# Almost commuting matrices, cohomology, and dimension

## TL;DR

This paper characterizes when commutative $C^*$-algebras are matricially semiprojective, linking algebraic stability under perturbations to topological dimension and cohomology of the underlying space.

## Contribution

It provides a complete characterization of matricial semiprojectivity for $C(X)$ based on dimension and cohomology, extending previous work on almost commuting matrices.

## Key findings

- $C(X)$ is matricially semiprojective iff $ ext{dim}(X) extless=2$ and $H^2(X;Q)=0$
- Identifies topological restrictions for stability of relations in matrices
- Applications to lifting problems in operator algebras

## Abstract

We investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative $C^*$-algebras $C(X)$ are matricially semiprojective. Extending the works of Davidson, Eilers-Loring-Pedersen, Lin and Voiculescu on almost commuting matrices, we identify the precise dimensional and cohomological restrictions for finite-dimensional spaces $X$ and thus obtain a complete characterization: $C(X)$ is matricially semiprojective if and only if $\dim(X)\leq 2$ and $H^2(X;\mathbb{Q})=0$. We give several applications to lifting problems for commutative $C^*$-algebras, in particular to liftings from the Calkin algebra and to $l$-closed $C^*$-algebras in the sense of Blackadar.