# Parametric connected sum

## 1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ and $N$$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 $n$$n$-disc. The parametric connected sum operation is more complicated than the usual connected sum operation since the isotopy classes of the embeddings of $T$$T$ into $M$$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$$M$ and $N$$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)$$(M, \phi)$ where $M$$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 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$$\psi$ (which of course includes the diffeomorphism types of $M$$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".