Parametric connected sum

Latest revision as of 02:04, 8 April 2020

[edit] 1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds ${{Stub}} == Introduction == <wikitex>; 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. </wikitex> == Connected sum along k-spheres == <wikitex>; 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}} </wikitex> === Applications === <wikitex>; 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. </wikitex> == Parametric connected sum along thickenings == <wikitex>; 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.<!--{{beginthm|Remark}} For an detailed exposition and extensive application of the modified surgery techniques of stable classifcaiton of 4-manifolds, see \cite{Teichner}. {{endthm}} --> </wikitex> == References == {{#RefList:}} <!-- --> [[Category:Theory]] [[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.

[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 2.2. 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

[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

• 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 $S^p \times S^q\to \Rr^m$ [Skopenkov2006], \S3.4, [Skopenkov2006a], \S3, [Skopenkov2015a].
• to construct an action of this group on the set of isotopy classes of embeddings of certain $(p+q)$-manifolds into $\Rr^m$ [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].

[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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• [Wall1966a] C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR0192509 (33 #734) Zbl 0149.20501