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I've looked around, but I haven't found anything in particular on Google or here, so I figure I'd ask. What are some solid prerequisites to be able to tackle Kirby Calculus?

I have a solid foundation in undergraduate analysis, working through graduate algebra now, and I have taken some courses in Statistics, along with a very brief introduction to topology. Thanks in advance!

kingdras
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  • Take a differential topology and an algebraic topology course. –  Mar 04 '16 at 21:38
  • I have a question: why are you interested in this topic? Is just naive curiosity. – Antonio Alfieri Mar 05 '16 at 13:35
  • Yeah, quite frankly. My curiosity has lead me down many of my paths today, and this seems to be the path it's leading me down next. Further, I'm interested in the mathematics surrounding General Relativity and Super Symmetry, and differential topology forms seems to gravitate around those aspects. – kingdras Mar 07 '16 at 03:58
  • Do you guys happen to know of any good books in algebraic or differential topology? – kingdras Mar 08 '16 at 01:28
  • @kingdras: I noticed this post after a couple of months. If you're still looking for such books, Hatcher's "Algebraic Topology" is justly the canonical reference for the first algebraic topology course, and Bott and Tu's "Differential Forms in Algebraic Topology" is possibly the best-written math book I've ever read. – anomaly May 11 '16 at 03:06

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The Kirby calculus is a way of coding how to generate $3$-manifolds from links and determine whether or not manifolds thus generated are equivalent. (There's also a $4$-manifold version with handlebody decompositions that's a bit more complicated. In extremely broad terms that shouldn't be taken seriously, the topology of dimensions $1$ and $2$ is trivial, dimensions $\geq 5$ are dominated by the cobordism theorems, $3$ is very different and geometric, and $4$ is just weird.) It's a bit odd to consider it before a general course in geometric or algebraic topology, since it's designed to answer more technical questions (e.g., constructing exotic $4$-manifolds) than arise at the level of, say, Hatcher's book on algebraic topology or Lee's on smooth manifolds. That having been said, if you're willing to ignore the motivation, you could probably get into the subject with a crash course of reading those two books.

In general, my response to questions about prerequisites for a subject is just to dive in and see how far you get. The worst case scenario is that you get stuck reading a book or paper, and you won't be able to understand everything the first time through; that doesn't mean that the time is wasted or that you can't get anything out of it. Especially at the undergrad or grad level, there's a huge amount of math out there to learn, and you shouldn't be dissuaded from something you're interested in.

anomaly
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  • The only reason I'm skeptical of this approach is I can't imagine a successful "crash course" of those two books that doesn't take quite a bit of time and energy. (The precise amount left unstated.) –  Mar 04 '16 at 21:59
  • @MikeMiller: It's an odd approach to take, and honestly the material in those books is so fundamental to topology that it would be better to delay the matter and go through them properly. I don't think that it would be productive to jump into Kirby calculus after, say, Munkres' book in point-set topology (which has one desultory chapter on $\pi_1$), but it might be feasible to jump back and forth between something like Rolfsen's book and Hatcher. It's not the way I would do it, but it could work. – anomaly Mar 04 '16 at 22:03
  • Sure, that's not so bad an idea. If one really wants to be thinking of geometric topology from the start of their topology education, Hatcher + Rolfsen might be the best possible approach. –  Mar 04 '16 at 22:04
  • Thank you guys! The reason I chose this answer is because I would indeed like to jump directly into it to see how I will do! It sounds like fun :) Thank you for all the information. – kingdras Mar 05 '16 at 07:53
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Some experience with Poincare duality, (co)homology, the fundamental group, and vector bundles is probably necessary to get started (these should all be covered either in first year courses on differential or algebraic topology at most universities).

Some experience with knot theory, Morse functions, characteristic classes, and projective algebraic geometry (especially regarding blowing up the projective plane) can be very useful.

The first 3 chapters of Hatcher and a good portion of a book like Gulliemin-Pollack would probably be enough to give a book like Gompf-Stipsicz a try.

PVAL-inactive
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