Parametric connected sum

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For an detailed exposition and extensive application of the modified surgery techniques of stable classifcaiton of 4-manifolds, see \cite{Teichner}.
For an detailed exposition and extensive application of the modified surgery techniques of stable classifcaiton of 4-manifolds, see \cite{Teichner}.
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== References ==
== References ==

Latest revision as of 01:04, 8 April 2020

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds M and N equipped with codimension 0-embeddings \phi: T \to M and \psi : T \to N of a compact connected manifold T. It generalises the usual connected sum operation which is the special case when T = D^n is the n-disc. The parametric connected sum operation is more complicated than the usual connected sum operation since the isotopy classes of the embeddings of T into M 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 M and N it is sufficient to equip them with an isotopy class of embeddings of the k-disc. Moreover, the disjoint union D^n \sqcup D^n is the unique thickening of S^0. This motivates the following

Defintion 2.1. A manifold with an S^k-thickening, an S^k-thickened manifold for short, is a pair (M, \phi) where M is a compact connected manifold and \phi : S^k \times D^{n-k} \to \mathrm{int}(M) is an embedding.

Defintion 2.2. Let M = (M, \phi) and N = (N, \psi) by S^k-thickened manifolds. Define

\displaystyle  M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/\simeq

where \simeq is defined via the embeddings \phi and \psi.

It is clear that we have the following

Observation 2.3. The diffeomorphism type of M \sharp_k N depends only upon the the isotopy classes of the embeddings \phi and \psi (which of course includes the diffeomorphism types of M and N).

2.1 Applications

The operation of S^k-connected sum was used in [Ajala1984] and [Ajala1987] to describe the set of smooth structures on the product of spheres \Pi_{i=1}^r S^{n_i}. This construction also appears in [Sako1981].

The analogue of such a construction for embeddings, the S^k-parametric connected sum of embeddings, is used

3 Parametric connected sum along thickenings

Let B be a stable fibred vector bundle. A foundational theorem of modified surgery is

Theorem 3.1 Stable classification: [Kreck1985, Theorem 2.1, p 19], [Kreck1999], [Kreck2016, Theorem 6.2].

\displaystyle  NSt_{2n}(B) \cong \Omega_{2n}^B.

In particular, NSt_{2n}(B) 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 25-6] 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. This is described in more detail (for n>2) in [Kreck2016, Section 6] and uses Wall's theory of thickenings, developed in [Wall1966a]. More precisely, it depends on Wall's embedding theorem [Wall1966a, p 76] for the existence of (unique up to concordance) embedded thickenings of the (n-1)-skeleton of B, and Wall's classification of thickenings in the stable range [Wall1966a, Proposition 5.1] to ensure that two such embedded thickenings are diffeomorphic as B-manifolds, so that one may cut out their interiors and glue the resulting B-manifolds along the boundaries of the embedded thickenings. The special case of n=2 is discussed separately in [Kreck2016, Section 5] under the name "connected sum along the 1-skeleton".

4 References

$-skeleton". == References == {{#RefList:}} [[Category:Definitions]]M and N equipped with codimension 0-embeddings \phi: T \to M and \psi : T \to N of a compact connected manifold T. It generalises the usual connected sum operation which is the special case when T = D^n is the n-disc. The parametric connected sum operation is more complicated than the usual connected sum operation since the isotopy classes of the embeddings of T into M 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 M and N it is sufficient to equip them with an isotopy class of embeddings of the k-disc. Moreover, the disjoint union D^n \sqcup D^n is the unique thickening of S^0. This motivates the following

Defintion 2.1. A manifold with an S^k-thickening, an S^k-thickened manifold for short, is a pair (M, \phi) where M is a compact connected manifold and \phi : S^k \times D^{n-k} \to \mathrm{int}(M) is an embedding.

Defintion 2.2. Let M = (M, \phi) and N = (N, \psi) by S^k-thickened manifolds. Define

\displaystyle  M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/\simeq

where \simeq is defined via the embeddings \phi and \psi.

It is clear that we have the following

Observation 2.3. The diffeomorphism type of M \sharp_k N depends only upon the the isotopy classes of the embeddings \phi and \psi (which of course includes the diffeomorphism types of M and N).

2.1 Applications

The operation of S^k-connected sum was used in [Ajala1984] and [Ajala1987] to describe the set of smooth structures on the product of spheres \Pi_{i=1}^r S^{n_i}. This construction also appears in [Sako1981].

The analogue of such a construction for embeddings, the S^k-parametric connected sum of embeddings, is used

3 Parametric connected sum along thickenings

Let B be a stable fibred vector bundle. A foundational theorem of modified surgery is

Theorem 3.1 Stable classification: [Kreck1985, Theorem 2.1, p 19], [Kreck1999], [Kreck2016, Theorem 6.2].

\displaystyle  NSt_{2n}(B) \cong \Omega_{2n}^B.

In particular, NSt_{2n}(B) 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 25-6] 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. This is described in more detail (for n>2) in [Kreck2016, Section 6] and uses Wall's theory of thickenings, developed in [Wall1966a]. More precisely, it depends on Wall's embedding theorem [Wall1966a, p 76] for the existence of (unique up to concordance) embedded thickenings of the (n-1)-skeleton of B, and Wall's classification of thickenings in the stable range [Wall1966a, Proposition 5.1] to ensure that two such embedded thickenings are diffeomorphic as B-manifolds, so that one may cut out their interiors and glue the resulting B-manifolds along the boundaries of the embedded thickenings. The special case of n=2 is discussed separately in [Kreck2016, Section 5] under the name "connected sum along the 1-skeleton".

4 References

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