Topological Manifolds in Four Dimensions
The 3-sphere $S^3$ is the set of points in $\mathbb{R}^4$ equidistant from the origin — the natural higher-dimensional analog of the familiar 2-sphere. This tome will guide the scholar through its parametrizations, its decomposition into a pair of solid tori (the Hopf fibration), and the topological invariants that distinguish it from other 3-manifolds.