Parametric connected sum

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The operation of $S^k$-connected sum was used in \cite{Ajala1984} and \cite{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 \cite{Sako1981}.
The operation of $S^k$-connected sum was used in \cite{Ajala1984} and \cite{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 \cite{Sako1981}.
The analogue of such a construction for [[High_codimension_embeddings|embeddings]], the $S^k$-parametric connected sum of embeddings, is used
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The analogue of such a construction for [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification|embeddings]], the $S^k$-parametric connected sum of embeddings, is used
* to define, for $m\ge 2p+q+3$, a group stucture on the set $E^m(S^p \times S^q)$ of (smooth or PL) isotopy classes of embeddings $S^p \times S^q\to \Rr^m$ \cite{Skopenkov2006}, \cite{Skopenkov2015a}.
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* to define, for $m\ge 2p+q+3$, a group stucture on the set $E^m(S^p \times S^q)$ of (smooth or PL) isotopy classes of embeddings $S^p \times S^q\to \Rr^m$ \cite{Skopenkov2006}, \S3.4, \cite{Skopenkov2006a}, \S3, \cite{Skopenkov2015a}.
* to construct an action of this group on the set of isotopy classes of embeddings of certain $(p+q)$-manifolds into $\Rr^m$ \cite{Skopenkov2014}, 1.2.
* to construct an action of this group on the set of isotopy classes of embeddings of certain $(p+q)$-manifolds into $\Rr^m$ \cite{Skopenkov2014}, 1.2.
* to estimate the set of isotopy classes of embeddings \cite{Skopenkov2007}, \cite{Skopenkov2010}.
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* to estimate the set of isotopy classes of embeddings \cite{Cencelj&Repovš&Skopenkov2007}, \cite{Cencelj&Repovš&Skopenkov2008}, \cite{Skopenkov2007}, \cite{Skopenkov2010}, \cite{Skopenkov2015}, \cite{Skopenkov2015a}, \cite{Crowley&Skopenkov2016} and unpublished paper \cite{Crowley&Skopenkov2016a}.
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Let $B$ be a [[stable fibred vector bundle]]. A foundational theorem of modified surgery is
Let $B$ be a [[stable fibred vector bundle]]. A foundational theorem of modified surgery is
{{beginthm|Theorem|Stable classification: \cite{Kreck1985}, \cite{Kreck1999}}}
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{{beginthm|Theorem|Stable classification: \cite{Kreck1985|Theorem 2.1, p 19}, \cite{Kreck1999}, \cite{Kreck2016|Theorem 6.2}}}
$$ NSt_{2n}(B) \cong \Omega_{2n}^B.$$
$$ NSt_{2n}(B) \cong \Omega_{2n}^B.$$
{{endthm}}
{{endthm}}
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 \cite{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.<!--{{beginthm|Remark}}
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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 \cite{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 \cite{Kreck2016|Section 6} and uses Wall's theory of thickenings, developed in \cite{Wall1966a}. More precisely, it depends on Wall's embedding theorem \cite{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 \cite{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 \cite{Kreck2016|Section 5} under the name "connected sum along the $1$-skeleton".
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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{{endthm}} -->
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<!--{{beginthm|Remark}}
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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}.
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{{endthm}}-->
== References ==
== References ==

Latest revision as of 02:04, 8 April 2020

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

Contents

[edit] 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
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. 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
Tex syntax error
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.

[edit] 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).

[edit] 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

[edit] 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".

[edit] 4 References

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