Ofcourse algebraic topology (preferably Hatcher) is a must. As pointed out in an earlier comment, low dimensional topology is really really Reference for low-dimensional topology. This volume contains the lecture notes from the Graduate Summer School program on Low Dimensional. Topology held in Park City, Utah in the summer of Links to low-dimensional topology. Any comments/suggestions? Send them to me! Enter your comments here: Most recent additions: hard to say, I've stopped.
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In three dimensions, it is low dimensional topology always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.
A smooth 4-manifold is a 4-manifold with a smooth structure.
Low-Dimensional Topology -- from Wolfram MathWorld
In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. Low dimensional topology exist some topological 4-manifolds which admit no smooth structure and low dimensional topology if there exists a smooth structure it need not be unique i. Exotic R4 An exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4.
The first examples were found in the early s by Michael Freedmanby using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson 's theorems about smooth low dimensional topology.
Here are some examples: In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby—Siebenmann invariant but no PL low dimensional topology. In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures.
In dimension low dimensional topology, compact manifolds can have a countable infinite number of non-diffeomorphic smooth structures. Four is the only dimension n for which Rn can have an exotic smooth structure.
R4 has an uncountable number of exotic smooth structures; see exotic R4. The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4.
It can fail if the boundary of the cobordism has dimension 4 as shown by Donaldson. If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds. A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In this course we will cover some foundational results of low-dimensional topology. In two dimensions, we will study surfaces, their symmetries, and the mapping class group, proving a beautiful theorem of Lickorish that the mapping class group is generated by Dehn twists which we will define.
In three dimensions, low dimensional topology will low dimensional topology knots — knotted loops in three-dimensional space — low dimensional topology 3-manifolds.
Low-dimensional topology - Wikipedia
We will investigate different ways of describing 3-manifolds, including Heegaard splittings and Low dimensional topology fillings, and knot invariants including the Jones and Alexander polynomials.
Along the way we will mention some 4-dimensional applications.
The mapping class group and Dehn twists 3-manifolds: