# Prime decomposition theorem in high dimensions

## 1 Problem

Every closed topological oriented manifold $M$$\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$ has a prime decomposition

$\displaystyle M \cong N_1 \sharp \dots \sharp N_k,$

where the $N_i$$N_i$ are prime manifolds. Recall that a manifold is prime if for any decomposition as a connected sum $M_0 \sharp M_1$$M_0 \sharp M_1$ one of the summands $M_0$$M_0$ or $M_1$$M_1$ is homeomorphic to $S^3$$S^3$.

For $3$$3$-manifolds, it was shown in [Milnor1962a] that the decomposition is unique. For high-dimensional manifolds, there is no notion of prime decomposition of smooth manifolds, but there is a notion of prime decomposition for topological manifolds. Nevertheless, the decomposition is not unique.

In [Kreck&Lück&Teichner1994] it is proven that a $4$$4$-dimensional stable version of Kneser's conjecture on the splitting of three-manifolds as connected sums. The result clearly doesn't work non-stably in dimension $4$$4$ and this paper gives some counterexamples. Another counterexample to the uniqueness of the decomposition is as follows.

Example 1.1. $\mathbb{C}P^2$$\mathbb{C}P^2$ has a homotopy equivalent twin $\star \mathbb{C}P^2$$\star \mathbb{C}P^2$. The following decompositions provide a counterexample to uniqueness.

$\displaystyle \star \mathbb{C}P^2 \sharp \star \mathbb{C}P^2 \cong \mathbb{C}P^2 \sharp \mathbb{C}P^2.$

Question: Show that given any $n\geq 4$$n\geq 4$ there exist closed topological $n$$n$-manifolds with non-unique prime decomposition.

Question: Give simply connected examples and give aspherical manifolds.

This question was posed by Stefan Friedl at the MATRIX meeting on Interactions between high and low dimensional topology.