nLab unstable K-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

Given that the (degree-shifted, reduced) topological K-theory group of a topological space XX can be computed as a colimit over sets of homotopy classes of maps (cf. at stable unitary group)

K˜ 1(X)=[X,U]=limn[X,U(n)] \tilde{K}^1(X) \;=\; [X,U] \;=\; \underset{\underset{n}{\longrightarrow}}{lim} \, \big[X,U(n)\big] \,

one may think of this as a stabilization of generalized nonabelian cohomology theories (in the sense of FSS23, §2) U n(X)U_n(X) whose classifying spaces are the unitary groups U(n)U(n). For this reason, we may call the groups [X,U(n)]π 0Map(X,U(n))[X,U(n)] \,\coloneqq\, \pi_0 Map\big(X,U(n)\big) the unstable K˜ 1\tilde{K}^1-theory groups of XX, at stage nn.

Definition

Definition

Let XX be a topological space (CW-complex), and U(n)U(n) the unitary group in dimension nn. The nn-unstable K˜ 1\tilde{K}^1-theory group U n(X)U_n(X) of XX is defined as the homotopy classes of maps, hence the connected components of the space of maps from XX to (the underlying topological space of) U(n)U(n):

U n(X)[X,U(n)]π 0Map(X,U(n)). U_n(X) \;\coloneqq\; [X,U(n)] \,\coloneqq\, \pi_0 Map\big(X,\, U(n)\big) \,.

From this definition one can see that, if XX is a finite-dimensional CW complex then U n(X)=K˜ 1(X)U_n(X) = \tilde{K}^1 (X) for sufficiently large nn.

Properties

The following properties are proven in Hamanaka and Kono 2003:

Proposition

(Theorem 1.1 of op.cit.)
Let dim(X)2n\text{dim}(X)\leq 2n. Then there exists an exact sequence of the form:

K˜ 0(X)ΘH 2n(X;)U n(X)K˜ 1(X)1, \tilde{K}^0(X) \xrightarrow{\Theta} H^{2n}(X;\mathbb{Z}) \to U_n(X) \to \tilde{K}^1 (X) \to 1 \,,

or, put differently, defining N n(X)coker(Θ)N_n (X) \coloneqq coker(\Theta) (see Section 3 of op.cit. for the definition of Θ\Theta):

1N n(X)U n(X)K˜ 1(X)1. 1 \to N_n (X) \to U_n(X) \to \tilde{K}^1 (X) \to 1 \,.

In particular, Prop. shows that in general U n(X)U_n(X) is neither abelian nor does it inject into K˜ 1(X)\tilde{K}^1 (X).

Proposition

(Theorem 1.2 of op.cit.)

For dim(X)2ndim(X)\leq 2n, the cokernel group N n(X)N_n(X) is a finite abelian group where the order of any element divides n!n!.

An example where not only do the unstable and stable K theory groups not coincide but the latter actually vanishes is provided by the even-dimensional spheres.

Proposition

(Lemma 4.1 of Hamanaka 2003)

For n3n\geq 3, the nn unstable K-theory group of the 2n2n-dimensional sphere S 2nS^{2n} is

U n(S 2n)=/n!, U_n (S^{2n} ) = \mathbb{Z}/ n!\mathbb{Z},

whereas K˜ 1(S 2n)=0\tilde{K}^1 (S^{2n} ) =0.

Algebraic K-theory

For the notion of unstable algebraic K-theory, see Jansen 2024.

References

On unstable algebraic K-theory:

  • Mikala Ørsnes Jansen. Unstable algebraic K-theory: homological stability and other observations (2024). (arXiv:2405.02065).

Last revised on May 18, 2024 at 21:28:56. See the history of this page for a list of all contributions to it.