17
votes
4answers
1k views

Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory? However, to my taste, the answers there consider the subject from a more modern point of view. When I open a book ...
1
vote
1answer
183 views

Additivity theorem for algebraic L-theory?

There is an additivity theorem for algebraic K-theory. My question is is there an additivity theorem for algebraic L-theory?
7
votes
1answer
355 views

Algebraic K-theory of odd-dimensional spheres

Let $A(X)$ denote the Waldhausen's algebraic K-theory of a space $X$, and let $n$ be odd. Are the rational homotopy groups of $A(S^n)$ known? Is the group $\pi_{2k}(A(S^n))$ finite for all ...
6
votes
0answers
112 views

Torsion in Whitehead group

Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?
6
votes
2answers
379 views

Algebraic K-theory and Homotopy Sheaves

Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory ...
8
votes
1answer
353 views

Waldhausen $K$-theory for $G$-spaces

I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it? Let $G$ be a finite group. We know that ...
33
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 ...
1
vote
1answer
112 views

homology of $B S^{-1} S$ computation in the proof that $+ = Q$

Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that $BS^{-1}S \sim B \text{GL}(R)^+ \times K_0(R)$ In proving this, in Srinivas' algebraic ...
2
votes
1answer
419 views

Intuition as to why the K-theory of a ring should be the homotopy theory of an H-space

For an ideal $I \subset R$ with relative K-groups $K_i(R,I)$ we have an exact sequence $K_2(R) \to K_2(R,I) \to K_2(R/I) \to K_1(R) \to K_1(R,I) \to K_1(R/I)$ $\to K_0(R) \to K_0(R,I) \to K_0(R/I)$ ...
1
vote
1answer
189 views

Simplicial sets from bisimplicial sets, and their realisations.

From a bisimplicial space $T$, one can consider the simplicial spaces $\underline p \mapsto T_{pp} $, $\underline p \mapsto |\underline q \mapsto T_{pq} |$, and $\underline q \mapsto |\underline p ...
2
votes
1answer
195 views

Path components of a monoidal category form a monoid?

In Grayson's 'Higher Algebraic K-theory II', leading up to the categorical generalisation of the plus construction, he considers $\pi_0(S) = \pi_0(BS)$, where $S$ is a (small, symmetric) monoidal ...
10
votes
3answers
632 views

Plus construction considerations.

In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction: Recall that $K_1(R) = ...
6
votes
1answer
309 views

Waldhausen Additivity in a More General Context

The following arose when I was thinking about a talk at the Midwest Topology Seminar: Background I want to consider a generalization of a Waldhausen-like structure on a category $C$ with 0-object ...
21
votes
1answer
695 views

Is every ''group-completion'' map an acyclic map?

I start with a longer discussion which will result in a precise version of the question. A am puzzled about an issue with the Quillen plus construction. I have seen outstanding experts being confused ...
10
votes
1answer
479 views

K-Theory space of finite abelian groups

Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a ...
2
votes
1answer
172 views

Cube of cofibrations II

Let $\mathcal{C}$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(\mathcal{C})$ the category with cofibrations consisting of sequences of ...
13
votes
2answers
897 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 ...
7
votes
1answer
295 views

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

Is the fibered K-theoretic farrell-jones conjecture true for cat(0)-groups?
4
votes
1answer
261 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
247 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
2answers
330 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 $G$?
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 ...
40
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 ...
5
votes
2answers
428 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 ...
6
votes
0answers
256 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 ...
4
votes
1answer
290 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?
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. ...
0
votes
2answers
403 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. ...
7
votes
2answers
447 views

(Co-) Homology associated to Waldhausen K-Theory

Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...
53
votes
10answers
9k 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 ...