Prime decomposition theorem in high dimensions

Revision as of 21:58, 8 January 2019

1 Problem

Every closed topological oriented manifold $== Problem == <wikitex>; Every closed topological oriented manifold $M$ has a prime decomposition $$M \cong N_1 \sharp \dots \sharp N_k,$$ 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 $-manifolds, it was shown in {{cite|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 {{cite|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. {{beginthm|Example|}} $\mathbb{C}P^2$ has a homotopy equivalent twin $\star \mathbb{C}P^2$. The following decompositions provide a counterexample to uniqueness. $$\star \mathbb{C}P^2 \sharp \star \mathbb{C}P^2 \cong \mathbb{C}P^2 \sharp \mathbb{C}P^2.$$ {{endthm}} ''Question'': Show that any prime decomposition of a topological $n$-manifold of dimension $n \geq 4$ is not unique. This question was posed by Stefan Friedl at the [[:Category:MATRIX 2019 Interactions|MATRIX meeting on Interactions between high and low dimensional topology.]] </wikitex> == References == {{#RefList:}} [[Category:Questions]] [[Category:Research questions]]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 any prime decomposition of a topological $n$-manifold of dimension $n \geq 4$ is not unique.

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

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 $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR0142125 Zbl 0108.36501