2 added clarification on group structure

This may be a fairly simple question. Suppose G is a (T0) topological group. Assume that G is path-connected, locally path-connected, and semilocally simply connected, so that covering space theory applies.

Question: Is it true that for any element of $\pi_1(G,e)$ (where e is the identity element of G), there exists a [ADDED: continuous] homomorphism from $S^1$ to $G$ having that element of $\pi_1(G,e)$ as its homotopy class?

Another way of formulating this is that there is a set map:

$$\operatorname{Hom}(S^1,G) \operatorname{Hom}_{cts}(S^1,G) \to \pi_1(G,e)$$

The subscript cts is to indicate continuous.

(when G is abelian, the left side has a group structure too [ADDED: under pointwise multiplication], and the Eckmann-Hilton principle tells us that we get a group homomorphism).

1. Is the set map surjective in all cases (regardless of whether G is abelian)?
2. Does the image of $\operatorname{Hom}(S^1,G)$ generate $\pi_1(G,e)$ as a group (this is equivalent to surjectivity when $G$ is abelian)?
3. Does surjectivity work for Lie groups? Compact Lie groups?
4. Does the weaker formulation (2) work for Lie groups?

I have a sketch of an argument/proof that may show (4) (basically, using properties of one-parameter subgroups), but I'm hoping somebody will have a clean proof that works in general for topological groups.

1

# Each element of fundamental group of a topological group represented by homomorphism?

This may be a fairly simple question. Suppose G is a (T0) topological group. Assume that G is path-connected, locally path-connected, and semilocally simply connected, so that covering space theory applies.

Question: Is it true that for any element of $\pi_1(G,e)$ (where e is the identity element of G), there exists a homomorphism from $S^1$ to $G$ having that element of $\pi_1(G,e)$ as its homotopy class?

Another way of formulating this is that there is a set map:

$$\operatorname{Hom}(S^1,G) \to \pi_1(G,e)$$

(when G is abelian, the left side has a group structure too, and the Eckmann-Hilton principle tells us that we get a group homomorphism).

1. Is the set map surjective in all cases (regardless of whether G is abelian)?
2. Does the image of $\operatorname{Hom}(S^1,G)$ generate $\pi_1(G,e)$ as a group (this is equivalent to surjectivity when $G$ is abelian)?
3. Does surjectivity work for Lie groups? Compact Lie groups?
4. Does the weaker formulation (2) work for Lie groups?

I have a sketch of an argument/proof that may show (4) (basically, using properties of one-parameter subgroups), but I'm hoping somebody will have a clean proof that works in general for topological groups.