Background: I'm reading Chapter 9 of Munkres Topology on the fundamental group, and Lemma 55.3. which is referenced many times states that, given a continuous map $h:S^1 \to X$, the following are equivalent
(1) $h$ is null-homotopic
(2) $h$ extends to a continuous map $k:B^2 \to X$
(3) $h_*$ is the trivial homomorphism of fundamental groups
Additionally, I proved earlier that if $h:(A,a_0)\to (Y,y_0)$ with $A\subset X$ and $X$ is simply connected, then if $h$ extends to a map $\tilde{h}:(X,a_0) \to (Y,y_0)$, then $h_*$ is the trivial homomorphism. The idea in this proof was to suppose there is an extension $\tilde{h}$, thus $h = \iota|_A \circ \tilde{h}$, and $h_* = \iota_{A,*} \circ \tilde{h}_*$, and since $\iota|_{A,*}:\pi_1(A,a_0) \to \pi_1(X,a_0) = 0$, we have that $h_*$ is the trivial homomorphism.
With this in mind, I have two questions:
1) Given a map $f:(X,x_0) \to (Y,y_0)$ with $\pi_1(X,x_0) \neq [1_{x_0}]$, if $f_*:\pi_1(X,x_0) \to \pi_1(Y,y_0)$ is trivial, then is $f$ null-homotopic?
2) If the answer to 1) is yes, is the following generalization of the lemma above is true?
Let $A \subset X$ and $h:(A,a_0)\to (Y,y_0)$ with $\pi_1(A,a_0) \neq 0$ and $X$ simply connected. Then TFAE:
(a) $h$ is null-homotopic
(b) $h$ extends to a continuous map $\tilde{h}:(X,a_0) \to (Y,y_0)$
(c) $h_*$ is the trivial homomorphism
All I can prove so far is $(a) \implies (c)$ and $(b)\implies (c)$.
Definitely not. You can, for example, without too much difficulty find a degree-one mapping of $S^1\times S^1$ to $S^2$. Of course, $S^2$ is simply connected, but the mapping cannot be null-homotopic.