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lol i'm a dummy for never realizing that SO(3) isn't homeomorphic (the diffeo part isn't necessary to prove this...) to T^3 (which i naively thought because s i
by anon_tor_12345 5y ago
lol i'm a dummy for never realizing that SO(3) isn't homeomorphic (the diffeo part isn't necessary to prove this...) to T^3 (which i naively thought because s in SO(3) seemingly has 3 free parameters). surprise surprise SO(3) is actually homeomorphic to P^3 lol.
- hanche 5y agoYou can drop “seemingly”: SO(3) is indeed tree-dimensional. And SO(3) a.k.a. P^3 has fundamental group Z_2 a.k.a. GF(2), whereas T^3 has fundamental group Z^3, so they are quite different beasts indeed.
- anon_tor_12345 5y agoyea you're right. kudos for explaining it so succinctly.