AUTOMORPHISMS OF SPHERES, NUMBER

FIELDS, MOTIVES AND SPECTRA

John Rognes

June 3rd 2005

Spheres and discs.

Split fiber sequence

B Diff(D^{n}) → B Diff(S^{n}) → BO(n + 1)

of diffeomorphisms rel ∂, and fiber sequence

B Diff(D^{n+1}) → BC(D^{n}) → B Diff(D^{n})

where C(D^{n}) = Diff(D^{n} × I rel D^{n} × 0) has a geometrically significant involution. For n large, approximate fiber sequence

BC(D^{n}) → Q(D_{+}^{n}) → A(D^{n})

and rational equivalence A(D^{n}) → K(ℤ). From Borel’s theorem

K_{i}(ℤ) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & for i = 0 \\ ℚ & for i = 4j + 1, j ≥ 1 \\ 0 & otherwise, \end{matrix}

so for n large

π_{i}BC(D^{n}) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & for i = 4j, j ≥ 1 \\ 0 & otherwise. \end{matrix}

The sign of the involution depends on the parity of n:

π_{i}B Diff(D^{n}) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & for n odd and i = 4j, j ≥ 1 \\ 0 & otherwise. \end{matrix}

The proof of Borel’s theorem involves the real analysis of SL_{n}(ℤ)-invariant differential forms on the symmetric space SL_{n}(ℝ)/O(n). This explains K(ℤ) from the “inside” of the matrix group GL_{n}(ℤ). Can we understand it from the “outside”, by descent from the extensions of ℤ ⊂ ℚ? The following is a brief extract from papers by Jack Morava, Alain Connes and Matilde Marcolli.

Typeset by AMS-TEX

Number fields.

ℚ ⊂ ℚ^{ab} ⊂ \overline{ℚ}

Gal(\overline{ℚ}/ℚ^{ab}) → Gal(\overline{ℚ}/ℚ) → Gal(ℚ^{ab}/ℚ)

Kronecker–Weber: Gal(ℚ^{ab}/ℚ) ≅ Aut(ℚ/ℤ) = ℤ̂^{∗}.

Shafarevich conjecture: Gal(\overline{ℚ}/ℚ^{ab}) is a free profinite group.

Covering spaces.

Let X be a topological space, and let Cov(X) be the category of covering spaces π: Y → X. Let

ω: Cov(X) → Sets

be the fiber functor at a point x_{0} ∈ X: ω(Y) = π^{−1}(x_{0}). The set of natural automorphisms g: ω → ω of the functor ω forms a group under composition, naturally isomorphic to the fundamental group π_{1}(X, x_{0}).

Motives.

ℚ-linear category MT (ℤ) of Tate motives of Spec(ℤ) is Tannakian (an abelian tensor category with rigid duality), thus equivalent to the category of representations of a pro-algebraic group, the motivic Tate Galois group Gal_{mot}(ℤ) of ℤ. The singular cohomology of the complex points (Betti cohomology) is a fiber functor

ω: MT (ℤ) → Vect

that respects direct sums and tensor products, and Gal_{mot}(ℤ) is the pro-algebraic group of natural automorphisms of this functor ω.

𝔉𝔯_{odd} → Gal_{mot}(ℤ) → 𝔾_{m}

𝔉𝔯_{odd} is pro-unipotent, with Lie algebra 𝔣𝔯_{odd} the free Lie algebra with one generator Z_{k} in each odd weight (or half-degree) k ≥ 3. The action by the multiplicative group 𝔾_{m} in the extension specifies the grading.

Motivic realization.

There is a motivic realization functor from commutative rings to motives. Symmetries of rings become symmetries of motives, defining a map of extensions from Gal(\overline{ℚ}/ℚ) to Gal_{mot}(ℤ).

The profinite group ℤ̂^{∗} includes as a Zariski dense subset of the pro-algebraic group 𝔾_{m}.

The Galois group Gal(\overline{ℚ}/ℚ^{ab}) also injects into a pro-algebraic group GT, the Grothendieck–Teichmueller group of Drinfel’d, which conjecturally is the same as 𝔉𝔯_{odd}. This concerns Deligne’s action on the motivic fundamental group of ℙ^{1} \ {0, 1, ∞}, and his conjecture on Hochschild (co-)homology.

Algebraic K-theory.

The algebraic K-theory K(ℤ) should only depend on the motive of ℤ. Expect descent spectral sequence

E_{s,t}^{2} = H_{mot}^{−s}(ℤ; K_{t}^{mot}) = H^{−s}(Gal_{mot}(ℤ); K_{t}^{mot}) ⟹ K_{s+t}(ℤ)

that collapses rationally. Here K_{∗}^{mot} = ℤ[β] with |β| = 2. We compute the E^{2}-term with the Serre spectral sequence

E_{2}^{p,q} = H^{p}(𝔾_{m}; H^{q}(𝔉𝔯_{odd}; K_{t}^{mot})) ⟹ H^{p+q}(Gal_{mot}(ℤ); K_{t}^{mot}) .

