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13
votes
2answers
944 views

Can anyone explain to me what is an assembly map?

Or can you give me a good place to read about things related to assembly map, besides wikipedia? I am specially interested in the case of algebraic K-theory. Would appreciated if you could provide ...
2
votes
0answers
739 views

Online Number Theory Video?

Are there any graduate level number theory course available on line ? The only video series I am aware of are some MSRI videos, and Ted Chinburg's courses http://www.math.upenn.edu/~ted/noframes.html, ...
7
votes
1answer
322 views

The K-theoretic Farrell-Jones conjecture for cat(0) groups

Is the fibered K-theoretic farrell-jones conjecture true for cat(0)-groups?
5
votes
1answer
309 views

When is a cube of cofibrations are “lattice”?

Let $C$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(C)$ the category with cofibrations consisting of sequences of $n$ cofibrations ...
10
votes
1answer
375 views

Eilenberg-Mazur swindle for higher K groups

The Eilenberg-Mazur swindle shows that the Grothendieck group of an additive category with countable coproducts is trivial. The strategy is to observe that any "Euler characteristic" $\chi$ on such a ...
4
votes
0answers
227 views

Vanishing of Nisnevich cohomology of K-theory over a one-dimensional local ring.

Let $A$ be a one-dimensional, Noetherian, local ring, and let $\mathcal{K}_n$ denote the sheafification of $K_n$ (degree $n$ $K$-theory) on the Nisnevich site of $X:=\mbox{Spec }A$. Then is it true ...
4
votes
1answer
271 views

A Reference on Multicategories with “Internal Hom”

The multicategory of Waldhausen categories is "enriched over itself": the Hom-set of $k$-exact functors can be given a Waldhausen category structure by letting the morphisms be natural ...
0
votes
0answers
262 views

A modified version of K-theory for manifolds ?

If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
8
votes
1answer
405 views

Nonnegative additive functions on coherent sheaves

Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...
3
votes
0answers
318 views

K-theory of differential graded modules over differential graded algebras

Suppose you have a smooth vector bundle $E$ over a smooth manifold $X$. If you consider the algebra $ \Omega^\ast (E)$ of differential forms on $E$, it will be homotopy equivalent to the algebra of ...
5
votes
1answer
333 views

K-theory and regular rings

Let $R$ be a noetherian (commutative) ring. It is a well-known fact that for $R$ regular, $K$-theory of (finitely generated) projective modules and $K$-theory of arbitrary (f.g.) modules agree. Does ...
15
votes
1answer
422 views

When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
9
votes
2answers
375 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} = ...
9
votes
1answer
653 views

Current status of a conjecture of Bloch

In the seminal paper $K_2$ and algebraic cycles, Bloch make the following conjecture : Suppose $A$ is a local Noetherian integral domain with quotient field $F$ $K_2(A)$ → $K_2(F)$ is ...
2
votes
1answer
243 views

Gersten for homotopy invariant K-theory of non-singular varieties.

Here is the question: if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective? If $X$ is ...
8
votes
2answers
337 views

Algorithm to calculate $Wh(G)$ for finitely presented group $G$?

Let $G$ be a finitely presented group. Are there any algorithm to calculate whitehead group $G$, $Wh(G)$ in terms of presentation of $G$?
1
vote
2answers
544 views

Nerves of simplicial objects in categories/Waldhausen's S-construction

Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences? To give this some context, I'd ...
28
votes
3answers
2k views

How much linear algebra can be done with graphs?

Let G be a finite directed acyclic graph, with sources $A=\{a_1,\ldots,a_n\}$ and sinks $B=\{b_1,\ldots,b_n\}$, with edge weights $w_{ij}$. The weight of a directed path P is the product of weights of ...
10
votes
1answer
688 views

When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims. On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams ...
12
votes
2answers
1k views

Why was it reasonable to ask what the higher K-groups are?

To say I am a novice in $K$-theory is to overstate my experience with the field. I've been reading the various wiki articles so as to have some preparation before jumping in, and I couldn't answer the ...
11
votes
1answer
761 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 ...
41
votes
6answers
3k views

Which of Quillen's Papers Should I read?

I just heard that Dan Quillen passed on. I am not familiar with his work and want to celebrate his life by reading some of his papers. Which one(s?) should I read? I am an algebraic geometry who is ...
6
votes
1answer
578 views

Galois descent for K-groups (or for étale cohomology groups)

Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$. When is $$ ...
15
votes
7answers
1k 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 ...
3
votes
2answers
311 views

Uses of the Chern--Connes Pairing?

