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The Homotopy Type of Two-regular K-theory

01 Jan 2003-pp 167-178
TL;DR: In this paper, the authors identify the 2-adic homotopy type of the algebraic K-theory space for rings of integers in two-regular exceptional number fields, in terms of well-known spaces considered in topological Ktheory.
Abstract: We identify the 2-adic homotopy type of the algebraic K-theory space for rings of integers in two-regular exceptional number fields. The answer is given in terms of well-known spaces considered in topological K-theory.

Summary (1 min read)

1. Introduction.

  • For the definition of the algebraic K-theory space K(OE) (resp. K(RE)) the authors refer to [Q1]; it is the usual ‘plus construction’ on the stable classifying space BGL(OE) (resp. BGL(RE)).
  • In particular, the authors have: (a) the spaces K(RE) and K(RE)ét are the same on zero-connected components at the prime 2, and (b) in consequence one can try to adopt the étale homotopy methods from [D-F1] to determine the 2-completed homotopy type of K(RE) (= K(OE) on one-connected components).
  • For a discussion, and further criteria, see proposition 2.2 of [R-Ø].
  • Any real number field is exceptional, while any 2-cyclotomic field Q(ζ2r ), r ≥ 2, is nonexceptional.
  • The authors main result is as follows: Theorem 1.1.

2. The totally imaginary case.

  • E will denote a number field which is 2-regular, exceptional, and totally imaginary.
  • The key adjustments to be made concern their images under φ.
  • The following result is now not surprising, given its similarity to the foregoing: Lemma 2.3.
  • Now the theory of [D-F2] tells us that K(V2) can be identified with the ‘homotopy fixed points’.

3. The case where r1 > 0.

  • The case where E admits real embeddings is (contrary to what one might expect) simplified by the fact that the ‘Z/2 part’ of the model space X can be absorbed into the real embeddings and dealt with there.
  • Parts (i), (ii) require essentially the same as the proof of lemma 2.2.
  • First note that the copies of Spec R are already taken care of by lemma 3.2.
  • Now as before consider the homomorphism from H1(RE) onto Λ ⊂ Aut(µ∞).
  • The authors now have once again to find the space K(X), representing wedges by fibred products over BU .

4. The homotopy groups.

  • As a check on the correctness of the preceding results, it makes sense to show that they imply the results on Ki(RE) = πi(K(RE)) proved in [R-Ø].
  • Z2̂ for i odd and 0 for i even, the question reduces to finding the homotopy groups of the factor the authors have called Jc(q), the homotopy fibre of the composite Fψ−1 j→BU ψ q −1−→ BU where j is the fibre inclusion.
  • From the above, the authors can deduce: Proposition 4.1.
  • This agrees with their result provided the authors can identify w2k with the 2-part of q 2k−.
  • Once again, these give exactly the results computed in [R-Ø].

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THE HOMOTOPY TYPE OF TWO-REGULAR K-THEORY
Luke Hodgkin and Paul Arne Østvær
Abstract. We identify the 2-adic homotopy type of the algebraic K-theory space
for rings of integers in two-regular exceptional number fields. The answer is given in
terms of well-known spaces considered in topological K-theory.
1. Introduction.
Let E be a number field, O
E
its ring of algebraic integers, and R
E
= O
E
[
1
2
] the
corresponding ring of 2-integers. For the definition of the algebraic K-theory space
K(O
E
) (resp. K(R
E
)) we refer to [Q1]; it is the usual ‘plus construction’ on the
stable classifying space BGL(O
E
) (resp. BGL(R
E
)). A related, homotopically
more accessible space is the ´etale topological K-theory space K(R
E
)
´et
of R
E
, see
[D-F1]. The purpose of this paper is to pin down the 2-adic homotopy type of
K(R
E
) for some special E. Our results rely on the following recent advances.
In [R-W1], Rognes and Weibel determined up to extensions the 2-completed groups
K
n
(R
E
)
2
ˆ (= K
n
(O
E
)
2
ˆ for n 2). Their computation is expressed in terms of
the ´etale cohomology groups H
(R
E
; Z
2
ˆ (i)) of R
E
with coefficients twisted by the
action of the roots of unity. A case where the extension problems disappear is the
‘2-regular case’, (see definition below) for which Rognes and Østvær [R-Ø] gave
a complete description of K
n
(R
E
)
2
ˆ , for all n. The above results are among the
consequences of Voevodsky’s solution of the Milnor conjecture [V], as developed in
subsequent work [S-V], [B-L]. A particular interesting end-product is ´etale descent
for K-theory of number fields at 2’, i.e. the strong Quillen-Lichtenbaum conjecture
is true for number fields at the prime 2. See [R-W2] for the case of totally imaginary
number fields, and [Ø1] for the case of real number fields. In particular, we have:
(a) the spaces K(R
E
) and K(R
E
)
´et
are the same on zero-connected components at
the prime 2, and
(b) in consequence one can try to adopt the ´etale homotopy methods from [D-F1] to
determine the 2-completed homotopy type of K(R
E
) (= K(O
E
) on one-connected
components).
Part (b) was solved in the case where E is 2-regular and ‘non-exceptional’ (see
definition below), in [Ø2]; the result is satisfyingly simple, in that the homotopy
type is a product of well-known components. In this paper we apply a similar
analysis to the more complicated ‘exceptional’ case. Here the product structure is
replaced by a twisting (fibration), but the degree of complication is minimal, and
1991 Mathematics Subject Classification. 19D99, 55N15, 55P15.
Key words and phrases. Homotopy type of K-theory spaces, rings of integers in number fields.
The second author was supported in part by NSF.
Typeset by A
M
S-T
E
X
1

