# Borel Conjecture for compact aspherical 4-manifolds

## 1 Problem

Let $M_0$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M_0$ and $M_1$$M_1$ be a compact aspherical $4$$4$-manifolds with boundary. The Borel Conjecture in this setting states that a homotopy equivalence of pairs $f \colon (M_0, \partial M_0) \to (M_1, \partial M_1)$$f \colon (M_0, \partial M_0) \to (M_1, \partial M_1)$ which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism.

To apply topological surgery to attack this conjecture, assume that the fundamental group $\pi = \pi_1(M_0) \cong \pi_1(M_1)$$\pi = \pi_1(M_0) \cong \pi_1(M_1)$ is good. One now proceeds to the following problems:

1. Decide which good $\pi$$\pi$ are the fundamental groups of compact $4$$4$-manifolds.
2. Determine the homeomorphism type of the boundaries which can occur for each group in Part 1.
3. For a fixed boundary and $\pi$$\pi$, prove the conjecture via surgery and the Farrell-Jones Conjecture

One can of course formulate the above in the smooth category. There are no known smooth counterexamples for closed manifolds; in particular, there is no known exotic smooth structure on $T^4$$T^4$. There are smooth counterexamples for manifolds with boundary; in particular, see Akbulut, "A fake compact contractible 4-manifold."

This problem was posed by Jim Davis, following discussions with Jonathan Hillman, Monday January 14th at MATRIX.