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:

How one can 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?

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?

I apologize if the setting for this question is rather vague.

How one can 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?

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:

How one can 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?