Connected sum

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== Introduction ==
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== Connected sum of smooth manifolds ==
<wikitex>;
<wikitex>;
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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 $M$.
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<!-- 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
If $M$ and $N$ are locally oriented n-manifolds then their [[Wikipedia:Connected_sum|connected sum]] is defined by
$$ M \sharp N = ((M - m) \cup (N - n))/ \simeq$$
$$ M \sharp N = ((M - m) \cup (N - n))/ \simeq$$
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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 $f_0: D^n \to M$ and $f_1 : D^n \to M$, that $f_0$ is isotopic to $f_1$.
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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 $f_1$. -->
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Let $M_0$ and $M_1$ be oriented closed smooth connected $n$-manifolds. Their connected sum is an oriented closed smooth connected $n$-manifold
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\[ M_1 \sharp M_2 \]
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which is defined as follows (c.f. \cite{Kervaire&Milnor1963|Section 2}. Choose smooth embeddings
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\[ i_0 \colon D^n \to M_0 \quad \text{and} \quad i_1 \colon D^n \to M_1 \]
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where $i_1$ preserves orientations and $i_2$ reverses orientations. The connected sum is formed from the
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disjoint union
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\[ (M_0 - i_0(0)) \sqcup (M_1 - i_1(0) \]
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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
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$M_0 \sharp M_1$ is obtain from the charts on $M_0 - i_0(0)$ and $M_1 - i_1(0)$. The orientation on
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$M_0 \sharp M_1$ is chosen to be the one compatible with the orientation of $M_0$ and $M_1$.
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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
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A fundamental lemma of differential topology, \cite{Palais1959|Theorem 5.5} \cite{Cerf1961} states that any
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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.
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{{beginthm|Lemma|\cite{Kervaire&Milnor1963|Lemma 2.1}}}
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The connected sum operation is well defined, associative and commutative up to
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orientation preserving diffeomoprhism. The sphere $S^n$ serves as the identity element.
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{{endthm}}
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The connected sum operation also descends to give well-defined operations on larger equivalence classes
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of oriented manifolds.
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{{beginthm|Lemma|c.f. \cite{Kervaire&Milnor1963|Lemma 2.2}}}}
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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 $M_0' \sharp M_1$.
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{{endthm}}
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<!-- If $M$ and $N$ are oriented manifolds the connected sum $M \sharp N$ is a well-defined up to diffeomorphism. Note that orientation matters! -->
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== Exmaples==
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The orientation of the manifolds is important in general. The canonical example is
$$ \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$$
$$ \CP^2 \sharp \CP^2 \neq \CP^2 \sharp (-\CP^2).$$
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.
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.
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{{beginthm|Lemma}}
{{beginthm|Lemma}}
Let $M$ and $N$ be locally oriented manifolds such that there is a diffeomoprhism $N \cong -N$, then $M \sharp N \cong M \sharp (-N)$.
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}}
{{endthm}}
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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 $H^4(S^4; \Z) \cong \Z$.
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{{beginthm|Lemma|c.f.\cite{Wilkens1974/75|Theorem 1} }}
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There are diffeomorphisms
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# $M \sharp M \cong M \sharp (S^3 \times S^4)$
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# $M \sharp \Sigma \cong M$ for any homotopy sphere $\Sigma$. (Recall that the
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group of [[Exotic spheres|homotopy 7-spheres]], $\Theta_7$ is isomorphic to $\Z/28$.)
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{{endthm}}
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{{beginproof}}
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1.)
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2.) From
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{{endproof}}
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</wikitex>
</wikitex>

Revision as of 17:11, 21 February 2013

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

1 Connected sum of smooth manifolds

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. [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 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] [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.

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 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' 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.

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).

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 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

  1. M \sharp M \cong M \sharp (S^3 \times S^4)
  2. 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.)

Proof. 1.)

2.) From

\square





2 References


3 External links

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