Rob Kusner recently pointed out to me that the 4-sphere has a very natural differential-geometric decomposition as a double mapping cylinder . Here is the group in the unit quaternions and is the real projective plane. Another way to say this is take the Voronese projective plane in , a regular neighbourhood of it is a mapping cylinder . Moreover, the *complement* of that regular neighbourhood is another such mapping cylinder.
This decomposition comes about via an elementary linear algebra argument. Consider the space of traceless symmetric real matrices. acts on this vector space (by conjugation), and so it acts on the unit sphere in that vector space, which is . The orbit-type decomposition of this action is the above double mapping cylinder construction. It’s a remarkably symmetric decomposition of the 4-sphere.
Some other similarly symmetric-looking decompositions of the 4-sphere have been coming up in some computations I’ve been working on recently. Take a knot in the 3-sphere. The complement is a homology . So if you glue two knot complements together, longitude to meridian, you get a homology sphere. It appears that many of these homology spheres admit smooth embeddings into , separating in a somewhat (maybe very) symmetric way. I don’t actually know what these decompositions are yet since I have only a moderately constructive embedding of these manifolds into . But it appears to be cutting in a fairly symmetric way. The 4-manifolds that these manifolds separate into are quite “small” so it would be difficult for the decomposition not to be symmetric.
Do other people have any symmetric decompositions of the 4-sphere?