Connected sum of manifolds

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Definition

Let M_1 and M_2 be connected manifolds. A connected sum of M_1 and M_2, denoted M_1 \# M_2, is constructed as follows. Let f_i:\R^n \to U_i be homeomorphisms where U_i are open subsets of M_i. Let M_i' denote the complement in M_i of the image of the open unit ball in \R^n, under f_i. Then the connected sum is the quotient of M_1 \sqcup M_2 under the identification of the boundary S^{n-1}s with each other, via the composite f_2 \circ f_1^{-1}.

In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together. Further information: homotopy type of connected sum depends on choice of gluing map

Homology

Further information: Homology of connected sum

The homology of the connected sum can be computed using the Mayer-Vietoris homology sequence for open sets obtained by enlarging the M_i's slightly, and using the fact that M_i' is a strong deformation retract of M_i minus a point.

The interesting phenomena occur at n and n-1, because this is where the gluing is occurring.

Fundamental group

Related notions