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When I was studying topology I remember being able to demonstrate that the set of topological surfaces with any number of punctures (including the projective plane, Klein bottle, Moebius strip, double torus, etc.), together with their associated inverses, was not a group with respect to the operations of connection and disconnection. I could show with a sequence of drawings that it is possible to connect two Klein bottles, deform the surface, and disconnect a torus from a single remaining Klein bottle, i.e. K+K = K+T, so inverses are not well defined even though the operator is otherwise associative on that set, has an identity (the sphere), and is also Abelian.

I never made much progress in my understanding beyond this point so I am interested in any answer that can simply explain what is going on here (am I misinterpreting something else as a topological property?), else provide a reference (I'm not even sure what branch of topology addresses this). One slightly more specific question I have is: what is an example of a collection of topologies which does form a group under connection and is this ever a useful property to have?

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I can answer my own more specific question... Z^2 can be generated in this way from a torus, a pancake, and their inverses. Still I am curious about other examples and what is the underlying meaning. –  Dan Brumleve Aug 19 '10 at 7:33
    
You shouldn't say that the "set" is associative, but that the binary operation on it is. –  Qiaochu Yuan Aug 19 '10 at 7:51
    
I know that there's no such thing as a "negative torus" (or is there?) but I'm extending the set with inverses whenever necessary to create a group. (Can this be done with any commutative monoid?) –  Dan Brumleve Aug 19 '10 at 8:07
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There's a formal process called the "group completion" that turns a monoid into a group. It can be disasterous -- it can turn a non-trivial monoid into a trivial group. But it's the only thing of that form available. The group completion of the surface monoid is just the integers (the torus gets identified with the klein bottle). So the surface monoid does not embed in its group completion. Group completions are sometimes called "Grothendieck groups" but IMO this is ahistorical. –  Ryan Budney Aug 19 '10 at 8:41
    
Ryan, I am understanding that the set of all surfaces is a commutative monoid but is not free, and if I want to build a group with virtual inverses then I have to start with a free commutative monoid. Is there any significance to the fact that the FC monoid I described as being generated by the pancake and the torus is exactly the set of surfaces embeddable in R^3, but the entire set (embeddable only in R^n with n >= 4) is not an FC monoid? –  Dan Brumleve Aug 20 '10 at 5:47
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Right, surfaces under the connect-sum operation form a commutative monoid, but it's not a free commutative monoid. It's the quotient of the free commutative monoid on two generators (projective space, torus) by the relation you mentioned (in these generators you'd express the klein bottle as the connect-sum of two projective spaces).

This is a monoid where the only invertible element is the sphere. The proof is that the "genus" is additive under connect sum, and is only zero for the sphere. Here the 'genus' is the sum of < the number of projective-space summands > + < twice the number of torus summands > (for this to make sense you have to express your surface as a connect-sum of only projective spaces and tori.

I'm not quite sure what your question is. In the sense that there's no question mark in your 1st paragraph.

I think there's a terminology problem. "Collection of topologies" literally means "a collection of subsets of various sets satisfying some rules".

If you replace surfaces with various other objects, the connect-sum operation can be a group operation. For example, if your objects are smooth embeddings of $S^3$ in $S^6$, taken up to isotopy, then the connect-sum operation turns this collection into an infinite-cyclic group.

Strangely enough, embeddings of $S^j$ in $S^n$, taken up to isotopy and with operation given by connect sum form a group whenever $n > j+2$. Sometimes it's a group for silly reasons -- there is only one such embedding up to isotopy provided $2n-3j-3>0$. But when $2n-3j-3 \leq 0$ generally there are many.

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Connected sum usually gives a structure of a commutative monoid but rarely of a group. Absence of inverse elements is explained (e.g. for manifolds or for knots) by Mazur swindle (wiki, Terence Tao).

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