Prime decomposition theorem in high dimensions
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== Problem == | == Problem == | ||
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Every closed topological oriented manifold $M$ has a prime decomposition | Every closed topological oriented manifold $M$ has a prime decomposition | ||
$$M \cong N_1 \sharp \dots \sharp N_k,$$ | $$M \cong N_1 \sharp \dots \sharp N_k,$$ | ||
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$$\star \mathbb{C}P^2 \sharp \star \mathbb{C}P^2 \cong \mathbb{C}P^2 \sharp \mathbb{C}P^2.$$ | $$\star \mathbb{C}P^2 \sharp \star \mathbb{C}P^2 \cong \mathbb{C}P^2 \sharp \mathbb{C}P^2.$$ | ||
{{endthm}} | {{endthm}} | ||
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== References == | == References == |
Revision as of 06:30, 8 January 2019
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 by Milnor \url{https://www.jstor.org/stable/2372800?seq=1#metadata_info_tab_contents} 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 \url{https://www.him.uni-bonn.de/lueck/data/kneser2.pdf}, Kreck, Lueck and Teichner prove 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.