ALGEBRAIC K-THEORY,

SOME APPLICATIONS AND COMPUTATIONS

John Rognes

Topological K-theory

We begin by recalling the topological K-theory of Atiyah and Hirzebruch, to motivate the definition of algebraic K-theory.

Let X be a compact topological space, and consider a vector bundle p: E → X over X. This is a family of complex, finite dimensional vector spaces E_{x} = p^{−1}(x) for x ∈ X which is locally trivial, i.e. X can be covered by open U ⊂ X where we have fibrewise linear trivializations θ: p^{−1}(U) → U × V. Here V = E_{x} for some x ∈ U, and θ is a homeomorphism commuting with the projections to U.

p^{−1}(U) \overline{θ ; ≅ ; }U × V

[figure: p]

U^{proj}

We can embed E in ℂ^{∞} since X is compact, e.g. map f: E ↪ X × ℂ^{∞} in a manner respecting the projections to X

E ^{⊂} X × ℂ^{∞}

[figure: p]

X^{proj}

We think of E as describing a family of subspaces f(E_{x}) ⊂ ℂ^{∞} for all x ∈ X. This amounts to a map X → Grass(ℂ^{∞}) where Grass(ℂ^{∞}) is the space of finite dimensional sub vector spaces of ℂ^{∞}. It is the disjoint union of the subspaces Grass_{n}(ℂ^{∞}) of n-dimensional sub vector spaces of ℂ^{∞}, over all n ≥ 0.

We get a bijection

Vect(X) = {isomorphism classes of vector bundles over X}

[X, ∐_{n≥0} Grass_{n}(ℂ^{∞})] = {homotopy classes of maps X → ∐_{n≥0} Grass_{n}(ℂ^{∞})}^{≅}

Typeset by AMS-TEX

We think of the disjoint union as indexed over the isomorphism classes of finite dimensional sub vector spaces of ℂ^{∞}.

There is a fibration sequence

GL(ℂ^{n}) ⟶ Frames_{n}(ℂ^{∞}) ⟶ Grass_{n}(ℂ^{∞})

with fiber GL(ℂ^{n}), where Frames_{n}(ℂ^{∞}) is the space of n-frames in ℂ^{∞}, i.e. n linearly independent vectors (v_{1}, …, v_{n}) in ℂ^{∞}. The projection map is given by

(v_{1}, …, v_{n}) ⟼ lin{v_{1}, …, v_{n}}

taking a frame to the sub vector space it spans.

Frames_{n}(ℂ^{∞}) is (weakly) contractible, so by the Puppe sequence there is a homotopy equivalence Ω Grass_{n}(ℂ^{∞}) ≃ GL(ℂ^{n}). Here ΩX denotes the loop space of X. We write Grass_{n}(ℂ^{∞}) ≃ BGL(ℂ^{n}), where B(−) denotes a delooping of G, thought of as the inverse operation to Ω(−), when one exists.

Vect(X) ≅ [X, ∐_{ℂ^{n}} BGL(ℂ^{n})].

More generally, when M is a topological monoid (semigroup), we can naturally construct a sequence

M ⟶ EM ⟶ BM

with EM ≃ ∗. When M = G is a topological group, this is a principal fibration sequence, and ΩBG ≃ G. In general we only get a map M → ΩBM. Now ΩBG is a group up to homotopy, because reversing orientation of loops provides an inverse. So the map M → ΩBM provides a map from a monoid to a group, up to homotopy.

Vect(X) is an abelian monoid under Whitney sum of vector bundles. E⊕F → X has fibres (E ⊕ F)_{x} = E_{x} ⊕ F_{x}. On the representing space, this pairing is induced by a map

∐_{ℂ^{n}} BGL(ℂ^{n}) × ∐_{ℂ^{n}} BGL(ℂ^{n}) ⟶^{⊕} ∐_{ℂ^{n}} BGL(ℂ^{n})

making ∐_{ℂ^{n}} BGL(ℂ^{n}) a topological monoid.

