Wall's D(2) problem

Theorem 1.1: there is a finite connected three-dimensional CW complex $X$ with $H_i(\widetilde X;\mathbb Z)=0$ for $i>2$ and $H^3(X;M)=0$ for every $\mathbb Z[\pi_1X]$-module $M$ that is not homotopy equivalent to any finite complex of dimension at most two. The obstruction: on a finite 2-complex, a rank-one local system with vanishing twisted $H_2$ is trivial on every finite-order element of $\pi_1$. $\pi_1(X)$ is infinite. It does not settle the D(2) problem for finite fundamental groups, the relation gap problem, or the Eilenberg-Ganea problem.

Theorem 1.1: there is a finite connected three-dimensional CW complex $X$ with $H_i(\widetilde X;\mathbb Z)=0$ for $i>2$ and $H^3(X;M)=0$ for every $\mathbb Z[\pi_1X]$-module $M$ that is not homotopy equivalent to any finite complex of dimension at most two. The obstruction: on a finite 2-complex, a rank-one local system with vanishing twisted $H_2$ is trivial on every finite-order element of $\pi_1$. $\pi_1(X)$ is infinite. It does not settle the D(2) problem for finite fundamental groups, the relation gap problem, or the Eilenberg-Ganea problem.