3Covering spaces

II Algebraic Topology



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
can make the group act on something if the group acts non-trivially on our
thing, then clearly the group cannot be trivial.
These things we act on are covering spaces.

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