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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?

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  • $\begingroup$ I am told (see map.mpim-bonn.mpg.de/…) that connected sum is a well-defined operation in the category of oriented connected closed smooth manifolds of the same dimension, so assuming the OP means to include such fine print, the question seems clear enough to be reopened. I am tempted to add that fine print myself, but would prefer that the OP do the honors. $\endgroup$
    – Todd Trimble
    Commented Jan 11, 2015 at 2:00
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    $\begingroup$ @Todd: I believe it is a well-defined operation on the isomorphism classes of objects in that category; I certainly don't believe it is functorial, which is what one really ought to mean by an operation on the objects of a category. $\endgroup$
    – Qiaochu Yuan
    Commented Jan 11, 2015 at 8:35
  • $\begingroup$ @QiaochuYuan Thanks for the feedback. I haven't thought carefully about it, so let me ask you: is it also not functorial on the core groupoid (of orientation-preserving diffeomorphisms)? $\endgroup$
    – Todd Trimble
    Commented Jan 11, 2015 at 8:57
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    $\begingroup$ @Todd: I see no reason that it should be; the homotopy type of the space of possible choices of a disk in a manifold $M$ to cut out should be quite complicated. $\endgroup$
    – Qiaochu Yuan
    Commented Jan 11, 2015 at 9:00

1 Answer 1

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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.

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