Connected sum

Revision as of 15:59, 21 February 2013

1 Introduction

Let ${{Stub}} == Introduction == <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 $f_0: D^n \to M$ and $f_1 : D^n \to M$, that $f_0$ 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> == References == {{#RefList:}} [[Category:Theory]] [[Category:Definitions]]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 connected sum is defined by

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

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 [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 $f_1$.

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

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

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.

2 References

• [Hirsch] Template:Hirsch

3 External links

• Mathoverflow:Connected sum of topological manifolds