Background: I'm going to be a junior, having very strong Analysis and Algebra yearlong sequences, in addition to a very intense Topology class, and a graduate Dynamical Systems class. For this coming Fall, I'm sort-of registered for this class, titled "Manifolds and Topology I" (part of a sequence). I say "sort-of" because I'm not technically registered yet, but registration would be no problem (the class never fills up, and it doesn't create any scheduling conflicts). My problem: I hate Topology,or at least Algebraic Topology. The Topology class I took dealt a lot of Algebraic Topology, since the instructor was a Post-doc more interested in his research than point-set topology, and so he spent about 4-weeks on point-set then jumped into Algebraic Topology. The whole experienced left a bad taste in my mouth and has made just...hate the subject. Even though I came out relatively well (with a B+), I can't say I have a good grasp on the subject nor any interest to even touch it with a 10 ft pole. I've even tried reading some stuff out of Hatcher, and it just doesn't sit well with me. However, at least my school, ph.d students have to take a preliminary exam based on this class and I'd imagine a lot of schools would be same. So, despite my (possibly irrational) dislike of the subject, I'm not stupid enough to not realize it's a big deal in math. My question is, should I just suck it up and take the class despite my feelings towards it? Is there anything I can read to maybe motivate it or something? Will I truly need it or could I get by without it? I was hoping the class would be more about differentiable manifolds than algebraic stuff (since the professor originally had Lee's Introduction to Smooth Manifolds as the text for the class), but it seems this will not be the case, since he recently assigned Hatcher and delegated Lee as a secondary text. Should I just contact the professor? Thanks in advance for the advice.
Most graduate program qualifying exams either don't have a topology/geometry component (like how Pitt's exams are real analysis and linear algebra), or allow you to choose three exams from four possible subjects, usually real analysis, algebra, topology/geometry, or complex analysis. So more likely than not, it's not compulsory that you subject yourself to the class unless you are interested in having a graduate focus like differential geometry, which requires some topological notions (point-set, elementary homotopy theory, etc.). Since you're a junior and it seems you've prepared yourself for graduate studies, start looking at schools you want to apply to and see what criteria they have for qualifying exams and go from there. However, topology is a rich and deep subject, and it would be a shame if a bad first impression kept you from exploring it and expanding your toolkit. I'd contact the professor and ask what text he'll use for the course. Then skim through the book and see if it compels you. Topology interacting with geometry might just be interesting enough draw you back in.
Do take the topology class. Topology is a very important subject in math and you'll encounter it in many places. A good knowledge of topology should get you really far. And maybe you'll even start appreciating the algebraic topology things??