Parametric connected sum

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1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds $M$${{Stub}} == Introduction == <wikitex>; Parametric connected sum is an operation on compact connected n-manifolds $M$ and $N$ equipped with codimension 0-embeddings $\phi: T \to M$ and $\psi : T \to N$ of a compact connected manifold $T$. It generalises the usual [[Connected sum|connected sum]] operation which is the special case when $T = D^n$ is the $n$-disc. The parametric connected sum operation is more complicated than the usual connected sum operation since the isotopy classes of the embeddings of $T$ into $M$ may be significantly more complicated than the isotopy classes of embeddings of n-discs need for connected sum: these last are determined by (local) orientations. </wikitex> == Connected sum along k-spheres == <wikitex>; We say above that to define connected sum for connected k-manifolds $M$ and $N$ it is sufficient to equip them with an isotopy class of embeddings of the k-disc. Moreover, the disjoint union $D^n \sqcup D^n$ is the unique [[thickening]] of $S^0$. This motivates the following {{beginthm|Defintion}} A manifold with an $S^k$-[[thickening]], an $S^k$-thickened manifold for short, is a pair $(M, \phi)$ where $M$ is a compact connected manifold and $\phi : S^k \times D^{n-k} \to \mathrm{int}(M)$ is an embedding. {{endthm}} {{beginthm|Defintion}} Let $M = (M, \phi)$ and $N = (N, \psi)$ by $S^k$-thickened manifolds. Define $$ M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/\simeq$$ where $\simeq$ is defined via the embeddings $\phi$ and $\psi$. {{endthm}} It is clear that we have the following {{beginthm|Observation}} The diffeomorphism type of $M \sharp_k N$ depends only upon the the isotopy classes of the embeddings $\phi$ and $\psi$ (which of course includes the diffeomorphism types of $M$ and $N$). {{endthm}} </wikitex> === Applications === <wikitex>; The operation of $S^k$-connected sum was used in \cite{Ajala1984} and \cite{Ajala1987} to describe the set of smooth structures on the product of spheres $\Pi_{i=1}^r S^{n_i}$. This construction also appears in \cite{Sako1981}. The analogue of such a construction for [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification|embeddings]], the $S^k$-parametric connected sum of embeddings, is used * to define, for $m\ge 2p+q+3$, a group stucture on the set $E^m(S^p \times S^q)$ of (smooth or PL) isotopy classes of embeddings $S^p \times S^q\to \Rr^m$ \cite{Skopenkov2006}, \S3.4, \cite{Skopenkov2006a}, \S3, \cite{Skopenkov2015a}. * to construct an action of this group on the set of isotopy classes of embeddings of certain $(p+q)$-manifolds into $\Rr^m$ \cite{Skopenkov2014}, 1.2. * to estimate the set of isotopy classes of embeddings \cite{Cencelj&Repovš&Skopenkov2007}, \cite{Cencelj&Repovš&Skopenkov2008}, \cite{Skopenkov2007}, \cite{Skopenkov2010}, \cite{Skopenkov2015}, \cite{Skopenkov2015a}. </wikitex> == Parametric connected sum along thickenings == <wikitex>; Let $B$ be a [[stable fibred vector bundle]]. A foundational theorem of modified surgery is {{beginthm|Theorem|Stable classification: \cite{Kreck1985}, \cite{Kreck1999}}} $$ NSt_{2n}(B) \cong \Omega_{2n}^B.$$ {{endthm}} In particular, $NSt_{2n}(B)$ has the structure of an abelian group. The question of whether there is a geometric definition of this group structure is taken up in \cite{Kreck1985|Chapter 2, pp 26-7} where it is shown how to use parametric connected sum along thickenings to define an addition of stable diffeomorphism classes of closed 2n-B-manifolds.<!--{{beginthm|Remark}} For an detailed exposition and extensive application of the modified surgery techniques of stable classifcaiton of 4-manifolds, see \cite{Teichner}. {{endthm}} --> </wikitex> == References == {{#RefList:}} <!-- --> [[Category:Definitions]]M$ and $N$$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

$\displaystyle M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/\simeq$ $$\displaystyle M \sharp_k N = (M - \phi(S^k \times \{ 0 \}) \cup (N - \psi(S^k \times \{ 0 \})/\simeq$$

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].

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], [Kreck1999].

$\displaystyle NSt_{2n}(B) \cong \Omega_{2n}^B.$ $$\displaystyle NSt_{2n}(B) \cong \Omega_{2n}^B.$$

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 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.

4 References

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• [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.
• [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
• [Sako1981] Y. Sako, Connected sum along the cycle operation of $S^p \times \tilde S^{n-p}$$S^p \times \tilde S^{n-p}$ on $\pi$$\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}$$\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