Parametric connected sum
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Parametric connected sum is an operation on compact connected n-manifolds and equipped with codimension 0-embeddings and of a compact connected manifold . It generalises the usual connected sum operation which is the special case when is the -disc. The parametric connected sum operation is more complicated than the usual connected sum operation since the isotopy classes of the embeddings of into may be significantly more complicated than the isotopy classes of embeddings of n-discs need for connected sum: these last are determined by (local) orientations.
2 Connected sum along k-spheres
We say above that to define connected sum for connected k-manifolds and it is sufficient to equip them with an isotopy class of embeddings of the k-disc. Moreover, the disjoint union is the unique thickening of . This motivates the following
Defintion 2.1. A manifold with an -thickening, an -thickened manifold for short, is a pair where is a compact connected manifold and is an embedding.
Defintion 2.2. Let and by -thickened manifolds. Define
where is defined via the embeddings and .
It is clear that we have the following
Observation 2.3. The diffeomorphism type of depends only upon the the isotopy classes of the embeddings and (which of course includes the diffeomorphism types of and ).
The operation of -connected sum was used in [Ajala1984] and [Ajala1987] to describe the set of smooth structures on the product of spheres . This construction also appears in [Sako1981]. The analogue of such a construction for embeddings is used in [Skopenkov2006] to define, for , a group stucture on the set of (smooth or PL) isotopy classes of embeddings of into . In [Skopenkov2007] and [Skopenkov2010] the -connected sum of embeddings was used to estimate the set of embeddings.
3 Parametric connected sum along thickenings
Let be a stable fibred vector bundle. A foundational theorem of modified surgery is
In particular, has the structure of an abelian group. The question of whether there is a geometric definition of this group structure is taken up in [Kreck1985, Chapter 2, pp 26-7] where it is shown how to use parametric connected sum along thickenings to define an addition of stable diffeomorphism classes of closed 2n-B-manifolds.
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