An Introduction to Geometric Topology [pdf]

From the preface:

""" this book is an introduction to surfaces and three-manifolds, and to their geometrisation, due to Poincaré and Koebe in 1907 in dimension two and to Thurston and Perelmann in 2002 in dimension three. Therefore this is also a textbook on low-dimensional topology, except that we completely neglect four-manifolds, that form a relevant part of this area but which do not (yet?) fit in any geometrisation perspective."""

Warning: if you just want to have a look-see, skip over chapter 1. There's plenty of pictures starting in chapter 2 that give an idea of what this is all about.

there's a nice trick in Thurston's book on 3-manifolds where you glue a strip of paper at the ends into a Möbius band and cut along the middle to get a trefoil knot out of paper.

i did it with 3 and 5 twists and cut along the middle you get various knots this way

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http://library.msri.org/books/gt3m/

Nice book, my favourite math subject, I recommend start with a more classic book, that explains the fundamentals of topology, starting from its definition. There are a lot of people talking about "topologic spaces", but, they don't know the definition, so they don't know what are they talking about.

Unfortunately happens with a lot of other math concepts, for example a lot of people say "idempotent" but they don't know what does it mean and use the term for things that are not idempotents, hence the word loose its meaning.