# What mathematical background to i need in order to understand proofs of the h-cobordism theorem?

I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some studies on Morse theory, handle decompositions and finally the h-cobordism theorem.

However, my algebra background is not too solid (even though I am not hesitant to catch up)... I've always felt a bit more comfortable in analysis/topology and thus I'd love to learn what the basic requirements for these subjects (especially h-cobordism theorem) might be.

The reason i'm asking is because i'd love to do my bachelor-thesis on this subject but i'm concerned that i might naively choose a topic that's way above my mathematical skills and i eventually might fail. I'm eager to self-study what's necessary but i still have some concerns.

I hope this kind of question is not against any rules and tolerated.

Thank you very much for any of your help.

• You don‘t need Algebra to work through the h-cobordism theorem. Algebra only enters for the more general s-cobordism theorem (where manifolds are not assumed to be simply connected) in form of the Whitehead group of the fundamental group. (The Whitehead group is closely related to algebraic K-theory.) The arguments in the proof of the h-cobordism Theorem are mostly explicit geometric arguments. Commented Mar 3, 2019 at 11:48
• thank you very much @ThiKu. Would you consider the h-cobordism theorem to be a "possible candidate" for a bachelor-thesis or does it maybe require several years in advanced topology/differentialgeometry to be understood i.e. covered in a thesis? I know it's a very vague question, i am sorry.
– Zest
Commented Mar 3, 2019 at 12:05
• @Zest: this should be a question for your advisor, not for MO. If you want an answer here, describe what you know (e.g., do you understand the details of the classification of closed surfaces?), and how much time will be devoted to the thesis. A simple test would be to spend a couple of days reading the proof, say at maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf, and see how far you get. Commented Mar 3, 2019 at 13:33
• @ThiKu I was under the impression that a basic knowledge of homology was required (it certainly was used in the presentation I saw). Is there a way to avoid it? (not that I'd ever consider doing so, but for the purpose of this question...) Commented Mar 3, 2019 at 16:24
• You certainly need the basic properties of singular homology. I wouldn‘t call this higher algebra, though. Commented Mar 3, 2019 at 19:10