Connected sums where the separating “2”-sphere bounds a punctured ball

geometric-topology

I have a question about connected sums in 3-dimensional spaces in relation to normal surfaces.
I was reading the paper https://arxiv.org/abs/math/9712269 by Joel Hass, which at one point outlines a proof of "Knesers Theorem", which states that a triangulated $3$-dimensional manifold (without boundary) $M$ can be decomposed
non-trivially along $2$-spheres (in terms of the connected sum) up to a number of times bounded by a constant $k(M)$ dependent on $M$ and its triangulation.

For the proof, the notion of a normal surface was introduced, which is a surface embedded in $M$ which is a disjoint union of normal triangles or normal quadrilaterals, where the latter are intersecting a tetrahedron of $M$ in 3 or 4 edges and faces.

At Page 7 in the proof, the notion of a "punctured ball" is introduced, which is just defined as "a ball with some open balls removed". At Page 8 in the proof it is mentioned, that if we use a separating (normal) $2$-sphere $S$ to decompose $M$ into a connected sum and $S$ bounds a punctured ball, then one of the summands in the connected sum is trivial, meaning its homeomorphic to $S^3$. I can't get my head around the following things:

  1. Do we have specific extra conditions regarding the definition of a punctured ball? For example that we only allow for finitely many open balls cut out or that they have to be completely enclosed in the main ball?
  2. I can't really see why a $2$-sphere bounding a punctured ball leads to a $S^3$-summand. Is it a simple trick I don't see (maybe having to do with normal $2$-sphere) or is this maybe done by calculation of homology groups? I think I am missing something important here.

I appreciate any kind of help and ideas, thanks in advance for your time!

Best Answer

  1. Yes, there are finitely many and they have disjoint closures.

  2. The summands (by definition of the connect sum) are gotten by gluing 3-balls to each 2-sphere in a connected region in the complement of the 2-spheres, This results in a 3-manifold with no boundary for each component.

If one of the summands is a punctured 3-ball then gluing in 3-balls to each boundary 2-sphere (ie filling back in the punctures) results in a 3-sphere, which is a trivial summand.

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