We desire to consider formal or virtual vector bundles, i.e. expressions involving signed sums of vector bundles. This idea was used by Grothendieck to prove a Riemann–Roch theorem, and is necessary in index theory to describe invariant characteristic classes by formulae in vector bundles.

One way to achieve this is the “group completion” map M → ΩBM.

Definition.

K(X) = [X, ΩB(∐_{ℂ^{n}} BGL(ℂ^{n}))]

is the initial abelian group receiving a monoid map from Vect(X), called the topological K-theory of the space X.

One can prove that

ΩB(∐_{ℂ^{n}} BGL(ℂ^{n})) ≃ ℤ × colim_{n} BGL(ℂ^{n}) = ℤ × BGL(ℂ^{∞}) ≃ ℤ × BU.

ALGEBRAIC K-THEORY 3

Here U = colim_{n} U(n) is the infinite unitary group, with U(n) ⊂ GL(ℂ^{n}) the maximal compact subgroup. GL(ℂ^{n})/U(n) ≅ ℝ^{N} for some N, so the inclusion U(n) ↪ GL(ℂ^{n}) is a homotopy equivalence.

We now say that the space ℤ × BU represents K(X), in view of the definition above.

Algebraic K-theory

Next we want to consider the analog of K(X) for arithmetic bundles, with applications to number theory, and to a sort of manifold bundles, for use in geometric topology.

The space X played the role of a test space or dummy variable in the previous section, and we choose to focus our interest on the representing space instead

K = ΩB(∐_{ℂ^{n}} BGL(ℂ^{n})) ≃ ℤ × BU.

Let R be an associative ring with unit. We let the representing space for algebraic K-theory be

K(R) = ΩB(∐_{P} B Aut(P))

where P runs over the isomorphism classes av finitely generated projective Rmodules, and Aut(P) is the group of R-isomorphisms P → P. The projective R-modules are precisely the direct summands of free R-modules, and serve to generalize the sub vector spaces of ℂ^{∞} used in defining topological K-theory.

K(R) is by definition a loop space, so its homotopy groups π_{i}K(R) are groups for i ≥ 0. These were defined by Quillen to be the (higher) algebraic K-groups of the ring R.

Definition. The ith algebraic K-group of the ring R is

K_{i}(R) = π_{i}K(R) = [S^{i}, K(R)] = [S^{i}, ΩB(∐_{P} B Aut(P))],

where S^{i} is the i-sphere.

Number theory

Algebraic K-theory is linked to algebraic number theory by the following formulas.

When R = O_{F} is the ring of integers in a number field F (ℚ ⊆ F ⊂ ℂ)

K_{0}(O_{F}) ≅ ℤ ⊕ Cl(F)  and  K_{1}(O_{F}) ≅ O_{F}^{×}

where Cl(F) is the ideal class group of F, i.e. the group of fractional ideals (O_{F} - modules m ⊂ F satisfying b · m ⊆ O_{F} for some b ≠ 0 in O_{F}), modulo the subgroup of principal fractional ideals (O_{F} -modules m ⊂ F of the form a/b · O_{F} for some a/b ≠ 0 in F).

This group Cl(F) is always finite, and its order is called the class number of F. More generally, Quillen proved that K_{i}(O_{F}) is finitely generated for all i, and

Borel computed the rank of K_{i}(O_{F}) ⊗ ℚ for all i in terms of the number of real and complex embeddings of the number field F. In the principal example F = ℚ with O_{F} = ℤ, we have

K_{i}(ℤ) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & if i = 0 or i ≡ 1 mod 4, i ≠ 1, \\ 0 & otherwise. \end{matrix}

There are conjectures by Lichtenbaum and Quillen, extended by Dwyer and Friedlander, on the precise values of the groups K_{i}(O_{F}) for any number field F. In the case O_{F} = ℤ the order of the torsion in K_{i}(ℤ) can be expressed in terms of numerators and denominators of Bernoulli numbers, and generally by values of zeta functions at negative integers.

