first define λ_{t} : V ect(X) → V ect(X)[t] by λ_{t}(V) := ∑_{i=0}^{∞} Λ^{i}(V)t^{i}. We have

1. λ_{t}(1) = 1

2. λ_{t}(V + W) = λ_{t}(V) ⊗ λ_{t}(W)

3. λ_{t}(L) = 1 + tL, if dim(L) = 1.

We then observe that they extend to K^{0}(X). We the define the operations γ_{t}(V) = λ_{t/1−t}(V) and write γ_{t}(V) = ∑_{i=0}^{∞} γ^{i}(V)t^{i} ∈ K(X)[[t]]. We have 1. γ_{t}(1) = 1

2. γ_{t}(L) = 1 + t(L − 1). Let Op(K̃^{0}, K^{0}) denote the operations from reduced K-theory to K-theory (on finite pointed connected CW -complexes). Pointwise addition and multiplication makes operations into a ring. Proposition 5.1 (see Thm. 3.1.7 [5]) We have an isomorphism of rings

ℤ[γ^{1}, γ^{2}, …] →^{∼} Op(K̃^{0}, K^{0}) .

We define the Adams operations by

ψ_{t}(V) := ψ^{0}(X) − t\frac{d}{dt} log λ_{−t}(V) ∈ K(X)[[t]]

and write

ψ_{t}(V) = ∑_{i=0}^{∞} t^{i}ψ^{i}(V) .

Proposition 5.2 (see Prop. 3.2.1, 3.2.2[5]) We have 1. ψ^{k}(V + W) = ψ^{k}(V) + ψ^{k}(W),

2. ψ^{k}(V W) = ψ^{k}(V)ψ^{k}(W),

3. ψ^{k}(L) = L^{k} for a line bundle,

4. ψ^{k}(ψ^{l}(V)) = ψ^{kl}(V),

5. ψ^{p}(V) = V^{p} + p …,

6. ψ^{k}(U) = k^{n}U for the generator of K̃^{0}(S^{2n}). The multiplicative properties follow from the speciality of the λ-operations as discussed in [6] 5.1.2 Recall that the homotopy groups of BO are periodic and are given by

0 1  2 3 4 5 6 7 8
0 ℤ/2ℤ ℤ/2ℤ 0 ℤ/2ℤ 0 0 0 ℤ
1 γ γ^{2} y b

where we have given names to the generators. Using these generators we can calculated the action of ψ^{k}.

Lemma 5.3 We have

\begin{matrix}x & γ & γ^{2} & y & b \\ ψ^{k}(x); k odd & γ & γ^{2} & k^{2}y & k^{4}b \\ ψ^{k}(x); k even & 0 & 0 & k^{2}y & k^{4}b \end{matrix}

Proof: The result on y and b follows from the complex case which can be reduced to line bundles. Let us consider γ ∈ K̃O^{0}(S^{1}). It is given by the bundle γ = (L − 1). We have ψ^{k}(L − 1) = L^{k} − 1 = L − 1. This implies the result. The result for γ^{2} follows by squaring. □ Using the Lemma and the multiplicativity of ψ^{k} one can calculate the action of ψ^{k} on all of π_{∗}(BO). 5.1.3 We consider the restriction

\xymatrix{Ψ_{n}^{k} : BO(n) \ar[r] & BO \ar[r]^{ψ^{k}} & BO}

as an element of KO(BO(n)) ≅ R̂O(O(n))_{I}, where I ⊂ RO(O(n)) is the dimension ideal.

5.1.4 We have defined the Adams operations on the K^{0}-level. We now discuss their extension to stable operations. Bott periodicity (in the real case) is an isomorphism b : BO × ℤ → Ω^{8}(BO × ℤ). We call the pair (BO × ℤ, b) a periodic space. The category of periodic spaces is equivalent to the a category of periodic spectra. We have k^{4}b ∘ ψ^{k} = Ω^{8}ψ^{k} ∘ b. Let KO^{n} := Ω^{−n}(BO × ℤ). We can define a stable transformation by

Ψ^{k} : KO^{8n} → KO^{8n}

as Ψ^{k} := k^{−4n}Ω^{8n}ψ^{k} : KO^{8n}[1/k] → KO^{8n}[1/k]. In this way we define the Adams operation Ψ^{k} on KO[1/k] in the stable homotopy category. There is a functorial construction of connective covers [11]. Therefore we get an induced Adams operation Ψ^{k} : kO[1/k] → kO[1/k]. It is multiplicative in the weak sense. But actually we need a better result. We want that Ψ^{k} is a map of E_{∞}-ring spectra. This would imply that its restriction to F KO is an infinite loop map. 5.1.5 Let X_{(p)} denote the localization of X at p. If (p, k) = 1, then we consider the restriction of the Adams operations ψ^{k} : BSO_{⊗,(p)} → BSO_{⊗,(p)} (and in the complex case (ψ^{k} : BSU_{⊗,(p)} → BSU_{⊗,(p)}).

Proposition 5.4 ([14][Theorem 4.5) ] These operations are infinite loop maps.

5.1.6 We now use 4.8.2.

Corollary 5.5 The cannibalistic class c(ψ^{k}) : B(SF, BSO_{⊗,p}) → BSO_{⊗,p} is an infinite loop map.

5.2 Calculation of cannibalistic classes

Lemma 5.6 We have splittings BO ≅ BSO × BO(1) and BSO ≅ BSpin × B^{2}O(1).

Proof: We consider the sequence

\xymatrix{O(1) \ar[r] & BSO \ar[r] & BO \ar[r]^{π} & BO(1)} .

The inclusion BO(1) → BO is a left-inverse of π and induces the splitting. We further consider the sequence

\xymatrix{BO(1) \ar[r] & BSpin \ar[r] & BSO \ar[r] & B^{2}O(1)} .

The space B^{2}O(2) ≅ K(ℤ/2ℤ, 2) carries a complex line bundle L with c_{1}(L) ∈ H^{2}(B^{2}O(1), Z) ≅ ℤ/2ℤ a generator. The underlying real line bundle L_{ℝ} is oriented and has a non-trivial second Stiefel-Whitney class w_{2}(L_{ℝ}) ∈ H^{2}(B^{2}O(1), ℤ/2ℤ) ≅ H^{2}(B^{2}O(1), ℤ) ×_{ℤ} ℤ/2ℤ which is the reduction of c_{1}(L). The classifying map B^{2}O(1) → BSO of L_{ℝ} is non-trivial on π_{2} and induces the split. □

5.2.1 By these splittings we see that BSpin → BO is the two-connective cover. In particular we now have well-defined generators of π_{∗}(BSpin) given by 5.1.2 in all degrees ≥ 3, namely those which are mapped to the correponding generators of π_{∗}(BO). We consider the composition

\xymatrix{σ^{k} : BSpin \ar[r]^{g} & B(SF, BO_{⊗}) \ar[r]^{c(Ψ^{k})} & BO_{⊗}} .

We have an isomorphism of spaces (not respecting the infinite loop space structures)

\xymatrix{BO \ar[r]^{x↦1+x} & BO_{⊗}} .

The generators of the homotopy groups of BO yield generators of the homotopy groups of BO_{⊗}. It is now a well-defined problem to calculate the effect of σ^{k} on homotopy groups.

