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28
votes
Does Milnor K-Theory arise from Waldhausen K-Theory
Bob Thomason proved that there is no Milnor K-theory functor for schemes,
with a reasonable map to Quillen K-theory, in:
Le principe de scindage et
l'inexistence d'une $K$-theorie de
Milnor gl …
27
votes
Motivation/interpretation for Quillen's Q-construction?
Quillen's $Q$-construction naturally arises as a cleaned-up version of Segal's edgewise
subdivision of Waldhausen's $s_\bullet$-construction.
Waldhausen's $s_\bullet$-construction gives a rather
natur …
16
votes
Is there a simple relationship between K-theory and Galois theory?
A simpler statement may be that if $F \to E$ is a $G$-Galois extension, then
there is a map $K(F) \to K(E)^{hG}$ from the algebraic $K$-theory space of $F$
to the $G$-homotopy fixed points of the alge …
13
votes
Morava on Shafarevich conjecture
(3) The statement is that the map of ring spectra $S \to HZ$ induces a rational equivalence $K(S) \to K(Z)$. A reference is Proposition 2.2 in:
Waldhausen, Friedhelm: Algebraic
$K$-theory of to …
8
votes
Detection of stable homotopy by K-theory spectra
I would say that the historically correct place to start is Quillen's letter to Milnor on the image of $(\pi_i O \to \pi_i^s \to K_i\mathbb{Z})$, published in Springer LNM 551 (1976). There Quillen pr …
8
votes
Good reference for topological Hochschild homology
Ib Madsen's survey
MR1474979 (98g:19004) Madsen, Ib Algebraic K-theory and traces.
Current developments in mathematics, 1995 (Cambridge, MA), 191–321,
Int. Press, Cambridge, MA, 1994.
was written to …
7
votes
Waldhausen $K$-theory for $G$-spaces
I did a little work on this problem in Bielefeld in 1991, based on a suggestion by Waldhausen.
Let $G$ be a finite group. The algebraic $K$-theory $A^G(X)$ of the category of finite retractive $G$-sp …
6
votes
Accepted
Map between homotopy groups of O, related to J-homomorphism and K-theory of Z
The composition is trivial. Composition with $\eta$ acts trivially on the source and nontrivially on the target, for positive $s$, which implies the claim. This argument does not apply for $s=0$, beca …
6
votes
Can any suspension spectrum be realized as Waldhausen K-theory?
(1) Given a simplicial monoid $G$ let $R^0(*, G)$ be the Waldhausen category of pointed finite free $G$-simplicial sets weakly equivalent to $(\coprod^k G)_+$, for varying $k\ge0$. This is the special …
5
votes
Accepted
Why does the map $BG\to A(*)$ fail to split?
Question 1: There are several arguments.
In degree 2 there is a reference: the proof of corollary 3.7 of Waldhausen's "Algebraic K-theory of spaces, a manifold approach". See http://www.math.uni-bie …
5
votes
Accepted
Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$?
The infinite loop space/spectrum level statements were written down in
May, Quinn, Ray, Tornehave:
"$E_\infty$ Ring Spaces and $E_\infty$ Ring Spectra" (1977) http://www.math.uchicago.edu/~may/BOOKS/ …
5
votes
Accepted
Descent properties of topological Hochschild homology
In Theorem 1.2 of B. I. Dundas and J. Rognes: "Cubical and cosimplicial descent", Journal of the London Mathematical Society (2) 98 (2018) 439-460, DOI 10.1112/jlms.12141, we showed that for each $1$- …
5
votes
Accepted
Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?
A published reference for the claim (right after the question in boldface) is the proof given by Thomason on pages 1657-1658 of "First quadrant spectral sequences in algebraic K-theory via homotopy co …
5
votes
Group completion of a monoid (braid groups)
See Proposition 1 in
McDuff, D.; Segal, G.
Homology fibrations and the "group-completion'' theorem.
Invent. Math. 31 (1975/76), no. 3, 279–284.
4
votes
Injectivity of assembly in A-theory for $BC_2 = \mathbb R P^\infty$ in degree $4$
Not an answer, but a possible approach: Using Dundas' cartesian square with corners $A(*)$, $K(Z)$, $TC(*)$ and $TC(Z)$ you can see that $Wh^{Diff}_3(*) = Z/2$ comes from $TC_4(Z) = Z/8$ (plus odd tor …