# The Fundamental Semigroups of a Topological Object?

I had the following idea for a generalization of the "fundamental group" of a manifold.

So the idea basically was that we can consider a manifold $O$ which has boundary $\partial O$, and instead of looking at n-dimensional spheres constrained on the boundary, we instead have some enumerated set of ambient isotopic classes of manifolds (and it need not be finite but my consideration was on them being a finite list) ${M_1 ... M_K}$ which exist on $\partial O$ (a sphere being one such object).

From here we can then organize the set of such possible manifolds into homotopic equivalence classes, and create a notion of "natural gluing" for how two classes $E_1, E_2$ can be combined to yield another class (I do not specify how this gluing occurs, that can up to the desired definition) (the type of manifolds in "product class" may be larger than the original list originally supported) and in doing so equip the manifold with a "Fundamental SemiGroup" with respect to that collection of manifolds.

Now do these fundamental semigroups, encode any more information, than the traditional fundamental groups of the object? If not, then there should exist an algorithm that allows one to derive the fundamental semigroup of an object with respect to some non ambient isotopic objects $M_1 ... M_K$ given the fundamental group of the same object.

• Can you try to define your construction more precisely, or maybe give an example? How exactly is this a generalization of the fundamental group? – Eric Wofsey Jul 21 '16 at 1:20