# Uniqueness of spheres in prime decomposition of a 3-manifold

Let $M$ be a closed connected orientable 3-manifold. Then Kneser tells us that there is a decomposition $M = P_1 \sharp \cdots \sharp P_k$ of $M$ into prime manifolds. Milnor tells us that if $M = Q_1 \sharp \cdots \sharp Q_l$ then $k = l$ and the $Q_i$ are homeomorphic to the $P_j$ in some order.

Is there any theorem that tells us to what extent the spheres that give a prime decomposition are unique? They are not unique up to isotopy at all bit I was wondering if there was some set of moves for getting between any set of prime decomposing spheres.

• I think it would be hard to find better moves than what you get by following along with the proof of the uniqueness of the decomposition - i.e. whenever the proof replaces one sphere with another, that is an allowable move.
– Carl
May 2 '18 at 6:04

Theorem. Let $$M$$ be a compact, connected, orientable 3-manifold. Then any orientation-preserving homeomorphism of $$M$$ is isotopic to a composition of the following four types of homeomorphisms:
3. spins of $$S^1\times S^2$$ summands, and