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.

  • 1
    $\begingroup$ 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. $\endgroup$ Jul 4 '12 at 18:27

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.