Is it possible to create an "antigroup"? What I mean by this is, given some group G, and some "antigroup" H, then the "free product" of G and H, G*H will equal the "group" (vacuously a group) of no generators and no relations. Can I construct such a thing? It makes me think that I need "anti-generators" and "anti-relations", somehow the "union" of the generators and anti-generators causes cancellation, and same goes for the relations. Which now makes me think that I need to generalise the notion of "union".
And is it possible to construct an "anti-manifold"? Where if given some manifold J and some "anti-manifold" K, then the "connected-sum" J#K will equal the empty set?
I know that these sorts of constructions are external to what groups or manifolds are. That is an "anti-group" won't be a group, because the free product of groups will always add complexity, as does the connected sum of manifolds, unless the connected sum is with an n-sphere (of appropriate dimension).