Parametric connected sum
Parametric connected sum is an operation on compact connected n-manifolds and equipped with codimension 0-embeddings and of a compact connected manifold . It generalises the usual connected sum operation but is more subtle since the isotopy classes of the embeddings and 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
Let be a compact connected n-manifold with base point . Recall that that a local orientation for is a choice of orientation of , the tangent space to at . We write for with the opposition orientation at . Of course, if is orientable then a local orientation for defines an orientation on .
If and are locally oriented n-manifolds then their connected sum is defined by
where is defined using the local orientations to identify small balls about and . The diffeomorphism type of is well-defined: in fact is the outcome of 0-surgery on . The essential point is [Hirsch] which states, for any and any two compatibly oriented embeddings and , that is isotopic to .
If and are oriented manifolds the connected sum is a well-defined up to diffeomorphism. Note that orientation matters! The canoical example is
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 and be locally oriented manifolds such that there is a diffeomoprhism , then .
3 Connected sum along k-spheres
We say above that to define connected sum for connected k-manifolds and it is sufficient to equip them with an isotopy class of embeddings of the k-disc. Moreover, the disjoint union is the unique thickening of . This motivates the following
Defintion 3.1. An -oriented manifold is a pair where is a compact connected manifold and is an embedding.
Defintion 3.2. Let and by -oriented manifolds. Define
where is defined via the embeddings and .
Is is clear that we have the following
Observation 3.3. The diffeomorphism type of depends only upon the the isotopy classes of the embeddings and (which of course includes the diffeomorphism types of and ).
The operation of -connected sum was used in [Ajala1984] and [Ajala1987] to describe the set of smooth structures on the product of spheres . This construction also appears in [Sako1981]. The analogue of such a construction for embeddings is used in [Skopenkov2006] to define, for , groups stucture on the set of (smooth or PL) isotopy classes of embeddings of into . In [Skopenkov2007] the -connected sum of embeddings was used to estimate the set of embeddings.
4 Parametric connected sum along thickenings
Let be a stable fibred vector bundle. A foundational theorem of modified surgery is
In particular, 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.
- [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
- [Hirsch] Template:Hirsch
- [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 on -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.
- [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
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