The Connected Sum

This is a way of making new surfaces out of old ones

Definition
Given two surfaces S₁ and S₂ the connected sum S₁ # S₂ is constructed by removing a disc from each and then joining them along the boundaries of the holes.

Examples

1. The connected sum of two tori is a 2-holed torus.

   
   

2. The sphere is an identity for the connected sum operation: [S² - (disc) disc ]

   
   If we start with a projective plane and remove a disc we get a Möbius band.

   
   We can split a Klein bottle into a pair of Möbius bands.

   

   Hence a connected sum of Projective planes is a Klein bottle.