Uniqueness of the set of decomposing spheres in prime decomposition of a 3-manifold

Question

At the end of Section 1.1 of 3-manifold groups it is written that "the decomposing spheres are not unique up to isotopy, but two different sets of decomposing spheres are related by ‘slide homeomorphisms’" and I am trying to understand what the author meant. The given references do not treat the topic (maybe Theorem 3 of The homotopy type of homeomorphisms of 3-manifolds, but if it does, then it is written in a quite mysterious way for my knowledge).

I guess that "two different sets of decomposing spheres are related by ‘slide homeomorphisms’" means that for every two sets of decomposing spheres there exists a slide homemomorphism sending one set of spheres in the other one. However, it is clear that two sets of decomposing spheres containing a different number of spheres cannot be related by slide homeomorphism. Maybe the author meant that the two sets of decomposing spheres have to be minimal (but in this case, notice that the decomposing system of spheres treated in the mysterious article that I cited before are not minimal).

Can someone cite some article stating the exact result? (In the case the result I am looking for is Theorem 3 in the mysterious article, then can someone explain me what this theorem is saying exactly?)

Answer 1

Here is a simple example.

Suppose that $A$, $B$, and $C$ are closed, connected, oriented, prime three-manifolds, so that no two are homeomorphic and none are the three-sphere. (For example, lens spaces with fundamental groups of distinct sizes.) We define a manifold $M = A \ \# \ B \ \#\ C$. Implicit in this a pair of spheres: say $S$ separating $A$ from $B \ \#\ C$ and $S'$ separating $A \ \# \ B$ from $C$.

"Clearly" $M$ is homeomorphic to $B \ \# \ C \ \#\ A$. However, here we find a different pair of spheres: say $T$ separating $B$ from $C \ \#\ A$ and $T'$ separating $A$ from $B \ \# \ C$. "Clearly" there is no homeomorphism of $M$ sending $S \cup S'$ to $T \cup T'$ because the two systems have different separation properties.

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So reading "slide homeomorphisms" as "homeomorphism of the ambient space" is wrong. I believe that the offending sentence would be clearer if written as follows.

The decomposing spheres are not unique up to isotopy, but two different sets of decomposing spheres are related by ‘sphere slides’.

The proof is somewhat complicated, and relies on finding the correct "sphere surgery sequence". My suggested references are Hatcher's Notes on basic three-manifold topology and the second would be Casson's Three-dimensional topology.