I've looked around quite a while for an answer to this but I have yet to find any explanation what so ever nevertheless a simplistic one.
I know It's homology (as well as if a surface is open or closed) that "gives algebraic structure" to the geometric surface of manifolds but I have no clue in what way how, so what I seek is a simple understanding of that how.
How is algebraic structure given to the surface of a manifold? What of homology gives algebraic structure to the surface of a manifold? Is there a theorem or result for this?