5.2.2 We can interpret σ^{k} ∈ KO^{0}(BSpin) such that dim(σ^{k}) = 1. The fact that σ^{k} is an H-map translates to

+^{∗}σ^{k} = σ^{k} ⊠ σ^{k} ∈ KO(BSpin × BSpin) ,

where + : BSpin × BSpin → BSpin. Now note that KO(BSpin(n)) ≅ R̂O(Spin(n))_{I_{n}}, the completion of RO(Spin(n)) at the dimension ideal I_{n}, and that we have a natural map

KO(BSpin) → lim_{←} R̂O(Spin(n))_{I_{n}}^{4} .

Therefore σ^{k} gives rise to a compatible sequence (σ_{n}^{k}). In terms of the representation ring the multiplicative property reads

(σ_{n+m}^{k})_{|Spin(n)×Spin(m)} ≅ σ_{n}^{k} ⊗ σ_{m}^{k} .

5.2.3 For the moment let us neglect two-torsion. Then we can calculate in the complex representation ring. We restrict to even dimensions. Let U(1)^{n} ⊂ Spin(2n) be the standard maximal torus. It follows from the multiplicativity of the sequence (ρ_{n}^{k}) that its restriction to the torus is determined σ_{1}^{k} ∈ R̂U(1)_{I}^{ℤ/2ℤ} ≅ ℤ̂[x, x^{−1}]_{I}^{ℤ/2ℤ}. The generator of the Weyl group ℤ/2ℤ interchanges x and x^{−1}, so that we actually consider power series in 2 − (x + x^{−1}). Such power series correspond to characteristic classes. We define

bh(x) := \frac{e^{x/2} − e^{−x/2}}{x} .

Then we have the following results: Let ξ be a real vector bundle with structure group Spin(2n), and let U_{H} and U_{K} be the associated Thom isomorphisms in cohomology and K-theory.

1. We have

bh(ξ) = U_{H}^{−1} ∘ ch ∘ U_{K}(1) .

2. We have

U_{H}^{−1} ∘ ch ∘ U_{K}(x) = bh(ξ) ∪ ch(x) .

3. We have

U_{K}^{−1} ∘ ψ^{k} ∘ U_{K}(x) = ρ_{n}^{k}(ξ) ∪ ψ^{k}(x) .

4. We define ψ_{H}^{k} on H^{∗∗}(…; ℚ) as the multiplication by k^{n} in degree 2n. Then we have

ψ_{H}^{k} ∘ ch = ch ∘ ψ^{k} .

5. We have

U_{H}^{−1} ∘ ψ_{H}^{k} ∘ U_{H}(x) = k^{n}ψ_{H}^{k}(x) .

From this we get the calculation of ρ_{n}^{k}(ξ) as follows

ch(ρ_{n}^{k}(ξ)) = bh^{−1}(ξ) ∪ k^{n}ψ_{H}^{k}bh(ξ).

Indeed

bh(ξ) ∪ ch(ρ_{n}^{k}(ξ)) = (U_{H}^{−1} ∘ ch ∘ U_{K})(ch(ρ_{n}^{k}(ξ)))
= (U_{H}^{−1} ∘ ch ∘ U_{K})(U_{K}^{−1} ∘ ψ^{k} ∘ U_{K})(1)
= (U_{H}^{−1} ∘ ch ∘ ψ^{k} ∘ U_{K})(1)
= (U_{H}^{−1} ∘ ψ_{H}^{k} ∘ ch ∘ U_{K}^{−1})(1)
= (U_{H}^{−1} ∘ ψ_{H}^{k} ∘ U_{H})(U_{H}^{−1} ∘ ch ∘ U_{K}^{−1})(1)
= k^{n}ψ_{H} ∘ bh(ξ)

4Is it an isomorphism ?

It follows that the characteristic class ch(ρ_{n}^{k}(ξ)) is represented by the power series

\frac{e^{kx/2} − e^{−kx/2}}{e^{x/2} − e^{−x/2}} .

We define α_{s} by

log \frac{e^{x/2} − e^{−x/2}}{x} = ∑_{s=1}^{∞} α_{2s} \frac{x^{2s}}{(2s)!}

Then we have

log bh(ξ) = ∑_{s=0}^{∞} \frac{1}{2}α_{2s}ch_{2s}(ξ) .

Note that the Chern character is taken of the complexified real bundle ξ, this accounts for the factor \frac{1}{2}. 5.2.4 This allows us to calculate ρ^{k} on π_{4i}(BSpin). Thus let ξ → S^{4i} be zero-dimensional a virtual Spin-bundle. Then ch(ξ) = ch_{0}(ξ) + ch_{2n}(ξ) It follows that log bh(ξ) = 1 + \frac{1}{2}α_{2i}ch_{2i}(ξ) = bh(ξ). We now calculate

χ(ρ_{n}^{k}(ξ)) = bh(ξ)^{−1} ∪ ψ_{H}(bh(ξ)) = (1 − \frac{1}{2}α_{2i}ch_{2i}(ξ)) ∪ (1 + \frac{1}{2}k^{2i}α_{2i}ch_{2i}(ξ)) = (1 + \frac{1}{2}(k^{2i} − 1)α_{2i}ch_{2i}(ξ))

This implies that ρ^{k} : π_{4i}(BSpin) → BO_{⊗} is multiplication by \frac{1}{2}(k^{2i−1} − 1)α_{2i}. 5.2.5 We now explain the calculation of the action ρ^{k} on π_{8i+1}(BSpin) and π_{8i+2}(BSpin). We start with the calculation of K̃O(ℝP^{n}) ([2, Thm. 7.4]. Let L ∈ KO(ℝP^{n}) be the canonical line bundle. The L^{2} = 1 and λ := L − 1 is a generator of KO(ℝP^{n}) which is cyclic of order 2^{f}, where f is the number of integers s such that 0 < s ≤ n and [s] ∈ {[0], [1], [2], [4]} ⊂ ℤ/8ℤ. Since later we want to localize away from k we are only interested in odd k. The bundle 2λ is orientable, but not spin. The bundle 4λ is spin.

5.3 The Adams conjecture

5.3.1 Let us first state the conjecture. The zeroth stable cohomotopy theory Q_{∞}B of a finite complex B is the Grothendieck group of spherical fibrations Sph(B). For a complex B let K : KO^{0}(X) → Sph(X) the map which associates to a vector bundle its spherical fibration. ON the classifying space level it is given by j : BO → BF.

Theorem 5.7 (Quillen [23], Sullivan, Friedlander) Let B be a finite complex, k ∈ ℕ and x ∈ KO^{0}(B). Then there exists n ∈ ℕ such that k^{n}J(ψ^{k}(x) − x) = 0.

In this Subsection we sketch the proof given by [7]. 5.3.2 The idea is to reduc to low-dimensional vector bundles (one and two-dimensional). Let V → B be a 2n-dimensional vector bundle with a metric chosen. It represents x ∈ KO. Let O(V) denote the frame bundle. Let T ⊂ N ⊂ O(2n) be the maximal torus and its normalizer. The quotient W := N/T is the Weyl group. We let X := O(V)/N and λ : X → B be the natural projection.

Theorem 5.8 λ^{∗} : h(B) → h(X) is an isomorphism for any generalized cohomology theory.

