If I wanted to do research related to the topology and geometry of space or their applications in quantum mechanics or general relativity, would it be possible to do that from a Mathematician’s POV without the upper level undergad physics typically required?
Check the website of the program you're interested in. Here's an example.
Basic preparation should include courses in advanced calculus, linear algebra, modern algebra, complex variables, classical mechanics, electromagnetism, quantum mechanics, modern physics, thermodynamics, and statistical mechanics. Knowledge of the following fields is desirable: real analysis, differential equations, probability, topology, differential geometry, and functional analysis.
This makes it sound like you should be OK, but my judgement is not useful; you need to convince the admissions committee.
topology and geometry of space or their applications in quantum mechanics or general relativity
one needs to have a good understanding of tensor algebra, typically used in General Relativity (GR) courses. So if you want to study different topologies of space-time and its implications on GR, then you need to be able to perform matrix and tensor algebra elegantly :)
Research on this can be done from two perspectives - (a) mathematically, identifying the space-time behaviour e.g. conformal field theory, and (b) with a somewhat more physical approach, but that can be in so many different ways that it is peculiar to each project. There is always a non-zero overlap between these two approaches in this research area.
Also, since each programme is unique, they usually will mention the specific courses/skills required to have been taken before they accept the candidates into the project. If that is not mentioned, asking it in the PhD interview is recommended.
Hope this helps.