2 LUKE HODGKIN AND PAUL ARNE ØSTVÆR
almost all of the factors are untwisted, and again of a simple type. As a check, we
verify that our result gives the homotopy groups calculated in [R-Ø].
To state the results, we need some definitions and notation. Let r
1
respectively
r
2
denote the number of real embeddings respectively pairs of conjugate complex
embeddings of E. We say that E is 2-regular if any of the following equivalent
conditions is satisfied:
1) The ‘modified tame kernel’ of E vanishes. This latter can be identified with the
kernel of the natural surjection:
α
2
: H
2
(R
E
; Z
2
(2))
r
1
M
H
2
(R; Z
2
(2))
=
r
1
.Z/2
summed over the real embeddings of E.
2) The ideal (2) does not split in E and the narrow Picard group Pic
+
(R
E
) of R
E
has odd order.
3)
H
2
(R
E
; Z
2
(i)) =
(
r
1
.Z/2 (i 6= 0 even)
0 (i 6= 1 odd)
For a discussion, and further criteria, see proposition 2.2 of [R-Ø].
Next, let ζ
m
denote an mth root of unity and
ζ
m
its conjugate. For each r, we
have the cyclotomic extension E(ζ
2
r
). E is said to be exceptional if Gal (E(ζ
2
r
)/E)
is not cyclic for some r; otherwise it is non-exceptional. It is easy to check that
E is non-exceptional if and only if ζ
4
E or ζ
2
k + ζ
2
k
E for some k 3. Any
real number field is exceptional, while any 2-cyclotomic field Q(ζ
2
r
), r 2, is non-
exceptional. In our case (exceptional), the description of K(O
E
)
2
ˆ is complicated
by the above-mentioned twisting, and we require some names for the spaces which
will be our building blocks.
We consider all spaces completed at the prime 2, where not explicitly stated; we
hope that the reader will accept statements such as π
1
(S
1
) = Z
2
ˆ which result
from this. We also abusively write Spec R for what is properly the ´etale homotopy
type (Spec R)
´et
. As usual U, O are the stable unitary and orthogonal groups, and
BU, BO their classifying spaces. The complexification c maps O into U as a sub-
group, with quotient U/O. For q odd, let ψ
q
be the Adams operation on BU. Note
in particular that ψ
1
is the conjugation map. By Quillen’s fundamental result
[Q2], when q is an odd prime-power the K-theory space of the finite field F
q
is the
fibre of ψ
q
1 : BU BU; we shall denote this space by F ψ
q
. Similarly, we write
F ψ
1
for the fibre of ψ
1
1. We require two variants of F ψ
q
:
1) In analogy with a construction of okstedt, define JK(q) to be the fibre of the
composite:
BO
c
BU
ψ
q
1
BU
(This is K(Z) if q ±3 mod 8.)
2) Let j : F ψ
1
BU be the inclusion of the fibre, and define J
c
(q) to be the
fibre of the composite (ψ
q
1) j : F ψ
1
BU . (Note the analogy between this
construction and the previous one.)
Our next points concern Galois groups. Let µ
(E) respectively µ
be the group
of 2-primary roots of unity in E respectively C, and let Γ
0
E
= Gal(E(µ
)/E). The
natural action of Γ
0
E
on µ
gives a monomorphism
(1) φ : Γ
0
E
Aut(µ
)
=
Z
2
ˆ Z/2

