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connected sum (Definition)

Let $ M$ and $ N$ be two $ n$-manifolds. Choose points $ m\in M$ and $ n\in N$, and let $ U, V$ be neighborhoods of these points, respectively. Since $ M$ and $ N$ are manifolds, we may assume that $ U$ and $ V$ are balls, with boundaries homeomorphic to $ (n-1)$-spheres, since this is possible in $ \mathbb{R}^n$. Then let $ \varphi :\partial U\to\partial V$ be a homeomorphism. If $ M$ and $ N$ are oriented, this should be orientation preserving with respect to the induced orientation (that is, degree 1). Then the connected sum $ M\sharp N$ is $ M-U$ and $ N-V$ glued along the boundaries by $ \varphi $.

That is, $ M\sharp N$ is the disjoint union of $ M-U$ and $ N-V$ modulo the equivalence relation $ x\sim y$ if $ x\in\partial U$, $ y\in\partial V$ and $ \varphi (x)=y$.



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Cross-references: equivalence relation, disjoint union, degree, induced, orientation, oriented, homeomorphism, homeomorphic, boundaries, balls, manifolds, neighborhoods, points
There is 1 reference to this entry.

This is version 3 of connected sum, born on 2002-12-20, modified 2006-10-15.
Object id is 3800, canonical name is ConnectedSum2.
Accessed 2329 times total.

Classification:
AMS MSC: 57-00 (Manifolds and cell complexes :: General reference works )
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