Make a list of algebraic, topological and dynamical properties of the circle.

Question

For example, there is only one cyclic group of order $m$ in the circle, the circle is compact and connected, all its proper compact and connected subspaces are arcs, is a minimal dynamical system under irrational rotation.

What else?

Answer 1

If you want, the circle is the only topological manifold of dimension one that is connected and compact. It is also an oriented manifold.

The circle can be described as the projectif space of the vector space $\mathbb{R}$.

All the superior homotopy groups of the circle are zero.

There is exactly two non isomorphic line bundles on the circle : $\mathrm{Pic}(S^1)=\mathbb{Z}/2$.

Seeing the circle as an abstract or Lie group, $\mathrm{End}(S^1)= \mathbb{Z}$.