Parametric connected sum

1 Introduction

Parametric connected sum is an operation on compact connected n-manifolds $M$$== Introduction == <wikitex>; Parametric connected sum is an operation on compact connected n-manifolds $M$ and $N$ equipped with codimension 0-embeddings $\phi: T \to M$ and $\psi : T \to N$ of a compact connected manifold $T$. It generalises the usual connected sum operation but is more subtle since the isotopy classes of the embeddings $\phi$ and $\psi$ may be significantly more complicated than the isotopy classes of embeddings of n-discs need for connected sum: these last are determined by (local) orientations. </wikitex> == Connected sum == <wikitex>; Let $M$ be a compact connected n-manifold with base point $x \in \mathrm{int}$. Recall that that a local orientation for $M$ is a choice of orientation of $TM_m$, the tangent space to $M$ at $m$. We write $-M$ for $M$ with the opposition orientation at $m$. Of course, if $M$ is orientable then a local orientation for $M$ defines an orientation on $M$. If $M$ and $N$ are locally oriented n-manifolds then their [[Wikipedia:Connected_sum|connected sum]] is defined by $$ M \sharp N = ((M - m) \cup (N - n))/ \simeq$$ where $\simeq$ is defined using the local orientations to identify small balls about $k$ and $n$. The diffeomorphism type of $M \sharp N$ is well-defined: in fact $M \sharp N$ is the outcome of 0-surgery on $M \sqcup N$. The essential point is \cite{Hirsch} which states, for any $M$ and any two compatibly oriented embeddings $f_0: D^n_1 \to M$ and $\phi_1 : D^n \to M$, that $\phi_0$ is isotopic to $f_1$. If $M$ and $N$ are oriented manifolds the connected sum $M \sharp N$ is a well-defined up to diffeomorphism. Note that orientation matters! The canoical example is $$ \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$$ where $\simeq$ is defined via the embeddings $\phi$ and $\psi$. {{endthm}} Is is clear that we have the following {{beginthm|Observation}} The diffeomorphism type of $M \sharp_k N$ depends only upon the the isotopy classes of the embeddings $\phi$ and $\psi$ (which of course includes the diffeomorphism types of $M$ and $N$). {{endthm}} </wikitex> === Applications === <wikitex>; The operation of $S^k$-connected sum was used in \cite{Ajala1984} and \cite{Ajala1987} to describe the set of smooth structures on the product of spheres $\Pi_{i=1}^r S^{n_i}$. This construction also appears in \cite{Sako1981}. The analogue of such a construction for embeddings is used in \cite{Skopenkov2006} to define, for $m\ge 2p+q+3$, a group stucture on the set $E^m(S^p \times S^q)$ of (smooth or PL) isotopy classes of embeddings of $S^p \times S^q$ into $\Rr^m$. In \cite{Skopenkov2007} and \cite{Skopenkov2010} the $S^k$-connected sum of embeddings was used to estimate the set of embeddings. </wikitex> == Parametric connected sum along thickenings == <wikitex>; Let $B$ be a [[stable fibred vector bundle]]. A foundational theorem of modified surgery is {{beginthm|Theorem|Stable classification: \cite{Kreck1985}, \cite{Kreck1999}}} $$ NSt_{2n}(B) \cong \Omega_{2n}^B.$$ {{endthm}} In particular, $NSt_{2n}(B)$ has the structure of an abelian group. The question of whether there is a geometric definition of this group structure is taken up in \cite{Kreck1985|Chapter 2, pp 26-7} where it is shown how to use parametric connected sum along thickenings to define an addition of stable diffeomorphism classes of closed 2n-B-manifolds.<!--{{beginthm|Remark}} For an detailed exposition and extensive application of the modified surgery techniques of stable classifcaiton of 4-manifolds, see \cite{Teichner}. {{endthm}} --> </wikitex> == References == {{#RefList:}} [[Category:Theory]] {{Stub}}M$ and $N$$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$$T$. It generalises the usual connected sum operation but is more subtle since the isotopy classes of the embeddings $\phi$$\phi$ and $\psi$$\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.

2 Connected sum

Let $M$$M$ be a compact connected n-manifold with base point $x \in \mathrm{int}$$x \in \mathrm{int}$. Recall that that a local orientation for $M$$M$ is a choice of orientation of $TM_m$$TM_m$, the tangent space to $M$$M$ at $m$$m$. We write $-M$$-M$ for $M$$M$ with the opposition orientation at $m$$m$. Of course, if $M$$M$ is orientable then a local orientation for $M$$M$ defines an orientation on $M$$M$.

If $M$$M$ and $N$$N$ are locally oriented n-manifolds then their connected sum is defined by

$\displaystyle M \sharp N = ((M - m) \cup (N - n))/ \simeq$ $$\displaystyle M \sharp N = ((M - m) \cup (N - n))/ \simeq$$

where $\simeq$$\simeq$ is defined using the local orientations to identify small balls about $k$$k$ and $n$$n$. The diffeomorphism type of $M \sharp N$$M \sharp N$ is well-defined: in fact $M \sharp N$$M \sharp N$ is the outcome of 0-surgery on $M \sqcup N$$M \sqcup N$. The essential point is [Hirsch] which states, for any $M$$M$ and any two compatibly oriented embeddings $f_0: D^n_1 \to M$$f_0: D^n_1 \to M$ and $\phi_1 : D^n \to M$$\phi_1 : D^n \to M$, that $\phi_0$$\phi_0$ is isotopic to $f_1$$f_1$.

If $M$$M$ and $N$$N$ are oriented manifolds the connected sum $M \sharp N$$M \sharp N$ is a well-defined up to diffeomorphism. Note that orientation matters! The canoical example is

$\displaystyle \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$ $$\displaystyle \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$$

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$$M$ and $N$$N$ be locally oriented manifolds such that there is a diffeomoprhism $N \cong -N$$N \cong -N$, then $M \sharp N \cong M \sharp (-N)$$M \sharp N \cong M \sharp (-N)$.

3 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 3.1. 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 3.2. 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$.

Is is clear that we have the following

Observation 3.3. 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$).

3.1 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 is used in [Skopenkov2006] 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 of $S^p \times S^q$$S^p \times S^q$ into $\Rr^m$$\Rr^m$. In [Skopenkov2007] and [Skopenkov2010] the $S^k$$S^k$-connected sum of embeddings was used to estimate the set of embeddings.

4 Parametric connected sum along thickenings

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

Theorem 4.1 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.

5 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
• [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 $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.
• [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.

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