Parametric connected sum

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== Introduction ==
== Introduction ==
<wikitex>;
<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 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.
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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
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which is the special case when $T = D^n$ is the $n$-disc.
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The parametric connected sum operation is more complicated than the usual connected
== Connected sum ==
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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>;
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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 $M$.
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If $M$ and $N$ are locally oriented n-manifolds then their [[Wikipedia:Connected_sum|connected sum]] is defined by
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$$ M \sharp N = ((M - m) \cup (N - n))/ \simeq$$
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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 $f_1$.
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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
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$$ \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$$
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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.
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{{beginthm|Lemma}}
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Let $M$ and $N$ be locally oriented manifolds such that there is a diffeomoprhism $N \cong -N$, then $M \sharp N \cong M \sharp (-N)$.
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{{endthm}}
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=== Applications ===
=== Applications ===
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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 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.
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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}.
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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
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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}.
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* 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.
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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 ==
{{#RefList:}}
{{#RefList:}}
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[[Category:Theory]]
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[[Category:Definitions]]

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
Tex syntax error
and N equipped with codimension 0-embeddings \phi: T \to M and \psi : T \to N of a compact connected manifold
Tex syntax error
. 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
Tex syntax error
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
Tex syntax error
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
Tex syntax error
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
Tex syntax error
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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