Question

In each case, choose a base point in (C), describe generators for the fundamental groups of (C) and (S), and write down, in terms of these generators, the homomorphism of fundamental groups induced by the inclusion of (C) in (S).

a) Choose a base point in (C):
b) Generators for the fundamental group of (C):
c) Generators for the fundamental group of (S):
d) Homomorphism induced by inclusion:

Answer 1

To find the fundamental groups of (C) and (S) and the induced homomorphism, choose a base point in (C), describe generators for the fundamental groups, and write down the homomorphism induced by inclusion.

Step-by-step explanation:

To find the fundamental groups of (C) and (S) and the induced homomorphism, we need to consider the inclusion of (C) in (S). Let's choose a base point in (C) and call it x0.

a) Base point in (C): x0

b) Generators for the fundamental group of (C): The fundamental group of (C) is trivial, meaning it has only one generator, the identity element.

c) Generators for the fundamental group of (S): The fundamental group of (S) is isomorphic to the integers (Z), so it has one generator represented by an integer n.

d) Homomorphism induced by inclusion: The inclusion map takes the generator of (C) (which is the identity element) to the generator of (S) (which is represented by the integer 1). Therefore, the induced homomorphism is H: (C) -> (S) defined as H(x0) = 1.