Proof: This is a consequence of the existence of the transfer τ : h(X) → h(B), the calculation τ ∘ λ = χ(O(2n)/N), and the calculation χ(O(2n)/N) = 1. All this will be discussed later. □

5.3.3 The point now (and this was the new point of view in [7]) is that Sph(B) = Q_{∞}^{0}(B) the part of a cohomology theory. KO^{0}(X) is so, too. We consider the diagram

\xymatrix{KO^{0}(X) \ar[r]^{k^{n}J(ψ^{k}(x)−x)} & Sph(X) \\ KO^{0}(B) \ar[u]^{λ^{∗}} \ar[r]^{k^{n}J(ψ^{k}(x)−x)} & Sph(B) \ar[u]^{λ^{∗}}} .

The vertical maps are injective.

5.3.4 We have W = Σ_{n} ∫ O(2). This group has a natural homomorphism W → Σ_{n}. Let Γ ⊂ W be the preimage of Σ_{n−1}. We have Γ ≅ O(2) × Σ_{n−1} ∫ O(2). Let ℝ^{2} be the standard representation of O(2) and ℝ^{2n} be the representation of W via its embedding into O(2n). Then one can check that ℝ^{2n} ≅ ind_{Γ}^{W} ℝ^{2}.

This can be translated to bundles. We consider X̃ := O(V)/Γ →^{ξ} X and the bundle E := O(V) ×_{Γ} ℝ^{2}. Then we have λ^{∗}V ≅ ξ_{∗}E naturally, where the fibre of ξ_{∗}E over x ∈ X is given by ⊕_{x̃∈ξ^{−1}(x)}Ex̃. 5.3.5 We now use the following facts.

1. The transfer ξ_{∗} : KO(X̃) → KO(X) here defined geometrically is the transfer in the cohomology theory and also defined for Sph.

2. The transfer commutes with ψ^{k} and J. This is clear for J. For ψ^{k} we need to invert k. This we can do up to an equivalent reformulation of the Adams conjecture: “J(ψ^{k} − 1) : KO(X)[1/k] → Sph(X)[1/k] is trivial”.

3. The Adams conjecture is true for a two-dimensional bundle, in particular for E.

5.3.6 We describe the Becker-Gottlieb transfer. Let τ : E → B be a bundle with a compact smooth manifold as fibre. We choose a vector bundle V → B and an embedding

\xymatrix{E \ar[rr]^{⊂} \ar[dr] & & V \ar[dl] \\ & B} .

We let U → E be the fibre-wise normal bundle. We can extend the diagram as

[figure: U; ⊂; V; E; ⊂; V; B] .

We compactify fibrewise and get the clutching map B^{V} → E^{U} (for a vector bundle U → X let X^{U} denote the Thom space). Let T → E be the fibre-wise tangent bundle. Then we have T ⊕ U ≅ τ^{∗}V. We obtain the composition

\xymatrix{c : B^{V} \ar[r] & E^{U} \ar[r] & E^{U⊕T} \ar[r]^{≅} & E^{τ^{∗}V}} .

We choose a complement W → B,

\xymatrix{V ⊕ W \ar[dr] &  & B × ℝ^{N} \ar[ll]^{Φ} \ar[dl] \\  & B}

We get

\xymatrix{B^{ℝ^{N}} \ar[r]^{Φ} & B^{V} ∧ B^{W} \ar[r]^{c∧id} & E^{τ^{∗}V} ∧ B^{W} \ar[r]^{≅} & E^{τ^{∗}(V ⊕W)} \ar[r]^{τ^{∗}ϕ^{−1}} & E^{ℝ^{N}}} .

If interpretet as a map τ_{∗} : Σ^{∞}B_{+} → Σ^{∞}E_{+} in HS this map is independent of choices. Finiteness of B is good for the existence of the complement of V. Compactness of F seems to be good for the existence of V.

5.3.7 Here we list some properties:

Theorem 5.9 1. τ_{∗} ∘ τ_{∗} is multiplicaton by χ(F).

2. If τ is orientable in a cohomology H, then τ_{∗}(x) = τ_{!}(x ∪ χ), where χ ∈ H(E) is the Euler class of T.

Proof: The verifications are not complicated. Details are in [7]. □

5.3.8 If χ(F) = 1, then τ^{∗} is an inclusion onto a direct summand for any cohomology theory. We need this for F = O/N. Note that |W |χ(O(2n)/N) = χ(O(2n)/T) = |W | and hence χ(O(2n)/N) = 1. 5.3.9 We now discuss the Adams conjecture of O(2)-bundles along the lines of [1]. It is based on the so called mod-k Dold theorem which provides a criterium for equivalence of spherical bundles.

Theorem 5.10 Consider spherical bundles ξ_{i} : E_{i} → B over a finite complex. If there exists a fibrewise map f : E_{0} → E_{1} of fibrewise degree k, then there exists a non-negative integer e such that k^{e}E_{0} ≅ k^{e}E_{1}.

This can be reformulated as follows: We consider the map F → F̃. Let F̃_{k} ⊂ F̃ denote the union of components of degree nk-maps for all n ≥ 1. The assumption of the theorem provide two maps ξ_{0}, ξ_{1} : B → BF such that their composition with BF → BF̃_{k} become homotopic. The claim is now that k^{e}ξ_{0} and k^{e}ξ_{1} are homotopic. Actually we know more. F̃_{k} ⊂ F̃ is an L-subspace (the multiplicative structure of Q_{∞}S^{0}). It is shown in [20, V.II] that its group completion ΓF̃_{k} = ΩBF̃_{k} is equivalent as an infinite loop space to F [1/k]. This shows BF [1/k] ≅ BF̃_{k} and this implies the mod-k Dold theorem. The proof given in [1] is different. 5.3.10 We consider the restriction of the Adams operation

\xymatrix{Ψ^{k} : BO(2) \ar[r] & BO \ar[r]^{ψ^{k}} & BO}

as an element in Ψ^{k} : KO(BO(2)) ≅ R̂O(2)_{I}. Let μ_{r}, λ_{0}, λ_{r} be the representations on S^{r}(ℝ^{2}), the trivial representation, and the determinant. Then we have

Ψ^{k} = \{ \begin{matrix}μ_{k} & k odd \\ μ_{k} − λ_{2} + λ_{0} & k even \end{matrix} .^{5}

We have for y ∈ RO(n) that

(λ_{2} − λ_{0})(y) = (λ_{2}y) − 1 ,

where λ_{2}y is some one-dimensional representation. Now 2^{e}(λ_{2}y−1) = 0 (this follows from K̃O(ℝP^{n})[1/2] = 0). It follows that for even k we have k^{e}(λ_{2} − λ_{0}) = 0. The kth power map provides a degree-k fibre map between the spherical bundles μ_{k}y and y for y ∈ RO(n), some n. By the mod-k Dold theorem k^{e}J(μ_{k}y − y) = 0. This proves the Adams conjecture for O(2)-bundles. 5.3.11 The arguments above show that the localization away from k of the map J ∘ (ψ^{k} − 1) vanishes on finite complexes. How can one conclude that the map itself is zero. We have the sequence

\xymatrix{0 \ar[r] & lim\underleftarrow{1}[F_{n}BO, F] \ar[r] & [BO, BF] \ar[r] & lim_{←}[F_{n}BO, BF] \ar[r] & 0}

for some increasing filtration of BO by finite subcomplexes. Here we use the fact that (F, BF, …) is the beginning of a spectrum representing some cohomology theory.

