Wikipedia:Reference desk/Archives/Mathematics/2022 June 20

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June 20

orientability and simplicity

Can a non-orientable n-manifold be simply connected? If n=2 I think not, but how about other n? —Tamfang (talk) 02:41, 20 June 2022 (UTC)

I have no experience in this area, so I can't confirm whether or not this is what you were looking for, but I found a StackExchange answer which, if I'm interpreting it correctly, implies the answer is no. GalacticShoe (talk) 05:27, 20 June 2022 (UTC)

Is a Moebius strip n=2? 2601:648:8202:350:0:0:0:90B2 (talk) 06:09, 20 June 2022 (UTC)

Yes, see Möbius strip § Surfaces of constant curvature: "[The Möbius strip] is the quotient space of a plane by a glide reflection, and (together with the plane, cylinder, torus, and Klein bottle) is one of only five two-dimensional complete flat manifolds." The Klein bottle is also non-orientable. However, neither the Möbius strip nor the Klein bottle is simply connected. Every simply connected manifold is orientable,^[1] so this generalizes beyond surfaces.  --Lambiam 08:08, 20 June 2022 (UTC)

Oh I see. Yes I knew that the Moebius strip was 2-dimensional and had gotten confused about the concept of simply connectedness. I had been asking for confirmation that it is non-orientable. Thanks. 2601:648:8202:350:0:0:0:90B2 (talk) 17:54, 21 June 2022 (UTC)