= Four-sphere
{title2=$S^4$}
The four-sphere is
$$
S^4=\{x\in\mathbb R^5:x_1^2+\cdots+x_5^2=1\},
$$
with 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.
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