Prime decomposition theorem in high dimensions

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[edit] 1 Problem

Every closed topological oriented manifold M has a prime decomposition

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

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

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

Example 1.1. \mathbb{C}P^2 has a homotopy equivalent twin \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 there exist closed topological 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.

[edit] 2 References

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