# Parametric connected sum

## 1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds $M$${{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 operation but is more subtle since the isotopy classes of the embeddings \phi and \psi 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 == ; Let M be a compact connected n-manifold with base point x \in \mathrm{int}. Recall that that a local orientation for M is a choice of orientation of TM_m, the tangent space to M at m. We write -M for M with the opposition orientation at m. Of course, if M is orientable then a local orientation for M defines an orientation on M. If M and N are locally oriented n-manifolds then their [[Wikipedia:Connected_sum|connected sum]] is defined by M \sharp N = ((M - m) \cup (N - n))/ \simeq where \simeq is defined using the local orientations to identify small balls about k and n. The diffeomorphism type of M \sharp N is well-defined: in fact M \sharp N is the outcome of 0-surgery on M \sqcup N. The essential point is \cite{Hirsch} which states, for any M and any two compatibly oriented embeddings f_0: D^n \to M and f_1 : D^n \to M, that f_0 is isotopic to f_1. If M and N are oriented manifolds the connected sum M \sharp N is a well-defined up to diffeomorphism. Note that orientation matters! The canoical example is \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2). The manifolds are not even homotopy equivalent: the first has signature 2 the other signature 0. The following elementary lemma is often useful to remember. {{beginthm|Lemma}} Let M and N be locally oriented manifolds such that there is a diffeomoprhism N \cong -N, then M \sharp N \cong M \sharp (-N). {{endthm}} == 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 is used in \cite{Skopenkov2006} 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 of S^p \times S^q into \Rr^m. In \cite{Skopenkov2007} and \cite{Skopenkov2010} the S^k-connected sum of embeddings was used to estimate the set of embeddings. == 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:Theory]]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
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$T$. It generalises the usual connected sum operation but is more subtle since the isotopy classes of the embeddings $\phi$$\phi$ and $\psi$$\psi$ 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

Let $M$$M$ be a compact connected n-manifold with base point $x \in \mathrm{int}$$x \in \mathrm{int}$. Recall that that a local orientation for $M$$M$ is a choice of orientation of $TM_m$$TM_m$, the tangent space to $M$$M$ at $m$$m$. We write $-M$$-M$ for $M$$M$ with the opposition orientation at $m$$m$. Of course, if $M$$M$ is orientable then a local orientation for $M$$M$ defines an orientation on $M$$M$.

If $M$$M$ and $N$$N$ are locally oriented n-manifolds then their connected sum is defined by

$\displaystyle M \sharp N = ((M - m) \cup (N - n))/ \simeq$

where $\simeq$$\simeq$ is defined using the local orientations to identify small balls about $k$$k$ and $n$$n$. The diffeomorphism type of $M \sharp N$$M \sharp N$ is well-defined: in fact $M \sharp N$$M \sharp N$ is the outcome of 0-surgery on $M \sqcup N$$M \sqcup N$. The essential point is [Hirsch] which states, for any $M$$M$ and any two compatibly oriented embeddings $f_0: D^n \to M$$f_0: D^n \to M$ and $f_1 : D^n \to M$$f_1 : D^n \to M$, that $f_0$$f_0$ is isotopic to $f_1$$f_1$.

If $M$$M$ and $N$$N$ are oriented manifolds the connected sum $M \sharp N$$M \sharp N$ is a well-defined up to diffeomorphism. Note that orientation matters! The canoical example is

$\displaystyle \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$

The manifolds are not even homotopy equivalent: the first has signature 2 the other signature 0. The following elementary lemma is often useful to remember.

Lemma 2.1. Let $M$$M$ and $N$$N$ be locally oriented manifolds such that there is a diffeomoprhism $N \cong -N$$N \cong -N$, then $M \sharp N \cong M \sharp (-N)$$M \sharp N \cong M \sharp (-N)$.

## 3 Connected sum along k-spheres

We say above that to define connected sum for connected k-manifolds $M$$M$ and $N$$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 3.1. A manifold with an $S^k$$S^k$-thickening, an $S^k$$S^k$-thickened manifold for short, is a pair $(M, \phi)$$(M, \phi)$ where $M$$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 3.2. 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 3.3. 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$$\psi$ (which of course includes the diffeomorphism types of $M$$M$ and $N$$N$).

### 3.1 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 is used in [Skopenkov2006] 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 of $S^p \times S^q$$S^p \times S^q$ into $\Rr^m$$\Rr^m$. In [Skopenkov2007] and [Skopenkov2010] the $S^k$$S^k$-connected sum of embeddings was used to estimate the set of embeddings.

## 4 Parametric connected sum along thickenings

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

Theorem 4.1 Stable classification: [Kreck1985], [Kreck1999].

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

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 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.