5I do not see at the moment how one derives this fact.

5.4 The J-theory diagram

5.4.1 We fix an integer r. Then we start with the following part of the comparison diagram (2) for Spin → SF and the ring spectrum kO (using the notation SF kO =: BO_{⊗}).

\xymatrix{SF \ar[r]^{τ} \ar[d]^{χ} & SF/Spin \ar[r]^{q} \ar[d]^{f} & BSpin \ar[r]^{Bj} \ar[d]^{g} & BSF \ar[d]^{=} \\ SF \ar[r]^{e} & BO_{⊗} \ar[r]^{τ} & B(SF, BO_{⊗}) \ar[r]^{q} & BSF}

Here g is the Atiyah-Bott-Shapiro orientation which is perfect (see 4.5). This is a diagram of infinite loop spaces where the horizontal lines are fibre sequences. 5.4.2 We localize away from r and get

[figure: SF [1/r]; τ; χ; SF/Spin[1/r]; q; f; BSpin[1/r]; Bj; g; BSF [1/r]; SF [1/r]; e; BO_{⊗}[1/r]; τ; B(SF, BO_{⊗}[1/r])[1/r]; q; BSF [1/r]]

Here we use B(SF, BO_{⊗}[1/r])[1/r] ≅ B(SF, BO_{⊗})[1/r] (see [20, Lemma 2.4]). 5.4.3 Next we factorize the Adams operation. We consider

\xymatrix{ & BSpin[1/r] \ar[d] \\ BO[1/r] \ar@{.>}[ur]^{Ψ^{r}−1} \ar[r]^{Ψ^{r}−1} & BO[1/r]} .

If r is even, then BSpin[1/r] ≅ BO[1/r] and the dotted arrow is obvious. Let now r be odd. By Lemma 5.6 we have BSO ≅ BSpin × B^{2}O(1). We now consider

\xymatrix{ &  & BSO[1/r] \ar[d] \ar[r]^{pr} & BSpin[1/r] \\ BO(1)[1/r] \ar[r] \ar@{-->}[urr]^{l} & BO[1/r] \ar[r]^{Ψ^{r}−1} \ar@{.>}[ur] & BO[1/r] \ar[d] \\  &  & BO(1)[1/r]}

where the vertical sequence is a fibration. The arrow l exists since Ψ^{r} −1 of a line bundle is trivial. Hence the obstruction against the dotted arrow (which ca be seen on BO(1)[1/r]) vanishes. 5.4.4 Note that BO and BSpin are infinite loop spaces. The argument 5.4.3 does not show that that the lift Ψ^{r} − 1 : BO[1/r] → BSpin[1/r] is an infinite loop map. We actually have the following stronger result. Note that

[figure: BSpin; kO < 1 >; BSO; kO < 0 >; BO; kO < −1 >; kO; KO]

is the two-connected, one-connected, 0-connected and −1-connected covers of KO, i.e. we want to recall that BSpin etc also denote the spectra of the underlying infinite loop spaces.

Lemma 5.11 ([20], Lemma 5.15) There is a unique lift

\xymatrix{ & BSpin[1/r] \ar[d] \\ Ψ^{r} − 1 : kO[1/r] \ar[r]^{Ψ^{r}−1} \ar@{.>}[ur]^{Ψ^{r}−1} & kO[1/r]}

as a map of connective spectra (infinite loop spaces).

Proof: The triangle

\xymatrix{BO \ar[r] & kO \ar[r] & Hℤ}

induces an exact sequence

\xymatrix{[kO[1/r], Hℤ[1/r][−1]] \ar[r] & [kO[1/r], BO[1/r]] \ar[r] & [kO[1/r], kO[1/r]] \ar[r]^{α} & [kO[1/r], Hℤ[1/r]]} .

Now we check that α(Ψ^{r} − 1) = 0. Since kO[1/r] is connective we have [kO[1/r], Hℤ[1/r][−1]] = 0. This provides a unique lift Ψ^{r} − 1 : kO[1/r] → BO[1/r]. In order to lift from BO to BSO we consider the triangle

BSO → BO → Hℤ/2ℤ[1] .

It gives the exact sequence

\xymatrix{[kO[1/r], ΩBO[1/r]] \ar[r]^{β} & [kO[1/r], Hℤ/2ℤ[1/r]] \ar[r] & [kO[1/r], BSO[1/r]] \ar[r] & [kO[1/r], BO[1/r]] \ar[d] \\ & ℤ/2ℤ \ar[u]^{≅} & 0 \ar[r]^{≅} & [kO[1/r], Hℤ/2ℤ[1][1/r]]} .

We recall the calculation

Hℤ/2ℤ^{∗}kO ≅ A/ASq^{1} + ASq^{2}

where A is the mod2-Steenrood algebra ([3, p.336]). This shows that [kO, Hℤ/2ℤ] ≅ ℤ/2, [kO, Hℤ/2ℤ[1]] ≅ [kO, Hℤ/2ℤ[2]] ≅ 0. In [20, p. 121] an argument is given that β is onto. Therefore we get a unique lift Ψ^{r} − 1 : kO[1/r] → BSO[1/r]. We next consider the triangle

\xymatrix{BSpin \ar[r] & BSO \ar[r] & Hℤ/2ℤ[2]} .

It induces the exact sequence

\xymatrix{[kO[1/r], Hℤ/2ℤ[1/r][1]] \ar[r] & [kO[1/r], BSpin[1/r]] \ar[r] & [kO[1/r], BSO[1/r]] \ar[r] & [kO[1/r], Hℤ/2ℤ[2][1/r]] \\ 0 \ar[u]^{≅} & & & 0 \ar[u]^{≅}} .

By the cohomology calculation the middle map is an isomorphism. This yields the unique lift Ψ^{r} − 1 : kO[1/r] → BSpin[1/r]. ^{6} 5.4.5 We define BSpin_{⊗} as the fibre of w_{2} : BSO_{⊗} → K(ℤ/2ℤ, 2). We now discuss

6That this middle map is an isomorphism depends on the finite structure. It is not true on the level of spaces since the universal second Stiefel-Whitney class is non-trivial.

\xymatrix{SF [1/r] \ar[d]^{e} \\ BO_{⊗}[1/r] \ar[d]^{τ} \ar@{-->}[r]^{Ψ^{r}/1} & BSpin_{⊗}[1/r] \ar[d] \\ B(SF, BO_{⊗}[1/r])[1/r] \ar@{.>}[ur]^{c(Ψ^{r})} \ar[r]^{c(Ψ^{r})} & BO_{⊗}[1/r]} .

Again, if r is even then BSpin_{⊗}[1/r] ≅ BO_{⊗}[1/r] and the lift is ovious. We have the splitting BO_{⊗} ≅ BO(1) × BSO_{⊗}. The restriction of Ψ^{r}/1 to BO(1)[1/r] is trivial (evaluate on the canonical line bundle). Furthermore the composition

\xymatrix{BSO_{⊗}[1/r] \ar[r]^{Ψ^{r}/1} & BO_{⊗}[1/r] \ar[r]^{pr} & BO(1)}

is trivial since BSO_{⊗}[1/r] is simply connected. We therefore must discuss

\xymatrix{ & BSpin_{⊗}[1/r] \ar[d] \\ BSO_{⊗}[1/r] \ar@{.>}[ur]^{Ψ^{r}/1} \ar[r]^{Ψ^{r}/1} & BSO_{⊗}[1/r] \ar[d]^{l} \\ & K(ℤ/2ℤ, 2)} .

