Parametric connected sum

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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 [[High_codimension_embeddings|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}, \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{Skopenkov2007}, \cite{Skopenkov2010}.
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Revision as of 09:24, 28 April 2016

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

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 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.

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 2.1. 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.

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 M and N).

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], [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 [Skopenkov2007], [Skopenkov2010].

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], [Kreck1999].

\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 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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