3 Covering spaces

We can ask ourselves a question — what are groups? We can write down a

definition in terms of operations and axioms, but this is not what groups are.

Groups were created to represent symmetries of objects. In particular, a group

should be acting on something. If not, something wrong is probably going on.

We have created the fundamental group. So what do they act on? Can we

find something on which these fundamental groups act on?

An answer to this would also be helpful in more practical terms. So far we

have not exhibited a non-trivial fundamental group. This would be easy if we