We consider the composition l ∘ Ψ^{r}/1 on the π_{2}-level. Since ψ^{r} preserves the generator of K̃^{0}(S^{2}) we conclude that this composition is trivial. This provides the lift [figure]. Now e is an isomorphism on the level of π_{1} and π_{2}. For π_{1} we consider the first square of the comparison diagram (extended)

[figure: SO; τ; SO/Spin; BO(1); η; ζ; id; i; j; SO; τ; χ; SO/Spin; f; SO; e; BO_{⊗}; BO(1)] .

ζ is non-trivial on π_{1} and η is the natural inclusion. It follows that τ ∘ ζ = i. Now e is an isomorphism on the π_{1}-level since there is a non-trivial obstruction against kO-orientability of oriented bundles. Also since i is an isomorphism on π_{1}, j is so and hence the composition e ∘ χ ∘ j. This shows the claim for π_{1}. For π_{2} we take the square.

It follows that B(SF, BO_{⊗}) is two-connected. So there is no obstruction for the lift [figure: c(Ψ^{r})]. 5.4.6 We again want to lift the result of 5.4.5 to infinite loop spaces. First of all we have the sequence

\xymatrix{BSpin_{⊗} \ar[r] & BSO_{⊗} \ar[r]^{w_{2}} & Hℤ/2ℤ[2]}

which defines BSpin_{⊗} as an infinite loop space ^{7}.

7I see that w_{2} is a map of H-spaces.

Recall that X_{(p)} denotes the localization of a space or spectrum X at p. We choose p and r = r(p) such that r(p) generates the units in the p-adic numbers ℤ_{p} We know^{8} that Ψ^{r} : kO_{(p)} → kO_{(p)} is a map of E_{∞}-ring spectra. This implies by the remark 4.8.2 the maps in

\xymatrix{B(SF; SF kO_{(p)}) \ar[dr]^{l} \ar[rr]^{c(Ψ^{r})} &  & SF kO_{(p)} \ar[r]^{≅} & BO_{⊗,(p)} \\  & B(SF; SF kO_{(p)})_{(p)} \ar@{-->}[ur]^{c(Ψ^{r})}}

are infinite loop maps. In order to obtain the factorization through l we use that SF kO_{(p)} is p-local and that l is p-localization [20, Lemma 2.4]. We now discuss the diagram

\xymatrix{SF_{(p)} \ar[d]^{e} \\ BO_{⊗,(p)} \ar[d]^{τ} \ar@{-->}[r]^{Ψ^{r}/1} & BSpin_{⊗,(p)} \ar[d] \\ B(SF, BO_{⊗,(p)})_{(p)} \ar@{.>}[ur]^{c(Ψ^{r})} \ar[r]^{c(Ψ^{r})} & BO_{⊗,(p)}} .

In 5.4.5 we have see that B(SF, BO_{⊗,(p)})_{(p)} is two-connected. This provides the unique lift given by the dotted arrow as an infinite loop map. 5.4.7 The infinite loop analysis of [20, V.7] is simplified to a great deal because we now can use the fact that the Atiyah-Bott-Shapiro orientation is perfect, see 4.5. In particular, [20, Thm. V.7.11] is now immediate. 5.4.8 In the next step we add the Adams operations Ψ^{r} as follows and define their fibres J_{(p)} and J_{⊗,(p)} by the following diagram.

[figure: J_{(p)}; BO_{(p)}; Ψ^{r}−1; BSpin_{(p)}; SF_{(p)}; τ; χ; SF/Spin_{(p)}; q; f; BSpin_{(p)}; Bj; g; BSF_{(p)}; SF_{(p)}; e; BO_{⊗,(p)}; τ; B(SF, BO_{⊗})_{(p)}; q; c(Ψ^{r}); BSF_{(p)}; J_{⊗,(p)}; π; BO_{⊗,(p)}; Ψ^{r}/1; BSpin_{⊗,(p)}] .

This is a diagram of infinite loop spaces and the horizontal lines are fibration sequences. 5.4.9 We have

Theorem 5.12 (Adams Conjecture, [7], [23]) The composition

\xymatrix{BO[1/r] \ar[r]^{Ψ^{r}−1} & BSpin[1/r] \ar[r]^{Bj} & BSF [1/r]}

is homotopic to a constant map.

8At the moment I do not know this. I think, that May shows this for the completion. One must check this with the rational version and then use the arithmetic square.

It is known that it is not zero-homotopic as an infinite loop map (see [14]). A solution of the Adams conjecture gives a map γ^{r} and α^{r} (not infinite loop).

[figure: J_{(p)}; α^{r}; BO_{(p)}; γ; Ψ^{r}−1; BSpin_{(p)}; SF_{(p)}; τ; χ; SF/Spin_{(p)}; q; f; BSpin_{(p)}; Bj; g; BSF_{(p)}; SF_{(p)}; e; ϵ^{r}; BO_{⊗,(p)}; τ; B(SF, BO_{⊗})_{(p)}; q; c(Ψ^{r}); BSF_{(p)}; J_{⊗,(p)}; π; BO_{⊗,(p)}; Ψ^{r}/1; BSpin_{⊗,(p)}] .

The map ϵ^{r} is an infinite loop map since the second square in the lowest raw commutes. Note that γ is not unique. We have an inclusion

\xymatrix{BO(1) \ar[r] & BSO \ar[r] & SF \ar[r]^{τ} & SF/Spin} .

Therfeore we can change γ by a map \xymatrix{BO[1/r] \ar[r] & BO(1)[1/r] \ar[r] & SF/Spin[r]} . In particular we can prescribe the restriction of γ to the BO(1)-component in BO ≅ BO(1) × BSO. 5.4.10 We consider now the following collaps of the above diagram

\xymatrix{J_{(p)} \ar[r] \ar[d]^{ϵ^{r}χα^{r}} & BO_{(p)} \ar[d]^{fγ} \ar[r]^{Ψ^{r}−1} & BSpin_{(p)} \ar[d]^{c(Ψ^{r})g} \\ J_{⊗,(p)} \ar[r]^{π} & BO_{⊗,(p)} \ar[r]^{Ψ^{r}/1} & BSpin_{⊗,(p)}} .

Proposition 5.13 The second square induces a cartesian diagram

\xymatrix{BO_{(p)} \ar[d]^{fγ} \ar[r]^{Ψ^{r}−1} & BSpin_{(p)} \ar[d]^{c(Ψ^{r})g} \\ BO_{⊗,(p)} \ar[r]^{Ψ^{r}/1} & BSpin_{⊗,(p)}} .

Proof: We follow the argument [20, V.4]. We first consider the case that p is odd. In this case we gave an infinite loop space splitting BO_{(p)} ≅ W × W^{⊥} such that π_{i}(W) = ℤ_{p} in degree i = 2j(p − 1) and trivial otherwise. A theorem of Adams and Priddy [4] says that BSO_{(p)} has a unique infinite loop space structure. Since BO_{(p)} ≅ BSO_{(p)} ≅ BSO_{⊗,(p)} ≅ BO_{⊗,(p)} ≅ BSpin_{(p)} ≅ BSpin_{⊗,(p)} we have splittings as W × W^{⊥} of all these spaces as infinite loop spaces.

