Homology & covering spaces

I am looking at a question for a practice class tomorrow (2.2.12 from Hatcher):

Show that the quotient map collapsing the subspace to a point is not nullhomotopic by showing that it induces an isomorphism on . On the other hand, show via covering spaces that any map is nullhomotopic.

For the first part, I'm not sure what the induced map looks like, so I can't use it to form an isomorphism; for the second, I'm unsure how to use the covering space to show that the map is nullhomotopic.