Connected sum of manifolds

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.

Homology
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

 * Fundamental group of connected sum is free product of fundamental groups in dimension at least three: This fails in dimension two, because the circle $$S^1$$ has nontrivial fundamental group.
 * Connected sum of simply connected manifolds is simply connected

Related notions

 * Fiber sum
 * Symplectic sum
 * Knot sum