Lemma 5.14 The following compositions are equiavlences.

\xymatrix{W \ar[r] & BO_{(p)} \ar[r]^{fγ} & BO_{⊗,(p)} \ar[r] & W \\ W \ar[r] & BSpin_{(p)} \ar[r]^{c(Ψ^{r})g} & (BSpin_{⊗,(p)} \ar[r] & W \\ W^{⊥} \ar[r] & BO_{(p)} \ar[r]^{Ψ^{k}−1} & BSpin_{(p)} \ar[r] & W^{⊥} \\ W^{⊥} \ar[r] & BO_{⊗,(p)} \ar[r]^{Ψ^{k}/1} & BSpin_{⊗,(p)} \ar[r] & W^{⊥} \\ BO_{(p)} \ar[r]^{Δ} & BO_{(p)} × BO_{(p)} \ar[r]^{fγ×(Ψ^{r}−1)} & BO_{⊗,(p)} × BSpin_{(p)} \ar[r]^{+} & W × W^{⊥} \\ W × W^{⊥} \ar[r] & BSpin_{(p)} × BO_{⊗,(p)} \ar[r]^{c(Ψ^{r})g×Ψ^{k}/1} & BSpin_{⊗,(p)} × BSpin_{⊗,(p)} \ar[r]^{+} & BO_{⊗,(p)}}

If p = 2, then fγ : BO_{2} → (BO_{⊗})_{2} and c(Ψ^{3})g : BSpin_{2} → (BSpin_{⊗})_{2} are equivalences.

Proof: This follows from a calculation of the action of the corresponding maps on homotopy groups. Here the special choice of r(p) is employed. □ This Lemma implies Proposition 5.13. □

Corollary 5.15 The composition ϵ^{r}χα^{r} : J_{(p)} → J_{⊗,(p)} is an equivalence.

5.4.11 We now discuss the lifting to infinite loop maps.

Lemma 5.16 The composition fγ is an infinite loop map.

Proof: If p is odd we use the result [15, Prop. D] stating that the lower map in the following diagram is infinite loop. By Adams-Priddy the vertical equivalences are infinite loop.

\xymatrix{BO_{(p)} \ar[d]^{≅} \ar[r]^{fγ} & BO_{⊗,(p)} \ar[d]^{≅} \\ BSO_{(p)} \ar[r]^{ρ^{r}} & BSO_{⊗,(p)}} .^{9}

By Adams-Priddy the vertical equivalences are infinite loop. If p = 2, then the result is clear.

Corollary 5.17 The composition ϵ^{r}χα^{r} : J_{(p)} → J_{⊗,(p)} is an equivalence of infinite loop spaces.

5.5 Consequences

5.5.1 Globalization of local results. Let T be a set of primes and X → X_{T} be the localization at T. Let X → X_{0} be rationalization. We consider the category C_{T} with objects 0, p ∈ T and maps p → 0 for all p ∈ T. Then we have a functor p ∈ C_{T} ↦ X_{p}.

9I have not checked that the map ρ^{k} in [15] is fγ for an appropriate choice of γ.

Lemma 5.18 ([9], 6,7) For a space Y the natural map

[Y, X_{T}] → lim_{p∈C_{T}} [Y, X_{p}]

is an injection. Furthermore, if π_{∗}Y is of finite type, then it is an bijection. 5.5.2 We consider a finite complex X. Theorem 5.19 Let ξ : X → BSpin be a stable spin bundle. Then the following are equivalent:

1. ξ is trivial as a spherical bundle.

2. The composition

\xymatrix{X_{p} \ar[r]^{ξ} & BSpin_{p} \ar[r]^{Bj} & BSF_{p}}

is trivial for all primes p.

3. There exists a unit η ∈ KO(X) such that c(ψ^{r})gξ = ψ^{r}η/η.

4. For each prime p there exists a unit η_{p} ∈ KO_{p}(X) such that c(ψ^{r(p)})gξ = ψ^{r(p)}η_{p}/η_{p}.

5. For each prime p there exists an element η_{p} ∈ KO_{p}(X) such that ξ_{p} = ψ^{r(p)}η_{p} − η_{p}.

Proof:

1 → 2 clear 2 → 1 Lemma 5.18 and finite type of π_{∗}(J). 1 → 3 If Bj ∘ ξ is trivial, then there exists a factorization ξ′ : X → SF/Spin. We set η := f ∘ ξ using the

map f from the comparison diagram. The assertion now follows from the J-theory diagram. 3 → 4 by localization. 4 → 5 We use Proposition 5.13. 5 → 2 by the J-theory diagram.

□

5.6 The decomposition of SF

5.6.1 We define j_{p} by the fibre sequence

\xymatrix{j_{p} \ar[r] & kO_{p} \ar[r]^{Ψ^{r}−1} & BSpin_{p}} .

Theorem 5.20 ([20], VIII.3.2) The spectrum j_{p} is a ring spectrum. Proof: The idea is to construct a discrete model. We discuss the details in ??. □ We further define the ring spectrum

j := ×_{p}j_{p} .

We have

Ω_{0}j_{p} = ℤ_{p} × J_{p}

as an infinite loop space where J_{p} := (J_{(p)})_{p} is the p-completion.

5.6.2 Here is a point where the modern technology helps. By [10] we know that kO_{p} has a symmetric ring spectrum version and that ψ^{r} is an automorphism of ring spectra. Its fixed points is the ring spectrum j_{p}. At this point would have to verify that the latter structure is the same as the former. 5.6.3 We have

SF j_{p} ≅ J_{⊗,p} := (J_{⊗,(p)})_{p}

as infinite loop spaces. This is not at all obvious. The infinite loop space structure on J_{⊗,p} comes from its place in the J-theory diagram while the infinite loop space structure of SF j_{p} comes from the ring spectrum structure via discrete models. The argument can be found in [20, p. 228ff]. 5.6.4 If we would work in symmetric spectra with the fixed point model for j_{p} it would be clear that the infinite loop space structure of SF j_{p} is that of J_{⊗}, since the multiplicative structure of the algebraic Ktheory model for kO_{p} is isomorphic with the completion of the multiplicative structureof the topological model for kO by [20, Lemma 2.4]. We set

J_{⊗} := ×_{p}J_{⊗,p}

and have SF j := J_{⊗} as infinite loop spaces. 5.6.5 We define the infinite loop space C_{⊗,(p)} as the fibre of

\xymatrix{C_{⊗,(p)} \ar[r] & SF_{(p)} \ar[r]^{ϵ^{r(p)}} & J_{⊗,(p)}} .

