Homotopy Invariance of Algebraic K-Theory for Group Rings

Definitions and mathematical statement

For an abstract group $\Gamma$ and a compact Hausdorff space $X$, interpret $C(X)$ provisionally as the unital ring $C(X,{\mathbb{C}})$ of continuous complex-valued functions, with pointwise operations. The supplied record does not explicitly specify the scalar field; the formulation below uses this usual convention and requires confirmation against the source. The algebraic group ring $C(X)[\Gamma]$ consists of finite sums $\sum_{\gamma\in\Gamma}a_\gamma\gamma$, with coefficients in $C(X)$, coefficientwise addition, and multiplication determined by the group law and by coefficients commuting with group elements. No completion is taken.

For a unital ring $A$, let $K_0^{\mathrm{alg}}(A)$ be the Grothendieck group of isomorphism classes of finitely generated projective right $A$-modules, with relations $[P\oplus Q]=[P]+[Q]$. A unital ring homomorphism induces a homomorphism on $K_0^{\mathrm{alg}}$ by extension of scalars. For the empty space, use $C(\varnothing)=0$ and $K_0^{\mathrm{alg}}(0)=0$.

A continuous map $f\colon X\to Y$ induces the pullback ring homomorphism \[ f^*\colon C(Y)[\Gamma]\longrightarrow C(X)[\Gamma],\qquad \sum_\gamma a_\gamma\gamma\longmapsto \sum_\gamma(a_\gamma\circ f)\gamma. \] Write $F_\Gamma(X)=K_0^{\mathrm{alg}}(C(X)[\Gamma])$ and $F_\Gamma(f)=K_0^{\mathrm{alg}}(f^*)$; thus $F_\Gamma$ is contravariant.

Conjecture, conditional on the scalar convention above. For every group $\Gamma$, every pair of compact Hausdorff spaces $X,Y$, and every pair of continuous maps $f_0,f_1\colon X\to Y$ admitting a continuous homotopy $H\colon X\times[0,1]\to Y$ with $H(x,0)=f_0(x)$ and $H(x,1)=f_1(x)$ for all $x\in X$, one has \[ F_\Gamma(f_0)=F_\Gamma(f_1)\colon F_\Gamma(Y)\longrightarrow F_\Gamma(X). \] Equivalently, for every group $\Gamma$ and compact Hausdorff space $X$, the projection $p\colon X\times[0,1]\to X$ induces an isomorphism \[ K_0^{\mathrm{alg}}(p^*)\colon K_0^{\mathrm{alg}}(C(X)[\Gamma]) \xrightarrow{\ \cong\ } K_0^{\mathrm{alg}}(C(X\times[0,1])[\Gamma]). \] There are no torsion, finite-generation, or polyhedral restrictions in the supplied conjecture. The supplied partial-results assessment does not specify enough hypotheses to formulate its special cases precisely.

Short English statement

For every group $\Gamma$, ask whether homotopic maps between compact Hausdorff spaces induce the same map on the algebraic $K_0$-groups of their function group rings. Equivalently, adjoining an interval factor should induce an isomorphism. Here $C(X)$ is provisionally interpreted as complex-valued continuous functions; the scalar convention needs source confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001331$, source problem code catalogue entry, entitled Homotopy Invariance of Algebraic K-Theory for Group Rings, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied report citation: Noncommutative Geometry (2010), Oberwolfach Reports, pp. 2271--2334, DOI 10.4171/owr/2009/41. Source URL: https://doi.org/10.4171/owr/2009/41. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "original Oberwolfach report 4084": https://doi.org/10.4171/owr/2009/41.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified here. It labels the problem partially solved and reports proofs for important classes, including torsion-free groups under a Farrell--Jones hypothesis and a polyhedral setting, while reporting that the assertion for every group remains open. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and says that the mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification assertion, not an independent verification performed for this draft.