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Take the class of all compact, connected, boundaryless, smooth oriented $n$-dimensional manifolds, each taken up to orientation-preserving diffeomorphism. This is a commutative monoid with operation connect-sum.

For $n=1$ this is known to be the trivial monoid -- it only has one element.

For $n=2$ this is the free commutative monoid on one generator, the torus.

For $n=3$ this is the free commutative monoid on countably-many generators. This is an old theorem of Kneser and Milnor.

  • Q: What is known about the monoid structure for $n \geq 4$? And is there much known about the group completion of the monoid?

Elements with inverses are precisely the homotopy spheres (provided $n \neq 4$). But presumably there are more interesting relations in this monoid. I've seen plenty of relations in these monoids but I've never spent the time to put together an overview of what's going on. What relations are there, to whatever extent they're known?

This question is related to a similar, older question of mine:

Computing the structure of the group completion of an abelian monoid, how hard can it be?

which is motivated by similar questions about connect-sums of pairs $(S^n,K)$ where $K \subset S^n$ is a co-dimension two oriented knot.

edit: When $n=4$ it seems to be not known if the sub-monoid of homotopy spheres is a group. See Bruno's comment below.

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I saw Milnor give a talk last year that more or less implied that if one restricts to spheres it is not even known if the group is finite for N=4 if memory serves. –  Benjamin Steinberg Apr 8 '12 at 19:14
    
Right, with the current trends in 4-manifold theory it seems possible all compact $4$-manifolds may admit infinitely-many distinct smooth structures. –  Ryan Budney Apr 8 '12 at 19:20
    
Homotopy spheres in dimension 4 are not necessarily invertible, because they are not twisted spheres, right? –  Bruno Martelli Apr 8 '12 at 19:50
    
@Bruno, I think you're likely right. I was thinking of taking a homotopy $4$-sphere, then $M \# -M$ is the boundary of $I \times M$ with an open tubular neighbourhood of $I \times \{p\}$ drilled out. But this is a contractible $5$-manifold with a homotopy sphere on the boundary. It's not known if there are non-standard smooth structures on $D^5$. –  Ryan Budney Apr 8 '12 at 20:04

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