Prime decomposition theorem in high dimensions
1 Problem
Every closed topological oriented manifold has a prime decomposition
![\displaystyle M \cong N_1 \sharp \dots \sharp N_k,](/images/math/5/e/d/5ed43304975249f8c25cb4937cbfb855.png)
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.
![\displaystyle \star \mathbb{C}P^2 \sharp \star \mathbb{C}P^2 \cong \mathbb{C}P^2 \sharp \mathbb{C}P^2.](/images/math/d/d/7/dd7e23565910cfab8a9a496e1b2b010e.png)
Question: Show that any prime decomposition of a topological -manifold of dimension
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
-manifolds, Amer. J. Math. 84 (1962), 1–7. MR0142125 Zbl 0108.36501