Connected sum
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1 Introduction
Let be a compact connected n-manifold with base point . Recall that that a local orientation for is a choice of orientation of , the tangent space to at . We write for with the opposition orientation at . Of course, if is orientable then a local orientation for defines an orientation on .
If and are locally oriented n-manifolds then their connected sum is defined by
where is defined using the local orientations to identify small balls about and . The diffeomorphism type of is well-defined: in fact is the outcome of 0-surgery on . The essential point is [Hirsch] which states, for any and any two compatibly oriented embeddings and , that is isotopic to .
If and are oriented manifolds the connected sum is a well-defined up to diffeomorphism. Note that orientation matters! The canoical example is
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 1.1. Let and be locally oriented manifolds such that there is a diffeomoprhism , then .