Its first deloop BC_{⊗,(p)} can also be seen as the fibre of the cannibalistic class. Indeed, the J-theory diagram gives

[figure: BC_{⊗,(p)}; BC_{⊗,(p)}; BO_{(p)}; τ; B(SF, BO_{⊗,(p)})_{(p)}; c(Ψ^{r}); q; BSF_{(p)}; ϵ^{r}; BO_{(p)}; Ψ^{r}/1; BSpin_{⊗,(p)}; BJ_{⊗,(p)}^{r}]

5.6.6 We have represented the spectrum J_{⊗,p} as the spectrum of special units of a E_{∞}-ring spectrum j_{p}. We therefore can talk about specially J_{p}-oriented spherical bundles. We consider the following diagram:

[figure: BC_{⊗,p}; BSF_{p}; BJ_{⊗,p}; B(SF, J_{⊗,p})_{p}; BSF_{p}; BJ_{⊗,p}; B(SF, BJ_{⊗,p}); BSF; BJ_{⊗,p}]

where the middle row is the p-completion of the orientation sequence given by the last raw. We obtain the equivalence of infinite loop spaces presented by the dotted arrow (compare [20, Cor 3.3]). This gives a bunldle theoretic interpretation of BC_{⊗,p}. 5.6.7 It is known that ε : SF_{p} → J_{⊗,p} does not split as an infinite loop map for p = 2 (even as an H-map). On the other hand it splits on the level of homotopy groups. In fact

Theorem 5.21 If p is odd, then SF_{p} → J_{⊗,p} splits as an infinite loop map.

Proof: The natural candidate for the split is

\xymatrix{J_{⊗,p} \ar[r]^{≅} & J_{p} \ar[r]^{α} & SF_{p}}

where the first isomorphism of infinite loop spaces was already obtained in 5.17. The point is to show that α is an infinite loop map at odd primes. This is the content of [20, VIII.4]. □

We thus get for odd primes p the decomposition

SF_{p} ≅ B(SF, J_{⊗,p})_{p} ∨ J_{⊗,p} .

We shall see below that B(SF, J_{⊗,p})_{p} is K_{p}-accclic, and that SF_{p} → J_{⊗,p} is the K-localization. 5.6.8 Using more modern results one could simplyfy here. Namely it was shown by Friedlander and Seymour [17]^{10}, [16] that the Adams conjection holds true in the sense of infinite loop spaces at p. This immediately would give that α and γ in the J-theory diagram are infinite loop maps.

5.7 K-localization of the sphere spectrum

5.7.1 Let ℤ_{(p)} denote the localization of the integers at the prime p and let Sℤ_{(p)} be the corresponding Moore spectrum.

Definition 5.22 For a spectrum or space X we write X → X_{(p)} for the localization at Sℤ_{(p)}^{11}. We call Sℤ_{(p)}-local spectra p-local.

We have X_{(p)} ≅ Sℤ_{(p)} ∧ X and the localization map is induced by the unit S → Sℤ_{(p)}. We have π_{∗}(X_{(p)}) ≅ π_{∗}(X)_{(p)}. Let Sℤ/pℤ denote the Moore spectrum of ℤ/pℤ.

Definition 5.23 For a spectrum or space X we write X → X_{p} for the localization at Sℤ/pℤ^{12}. We call Sℤ/pℤ-local spectra p-complete.

We have X_{p} ≅ Sℤ/pℤ ∧ X and the localization map is induced by the unit S → Sℤ/pℤ. We have π_{∗}(X_{p}) ≅ ℤ_{p} ⊗ π_{∗}(X).

Lemma 5.24 ([8], Special case of Lemma 2.11) We have (X_{(p)})_{p} ≅ X_{p}.

Let X_{(0)} denote rationalization.

Lemma 5.25 For any spectrum X we have a homotopy fibre square

\xymatrix{X \ar[r] \ar[d] & ∏_{p} X_{p} \ar[d] \\ X_{(0)} \ar[r] & ∏_{p}(X_{p})_{(0)}} .

Let K be a second spectrum and let X_{K} denote the K-localization. Then this generalizes to

Lemma 5.26 For any spectrum X we have a homotopy fibre square

\xymatrix{X_{K} \ar[r] \ar[d] & ∏_{p} X_{K_{p}} \ar[d] \\ X_{K_{(0)}} \ar[r] & ∏_{p}(X_{K_{p}})_{K_{(0)}}} .

For all this see [8].

10Is there a problem with the proofs. The details of Seymours proof have not been published

11In [20, p. 37] it is defined as localization at the Eilenberg-MacLane spectrum Hℤ_{(p)}. Hopefully the same thing.

12Again, in [20, p. 37] it is defined as localization at the Eilenberg-MacLane spectrum Hℤ/pℤ. Hopefully the same thing.

5.7.2 For a spectrum X let X < n >→ X be the n-connective covering. We have BO×ℤ ≅ KO < −1 >, BO ≅ KO < 0 > and BSpin ≅ KO < 3 > and the same for the p-localizations. Therefore the upper two horizontal fibre sequences in the following diagram are just our previous definitions of J_{(p)} and j_{(p)}.

[figure: J_{(p)}; KO_{(p)} < 0 >; ψ^{r}−1; KO_{(p)} < 3 >; j_{(p)}; KO_{(p)} < −1 >; ψ^{r}−1; KO_{(p)} < 3 >; J_{(p)}; KO_{(p)}; ψ^{r}−1; KO_{(p)}] .

The last raw defines a p-local spectrum J_{(p)}. One can check that j_{(p)} and J_{(p)} are the −1-connective and 0-connective covers of J_{(p)}. 5.7.3 Let us calculate π_{−1}(J_{(p)}. We use the sequence

[figure: ℤ_{(p)}; 0; ℤ_{(p)}; ℤ_{(p)}; ≅; π_{0}(KO_{(p)}); ≅; (ψ^{r}−1)^{∗}; π_{0}(KO_{(p)}); ≅; ≅; π_{−1}(J_{(p)}); π_{−1}(KO_{(p)}); ≅; (ψ^{r}−1)^{∗}; π_{−1}(KO_{(p)}); 0] .

In [8], p. 268 it is claimed that there is a unique/natural map k : J_{(p)} → S^{−1}ℚ (the shifted Moorespectrum) which induces an isomorphism

ℚ ⊗ π_{−1}J_{(p)} →^{∼} ℚ ⊗ π_{−1}S^{−1}ℚ ^{13}

Using this map we define the triangle

\xymatrix{J̃_{(p)} \ar[r] & J_{(p)} \ar[r]^{k} & S^{−1}ℚ} .

We find a lift

\xymatrix{ & J̃_{(p)} \ar[d] \\ S \ar[r]^{e} \ar@{.>}[ur]^{λ} & J_{(p)}} .

Theorem 5.27 (Adams-Baird, Ravenel, Bousfield [8], Thm. 4.3)

\xymatrix{X \ar[r]^{≅} & X ∧ S \ar[r]^{1∧λ} & X ∧ J̃_{(p)}}

is K_{(p)}-localization.

13I think we have π_{−1}S^{−1}ℚ ≅ ℚ. Furthermore, ψ^{r} − 1 is trivial on π_{0}(KO_{(p)} and π_{−1}(KO_{(p)}) ≅ 0. Therefore π_{−1}(J_{(p)}) ≅ ℤ_{(p)}.

If we apply this to S we get

Corollary 5.28 S_{K_{(p)}} ≅ J̃_{(p)}

Corollary 5.29 The sequence

\xymatrix{C_{⊗,(p)} \ar[r] & SF_{(p)} \ar[r] & J_{⊗,(p)}}

is the K_{(p)}-localization triangle of SF_{(p)}.

