Homotopic to a constant map $f$ and the $f_{\#}$

Question

This question is somewhat extension of my former question

Homotpoic and $\#$ of two maps

First what i know is that if two maps are homotopic, then their $\#$, $i.e$, maps on $\pi_1$, are identical, $i.e$,

\begin{align} f \simeq g, \qquad \Rightarrow \qquad f_{\#} = g_{\#} \end{align} further what i know is

if $f$ is identity, then $f_{\#}$ is also identity.

1. I want to know the case for $f$ is constant, then what is $f_{\#}$?

2. Furthermore if $f \simeq \{ *\}$, i.e contractible, then what is $f_{\#}$?

Answer 1

If $f$ is constant, then $f$ sends everything to a point. All loops in $f$ are thus sent to a point, which is already a trivial loop, and so this homomorphism is just the homomorphism taking everything to identity.

If $f$ is null-homotopic, then the same comments can be made, after composing with a homotopy that actually contracts the space.