# Prime decomposition theorem in high dimensions

## [edit] 1 Problem

Every closed topological oriented manifold has a prime decomposition

where the are prime manifolds. Recall that a manifold is prime if for any decomposition as a connected sum one of the summands or is homeomorphic to .

For -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 -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 and this paper gives some counterexamples. Another counterexample to the uniqueness of the decomposition is as follows.

**Example 1.1.**
has a homotopy equivalent twin . The following decompositions provide a counterexample to uniqueness.

*Question*: Show that given any there exist closed topological -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.

## [edit] 2 References

- [Kreck&Lück&Teichner1994] M. Kreck, W. Lück and P. Teichner,
*Stable prime decompositions of four-manifolds*, Prospects in topology, Princeton Univ. Press, Princeton, NJ (1995), 251–269. MR1368662 Zbl 0928.57019 - [Milnor1962a] J. Milnor,
*A unique decomposition theorem for -manifolds*, Amer. J. Math.**84**(1962), 1–7. MR0142125 Zbl 0108.36501