Covering Space

>>8203430
Are you talking about the homomorphism of fundamental groups?

>>8203506
Yes, sorry

>>8203430
It's not. (the usual example of the reals and the circle).
It's injective, actually, because you can lift path homotopies. Look at the monodromy theorem.

>>8203507
Not necessarily surjective as the last poster said, though its injective,see http://www.maths.tcd.ie/~dwilkins/Courses/421/421S4_0809.pdf Corollary 4.2

>>8203515
>>8203516
I know that it is injective. I am looking at the real and circle example. Doesn't every loop f in circle has its origin in R via inverse of cover map? And does it not imply that every class of loops in circle has it's pair in R?

>>8203523
Circle fundamental group is the integers.
When you lift a loop in he circle you get a path in reals, which can have different end point from the initial point. The lift actually measures the amount of turns the loop has made.

>>8203526
>>8203523
Why are you guys lifting loops? The induced homomorphism still goes from pi_1(R) (trivial) to pi_1(S^1) (Z) which can never be surjective

>>8203535
Whoa, but then it can't be injective, right?

>>8203546
oops my bad. Ofc it is injective.

>>8203546
It's the trivial homomorphism of course its injective

and there's a proof for general case in the post above anyway

>>8203535
The lifting of loops is usually the way you find the fundamental group of the circle.