THE HOMOTOPY TYPE OF TWO-REGULAR K -THEORY 3
compare §1 of [Mi2]. By considering π
1
(Spec R
E
) as the Galois group of the
maximal unramified extension of R
E
, we see that the action of π
1
(Spec R
E
) on µ
factors through Γ
0
E
to give a composite:
(2) π
1
(Spec R
E
) Γ
0
E
φ
Aut(µ
)
which we shall call
ˆ
φ. Clearly the images of φ and
ˆ
φ in Aut(µ
) are the same
subgroup, say Λ. If E
0
= E(
1), it is a consequence of the ‘exceptional’ condition
on E that Γ
0
E
= Γ
E
× Z/2, where Γ
E
= Gal (E(µ
)/E
0
). Still following [Mi2], set
a
E
= ν
2
(|µ
(E
0
)|) (the 2-adic valuation). Then Λ is (topologically) generated by
elements q, σ where σ (order 2) is conjugation and q Z
2
ˆ is represented by any
integer such that q is ±1 mod 2
a
E
but not mod2
a
E
+1
. By
ˇ
Cebotarev’s theorem
we can always choose a prime P in R
E
such that the order q of the finite field
R
E
/(P) is an integer with these properties.
For future reference, we define numbers w
m
= w
m
(E) by: w
m
= 2 (m odd),
w
m
= 2
a
E
+ν
2
(m)
(m even). (Compare e.g. the definition in [R-W1], which is
equivalent in the exceptional case. Mitchell [Mi2] writes w
i
for the exponents,
rather than the powers of 2.)
Our main result is as follows:
Theorem 1.1. With the preceding notation, let E be 2-regular and exceptional.
Then at the prime 2:
(i) If E is totally imaginary (r
1
= 0), K(R
E
) is homotopy equivalent to the product
J
c
(q) ×
r
2
1
Y
U
(r
2
1 factors in the product)
(ii) If r
1
> 0, K(R
E
) is homotopy equivalent to the product
JK(q) ×
r
2
Y
U ×
r
1
1
Y
(U/O)
Our strategy in proving this result is based on ideas present in [D-F1], and is
as follows. We find a space X, and map f : X Spec R
E
which induces an
isomorphism on mod 2 homology. The space X is a wedge of r
2
+ 1 circles and
r
1
copies of the infinite projective space RP
. The key question is then how the
wedge components of X map via the composite:
(3) π
1
(X)
f
π
1
(Spec R
E
)
ˆ
φ
Aut(µ
)
Specifically, the condition required is that Im(
ˆ
φ f
) = Im(
ˆ
φ) = Λ. According to
[D-F1], in the given circumstances, K(R
E
) is homotopy equivalent to a space called
K(X); and to find K(X) we need to know the wedge components, and the way that
their fundamental groups map into Aut(µ
). We therefore find these, simplify as
much as possible, and our theorem will follow from Dwyer and Friedlander’s result.
As can be seen from the statement of the theorem, cases (i) and (ii) need separate
treatment; they are dealt with in sections 2, 3 respectively.