When F = ℚ(ξ_{p}) with p a prime and ξ_{p} a primitive pth root of unity, the order of Cl(ℚ(ξ_{p})) is denoted h_{p}.

p is called a regular prime if p |̸h_{p}, and Kummer proved Fermat’s last theorem for such prime exponents, i.e. that there are no positive integers x, y and z with x^{p} + y^{p} = z^{p}.

h_{p} factors as h_{p}^{+} · h_{p}^{−}, where h_{p}^{+} is the class number of the maximal real subfield of ℚ(ξ_{p}), which is ℚ(ξ_{p} + ξ_{p}^{−1}), and h_{p}^{−} is called the relative class number.

A classical unresolved conjecture of Kummer and Vandiver asserts that p |̸h_{p}^{+} for all p, i.e. that for irregular primes p, even if p divides h_{p} it only divides the relative part h_{p}^{−}. This conjecture is known to hold for p < 125.000, but this is not considered sufficient to believe the conjecture on statistical grounds.

The Galois group G = Gal(ℚ(ξ_{p})/ℚ) ≅ ℤ/(p − 1) acts on Cl(ℚ(ξ_{p})), and there are (p − 1) natural idempotents in ℤ_{p}[G] acting on the p-Sylow subgroup A of Cl(ℚ(ξ_{p})), inducing a splitting

A ≅ ∑_{0≤i<p−1} A^{[i]}

of the p-part of the ideal class group into certain eigenspaces. Here ℤ_{p} denotes the p-adic integers. A^{[0]} = A^{[1]} = 0 in general.

A refined version of the Kummer–Vandiver conjecture then asserts that all the even eigenspaces are trivial, i.e. A^{[i]} = 0 for i even. This was shown by Iwasawa to imply that the odd eigenspaces are cyclic groups, of specific orders.

Versions of the follwing theorem have been proved by Bloch–Kato, Kurihara, Banaszak–Gajda and Kolster.

Theorem. The Kummer–Vandiver conjecture holds for p if there is no p-torsion in K_{4i}(ℤ) for every i ≥ 1.

For p odd, the Kummer–Vandiver conjecture holds for p if there is no p-torsion in K_{4}(ℤ(ξ_{p} + ξ_{p}^{−1})).

The Lichtenbaum–Quillen conjecture asserts in particular that K_{4i}(ℤ) = 0 for all i > 0, and hence implies the Kummer–Vandiver conjecture for all primes p.

The theorem K_{4}(ℤ) = 0 then suffices to prove that the eigenspace A^{[p−3]} is trivial and that A^{[3]} is cyclic, for all p. This gives relatively little information on the Kummer–Vandiver conjecture for a general prime p, however.

Differentiable manifolds

By the work of Waldhausen there are applications of algebraic K-theory to the study of the topological group of diffeomorphisms of a smooth manifold.

Let X be a topological space. We view the loop space ΩX as a group up to homotopy under multiplication by composition of loops. We can form a ring up to homotopy

Q(ΩX_{+}) = colim_{n} Ω^{n}(S^{n} ∧ ΩX_{+}) = colim_{n} Map(S^{n}, S^{n} ∧ ΩX_{+})

with addition from loop sum in Ω^{n}, and product from the group up to homotopy ΩX. We think of this as a kind of group ring, and there is a natural map

Q(ΩX_{+}) ⟶ ℤ[π_{1}(X)]

to the genuine group ring on the fundamental group of X.

We cannot quite use our definition of algebraic K-theory to make sense of K(Q(ΩX_{+})), but there is an alternative definition making this precise, due to Waldhausen. It is called the algebraic K-theory of topological spaces, or A-theory

A(X) = “K(Q(ΩX_{+}))” ⟶^{l} K(ℤ[π_{1}(X)]),

and comes equipped with a “linearization” map to algebraic K-theory. The induced map on homotopy groups is a natural transformation from π_{i}A(X) to K_{i}(ℤ[π_{1}(X)]).

