Connected sum

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[edit] 1 Connected sum of smooth manifolds

Let ${{Stub}} == Connected sum of smooth manifolds == <wikitex>;  Let $M_0$ and $M_1$ be oriented closed smooth connected $n$-manifolds. Their connected sum is an oriented closed smooth connected $n$-manifold \[ M_1 \sharp M_2 \] which is defined as follows (c.f. \cite{Kervaire&Milnor1963|Section 2}. Choose smooth embeddings \[ i_0 \colon D^n \to M_0 \quad \text{and} \quad i_1 \colon D^n \to M_1 \] where $i_1$ preserves orientations and $i_2$ reverses orientations. The connected sum is formed from the disjoint union \[ (M_0 - i_0(0)) \sqcup (M_1 - i_1(0) \] by identifying $i_0(tu)$ with $i_1((1-t)u)$ for $u \in S^{n-1}$ and <script type="math/tex">M_0$ and $M_1$ be oriented closed smooth connected $n$ -manifolds. Their connected sum is an oriented closed smooth connected $n$ -manifold

$\displaystyle M_1 \sharp M_2$ $\displaystyle M_1 \sharp M_2$

which is defined as follows (c.f. [Kervaire&Milnor1963, Section 2]). Choose smooth embeddings

$\displaystyle i_0 \colon D^n \to M_0 \quad \text{and} \quad i_1 \colon D^n \to M_1$ $\displaystyle i_0 \colon D^n \to M_0 \quad \text{and} \quad i_1 \colon D^n \to M_1$

where $i_1$ preserves orientations and $i_2$ reverses orientations. The connected sum is formed from the disjoint union

$\displaystyle \bigl( M_0 - i_0(0) \bigr) \sqcup \bigl(M_1 - i_1(0) \bigr)$ $\displaystyle \bigl( M_0 - i_0(0) \bigr) \sqcup \bigl(M_1 - i_1(0) \bigr)$

by identifying $i_0(tu)$ with $i_1((1-t)u)$ for $u \in S^{n-1}$ and $0 < t < 1$ . The smooth structure on $M_0 \sharp M_1$ is obtain from the charts on $M_0 - i_0(0)$ and $M_1 - i_1(0)$ . The orientation on $M_0 \sharp M_1$ is chosen to be the one compatible with the orientation of $M_0$ and $M_1$ .

A fundamental lemma of differential topology, [Palais1959, Theorem 5.5] and [Cerf1961] states that any two orientation preserving smooth embeddings of the $n$ -disc into a closed oriented smooth $n$ -manifold are isotopic. As a consequence we have the following lemma.

The connected sum operation is well defined, associative and commutative up to orientation preserving diffeomoprhism. The sphere $S^n$ serves as the identity element.

The connected sum operation also descends to give well-defined operations on larger equivalence classes of oriented manifolds.

Let $M_0$ , $M_0'$ and $M_1$ be oriented closed connected smooth manifold. Suppose that $M_0$ is h-cobordant to $M_0'$ , resp. bordant to $M_0'$ then $M_0 \sharp M_1$ is h-cobordant, resp. bordant, to $M_0' \sharp M_1$ .

[edit] 2 Connected sum of topological manifolds

Connected sum is a well-defined operation up to orientation preserving homeomorphism for oriented closed connected topological $n$ -manifolds. However, there is no analogue of the Palais/Cerf result and so the proof is more complicated. See the mathoverflow discussion cited below.

[edit] 3 Examples

The orientation of the manifolds is important in general. The canonical 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.

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

Connected sum decompositions of manifolds are far from being unique. For example, let $M = S^3 \tilde \times S^4$ be the total space of the non-trivial 3-sphere bundle over $S^4$ with Euler class zero and Pontrjagin class four times a preferred generator of $H^4(S^4; \Z) \cong \Z$ .

Lemma 3.2 c.f.[Wilkens1974/75, Theorem 1] . There are diffeomorphisms

$M \sharp M \cong M \sharp (S^3 \times S^4)$
$M \sharp \Sigma \cong M$ for any homotopy sphere $\Sigma$ . (Recall that the group of homotopy 7-spheres, $\Theta_7$ is isomorphic to $\Z/28$ .)

1.) The manifold $M$ is the boundary of the total space of the corresponding disc bundle $W : = D^4 \tilde \times S^4$ and hence $M \sharp M$ is the boundary of $W \natural W$ . Compact $3$ -connected $8$ -manifolds were classified in [Wall1962a, Section 2]. Since the intersection form of $W \natural W$ is trivial, it is a simple consequence of Wall's classification that there is a diffeomorphism $f \colon W \natural W \cong W \natural (D^4 \times S^4)$ . Restricting $f$ to the boundary gives the desired diffeomorphism.

2.) This is a special case of [Wilkens1974/75, Theorem 1].

$\square$ $\square$

[edit] 4 Properties

Let $M$ be a closed connected $n$ -manifold and let

$\displaystyle M^\bullet : = M \setminus \textup{Int}(D^n)$ $\displaystyle M^\bullet : = M \setminus \textup{Int}(D^n)$

denote the compact manifold obtained from $M$ by deleting a small embedded open $n$ -disc. From the definition it is clear that

$\displaystyle (M_0 \sharp M_1)^\bullet = (M_0^\bullet) \natural (M_1^\bullet) \simeq M_0^\bullet \vee M_1^\bullet.$ $\displaystyle (M_0 \sharp M_1)^\bullet = (M_0^\bullet) \natural (M_1^\bullet) \simeq M_0^\bullet \vee M_1^\bullet.$

Here $\vee$ denotes the one point union of topological spaces and $\simeq$ indicates that two spaces are homotopy equivalent. Applying Van Kampen's Theorem we immediately obtain the following lemma.

