added to *KK-theory* brief remark and reference to relation to stable $\infty$-categories / triangulated categories

added to *equivariant K-theory* comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

(Also finally added references to Green and Julg at *Green-Julg theorem*).

This all deserves to be prettified further, but I have to quit now.

]]>The Karoubian envelope is also used in the construction of the category of pure motives,

and in K-theory.

Although there is a lot of online notes/courses available where is precisely explaned how taking Karoubian envelope

is involved in the construction of (pure) motives, there seems to be a serious lack of sources where is explained how

the Karoubian envolope is involved in construtions in K-theory.

(appart from the 'basic' construction of algebraic K-theory K_0 (A) for a ring A as K_0 (P_A), where P_A

is the category of finitely generated A-modules, where P_A can also be recognized as Karoubi completion of the

category F_A of finite generated free A-modules.

Nevertheless this construction of algebraic K_0 might be considered as a 'toy' example.

Is there in the quoted sentence above also referred to certain constructions in K-theory in more general setting (eg for K-groups of exact or Waldhausen-categories)

which make use of the Karoubian completion? ]]>

added details of the statement to *fundamental product theorem in topological K-theory*

(nothing like a proof yet)

One day when I have the leisure, I might follow up on my conjecture that under the translation of K-theory to D-brane physics, the fundamental product theorem in K-theory is the *Myers effect*.

I am looking for a decent account of the homotopy ring spectrum structure on $KU$ with $KU_0 = BU \times \mathbb{Z}$ that would be self-contained for a reader with good point-set topology background, but not involving $E_\infty$ or model category theory.

What I find in the literature is all sketchy, but maybe I am looking in the wrong places.

First, a discussion of the H-space structure on $BU\times \mathbb{Z}$ in the first place I find on p. 205 (213 of 251) in *A Concise Course in Algebraic Topology*. But for the crucial step it there only says:

we merely affirm that, by fairly elaborate arguments, one can pass to colimits to obtain a product

Is there a reference that would spell this out?

Next, for the proof of the homotopy ring spectrum structure on $KU$, the idea is indicated on the first page of

James McLure, *$H_\infty$-ring spectra via space-level homotopy theory* (pdf)

Is there a place where this would be spelled out in some detail?

]]>I have started at *topological K-theory* a section “For non-compact spaces” (here).

I have begun an entry

meant to contain detailed notes, similar in nature to those at *Introduction to Stable homotopy theory* (but just point-set topology now).

There is a chunk of stuff already in the entry, but it’s just the beginning. I am announcing this here not because there is anything to read yet, but just in case you are watching the logs and are wondering what’s happening. In the course of editing this I am and will be creating plenty of auxiliary entries, such as *basic line bundle on the 2-sphere*, and others.

I’ve seen the recent Lurie paper, *Rotation invariance in K-theory*; the main theorem states that the Waldhausen K-theory $\mathcal{C}\mapsto K(\mathcal{C})$ is invariant under the action of $S^1$. What’s the significance of this result? Can it be generalized to actions of $S^n$?

I am giving *fiber integration in K-theory* a dedicated entry.

One section *In operator K-theory* used to be a subsection of *fiber integration in generalized cohomology*, and I copied it over.

Another section *In terms of bundles of Fredholm operators* I have now started to write.

Created a stub at Milnor K-theory, which is now just an MO answer of Cisinski. To be expanded at some later point when I study this in more detail.

]]>added at *Beilinson regulator* a section *Geometric constructions* such as to finally give a canonical home to the pointer to Brylinki’s article that David Roberts keeps highlighting (in other threads).

created *Dirac induction* with a brief note on the relation to the orbit method, via FHT-II.