What books one must read and in what sequence to learn a low-dimensional topology at the grad level? The goal is to read in about a year at least something about Geometrization conjecture of Thurston. The background of the OP is merely Basic Topology by MA Armstrong and undergrad level abstract algebra. Please suggest a roadmap. Thank you.
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2$\begingroup$ Not sure about the roadmap, but you should download blms.oxfordjournals.org/content/15/5/401.full.pdf+html $\endgroup$– Mark GrantCommented Apr 22, 2013 at 13:58
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$\begingroup$ @MarkGrant, I do not have access to that particular document. $\endgroup$– Richard NashCommented Apr 22, 2013 at 14:25
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3$\begingroup$ It's Scott's survey on $3$-manifolds, also available here math.lsa.umich.edu/~pscott $\endgroup$– Mark GrantCommented Apr 22, 2013 at 14:29
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4$\begingroup$ mathoverflow.net/questions/93408/… . See also math.stackexchange.com/questions/128962/… $\endgroup$– Daniel MoskovichCommented Apr 22, 2013 at 14:36
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One of the best introductions to the subject is certainly Thurston's Three-dimensional Topology and Geometry, Vol.1 (not to be confused with his much harder lecture notes Three-dimensional Topology and Geometry). It has almost no prerequisits, but leads you right to the statement of the geometrization conjecture of Thurston and some surrounding mathematics.
A more topological view on 3-manifolds is presented in a set of notes by Hatcher: http://www.math.cornell.edu/~hatcher/3M/3Mfds.pdf
For mapping class groups, i.e. groups of homeomorphisms of surfaces, you may have a look at Farb's and Margalit's http://www.math.ethz.ch/~bgabi/Farb%20Magalit%20January%202011%20version.pdf . This is not directly related to the Geometrization Conjecture, but mapping class groups are extremely important both in 2-dimensional and 3-dimensional geometry.
Edit: At some point you have also to learn some differential topology to understand geometric topology. I myself learned differential topology (partly) from the book by Bröcker and Jänich, but this is a little bit terse - there might be better choices. But the nice thing is that Thurston's book does not really presuppose any deeper knowledge in differential topology.
I want also to comment than none of the above sources says anything about the proof of the geometrization conjecture; but I think, it would be unreasonable to try to understand the proof with your current background anyhow.
answered Apr 22, 2013 at 16:54
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$\begingroup$ A gentler introduction than the Thurston book is the new Bonahon book: Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots $\endgroup$ Commented Apr 22, 2013 at 18:07
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$\begingroup$ Thank you for the answer. I shall have my hands full with the references provided. $\endgroup$ Commented Apr 22, 2013 at 18:11
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$\begingroup$ But will my question get deleted? $\endgroup$ Commented Apr 22, 2013 at 18:11
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$\begingroup$ It is true that the Thurston book is sometimes a bit tough for people which do not have a brilliant geometric intuition; but on the other hand, it also really helps to build geometric intuition. $\endgroup$ Commented Apr 22, 2013 at 18:15