Observe that (Ω_{0}J̃_{(p)})_{0} ≅ (Ω_{0}j_{(p)})_{0} ≅ J_{(p)}.

Corollary 5.30 We have J_{(p)} ≅ (Ω_{0}S_{K_{(p)}})_{0}.

We have furthermore an isomorphism J_{⊗,(p)} ≅ SF S_{K_{(p)}} (the 1-component of the ring space). The ring spectrum structure of S_{K_{(p)}} give an infinite loop space structure on the right hand side and the question wether this isomorphism is an infinite loop map appears. 5.7.4 We have the following fibre sequences

\xymatrix{C_{⊗,p} \ar[r] \ar@{.>}[d]^{≅conjecture} & SF_{(p)} \ar[d]^{···−1} \ar[r]^{ε} & J_{⊗,p} \ar[d]^{≅ε∘α} \\ C_{(p)} \ar[r] & (Ω_{0}S_{(p)})_{0} \ar[r] & Ω_{0}J_{(p)} \ar[ul]^{α}}

The dotted arrow is a conjectural isomorphism. We have

SF_{(p)} ≅^{···−1} Ω_{0}^{∞}S_{(p)}^{0}

as spaces, but not as infinite loop spaces. The two horizontal compositions are conjectured to be two different extensions of the same H-spaces. 5.7.5 We now use that (X_{K_{(p)}})_{p} ≅ (X_{K})_{p} ([8, Prop. 2.11]). We define the p-completions J̃_{p} := (J̃_{(p)})_{p}. Then we have the homotopy cartesian square

\xymatrix{S_{K} \ar[r] \ar[d] & ∏_{p} J̃_{p} \ar[d] \\ S_{K_{(0)}} \ar[r] & ∏_{p}(J̃_{p})_{K_{(0)}}} .

We now apply [8, Thm. 3.1] to the extend that S_{K_{(0)}} ≅ S_{(0)} ≅ Hℚ (this uses that S is connected).

5.8 Discrete models

5.8.1 In this subsection we discuss the approach via discrete models to various questions:

1. Is Ψ^{r} : BO_{⊗} → BO_{⊗} an infinite loop map.

2. Is j_{p} a ring spectrum.

3. Does the map SF_{p} → J_{⊗,p} split.

5.8.2 We construct bipermutative categories C and then apply the delooping machine described in 3.6 to the associated E_{∞}-ring spaces BC in order to get E_{∞}-ring spectra. We produce E_{∞}-maps between those spectra by applying these machines to functors between these bipermutative categories. Here are the symbols. If X is an E_{∞}-ring space, then T X is the T X → Ω^{∞}T X is the associated E_{∞}-ring prespectrum and its spectrification.

5.8.3 The most primite bipermutative category is E its objects are the integers 0, 1, 2, 3, 4, 5, . . .. The morphisms E(n, m) are nonempty only if n = m, and in this case E(n, m) := Σ_{n}. The two monoidal structures on objects are n ⊕ m := n + m and n ⊗ m := nm. We view the morphisms as plane rectangles with n ordered inputs in the upper border and n outputs in the lower border and the morphism is givnen by a collection of lines each connecting and input with an output. Composition is vertical. The sum of two morphisms is by horizontal side-by side. In order to describe the product nm we thicken the m lines to rectangles and identify each with the rectangle with n inputs. Observe that distributivity is only one-sided, (n + m)r = nr + mr but r(n + m) ≠ rn + rm. The commutativity transformations are obvious but not trivial. The ring space is ⊔_{n≥0}BΣ_{n}. It is the free (Q, Q)-space generated by S^{0}. Therefore Ω^{∞}T BE is a model of the sphere spectrum. 5.8.4 Let A be a topological ring. Then we form the bipermutativ category ILA. Its objects are again the integers. The morphisms ILA(n, m) are non-empty iff n = m, and this case they are given by ILAGL(n, A). The monoidal structre is similar to that of E. Just connect all inputs with all outputs by lines. Then the matrix can be viewed as a lebeling of the lines with elements of A. Composition is vertical. The label of the line connecting an input with an output is obtained by summing the labels of all path where the label of the path obtained by concatenation of two labeled path is the product of the labels. In the multiplicative monoidal structure the matrix put into a thickened path is multiplied by the label of the path. We define the category IOA by only taking orthogonal marices (i.e. those with MM^{t} = 1). We denote the resulting E_{∞}-ring spectra by kA and kOA.

Lemma 5.31 Assume that A is discrete. The space (Ω_{0}kA)_{0} is homotopy equivalent to BGL(A)^{+} (Quillens +-construction).

Lemma 5.32 If A = ℂ or A = ℝ as a topological ring, then kA ≅ ku or kA ≅ ko as ring spectra in HS.

5.8.5 Let k be a perfect field of characteristic p. Then we have a Frobenious automorphism θ^{q} : k → k, x ↦ x^{p} (we need perfect in order to have surjectivity of θ^{p}). The Frobenious induces a functor θ^{p} : ILk → ILk and therefore an automorphism of the E_{∞} ring spectrum Ω^{∞}T BILk. We will compare this later with the Adams operations. There is a natural functor E → ILA given on morphisms by the mebdding Σ_{n} → GL(n, A) by permutation matrices. This yields the unit

S ≅ Ω^{∞}T BE → Ω^{∞}T BILA .

5.8.6 Let k be a finite field with q = p^{r} element. We define a functor of permutative categories (ILk, ⊕) → (E, ⊗). We choose bijection of sets k ≅ {1 …, q}. On objects it is given by n ↦ q^{n}. The matrix g ∈ ILk ≅ Gl(n, k) is mapped to the permutation which its action on k^{n} ≅ {1, …, q}^{n} induces. One checks that this map is functorial and monoidal. We obtain a map of infinite loop spaces

(Ω_{0}Kk)_{⊗} → (Ω_{0}S)_{⊕} .

The subscript indicates with relevant infinite loop space structure. This will be used in order to define the infinite loop map J_{⊗,p} → SF_{p} for odd primes which splits e : SF_{p} → J_{p}. 5.8.7 Let 𝔽_{p} be the field with p elements and 𝔽_{p} ⊂ 𝔽̄_{p} be its algebraic closure. We fix an embedding μ : F̄_{p}^{∗} → ℂ^{∗}^{14} Let ρ : G → GL(n, 𝔽̄_{p}) be a representation of a finite group. For g ∈ G we let ξ_{1}(g), …, ξ_{n}(g) ∈ 𝔽̄_{p} denote the eigenvalues of ρ(g). There exists a unique representation λ(ρ) : G → GL(n, ℂ) such that χ_{λ(ρ)}(g) = ∑_{i=1}^{n} μ(ξ_{i}(g)) (χ_{κ} is the character of κ). This is the assertion of the Theorem of Brauer. Quillen observed that if q is odd and ρ was orthogonal, then λ(ρ) comes from a unique real representation which we also denote by λ(ρ).

14Does such an ebedding exist? We adjoin the p^{r}-th roots of unity to ℚ and get a field K^{r}. Let q ⊂ O_{K^{r}} (the integers in K^{r}) be any prime factor of p. Then 𝔽_{p^{r}} := O_{K^{r}} /q and F̄_{p} := lim_{→} 𝔽_{p^{r}}. This allows to lift units from 𝔽̄_{p} to ℂ^{∗}. But how does one arrange a group homomorphism?

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