Connected sum

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1 Connected sum of smooth manifolds

Let $M_0$${{Stub}} == Connected sum of smooth manifolds == <wikitex>; <!-- Let $M$ be a compact connected n-manifold with base point $m \in \mathrm{int} M$. Recall that that a local orientation for $M$ is a choice of orientation of $TM_m$, the tangent space to $M$ at $m$. We write $-M$ for $M$ with the opposition orientation at $m$. Of course, if $M$ is orientable then a local orientation for $M$ defines an orientation on $M$. If $M$ and $N$ are locally oriented n-manifolds then their [[Wikipedia:Connected_sum|connected sum]] is defined by $$ M \sharp N = ((M - m) \cup (N - n))/ \simeq$$ where $\simeq$ is defined using the local orientations to identify small balls about $m$ 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 $f_1: D^n \to M$ and $f_1 : D^n \to M$, that $f_0$ is isotopic to $f_1$. --> 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$$M_1$ be oriented closed smooth connected $n$$n$-manifolds. Their connected sum is an oriented closed smooth connected $n$$n$-manifold \[ M_1 \sharp M_2 \] which is defined as follows (c.f. [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$$i_1$ preserves orientations and $i_2$$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)$$i_0(tu)$ with $i_1((1-t)u)$$i_1((1-t)u)$ for $u \in S^{n-1}$$u \in S^{n-1}$ and $0 < t < 1$$0 < t < 1$. The smooth structure on $M_0 \sharp M_1$$M_0 \sharp M_1$ is obtain from the charts on $M_0 - i_0(0)$$M_0 - i_0(0)$ and $M_1 - i_1(0)$$M_1 - i_1(0)$. The orientation on $M_0 \sharp M_1$$M_0 \sharp M_1$ is chosen to be the one compatible with the orientation of $M_0$$M_0$ and $M_1$$M_1$.

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

Lemma 1.1 [Kervaire&Milnor1963, Lemma 2.1]. The connected sum operation is well defined, associative and commutative up to orientation preserving diffeomoprhism. The sphere $S^n$$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.

Lemma 1.2 c.f. [Kervaire&Milnor1963, Lemma 2.2].} Let $M_0$$M_0$, $M_0'$$M_0'$ and $M_1$$M_1$ be oriented closed connected smooth manifold. Suppose that $M_0$$M_0$ is h-cobordant to $M_0'$$M_0'$, resp. bordant to $M_0'$$M_0'$ then $M_0 \sharp M_1$$M_0 \sharp M_1$ is h-cobordant, resp. bordant, to $M_0' \sharp M_1$$M_0' \sharp M_1$.

Exmaples

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.

Lemma 3.1. 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)$.

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

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

1. $M \sharp M \cong M \sharp (S^3 \times S^4)$$M \sharp M \cong M \sharp (S^3 \times S^4)$
2. $M \sharp \Sigma \cong M$$M \sharp \Sigma \cong M$ for any homotopy sphere $\Sigma$$\Sigma$. (Recall that the

group of homotopy 7-spheres, $\Theta_7$$\Theta_7$ is isomorphic to $\Z/28$$\Z/28$.)

Proof. 1.)

2.) From

$\square$$\square$

2 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
• [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)

3 External links

• Mathoverflow:Connected sum of topological manifolds

< 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.) 2.) From {{endproof}} == 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 \[ M_1 \sharp M_2 \] which is defined as follows (c.f. [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$$i_1$ preserves orientations and $i_2$$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)$$i_0(tu)$ with $i_1((1-t)u)$$i_1((1-t)u)$ for $u \in S^{n-1}$$u \in S^{n-1}$ and $0 < t < 1$$0 < t < 1$. The smooth structure on $M_0 \sharp M_1$$M_0 \sharp M_1$ is obtain from the charts on $M_0 - i_0(0)$$M_0 - i_0(0)$ and $M_1 - i_1(0)$$M_1 - i_1(0)$. The orientation on $M_0 \sharp M_1$$M_0 \sharp M_1$ is chosen to be the one compatible with the orientation of $M_0$$M_0$ and $M_1$$M_1$.

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

Lemma 1.1 [Kervaire&Milnor1963, Lemma 2.1]. The connected sum operation is well defined, associative and commutative up to orientation preserving diffeomoprhism. The sphere $S^n$$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.

Lemma 1.2 c.f. [Kervaire&Milnor1963, Lemma 2.2].} Let $M_0$$M_0$, $M_0'$$M_0'$ and $M_1$$M_1$ be oriented closed connected smooth manifold. Suppose that $M_0$$M_0$ is h-cobordant to $M_0'$$M_0'$, resp. bordant to $M_0'$$M_0'$ then $M_0 \sharp M_1$$M_0 \sharp M_1$ is h-cobordant, resp. bordant, to $M_0' \sharp M_1$$M_0' \sharp M_1$.

Exmaples

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.

Lemma 3.1. 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)$.

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

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

1. $M \sharp M \cong M \sharp (S^3 \times S^4)$$M \sharp M \cong M \sharp (S^3 \times S^4)$
2. $M \sharp \Sigma \cong M$$M \sharp \Sigma \cong M$ for any homotopy sphere $\Sigma$$\Sigma$. (Recall that the

group of homotopy 7-spheres, $\Theta_7$$\Theta_7$ is isomorphic to $\Z/28$$\Z/28$.)

Proof. 1.)

2.) From

$\square$$\square$

2 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
• [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)

3 External links

• Mathoverflow:Connected sum of topological manifolds