Omar Antolín on Nostr: npub1knzsu…ac73p npub12n4k2…c5dmn de Rham cohomology works half the time for this ...

npub1knzsux7p6lzwzdedp3m8c3c92z0swzc0xyy5glvse58txj5e9ztqpac73p (npub1knz…c73p) npub12n4k27w8jyj29f0jrpmupmzx8rpvttg6c4z0jws7xfu9kky0cp6q8c5dmn (npub12n4…5dmn) de Rham cohomology works half the time for this question! For SO(n) with n=2k+1 you have exterior generators in degrees 3, 7, ..., 4k-1; and the slightly different pattern you mentioned for even n is that if n=2k+2 then you have exterior generators in those same degrees 3, 7, ..., 4k-1 and an additional exterior generator in degree 2k+1. This means that when n is odd SO(n+1) has the same de Rham cohomology ring as SO(n)×S^n! (And when n is even the rings are non-isomorphic so the argument does work in those cases.)

When n=1 or 3, then SO(n+1) really is diffeomorphism to SO(n)×S^n, so I'm not sure what to expect for odd n.