Prime decomposition theorem in high dimensions

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1 Problem

Every closed topological oriented manifold

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$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == 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

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

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

For $3$$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 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$$4$-dimensional stable version of Kneser's conjecture on the splitting of three-manifolds as connected sums. The result clearly doesn't work non-stably in dimension $4$$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$$\mathbb{C}P^2$ has a homotopy equivalent twin $\star \mathbb{C}P^2$$\star \mathbb{C}P^2$. The following decompositions provide a counterexample to uniqueness.

$\displaystyle \star \mathbb{C}P^2 \sharp \star \mathbb{C}P^2 \cong \mathbb{C}P^2 \sharp \mathbb{C}P^2.$ $$\displaystyle \star \mathbb{C}P^2 \sharp \star \mathbb{C}P^2 \cong \mathbb{C}P^2 \sharp \mathbb{C}P^2.$$