The classifying space of a free group is a wedge of circles, one for each generator. Its reduced cohomology is concentrated in degree 1, with one summand for each generator. Likewise for 𝔉𝔯_{odd}:

H^{q}(𝔉𝔯_{odd}; ℤ[β]) = \{ \begin{matrix}ℤ[β] & for q = 0 \\ ℤ\{e_{2j+1} | j ≥ 1\}[β] & for q = 1 \\ 0 & otherwise \end{matrix}

Rationally the cohomology H^{p}(𝔾_{m}; −) consists only of the invariants (p = 0, roughly). Here β has weight 1 and e_{2j+1} weight −(2j + 1), so these are spanned by 1 in (s, t) = (0, 0) and e_{2j+1}β^{2j+1} in bidegree (s, t) = (−1, 4j + 2). Thus

K_{i}(ℤ) ≅ \{ \begin{matrix}ℤ\{1\} & for i = 0 \\ ℤ\{e_{2j+1}β^{2j+1}\} & for i = 4j + 1, j ≥ 1 \\ 0 & otherwise \end{matrix}

modulo torsion.

Renormalization of QFT.

Connes and Marcolli interpret an extension

𝔉𝔯 → U^{∗} → 𝔾_{m}

as a cosmic Galois group acting on the set of dimensionless coupling constants of physical theories, following Cartier. These “constants” are really sections in a bundle over a space of complexified dimensions, punctured at an integer dimension D = 4, and renormalization in quantum field theory amounts to extracting the regular part of meromorphic functions at D.

Here the pro-unipotent group 𝔉𝔯 has Lie algebra 𝔣𝔯 the free Lie algebra with one generator Z_{k} in each weight ≥ 1. It is the affine variety

𝔉𝔯 = Spec(QSymm)

of the commutative Hopf algebra of quasi-symmetric functions, which is dual to the cocommutative Hopf algebra

QSymm^{∗} = U(𝔣𝔯) = ℤ⟨Z_{k} | k ≥ 1⟩

with coproduct

Δ(Z_{k}) = ∑_{i+j=k} Z_{i} ⊗ Z_{j} .

Here U(−) denotes the enveloping algebra. The Z_{k} map to the Virasoro generators in the central extension of Diff(S^{1}).

The inclusion 𝔣𝔯_{odd} → 𝔣𝔯 corresponds to group homomorphisms 𝔉𝔯_{odd} → 𝔉𝔯 and

Gal_{mot}(ℤ) → U^{∗} .

Complex cobordism.

The maximal abelian quotient of U(𝔣𝔯) is

ℤ⟨Z_{k} | k ≥ 1⟩ → ℤ[c_{k} | k ≥ 1] = H^{∗}(BU)

where Z_{k} ↦ c_{k} goes to the k-th Chern class. This is a cocommutative Hopf algebra homomorphism, by the Cartan formula

Δ(c_{k}) = ∑_{i+j=k} c_{i} ⊗ c_{j} .

Dually, there is an injection

QSymm ⟵ H_{∗}(BU) = ℤ[b_{k} | k ≥ 1] = Symm

of commutative Hopf algebras.

At the level of algebraic groups, we interpret this as a homomorphism

𝔉𝔯 → Gal(MU/S) .

If S → MU were a finite G-Galois extension, there would be an action map α: G_{+} ∧ MU → MU. Or, by adjunction, we could write it as a coaction

β: MU → MU ∧ DG_{+} ,

where DG_{+} = F (G_{+}, S) is the functional dual. The diagonal and product on G make DG_{+} a commutative S-Hopf algebra. There is no such group G, but its functional dual DG_{+} exits as the commutative S-Hopf algebra S[BU] = Σ^{∞}BU_{+}, with product coming from the infinite loop space structure on BU.

The coaction map is then the Thom diagonal

β: MU → MU ∧ S[BU]

derived from the space-level analogue

T h(ξ) → T h(ξ) ∧ B_{+}

for a vector bundle ξ: E → B. This makes S → MU a Hopf–Galois extension, since the composite

MU ∧ MU ⟶^{1∧β} MU ∧ MU ∧ S[BU] ⟶^{μ∧1} MU ∧ S[BU]

is a weak equivalence, namely the Thom isomorphism in MU-homology. So we should interpret DG_{+} as S[BU]. In particular H_{∗}(BU) = H_{∗}(DG_{+}) = H^{∗}(G) is the ring of graded functions on G.

The last two homomorphisms

Gal(\overline{ℚ}/ℚ^{ab}) → 𝔉𝔯_{odd} → 𝔉𝔯 → Gal(MU/S)

suggest that there is an S-algebraic realization of motives, lifting the motive of ℤ to the sphere spectrum S, up to roots of unity.