Parametric connected sum

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{{Stub}}
== 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
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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>
</wikitex>
== Connected sum ==
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== Connected sum along k-spheres ==
<wikitex>;
<wikitex>;
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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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
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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 $f_0: D^n_1 \to M$ and $\phi_1 : D^n \to M$, that $\phi_0$ 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 $N$ there is a diffeomoprhism $N \cong -N$, then $M \sharp N \cong M \sharp (-N)$.
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{{endthm}}
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</wikitex>
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== Parametric connected sum along k-spheres ==
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<wikitex>;
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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. Moreove, the disjoint union $D^n \sqcup D^n$ is the unique [[thickening]] of $S^0$. This motivates the following
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{{beginthm|Defintion}}
{{beginthm|Defintion}}
An $S^k$-oriented manifold 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.
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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}}
{{endthm}}
{{beginthm|Defintion}}
{{beginthm|Defintion}}
Let $M = (M, \phi)$ and $N = (N, \psi)$ by $S^k$-oriented manifolds. Define
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Let $M = (M, \phi)$ and $N = (N, \psi)$ by $S^k$-thickened manifolds. Define
$$ M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/simeq$$
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$$ M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/\simeq$$
where $\simeq$ is defined via the embeddings $\phi$ and $\psi$.
where $\simeq$ is defined via the embeddings $\phi$ and $\psi$.
{{endthm}}
{{endthm}}
Is is clear that we have the following
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It is clear that we have the following
{{beginthm|Observation}}
{{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 $N$).
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}}
{{endthm}}
</wikitex>
</wikitex>
== Applications of k-sphere conneted sum ==
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=== Applications ===
<wikitex>;
<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 $\Pi_{i=1}^r S^{n_i}$. It is also used in \cite{Skopenkov} to define, for appropriate values of $p, q$ and m groups stuctures on $E^m(S^p \times S^q)$ the set of smooth isotopy classes of embeddings of $S^p \times S^q$ into $\Rr^m$. It also appears in \cite{Sako1981}.
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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}.
</wikitex>
</wikitex>
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== Parametric connected sum along thickenings ==
== Parametric connected sum along thickenings ==
<wikitex>;
<wikitex>;
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{Kreck1999} \cite{Kreck1985}}}
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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 close 2n-B-manifolds.
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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".
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</wikitex>
<!--{{beginthm|Remark}}
<!--{{beginthm|Remark}}
For an detailed exposition and extensive application of the modified surgery techniques of stable classifcaiton of 4-manifolds, see \cite{Teichner}.
For an detailed exposition and extensive application of the modified surgery techniques of stable classifcaiton of 4-manifolds, see \cite{Teichner}.
{{endthm}} -->
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{{endthm}}-->
</wikitex>
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== References ==
== References ==
{{#RefList:}}
{{#RefList:}}
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<!-- -->
[[Category:Theory]]
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[[Category:Definitions]]
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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 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
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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