nLab unstable K-theory





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

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


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topological homotopy theory



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.



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.


The following properties are proven in Hamanaka and Kono 2003:


(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).


(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.


(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.


On unstable algebraic K-theory:

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

Last revised on June 15, 2024 at 17:43:59. See the history of this page for a list of all contributions to it.