Rough Guides to Topology

The category "topology" includes the following articles:

Knot Theory

Knot theory is the study of loops in space…


knot-theory level3 msc57 topology

last edited 19 Sep 2009 23:25

3-Manifold Topology

Studying 3-manifolds is kind of like studying surfaces, or 2-manifolds. We can classify surfaces, so why not 3-manifolds? We also like to look at ways we embed closed curves in surfaces (homotopy theory). This is an extremely useful way to get information about the surface. The analog with 3-manifolds is embedding closed surfaces in the 3-manifolds. Of course, there are a lot more of these, namely all the handlebodies, so 3-manifold theory turns out to be a lot more interesting and complex than 2-manifold theory.


level5 manifolds msc57 topology

last edited 19 Sep 2009 22:50

Homology Theory



algebraic-topology homology level5 msc55 topology

last edited 07 Oct 2009 15:31

The Fundamental Group

The fundamental group is a tool used to study topological spaces. It is a topological invariant, which means that it is the same for homeomorphic spaces. Because of this, it is frequently used to determine when two spaces are not homeomorphic.


algebraic-topology homotopy level4 msc55 topology

last edited 03 Oct 2009 00:52

Riemannian Geometry

In general, geometry is the study of spaces which have some notion of distance. Differentiable geometry adds such a notion to topological spaces by requiring the spaces to locally "look like" $\mathbb{R}^n$. One can then analyze the space (called a manifold) by extending results on $\mathbb{R}^n$ to the manifold. This becomes especially fruitful if the manifold is given a \emph{Riemannian metric}, which intuitively speaking is a notion of distance. This allows ‘calculus’ on the manifold, and forms the basis for \emph{Riemannian geometry}.


level4 manifolds msc53 topology

last edited 27 Sep 2009 16:58

Curves And Surfaces

The simplest objects of study in differential geometry are curves and surfaces in 3-space. These inherit notions of distance and area from the ambient space.


differential-geometry geometry level3 manifolds msc53 topology

last edited 27 Sep 2009 16:57

Point-Set Topology

Point-set topology is the study of the intrinsic properties of surfaces that are independent of distance. The classic example is the donut and the coffee cup, which, from our point of view, will be the same object.


general-topology level3 msc54 topology

last edited 19 Sep 2009 21:33

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