4 LUKE HODGKIN AND PAUL ARNE ØSTVÆR
It should be noted that all of these computations could have been done at the time
of the original paper [D-F1], modulo replacing the K-theory spaces by their ´etale
topological versions; indeed, our reliance on the methods of Dwyer and Friedlander
is substantial. However, the ´etale descent results we mentioned in the beginning
make it possible to state the results for the K-theory spaces themselves.
A general solution to the problem of finding K-theory spectra of number rings at
the prime 2 has been undertaken by Mitchell [Mi1, Mi2]. The results are fuller,
and give in fact the 2-adic homotopy type when combined with [R-W2] and [Ø1].
But their interpretation requires knowledge of the ’Iwasawa module’, which appears
difficult in general.
2. The totally imaginary case.
In this section, E will denote a number field which is 2-regular, exceptional, and
totally imaginary. We can deduce immediately (cf. [R-Ø]):
Lemma 2.1. The mod 2 homology of R
E
is given by:
H
1
(R
E
; Z/2) = (r
2
+ 1).Z/2
H
i
(R
E
; Z/2) = 0 (i > 1).
However, in this case we can do better, since the 2-completed homology is also
simple; for this we write H
i
( ), omitting the coefficients.
Lemma 2.2. The 2-completed homology of R
E
is given by:
H
1
(R
E
) = (r
2
+ 1).Z
2
ˆ
H
i
(R
E
) = 0 (i > 1).
Proof. Let α be the unique prime above 2 in O
E
. Then by theorem 2.2 of [D-F1],
H
1
(R
E
) is a quotient of (O
E
)
α
ˆ (since the narrow Picard group vanishes); that is,
of a direct sum of copies of Z
2
ˆ and 2-torsion groups. From the description of the
mod 2 homology in lemma 2.1, there is no 2-torsion, so H
1
is a sum of copies of
Z
2
ˆ , and the rank, again from lemma 2.1, is r
2
+ 1.
Next, recall from §1 the homomorphism
ˆ
φ : π
1
(Spec R
E
) Aut(µ
). In [Ø2]
(following a model from [D-F1]) it was shown that in the non-exceptional case
there is a map, f : X Spec R
E
, inducing an isomorphism on mod 2 homology,
where:
(i) X is a wedge of r
2
+ 1 circles;
(ii) The circles can be chosen in such a way that the first one maps to a topological
generator of Z
2
ˆ , and the rest map trivially.
Our first result in this section is parallel to this, if slightly more complicated. It
states:
Proposition 2.1. In the exceptional totally imaginary case, there is a map f :
X Spec R
E
such that:
(i) X is a wedge of r
2
+ 1 circles;
(ii) The first circle, considered as an element of π
1
(X), maps under
ˆ
φf
to q Λ;
the second to the non-trivial element σ of order 2 in Λ; and the rest (if any) map
trivially;

THE HOMOTOPY TYPE OF TWO-REGULAR K -THEORY 5
(iii) f induces an isomorphism on mod 2 homology.
Proof. This is essentially elementary topology, using what we know of the homol-
ogy of Spec R
E
. In fact, we can clearly choose r
2
+ 1 maps from S
1
to Spec R
E
whose images under the Hurewicz map generate H
1
(R
E
) (Z
2
ˆ or Z/2 coefficients).
The key adjustments to be made concern their images under φ. We know the
structure of Im(
ˆ
φ) = Λ, and since the latter is Abelian,
ˆ
φ factors through H
1
(R
E
).
Hence we can find maps f
1
, f
2
: S
1
Spec R
E
which represent generators of H
1
,
such that f
1
maps under φ to q, and f
2
maps to σ.
Now let f
3
, . . . , f
r
2
+1
: S
1
Spec R
E
represent the remaining generators of H
1
.
We can multiply (in π
1
) by suitable powers of f
1
, f
2
to obtain maps g
3
, . . . , g
r
2
+1
which still define a basis of H
1
together with f
1
, f
2
, and map trivially under
ˆ
φ. We
can now use f
1
, f
2
, g
3
, . . . , g
r
2
+1
to construct a map f from the wedge X of r
2
+ 1
circles to Spec R
E
which has the properties claimed in proposition 2.1.
Now proposition 3.2 of [D-F1] tells us that f induces a homotopy equivalence from
K(R
E
) to a space K(X), whose definition strictly depends not on X as space but
on the composite map
X
f
Spec R
E
Spec Z[
1
2
].
Again using the methods of [D-F1], we write X = V
1
V
2
W , where V
1
, V
2
are the
first two circles and W is the wedge of the remaining ones. There is a fibre square
K(X)
//
K(V
1
V
2
)
K(W )
//
K() = BU
and K(W ), by the arguments of [D-F1] proposition 4.5, is the unpointed function
space BU
W
= BU ×
Q
r
2
1
U. It follows easily that:
Proposition 2.2. The space K(X) is homotopy equivalent to the product:
K(V
1
V
2
) ×
r
2
1
Y
U
And our main challenge is to identify the space K(V
1
V
2
). For this we have a
second fibre square:
K(V
1
V
2
)
//
K(V
1
)
K(V
2
)
//
BU
The space K(V
1
) is essentially well-known, and often used, being the ‘finite fields’
K-theory space F ψ
q
of Quillen [Q2]. It follows from the choice of the prime P and
the integer q in §1 that the composite
S
1
Spec R
E
/(P) Spec R
E

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  • ...For a fuller discussion see Hodkin and Østvær [137]....

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Cites background or methods from "The Homotopy Type of Two-regular K-..."

