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.	+	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
−	</wikitex>	+	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
−	== Connected sum ==	+	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>;	+
−	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}}	+
	</wikitex>		</wikitex>
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	=== Applications ===		=== 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 $\Pi_{i=1}^r S^{n_i}$. 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.	+	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_in_Euclidean_space:_an_introduction_to_their_classification|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 $S^p \times S^q\to \Rr^m$ \cite{Skopenkov2006}, \S3.4, \cite{Skopenkov2006a}, \S3, \cite{Skopenkov2015a}.
		+	* 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.
		+	* 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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	<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{Kreck1985}, \cite{Kreck1999}}}	+	{{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}}	+	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".
		+	</wikitex>
		+	<!--{{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}} -->	+	{{endthm}}-->
−	</wikitex>	+
	== References ==		== References ==
	{{#RefList:}}		{{#RefList:}}
−		+	<!-- -->
−	[[Category:Theory]]	+	[[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 1 Introduction 2 Connected sum along k-spheres 2.1 Applications 3 Parametric connected sum along thickenings 4 References

1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds

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${{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 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. </wikitex> == Connected sum == <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 $M$. 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}} </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]]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

Tex syntax error

$M$ 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

Tex syntax error

$M$ 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

It is clear that we have the following

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].

3 Parametric connected sum along thickenings

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

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".

4 References

• [Ajala1984] S. O. Ajala, Differentiable structures on products of spheres, Houston J. Math. 10 (1984), no.1, 1–14. MR736571 (85c:57032) Zbl 0547.57026
• [Ajala1987] S. O. Ajala, Differentiable structures on a generalized product of spheres, Internat. J. Math. Math. Sci. 10 (1987), no.2, 217–226. MR886378 (88j:57028) Zbl 0627.57022
• [Cencelj&Repovš&Skopenkov2007] M. Cencelj, D. Repovš and M. Skopenkov, Homotopy type of the complement of an immersion and classification of embeddings of tori., Russ. Math. Surv.62 (2007), no.5, 985-987. Zbl 1141.57009
• [Cencelj&Repovš&Skopenkov2008] M. Cencelj, D. Repovš and M. Skopenkov, Classification of knotted tori in the 2-metastable dimension, Mat. Sbornik, 203:11 (2012), 1654-1681. Available at the arXiv:0811.2745.
• [Crowley&Skopenkov2016] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, I. Classification modulo knots, Moscow Math. J., 21 (2021), 43--98. arXiv:1611.04738.
• [Crowley&Skopenkov2016a] D. Crowley and A. Skopenkov, Embeddings of non-simply-connected 4-manifolds in 7-space, II. Smooth classification. Proc. A of the Royal Soc. of Edinburgh, to appear. arXiv:1612.04776
• [Kreck1985] M. Kreck, An extension of the results of Browder, Novikov and Wall about surgery on compact manifolds, preprint Mainz (1985).

• [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
• [Kreck2016] M. Kreck, Thoughts about a good classification of manifolds, Proceedings of the 22nd Gökova Geometry-Topology Conference (2016), 187–201. MR3526843 ()
• [Sako1981] Y. Sako, Connected sum along the cycle operation of $S^p \times \tilde S^{n-p}$ on $\pi$-manifolds, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), no.10, 499–502. MR640259 (83a:57043) Zbl 0505.57010
• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2006a] A. Skopenkov, Classification of embeddings below the metastable dimension. Available at the arXiv:0607422.
• [Skopenkov2007] A. Skopenkov, A new invariant and parametric connected sum of embeddings, Fund. Math. 197 (2007), 253–269. arXiv:math/0509621. MR2365891 (2008k:57044) Zbl 1145.57019
• [Skopenkov2010] A. Skopenkov, Embeddings of k-connected n-manifolds into $\Rr^{2n-k-1}$, Proc. AMS, 138 (2010) 3377--3389. Available at the arXiv:0812.0263.
• [Skopenkov2014] A. Skopenkov, How do autodiffeomorphisms act on embeddings, Proc. A of the Royal Society of Edinburgh, 148:4 (2018) 835--848.
• [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878
• [Skopenkov2015a] A. Skopenkov, A classification of knotted tori, Proc. A of the Royal Society of Edinburgh, 150:2 (2020), 549-567. Full version: http://arxiv.org/abs/1502.04470

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