What is the geometric object obtained by gluing together all pairs of antipodal points of an n-sphere?

Question

The resulting object locally looks like an n-hyperplane, which I understand to mean that it's a manifold. As for the global picture, I have no idea.

This is purely out of personal curiosity.

Answer 1

Identifying antipodal points on a sphere results in a projective space.

One common construction of (the set of points in) a projective space is as

$$\frac{\mathbb{R}^{d+1}\setminus\{0\}}{\mathbb{R}\setminus\{0\}}$$

i.e. you take all $(d+1)$-dimensional vectors except for the null vector, but then you identify (“glue together”) non-zero multiples of the same vector, forming equivalence classes which are called homogeneous coordinates. Each such equivalence class can be seen as a line through $\mathbb{R}^{d+1}$, although technically it's a line minus the origin. Each such line will intersect a $d$-sphere in two antipodal points. So there is a one to one correspondence between pairs of antipodal points and points in a projective space.