I am working through Massey's introduction to algebraic topology, https://www.springer.com/gp/book/9780387902715 and I have come to the section on covering spaces where it mentions that every induced homomorphism from the fundamental group of a covering space to the fundamental group of the space covered is injective (Chapter 5, section 4, theorem 4.1)
My question is then, does the torus cover a punctured disc, and if so, is the induced homomorphism injective? The torus' fundamental group is clearly larger than that of the punctured disc and so I can't reconcile this with theorem 4.1. I can only then assume that the torus doesn't actually cover the punctured disc but I can't see how that is so.
Any help with where I have gone wrong in my thinking would be greatly appreciated.
No, it does not. As pointed out in the comments to my question, the covering map between a space and it's covering space has to be a continuous map, which must therefore preserve the compactness of the space. Since a torus is compact and a punctured disc is not, the covering map doesn't preserve compactness and is therefore not a continuous map.
Thank you to the commenters for pointing this out.