By work of Farell and Hsiang, this linearization map is a rational equivalence, i.e. induces an isomorphism on π_{i}(−) ⊗ ℚ. So for a disc D^{n}

π_{i}A(D^{n}) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & if i = 0 or i ≡ 1 mod 4, i ≠ 1, \\ 0 & otherwise. \end{matrix}

Waldhausen proved the following splitting theorem when M is a smooth manifold

A(M) ≃ Q(M_{+}) × Wh^{Diff}(M),

where Ω^{2} Wh^{Diff}(M) ≃ P(M) is the stable smooth pseudoisotopy (concordance) space.

The pseudoisotopy space P (M) is the space of diffeomorphisms of M × I leaving the subspace of the boundary M × {0} ∪ ∂M × I fixed. Here I = [0, 1].

There is a stabilization map P (M) → P(M) which induces an isomorphism on homotopy groups in a stable range, e.g. on π_{i} when i ≪ dim(M)/3.

For example π_{i}P (D^{n}) ⊗ ℚ ≅ ℚ when i ≡ 3 mod 4 and i ≪ n/3. To see this, note that in this range π_{i}P (D^{n}) ≅ π_{i}P(D^{n}) ≅ π_{i+2} Wh^{Diff}(D^{n}) ≅ π_{i+2}A(D^{n}), which rationally is the same as K_{i+2}(ℤ) as given by Borel’s calculation.

There is a map

P (M) ⟶ Diff(M, ∂M)

to the topological group of diffeomorphisms of M leaving the boundary fixed, given by restricting a pseudoisotopy to (M × {1}, ∂M × {1}). With suitable models for these spaces, this is a fibration with fiber the space of diffeormorphisms of M × I relative to the boundary.

Consider the case M = D^{n}. We get a fibration sequence

Diff(D^{n} × I, ∂) ⟶ P (D^{n}) ⟶ Diff(D^{n}, ∂),

and a long exact sequence of homotopy groups

π_{i} Diff(D^{n} × I, ∂) ⟶ π_{i}P (D^{n}) ⟶ π_{i} Diff(D^{n}, ∂) ⟶^{∂} π_{i−1} Diff(D^{n} × I, ∂).

Here ∂ denotes the boundary of the relevant manifold, or the connecting homomorphism, and D^{n} × I ≅ D^{n+1}.

In particular either π_{i} Diff(D^{n}, ∂)⊗ℚ or π_{i} Diff(D^{n}×I, ∂)⊗ℚ must be nontrivial when i ≡ 3 mod 4. Further work yields :

Theorem (Farrell–Hsiang–Jahren). Let i ≪ n/3 be in the pseudoisotopy stable range. Then

π_{i} Diff(D^{n}, ∂) ⊗ ℚ ≅ \{ \begin{matrix}ℚ & when n is odd and i ≡ 3 mod 4, \\ 0 & otherwise. \end{matrix}

π_{i} Diff(S^{n}) ⊗ ℚ ≅ \{ \begin{matrix}ℚ ⊕ ℚ & when n is odd and i ≡ 3 mod 4, \\ ℚ & when n is even and i ≡ 3 mod 4, \\ 0 & otherwise. \end{matrix}

More generally one can use algebraic K-theory to get information about

π_{i} Diff(M, ∂M) ⊗ ℚ

when M is e.g. a smooth manifold with contractible universal cover, and dim(M) large with respect to i.

Computations

We now turn to the question of determining torsion information about K_{i}(R).

Theorem.

K_{∗}(ℤ) ≅ (ℤ, ℤ/2, ℤ/2, ℤ/48, 0, ℤ ⊕ torsion, …).

Here the complete calculation of K_{2}(ℤ) is due to Milnor, K_{3}(ℤ) to Lee and Sczcarba, K_{4}(ℤ) to the author, and the rational information on K_{5}(ℤ) is due to Borel. It should be noted that Lee and Sczcarba had determined K_{4}(ℤ) up to 2– and 3–torsion.

