Prime decomposition (topology)

This article deals with the prime decomposition of manifolds. For the prime decomposition of nodes, see Compound Node .

In topology , a branch of mathematics , prime decomposition denotes a decomposition of manifolds into "prime components".

Prim manifolds

A closed, connected -dimensional manifold is a prime manifold if it cannot be decomposed as a connected sum, i.e. if from ${\displaystyle d}$ ${\displaystyle M}$

${\displaystyle M=M_{1}\sharp M_{2}}$

it follows that or is homeomorphic to the sphere . ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$ ${\displaystyle S^{d}}$

Prim decomposition

The prime decomposition of a closed, connected -dimensional manifold is a decomposition as a connected sum of finitely many prime manifolds, i.e. ${\displaystyle d}$ ${\displaystyle M}$

${\displaystyle M=M_{1}\sharp \ldots \sharp M_{k}}$

with prime manifolds (the prime components ). ${\displaystyle M_{1},\ldots ,M_{k}}$

existence

From the Poincaré conjecture it follows that every closed connected 3-manifold has a prime decomposition. In fact, according to Grushko-Neumann's theorem, every finitely generated group can be decomposed as a free product of indivisible groups. Because (in dimensions ) the fundamental group of the connected sum is the free product of the fundamental groups of the individual summands, one can then decompose every 3-manifold as a connected sum of finitely many manifolds of a nontrivial fundamental group with (a priori) further simply connected manifolds, but the latter must be broken down be homeomorphic to the sphere according to the Poincaré conjecture. ${\displaystyle d\geq 3}$

In the case of 3-dimensional manifolds, the existence of a prime decomposition had already been proven by Kneser in 1924, long before the Poincaré conjecture was proven. His methods were later generalized by Haken to prove the finiteness of hierarchies of incompressible surfaces in hook manifolds .

Kneser proved to each decomposition that the fundamental group of a closed 3-manifold as a free product by a coherent sum to be realized. The analogous problem in higher dimensions was known as the Kneser conjecture , but there are counterexamples to this conjecture in all dimensions . ${\displaystyle \pi _{1}M=\Gamma _{1}*\Gamma _{2}}$ ${\displaystyle M=M_{1}\sharp M_{2}}$ ${\displaystyle \pi _{1}M_{i}=\Gamma _{i},i=1,2}$ ${\displaystyle d\geq 4}$

The prime decomposition plays an important role in the geometrization of 3-manifolds .

Uniqueness

The prime decomposition of closed, orientable 3-manifolds is unambiguous (except for rearrangement and homeomorphisms), as was proven by Milnor in 1962.

In higher dimensions the uniqueness does not apply, for example is

${\displaystyle \mathbb {C} P^{2}\sharp {\overline {\mathbb {C} P^{2}}}\sharp {\overline {\mathbb {C} P^{2}}}\simeq (S^{2}\times S^{2})\sharp {\overline {\mathbb {C} P^{2}}}}$.

The uniqueness of the prime decomposition does not apply to non-orientable manifolds either, counterexamples already exist in dimension 2.

literature

• Hellmuth Kneser : A topological decomposition theorem . Proc. Konink. Nederl. Akad. Wetensch. 27: 601-616 (1924).
• John Milnor : A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84: 1-7 (1962).

Web links

• Allen Hatcher: Notes on Basic 3-Manifold Topology (PDF; 385 kB) 2000 (English).

Individual evidence

1. ↑ Kreck, Matthias; Lück, Wolfgang; Teichner, Peter: Counterexamples to the Kneser conjecture in dimension four. Comment. Math. Helv. 70 (1995), no. 3, 423-433, doi: 10.1007 / BF02566016 .
2. ^ Cappell, Sylvain E .: On connected sums of manifolds. Topology, 13: 395-400 (1974).