Connected sum of manifolds
Let and be connected manifolds. A connected sum of and , denoted , is constructed as follows. Let be homeomorphisms where are open subsets of . Let denote the complement in of the image of the open unit ball in , under . Then the connected sum is the quotient of under the identification of the boundary s with each other, via the composite .
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
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 s slightly, and using the fact that is a strong deformation retract of minus a point.
The interesting phenomena occur at and , because this is where the gluing is occurring.
- Fundamental group of connected sum is free product of fundamental groups in dimension at least three: This fails in dimension two, because the circle has nontrivial fundamental group.
- Connected sum of simply connected manifolds is simply connected