The algebraic K-groups K_{i}(ℤ) are, as we have seen, the homotopy groups of the loop space K(ℤ). In fact K(ℤ) is an infinite loop space, in the sense that we can construct spaces B^{n}K(ℤ) for every n ≥ 0 with Ω^{n}B^{n}K(ℤ) ≃ K(ℤ). This sequence of spaces {B^{n}K(ℤ)} assembles into a spectrum in the sense of algebraic topology, which we denote K(ℤ). We think of K(ℤ) as a space with additional structure analogous to a topological abelian group up to homotopy.

We can now construct a rank filtration of spectra

∗ ≃ F_{0}K(R) ⟶ F_{1}K(R) ⟶ … ⟶ F_{k}K(R) ⟶ … ⟶ K(R)

for quite general rings R, where F_{k}K(R) is a subspectrum of K(R) built using only R-modules of rank less than or equal to k.

Next we recover the filtration subquotient spectra as

F_{k}K(R)/F_{k−1}K(R) ≃ EGL_{k}(R)_{+} ∧_{GL_{k}(R)} D(R^{k})

where D(R^{k}) is a suspension spectrum with GL_{k}(R)-action called a stable building. This is combinatorially defined in terms of cubical diagrams of R-modules.

From the model for the stable building we obtain a spectral sequence

E_{∗∗}^{1} ⟹ H_{∗}^{spec}(F_{k}K(R)/F_{k−1}K(R))

going from an E^{1}-term given by group homology of certain parabolic subgroups of GL_{k}(R), with coefficients twisted by permutation representations. The target is the spectrum homology of the kth subquotient spectrum of the rank filtration.

In the case R = ℤ we can compute these spectral sequences for k ≤ 3, and resolve the extension problems, to find

H_{∗}^{spec}(F_{3}K(ℤ)) ≅ (ℤ, 0, 0, ℤ/2, 0, ℤ ⊕ X, …).

Here X is a group of order dividing four. These computations rely on Soulé’s determination of the group homology of SL_{3}(ℤ).

Further we can prove that for R a principal ideal domain, the stable building D(R^{k}) is at least (k − 2)-connected, and at least (k − 1)-connected when k ≥ 2. Finally we can show that D(R^{4}) is at least 4-connected.

Since the reduced Borel construction EGL_{k}(R)_{+} ∧_{GL_{k}(R)} (−) preserves connectivity, we find F_{3}K(ℤ) → K(ℤ) is at least 4–connected. Hence

H_{∗}^{spec}(K(ℤ)) ≅ (ℤ, 0, 0, ℤ/2, 0, ℤ⊕?, …).

Now there is an Atiyah–Hirzebruch spectral sequence for stable homotopy theory, viewed as a generalized homology theory :

E_{s,t}^{2} = H_{s}^{spec}(K(ℤ); π_{t}^{S}) ⟹ π_{s+t}K(ℤ) = K_{s+t}(ℤ).

It appears as follows in low degrees.

6  ℤ/2  0  0  ℤ/2  ℤ/2  ℤ/2⊕?
5  0  0  0  0  0  0
4  0  0  0  0  0  0
3  ℤ/24  0  0  ℤ/2  ℤ/2  ℤ/24⊕?
2  ℤ/2  0  0  ℤ/2  ℤ/2  ℤ/2⊕?
1  ℤ/2  0  0  ℤ/2  ℤ/2  ℤ/2⊕?
0  ℤ  0  0  ℤ/2  0  ℤ⊕?
0  1  2  3  4  5

8 JOHN ROGNES

By comparison with a model for étale K-theory at two, considered by Bökstedt, we can prove that there is a nontrivial extension in K_{3}(ℤ)

ℤ/24 ⟶ ℤ/48 ⟶ ℤ/2

and that the first nontrivial differential is the surjection

d_{5,0}^{2} : E_{5,0}^{2} ≅ ℤ⊕? ⟶ E_{3,1}^{2} ≅ ℤ/2.

This proves the theorem K_{∗}(ℤ) ≅ (ℤ, ℤ/2, ℤ/2, ℤ/48, 0, ℤ ⊕ torsion, …).