Let the dimension $n$ be three or greater. Then the fundamental group of a connected sum is the free product of the fundamental group of the components:

$\displaystyle \pi_1(M_0 \sharp M_1) \cong \pi_1(M_0) \ast \pi_1(M_2) .$ $\displaystyle \pi_1(M_0 \sharp M_1) \cong \pi_1(M_0) \ast \pi_1(M_2) .$

[edit] 5 References

[Cerf1961] J. Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227–380. MR0140120 (25 #3543) Zbl 0101.16001
[Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
[Palais1959] R. S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125–141. MR0116352 (22 #7140) Zbl 0092.30802
[Wall1962a] C. T. C. Wall, Classification of $(n-1)$ -connected $2n$ -manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR0145540 (26 #3071) Zbl 0218.57022
[Wilkens1974/75] D. L. Wilkens, On the inertia groups of certain manifolds, J. London Math. Soc. (2) 9 (1974/75), 537–548. MR0383435 (52 #4316)

[edit] 6 External links

Mathoverflow:Connected sum of topological manifolds
The Encyclopedia of Mathematics article on connected sum
The Wikipedia page about connected sum

< t < 1$. The smooth structure on $M_0 \sharp M_1$ is obtain from the charts on $M_0 - i_0(0)$ and $M_1 - i_1(0)$. The orientation on $M_0 \sharp M_1$ is chosen to be the one compatible with the orientation of $M_0$ and $M_1$. A fundamental lemma of differential topology, \cite{Palais1959|Theorem 5.5} \cite{Cerf1961} states that any two orientation preserving smootgh embeddings of the $n$-disc into a closed oriented smooth $n$-manifold are isotopic. As a consequence we have the following lemma. {{beginthm|Lemma|\cite{Kervaire&Milnor1963|Lemma 2.1}}} The connected sum operation is well defined, associative and commutative up to orientation preserving diffeomoprhism. The sphere $S^n$ serves as the identity element. {{endthm}} The connected sum operation also descends to give well-defined operations on larger equivalence classes of oriented manifolds. {{beginthm|Lemma|c.f. \cite{Kervaire&Milnor1963|Lemma 2.2}}}} Let $M_0$, $M_0'$ and $M_1$ be oriented closed connected smooth manifold. Suppose that $M_0$ is [[h-cobordism|h-cobordant]] to $M_0'$, resp. bordant to $M_0'$ then $M_0 \sharp M_1$ is h-cobordant, resp. bordant, to $M_0' \sharp M_1$. {{endthm}} == Exmaples== The orientation of the manifolds is important in general. The canonical 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}} Connected sum decompositions of manifolds are far from being unique. For example, let $M = S^3 \tilde \times S^4$ be the total space of the non-trivial 3-sphere bundle over $S^4$ with Euler class zero and Pontrjagin class four times a preferred generator of $H^4(S^4; \Z) \cong \Z$. {{beginthm|Lemma|c.f.\cite{Wilkens1974/75|Theorem 1} }} There are diffeomorphisms # $M \sharp M \cong M \sharp (S^3 \times S^4)$ # $M \sharp \Sigma \cong M$ for any homotopy sphere $\Sigma$. (Recall that the group of [[Exotic spheres|homotopy 7-spheres]], $\Theta_7$ is isomorphic to $\Z/28$.) {{endthm}} {{beginproof}} 1.) The manifold $M$ is the boundary of the total space of the corresponding disc bundle $W : = D^4 \tilde \times S^4$ and hence $M \sharp M$ is the boundary of $W \natural W$. Compact $-connected $-manifolds were classified in \cite{Wall1962a|Section 2}. Since the intersection form of $W \natural W$ is trivial, it is a simple consequence of Wall's classification that there is a diffeomorphism $f \colon W \natural W \cong W \natural (D^4 \times S^4)$. Restricting $f$ to the boundary gives the desired diffeomorphism. 2.) This is a special case of \cite{Wilkens1974/75|Theorem 1}. {{endproof}} == Properties == Let $M$ be a closed connected $n$-manifold and let $$M^\bullet : = M \setminus {\rm Int}(D^n)$$ denote the compact manifold obtained from $M$ by deleting a small embedded open $n$-disc. From the definition it is clear that $$ (M_0 \sharp M_1)^\bullet = (M_0^\bullet) \natural (M_1^\bullet) \simeq M_0^\bullet \vee M_1^\bullet.$$ Here $\vee$ denotes the one point union of topological spaces and $\simeq$ indicates that two spaces are homotopy equivalent. Applying Van Kampen's Theorem we immediately obtain the following lemma. {{beginthm|Lemma}} Let the dimension $n$ be three or greater. Then the fundamental group of a connected sum is the free product of the fundamental group of the components: $$ \pi_1(M_0 \sharp M_1) \cong \pi_1(M_0) \ast \pi_1(M_2) .$$ {{endthm}} == References == {{#RefList:}} == External links == * [http://mathoverflow.net/questions/121571/connected-sum-of-topological-manifolds Mathoverflow:Connected sum of topological manifolds] [[Category:Theory]] [[Category:Definitions]]M_0 and $M_1$ $M_1$ be oriented closed smooth connected $n$ $n$ -manifolds. Their connected sum is an oriented closed smooth connected $n$ $n$ -manifold