Is homomorphism induced by covering map surjective? Does it naturally comes from surjectivity of covering map? Why is Hatcher's book not readable?
>>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
>>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.
>>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.