The backbone of Connes' approach to noncommutative geometry is the Chern--Connes pairing. By discovering the cyclic homology of an algebra and then pairing it the $K$-theory of that algebra, Connes ...
3
votes
1answer
230 views

A detail in the construction of the coarse index of a Dirac operator in “Roe: An Index Theorem on Open Manifold, I”

Hi, I'm currently wreading "Roe: An Index Theorem on Open Manifolds, I, J. Differential Geometry 27 (1988), p. 87-113" and there is a detail in the construction of the coarse index of a Dirac ...
8
votes
1answer
833 views

Explicit description of boundary map in algebraic K-theory

Recall that for a DVR A with fraction field F and residue field k, there is a "localization" fiber sequence in algebraic K-theory, $$K(k) \rightarrow K(A) \rightarrow K(F).$$ In Remark 5.17 of his ...
4
votes
1answer
268 views

A question on equivariant K-theory

Let $G$ be a connected linear algebraic group over $\mathbb{C}$ and $X$ a $G$-variety. Let $K_G^0(X)$ be the Grothedieck group of coherent $G$-equivariant sheaves on $X$, and $K_G^i(X)$ for $i>0$ ...
4
votes
1answer
735 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 ...
7
votes
3answers
2k views

“Must read” papers in algebraic K-theory?

I'm mainly interested (graduate student) in surgery theory and geometric topology. If I have a chance to suggest "must read" papers in geometric topology for beginner, I'm very glad to suggest ...
5
votes
2answers
437 views

Why does the map $BG\to A(*)$ fail to split?

There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus ...
4
votes
1answer
266 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 ...
4
votes
1answer
339 views

Is there a clean definition of the residue map in Milnor K-theory?

If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of ...
3
votes
0answers
422 views

Algebraic description of double vector bundles.

It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated ...
5
votes
1answer
974 views

Geometric Realization of a Simplicial Category

Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, ...
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 ...
3
votes
1answer
287 views

Explain the relation between $K_0$ and morphisms of Chow motives

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to ...
6
votes
0answers
264 views

homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence

Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
5
votes
0answers
287 views

Inverse Galois Problem…and parallelizable vector fields?

Usual approaches to the inverse Galois problem start with realizations of a group $G$ over a larger field, and then try to specialize to ${\Bbb Q}$. One could also start by building suitable objects ...
4
votes
1answer
293 views

Algebraic K-groups and braids

This is (I think) a reference request: Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
5
votes
3answers
650 views

Group rings of infinite products of groups

Given a infinite family of groups $(G_i)$ for $i\in I$. Is there a ring theoretic construction, that produces $R[\prod_{i\in I} G_i]$ using only the rings $(R[G_i])_{i\in I}$ ? For the case of a ...
9
votes
2answers
1k views

Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules?

I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of ...
2
votes
1answer
853 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 ...
11
votes
1answer
357 views

Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences

A nice property of $\mathbb P^n$ is: Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect. (for example, any 2 curves in $\mathbb ...
21
votes
1answer
1k views

Morava on Shafarevich

Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture. The Shafarevich Conjecture states: $Gal(\bar Q / Q_{cycl})$ is free. ...
10
votes
1answer
1k views

Values of zeta at odd positive integers and Borel's computations

Someone recently quoted to me this recent article that claims to prove that $\zeta(2n+1) \notin (2\pi )^{2n+1} \mathbb{Q}$. I always assumed this was well known. More precisely I thought this result ...
0
votes
2answers
422 views

Definition for fundamental group (higher homotopy groups) for a category?

How to define homotopy groups in categories as in Quillen's definition for Higher algebraic K-theory: K_i(M)=\pi_{i+1}(BQM, 0), where M is a small category and BQM is the classifying space of QM. ...
16
votes
1answer
1k views

What is a path in K-theory space?

In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement: In brief, the current view is ...
5
votes
1answer
315 views

On $\gamma$-graded pieces of the localization sequence for G-theory (i.e. for K'-theory)

There is a well-known Quillen's localization sequence for (algebraic) K-theory: $\dots\to K_p^Y(X)\to K_p(X)\to K_p(X-Y)\to \dots$, where $Y\to X$ is a closed embedding of schemes. Now suppose that ...
5
votes
2answers
795 views

Relation between motivic cohomology and Quillen K-theory

What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?