Parametric connected sum

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1 Introduction

2 Connected sum

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 connected sum is defined by

\displaystyle  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 [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.

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

\displaystyle  \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.

Lemma 2.1. 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).

3 Parametric 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. Moreove, the disjoint union D^n \sqcup D^n is the unique thickening of S^0. This motivates the following

Defintion 3.1. 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.

Defintion 3.2. Let M = (M, \phi) and N = (N, \psi) by S^k-oriented 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.

Is is clear that we have the following

Observation 3.3. 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).

4 Applications of k-sphere conneted sum

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}. It is also used in [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 [Sako1981].

5 Parametric connected sum along thickenings

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

Theorem 5.1 Stable classification: [Kreck1999] [Kreck1985].

\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 close 2n-B-manifolds.

6 References

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

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