pi-base
Space Suggestion: $S^3$, Three-dimensional sphere, Unit quaternions
Space Suggestion
Let $X$ be $S^3$ = Three-dimensional sphere = Unit quaternions. Defined as in Sphere.
- For a reference to the unit quaternions / group structure of $S^3$, I like section 2.7 of Thurston's 1997 book Three-Dimensional Geometry and Topology Volume 1, but there's probably a more widely available reference. This book is very popular, but its explicit heterodox style could be viewed unfavorably here. Not sure. Anybody have a favorite reference for this?
Rationale
Continuing the project of getting spheres into pi-base. I believe $\mathbb{R}^3$ is not yet pi-base-distinguishable from $\mathbb{R}^2$, otherwise I would suggest that one first.
Filling in the traits is mostly a matter of modifying the traits from Sphere, so should be a fairly manageable task. I think the argument of Sphere is not Contractible from Corollary 11.13 of Bredon's book Topology and Geometry is maybe a bit more in the spirit of reducing references to theories not yet included in pi-base, since it doesn't reference homology. It does involve differential topology though, so maybe this is just swapping one theory not in pi-base for another theory not in pi-base.
Relationship to other spaces and properties
This space provides an example satisfying the search π-Base, Search for locally n-euclidean + has a group topology + compact + Simply connected + Has multiple points
