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I am currently studying Artin Algebra and Topology by Munkres. I have previously studied analysis. After reading few lecture notes on knot theory, I have developed a genuine interest in Low dimensional Topology. So, I am trying to know about how should I self study it. I would like to know about the subjects I should cover in order to start research in some area of low dimensional topology. I would also want to know about the books I should refer to cover all the subjects needed for it. In short, I want to know the pathway to low dimensional topology to a level where I can start reading research papers and also try to develop something of my own. Thank you :))

Abir Mukherjee
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    Which dimension? Low dimensional topology normally refers to either 3 or 4 dimensional topology, and the techniques are quite different in each of these dimensions. – Cheerful Parsnip Mar 11 '14 at 05:18
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    Actually I am yet actually understand the subfields. I have studied some knot theory notes and I like the subject matter. So I would firstly like to study about every sub-field of low dimensional topology. Right now I am studying Topology by Munkres. I would like to know how I should proceed towards specialising in a sub-field in low dimensional topology. Thanks. :) – Abir Mukherjee Mar 11 '14 at 05:33
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    For 4-manifold topology, you might check out "The Wild World of 4-manifolds," by Scorpan. Freedman and Quinn's book is also good. Hempel and Jaco have both written classic texts on 3-manifold theory. Thurston's book "The geometry and topology of 3-manifolds" is also quite good. – Cheerful Parsnip Mar 11 '14 at 06:52

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Ofcourse algebraic topology ( preferably Hatcher ) is a must. You also need to study manifold theory for which you may read either Spivak Vol.1 or John Lee's Introduction to smooth manifolds. But this would be just bare minimum preparation.

As pointed out in an earlier comment, low dimensional topology is really really vast and you can spend more than a lifetime reading literature in either dimension 3 or 4. So, try to get some idea from Manolescu's site who is a renowned topologist and focus on a particular topic.

Sandeep Thilakan
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Allen Hatcher's notes are frequently recommended and available online.

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Drouglas Crash
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