  • ...When this is combined with the fact that the corresponding algebraic K-theory square (1) is homotopy cartesian (which is shown in both [26] and [38], see also Appendix A), and the induction methods for hermitian K-groups in [8, §3], we deduce that the morphism of spectra (6) induces an isomorphism of all homotopy groups both rationally and with finite 2-group coefficients....

    [...]

  • ...introduced by Bökstedt for the rational case (r = 1) in [10], and in [26], [38] for the general case; see also Appendix A....

    [...]

  • ...In this appendix we deduce the homotopy cartesian square of K-theory spectra (1) using the space level results given in [26]....

    [...]

  • ...Similarly, the K-theoretic theorem of [26] and [38] in case (i) asserts that (i) implies (iii)....

    [...]

  • ...0 → W (Fq) ⊕ r ⊕ W ′(R) → 1KQ0(Fq) ⊕ r ⊕ 1KQ0(R) → K0(Fq) ⊕ r ⊕ K0(R) → 0 ↓ ↓ ↓ 0 → r ⊕ W ′(C) → r ⊕ 1KQ0(C) → r ⊕ K0(C) → 0 By [26] and [38], K0(RF ) is the kernel of the rightmost vertical map....

    [...]

References
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Book ChapterDOI
01 Jan 1973

2,106 citations

Book ChapterDOI
14 Dec 1999

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"The Homotopy Type of Two-regular K-..." refers background in this paper

  • ...Anderson in his thesis from 1963, but see also Atiyah’s paper ‘K-theory and Reality’ [At]....

    [...]

  • ...The homotopy groups of Fψ−1 (the classifying space of self-conjugate K-theory) were computed by D. Anderson in his thesis from 1963, but see also Atiyah’s paper ‘K-theory and Reality’ [At]....

    [...]

Book ChapterDOI
01 Jan 2000
TL;DR: In this paper, it was shown that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of smooth (smooth) varieties with finite coefficients is equivalent to the Bloch-KatoConjecture, relating Milnor K-theory to Galois cohomologies.
Abstract: In this paper we show that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of (smooth) varieties with finite coefficients is equivalent to the Bloch-Kato Conjecture, relating Milnor K-theory to Galois cohomology The latter conjecture is known to be true in weight 2 for all primes [M-S] and in all weights for the prime 2 [V 3]

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Frequently Asked Questions (9)
Q1. What contributions have the authors mentioned in the paper "The homotopy type of two-regular k-theory" ?

In this paper, the authors showed that the 2-adic homotopy type of the algebraic K-theory space for rings of integers in two-regular exceptional number fields is known. 

The homotopy groups of Jc(q) are zero in even dimensions; for the odd ones the authors have:π4k−1(Jc(q)) = Z2̂ ⊕ (Z/(q2k − 1))2and there is a split exact sequence0 → Z2̂ → π4k+1(Jc(q)) → Z/2 → 0Proof. 

By Quillen’s fundamental result [Q2], when q is an odd prime-power the K-theory space of the finite field Fq is the fibre of ψq − 1 : BU → BU ; the authors shall denote this space by Fψq. 

The space K(X) is homotopy equivalent to the product:K(V1 ∨ V2) × r2−1 ∏ UAnd their main challenge is to identify the space K(V1 ∨ V2). 

Their computation is expressed in terms of the étale cohomology groups H∗(RE ; Z2̂ (i)) of RE with coefficients twisted by the action of the roots of unity. 

Then at the prime 2: (i) If E is totally imaginary (r1 = 0), K(RE) is homotopy equivalent to the productJc(q) × r2−1 ∏ U(r2 − 1 factors in the product) (ii) If r1 > 0, K(RE) is homotopy equivalent to the productJK(q) × r2 ∏ U × r1−1 ∏ (U/O)Their strategy in proving this result is based on ideas present in [D-F1], and is as follows. 

By Čebotarev’s theorem the authors can always choose a prime P in RE such that the order q of the finite field RE/(P) is an integer with these properties. 

¿From this it is clear that K(V1 ∨ V2) is the fibre of the composite (ψq − 1) ◦ j : Fψ−1 → BU , which is the space denoted Jc(q) in §1. 

The homotopy groups of JK(q) are as follows:π8k = 0π8k+2 = Z/2π8k+3 = Z/2w4k+2π8k+4 = 0π8k+5 = Z2̂π8k+6 = 0π8k+7 = Z/w4k+4and there is a split short exact sequence0 → Z2̂ → π8k+1 → Z/2 → 0