TITLE: The function algebra $C^{\infty}(M\#N)$ of the connected sum of two spaces
QUESTION [6 upvotes]: Operations such as taking union or Cartesian products of spaces have direct analogues in term of algebra of functions on them (direct sum and tensor product, respectively),
my question is:
Is there a way to determine the structure of $C^{\infty}(M\#N)$ in terms of the function spaces $C^{\infty}(M)$ and $C^{\infty}(N)$?
OR 
How is the operation of taking the connected sum of two closed manifold reflected in the algebra of observables?
Are there any similar methods used in algebraic geometry or homotopy theory?

REPLY [7 votes]: In full generality, the connected sum of $M$ and $N$ depends on more data than just $M$ and $N$; you also need to choose a disk in $M$ and a disk in $N$ to cut out. For example, this choice clearly matters if either $M$ or $N$ has more than one connected component. 
So to me it's better to think about the connected sum as a special case of the composition of two cobordisms, one from the empty manifold to $S^{n-1}$, and one from $S^{n-1}$ to the empty manifold. The composition of cobordisms is in particular a kind of pushout, and smooth functions on a pushout is a pullback. And of course pushouts are used all the time in algebraic geometry and homotopy theory.