 Four-sphere

The four-sphere iswith its standard topology and smooth structure. It is a compact smooth four-manifold without boundary, also the boundary of the five-dimensional closed disk. Its dimension is four even though its displayed embedding uses five coordinates.

Stereographic projection identifies the complement of either pole with ${\mathbb{R}}^4$. Adding the missing pole recovers the one-point compactification of ${\mathbb{R}}^4$. These charts give a concrete manifold model rather than merely a formal dimension label.

Its integral homology is $\mathbb{Z}$ in dimensions zero and four and zero otherwise; in particular its Euler characteristic is two and its degree-two intersection form is trivial. It is simply connected. The standard sphere is an essential comparison object in four-dimensional topology, where homology, topology and smooth structure record different information.