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Let $\pi_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the fundamental group functor and let $H_{1}: \text{Top}^* \rightarrow \text{Grp}$ denote the first homology group functor. We can then define …

The configuration space of $n$ points in a topological space $X$ is usually defined to be, $$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) \in X^{n} \; | \; z_{i} \neq z_{j} \; \text{if} \; i\neq j \}$$ A th …

I'm currently researching the relationship between Moise's Theorem (Every closed $3$-manifold has a triangulation) and other properties of manifolds. In particular, I'm trying to learn about the funda …

There are three consistent ways to glue opposite faces of a dodecahedron to get a closed 3-manifold. With a $\frac{1}{10}$ twist we receive the Poincaré dodecahedral space, with a $\frac{3}{10}$ twist …

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm …