# Kneser Milnor decomposition in higher dimensions

Is there any sort of Kneser Milnor decomposition in dimension d larger than or equal to 4?, I mean a family of d-1 dimensional spheres disconnecting the d-dimensional manifold into irreducible components and with some uniqueness?, If so, any reference so far?. Thanks.

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This is the topic of a previous MO question: mathoverflow.net/questions/93512/… In short, oriented manifolds up to diffeo with connect sum operation certainly is not a free commutative monoid in high dimensions since you have torsion elements -- homotopy spheres. But there are many things known about this commutative monoid. –  Ryan Budney Jul 4 '12 at 18:27
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## 1 Answer

Some results are available in dimension 4, although a "complete" answer is not known. As expected, the known results depend on the category: smooth/topological. Here are some references: first two papers by Matthias Kreck, Wolfgang Lück and Peter Teichner: "Counterexamples to the Kneser conjecture in dimension four", Comment. Math. Helv. 70 (1995), 423–433. "Stable prime decompositions of four-manifolds". Prospects in topology, 251–269, Ann. of Math. Stud., 138, Princeton Univ. Press, Princeton, NJ, 1995.

The first paper gives various counterexamples to the "naive" guess for 4-dimensional Kneser's conjecture: that a free product decomposition of the fundamental group would lead to a connected sum decomposition. The second paper gives a proof of this "naive" prime decomposition theorem (including uniqueness) stably, i.e. up to a connected sum with copies of $S^2\times S^2$. (The existence of such a stable decomposition is also proved by Jonathan Hillman, "Free products and 4-dimensional connected sums" Bull. London Math. Soc. 27 (1995), 387–391.)

Some further results are known in the topological category, these hold up to s-cobordism (given that topological surgery and s-cobordism conjectures are not known for "large" fundamental groups, at this point one doesn't expect an actual result up to a homeomorphism). These theorems are of the following flavor: under some restrictions assume the fundamental group splits as a free product and also assume the second homotopy group and the intersection pairing "look like" they correspond to a connected sum decomposition (these are natural algebraic assumptions).Then up to an s-cobordism the manifold splits as a connected sum. Some of the references are: A. Cavicchioli, F. Hegenbarth, On 4-manifolds with free fundamental group, Forum Math. 6 (1994), 415-429; V. Krushkal, R. Lee, Surgery on closed 4-manifolds with free fundamental group, Math. Proc. Cambridge Philos. Soc. 133 (2002), 305–310; Q. Khan, Homotopy invariance of 4-manifold decompositions: connected sums, arXiv:0907.0308

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