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?