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See Proposition 1 in McDuff, D.; Segal, G. Homology fibrations and the "group-completion'' theorem. Invent. Math. 31 (1975/76), no. 3, 279–284.

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 …

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 …

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 …

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 $1$- …

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/ …

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 …

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 …

(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 …

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 …

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 …

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 …

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 …

(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 …

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 …