# Parametric connected sum

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## 1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds
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${{Stub}} == 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|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. == 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 {{beginthm|Defintion}} 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. {{endthm}} {{beginthm|Defintion}} Let M = (M, \phi) and N = (N, \psi) by S^k-thickened manifolds. Define 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. {{endthm}} It is clear that we have the following {{beginthm|Observation}} 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). {{endthm}} === Applications === ; 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 [[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}, \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 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}. == Parametric connected sum along thickenings == ; Let B be a [[stable fibred vector bundle]]. A foundational theorem of modified surgery is {{beginthm|Theorem|Stable classification: \cite{Kreck1985}, \cite{Kreck1999}}} NSt_{2n}(B) \cong \Omega_{2n}^B. {{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. == References == {{#RefList:}} [[Category:Definitions]]M$ and $N$$N$ equipped with codimension 0-embeddings $\phi: T \to M$$\phi: T \to M$ and $\psi : T \to N$$\psi : T \to N$ of a compact connected manifold $T$$T$. It generalises the usual connected sum operation which is the special case when $T = D^n$$T = D^n$ is the
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$n$-disc.

The parametric connected sum operation is more complicated than the usual connected

sum operation since the isotopy classes of the embeddings of $T$$T$ into
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$M$ 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
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$M$ and $N$$N$ it is sufficient to equip them with an isotopy class of embeddings of the k-disc. Moreover, the disjoint union $D^n \sqcup D^n$$D^n \sqcup D^n$ is the unique thickening of $S^0$$S^0$. This motivates the following

Defintion 2.1.

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

Defintion 2.2. Let $M = (M, \phi)$$M = (M, \phi)$ and $N = (N, \psi)$$N = (N, \psi)$ by $S^k$$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$$\simeq$ is defined via the embeddings $\phi$$\phi$ and $\psi$$\psi$.

It is clear that we have the following

Observation 2.3.

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

### 2.1 Applications

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

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

• to define, for $m\ge 2p+q+3$$m\ge 2p+q+3$, a group stucture on the set $E^m(S^p \times S^q)$$E^m(S^p \times S^q)$ of (smooth or PL) isotopy classes of embeddings $S^p \times S^q\to \Rr^m$$S^p \times S^q\to \Rr^m$ [Skopenkov2006], \S3.4, [Skopenkov2006a], \S3, [Skopenkov2015a].
• to construct an action of this group on the set of isotopy classes of embeddings of certain $(p+q)$$(p+q)$-manifolds into $\Rr^m$$\Rr^m$ [Skopenkov2014], 1.2.
• to estimate the set of isotopy classes of embeddings [Cencelj&Repovš&Skopenkov2007], [Cencelj&Repovš&Skopenkov2008], [Skopenkov2007], [Skopenkov2010], [Skopenkov2015], [Skopenkov2015a], [Crowley&Skopenkov2016] and unpublished paper [Crowley&Skopenkov2016a].

## 3 Parametric connected sum along thickenings

Let $B$$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)$$NSt_{2n}(B)$ has the structure of an abelian group. The question of whether there is a geometric definition of this group structure is taken up in [Kreck1985, Chapter 2, pp 25-6] where it is shown how to use parametric connected sum along thickenings to define an addition of stable diffeomorphism classes of closed $2n$$2n$-$B$$B$-manifolds. This is described in more detail (for $n>2$$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)$$(n-1)$-skeleton of $B$$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$$B$-manifolds, so that one may cut out their interiors and glue the resulting $B$$B$-manifolds along the boundaries of the embedded thickenings. The special case of $n=2$$n=2$ is discussed separately in [Kreck2016, Section 5] under the name "connected sum along the $1$$1$-skeleton".