Prime decomposition theorem in high dimensions

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	sum $M_0 \sharp M_1$ one of the summands $M_0$ or $M_1$ is homeomorphic to $S^3$.		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 by Milnor \url{https://www.jstor.org/stable/2372800?seq=1#metadata_info_tab_contents} that the decomposition is unique.	+	For $3$-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.		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.		Nevertheless, the decomposition is not unique.
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	{{endthm}}		{{endthm}}
	</wikitex>		</wikitex>
		+	== References ==
		+	{{#RefList:}}
	== References ==		== References ==

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Revision as of 06:39, 8 January 2019

1 Problem

Every closed topological oriented manifold

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$== 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 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. {{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}} </wikitex> == References == {{#RefList:}} [[Category:Questions]] [[Category:Research questions]]M$ has a prime decomposition

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 \url{https://www.him.uni-bonn.de/lueck/data/kneser2.pdf}, Kreck, Lueck and Teichner prove 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.

2 References

• [Milnor1962a] J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR0142125 Zbl 0108.36501

3 References

• [Milnor1962a] J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR0142125 Zbl 0108.36501