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61
votes
10answers
11k views

Motivation for algebraic K-theory?

I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...
37
votes
2answers
2k views

What arithmetic information is contained in the algebraic K-theory of the integers

I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
13
votes
2answers
2k views

What's about “quantum modular forms”?

Do you know where one could read on "Modular Forms, K-theory and Knots"? The combination of themes sounds thrilling! Edit: Zagier's paper on "quantum modular forms" will be published in Clay's ...
15
votes
7answers
2k views

universal cover of SL2(R): does it admit central extensions?

Is it true that the universal cover of SL2(ℝ) has no non-trivial central extensions... as an abstract group? (that's certainly true as a Lie group) Motivation: I have a projective action of ...
12
votes
1answer
780 views

Rationalised K-theory of number fields

Let $A$ be the ring of integers in a number field, and consider the rationalised algebraic $K$-theory groups $\mathbb{Q}\otimes K_*(A)$. A theorem of Borel calculates the ranks of these groups; the ...
4
votes
1answer
846 views

Atiyah class for non-locally free sheaf

Let $E$ be a holomorphic vector bundle over a compact complex manifold (or projective algebraic variety) $X$. The Atiyah class of $E$, $a(E)\in Ext^1(T_X,End(E))$, is defined to be the class of the ...
11
votes
3answers
865 views

What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?

More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...
6
votes
3answers
1k views

The localisation long exact sequence in K-theory over an arbitrary base

If I work over a field k,write D for the formal disk k[[t]] and Dx for the formal punctured disk k((t)), then there is an associated long exact sequence in algebraic K-theory ... Kn+1(Dx) --> Kn(k) ...
9
votes
2answers
418 views

Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that $$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$ i.e. the connected ...
8
votes
2answers
467 views

is there a p-adic Borel theorem?

Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. ...
1
vote
1answer
928 views

Z/48 and Moonshine Beyond the Monster

I am interested in pursuing an understanding of K-theory. Primarily, the $K_3(\mathbb{Z})$ algebraic K-group over ring of integers of an algebraic number field and its relationship to the ...
9
votes
2answers
399 views

Genus of smooth varieties with small Chow group

Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = ...
5
votes
6answers
2k views

Differences between reflexives and projectives modules

Let R be a normal noetherian domain. What is the difference between a finitely generated reflexive module and a finitely generated projective module? Can anybody recommend any references or make ...
4
votes
1answer
278 views

Adams graded parts of rational K-theory of a number field.

Let $F$ be a number field and $r_{1}$ and $r_{2}$ the numbers of real and pairs of complex embeddings respectively of $F$. Then Borel computed that for $n\geq 2$ $$ K_{n}(F)_{\mathbb{Q}}\simeq ...
2
votes
0answers
162 views

K-theory and completion

I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community. I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...