# A Universal Property Defining Connected Sums

I once read (I believe in Ravi Vakil's notes on Algebraic Geometry) that the connected sum of a pair of surfaces can be defined in terms of a universal property. This gives a slick proof that the connected sum is unique up to homeomorphism. Unfortunately, I am unable to find where exactly I read this or remember what exactly universal property was; if anyone could help me out in either regard it would be much appreciated.

• A quick search of Foundations of Algebraic Geometry reveals nothing. Certainly one can define the wedge sum by a universal property: it is the coproduct in the category of pointed topological spaces, or equivalently the pushout of $X \leftarrow \{ * \} \rightarrow Y$ in the category of topological spaces. Perhaps something similar works for connected sums? Jul 7, 2011 at 3:44
• Well, that does not quite give what you're looking for, but by definition the connected sum is the push-out of $(M\smallsetminus D) \gets \partial D \to (N \smallsetminus D)$ where $D$ is an embedded open disk in $M$ and $N$.
– t.b.
Jul 7, 2011 at 4:36
• I am afraid that the complex structure or algebraic structure will be totally ruined in general cases when you do a surgery like that in the category of topological spaces or smooth manifolds. However, a definition can always be made via the universal property, regardless of the existence of such an object. Oct 1, 2012 at 11:20
• Even if one found a universal property that characterized the direct sum, you'd have to prove that whatever disc you removed satisfied that property at the end of the day. For this I can think of no way to avoid the use of the disc theorem: en.wikipedia.org/wiki/Disc_theorem which is hard. But, if you've gotten this far, then you've already proven that the connected sum is well-defined up to homeomorphism. So I don't know if the end result would be "slick." Nov 29, 2012 at 14:06