The Homotopy Type of Two-regular 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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Citations
177 citations
Cites background from "The Homotopy Type of Two-regular K-..."
...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....
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...introduced by Bökstedt for the rational case (r = 1) in [10], and in [26], [38] for the general case; see also Appendix A....
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...In this appendix we deduce the homotopy cartesian square of K-theory spectra (1) using the space level results given in [26]....
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...Similarly, the K-theoretic theorem of [26] and [38] in case (i) asserts that (i) implies (iii)....
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...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....
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References
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652 citations
"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]....
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...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]....
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Frequently Asked Questions (9)
Q2. what is the homotopy group of jc(q)?
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.
Q3. What is the simplest way to denote the fibre of a finite field?
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.
Q4. What is the main challenge to identify the space K(X)?
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).
Q5. What is the étale cohomology of RE?
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.
Q6. What is the simplest way to prove this?
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.
Q7. What is the simplest way to determine the order of the finite field RE?
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.
Q8. What is the definition of the fibre of the composite?
¿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.
Q9. What is the exact sequence of the jo?
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