nLab unstable K-theory

Idea

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

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

one may think of this as a stabilization of generalized nonabelian cohomology theories (in the sense of Lurie 14, Def. 6, FSS23, §2) $U_n(X)$ whose classifying spaces are the unitary groups $U(n)$. For this reason, one may call the groups $[X,U(n)] \,\coloneqq\, \pi_0 Map\big(X,U(n)\big)$ the unstable $\tilde{K}^1$-theory groups of $X$, at stage $n$ (Hamanaka & Kono 2003, 2004).

Similarly, the infinite loop space representing algebraic K-theory of a suitable ring $R$ is the colimit

$$K_0(R) \;\times\; \underset{\underset{n}{\longrightarrow}}{\lim} B GL(n,R)^+$$

of the Quillen plus construction on the classifying spaces of the general linear groups $GL(n,R)$, and considering this instead for finite $n$ is the topic of unstable algebraic K-theory (vd Kallen 1980 §1.2, Clausen & Jansen 2024 §1.3).

Definition

Definition

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

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

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

Properties

The following properties are proven in Hamanaka & Kono 2003:

Proposition

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

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

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

$$1 \to N_n (X) \to U_n(X) \to \tilde{K}^1 (X) \to 1 \,.$$

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

Proposition

(Theorem 1.2 of op.cit.)

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

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 $n\geq 3$, the $n$ unstable K-theory group of the $2n$-dimensional sphere $S^{2n}$ is

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

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

Related concepts

• topological K-theory

• unstable Cohomotopy

• D-brane charge quantization in K-theory

References

On unstable topological K-theory:

• Hiroaki Hamanaka, Akira Kono: On $[X, U(n)]$ when $\text{dim}(X)$ is $2n$, Journal of Mathematics of Kyoto University 43 2 (2003) 333-348 [doi:10.1215/kjm/1250283730]

• Hiroaki Hamanaka: Adams $e$-invariant, Toda bracket and $[X, U (n)]$, Journal of Mathematics of Kyoto University 43 4 (2003) 815-827 [doi:10.1215/kjm/1250281737]

• Hiroaki Hamanaka, Akira Kono: An application of unstable K-theory, Journal of Mathematics of Kyoto University 44 2 (2004) 451-456 [doi:10.1215/kjm/1250283560]

On unstable algebraic K-theory:

• Wilberd van der Kallen: Homology stability for linear groups, Invent Math 60 (1980) 269–295 [doi:10.1007/BF01390018]

• Soren Galatius, Alexander Kupers, Oscar Randal-Williams: $E_\infty$-cells and general linear groups of infinite fields, Duke Math. J. 174 14 (2025) 2927–3046 [arXiv:2005.05620]

• Dustin Clausen, Mikala Ørsnes Jansen: The reductive Borel-Serre compactification as a model for unstable algebraic K-theory, Sel. Math. New Ser. 30 10 (2024) [arXiv:2108.01924 math.KT, doi:10.1007/s00029-023-00900-8]

• Mikala Ørsnes Jansen: Unstable algebraic K-theory: homological stability and other observations [arXiv:2405.02065]