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Results tagged with kt.k-theory-and-homology
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user 3931
Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
30
votes
Accepted
Plus construction considerations.
Here are some thoughts, gathered from reading many texts about algebraic K-theory. Let me start with some historical remarks, then try to give a more revisionist motivation of the plus construction.
…
- 9,283
18
votes
Adams' theorems on the Hopf-Whitehead J-homomorphism
This answer probably won't be coming from the perspective that you want, since it'll use even more stable homotopy theory than Adams did. But I think it's pretty clear in its own way.
First let me m …
- 9,283
15
votes
Maps from mod-$p$ Eilenberg-MacLane spectrum to connective $K$-theory spectrum
You can also cheat: since HZ/p is connective,$$ [HZ/p,ku] = [HZ/p,KU] = [HZ/p \wedge KU,KU]_{KU-modules} = 0$$ since $HZ/p \wedge KU = 0$.
Edit: Let me stress that the other answers give more informa …
- 9,283
12
votes
Accepted
Why does one invert $G_m$ in the construction of the motivic stable homotopy category?
I guess there are "internal" and "external" motivations. External for instance -- most natural examples of functors we have from the stable motivic homotopy category to some other category invert G_m …
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11
votes
Accepted
K-Theory space of finite abelian groups
Everything is known. In fact as spectra we have canonically $K(\mathsf{finAb}) = \vee_p K(\mathbb{F}_p)$, and the spectra $K(\mathbb{F}_p)$ are identified in the work of Quillen (see e.g. http://www. …
- 9,283
10
votes
Accepted
Is there a categorification of topological K-theory?
The abelian group $K^0(X)$ has a natural $(\infty,0)$-categorification'', meaning a spectrum $K(X)$ whose $\pi_0$ recovers $K^0(X)$: take $K(X)$ to be the spectrum of maps from (the suspension spectru …
- 9,283
8
votes
When do non-exact functors induce morphisms on $K$-theory?
I think the idea should be that "polynomial maps" from A to B induce maps on the K-theory spaces. Here's one possible suggestion of a way to implement this, based on Segal's proof of the Kahn-Priddy …
- 9,283
8
votes
Accepted
K-theory, monoidal vs. exact
As you note, the classifying space of any exact category is contractible because of the presence of an initial object, so your second K(D) (which I take to be the group completion of the classifying s …
- 9,283
5
votes
Accepted
Algebraic K-theory "with proper support"
Actually, the possibility of defining such a thing was one of my motivations for studying this condensed mathematics in the first place. That said, the story is far from complete.
First, I guess a re …
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3
votes
Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles
You might have luck with the "universal" choice of $h$, namely the cohomology theory given by the connective spectrum corresponding to the $E_{\infty}$-space of stable spheres under smash product (as …
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