From: Robin Chapman Subject: Re: newbie question: Euler characteristic Date: Mon, 24 Jan 2000 08:36:56 GMT Newsgroups: sci.math Summary: [missing] In article , barnabyfinch@pe.net (Barnaby Finch) wrote: > I'm am the proud owner of a shiny new Klein bottle a friend gave me for my > birthday. I'm trying to learn more about it, and other manifolds. For > example, I know that my Klein bottle is an immersion ( the KB being > non-embeddable in R^3), and that it is closed, boundaryless, and > non-orientable (one-sided). I also know that it possesses Euler > characteristic = 0. Upon looking up said characteristic at > mathworld.wolfram.com, I learn that the Euler characteristic is the term > added to make a generalization of the old Polyhedral Formula; V - E + F = > 2. In the new formula, the Poincare Formula, the "2" is replaced with "2 - > 2g", where "g" is the genus. This "2 - 2g" is the Euler characteristic. Am > I getting it so far? Yes, but this formula is only valid for *orientable* manifolds. > This means the sphere (genus 0) has E.C. = 2, and the Klein bottle and the > torus, each with genus 1, each have E.C. = 0. So, does this mean that a > closed, genus 2 surface would have E.C = -2? Also, I see that the If it's *orientable*. > projective plane ( and its homotopies - the Boy's surface, the Roman > surface, and the Crosscap ) have E.C. = 1. That gives genus 1/2 !!. > Clearly I'm not getting something here... perhaps I need to be thinking in > terms of the integral curvature, or the Betti number of the space. A compact nonorientable surface is usually said to have genus g if it is a sphere with g crosscaps pasted in. Its Euler characteristic is then 2-g. -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: mathwft@math.canterbury.ac.nz (Bill Taylor) Subject: Re: newbie question: Euler characteristic Date: 31 Jan 2000 09:23:42 GMT Newsgroups: sci.math barnabyfinch@pe.net (Barnaby Finch) writes: |> This means the sphere (genus 0) has E.C. = 2, and the Klein bottle and the |> torus, each with genus 1, each have E.C. = 0. So, does this mean that a |> closed, genus 2 surface would have E.C = -2? Fine so far. |> Also, I see that the |> projective plane ( and its homotopies - the Boy's surface, the Roman |> surface, and the Crosscap ) have E.C. = 1. That gives genus 1/2 !!. Someone else mentioned that genus strictly only applies to orientable surfaces, (which would actually exclude the KB); whether this is standard I don't know, but I'd think it unlikely anyone would so confuse the issue in a text by putting the genus of a non-orientable surface as twice that amount, in effect, as he suggested. It would be far more natural to keep genus = 1 - (E_C)/2 . And then yes! - the projective plane *would* have genus 1/2, by that count. This amounts to calling the genus the number of handles (if orientable) or half the number of crosscaps (if not). Note that two cross-caps (in the presence of another crosscap) are fully equivalent to a handle. Even not in that environement, two cross-caps are *almost* the same as a handle - as in the Klein bottle, a sphere with two crosscaps, and almost a sphere with a handle - it has that inside-out handle. If you ever get to see a sphere with a crosscap, (I've got a nice polystyrene one here in my office!), maybe in a book, you'll see that the crosscap *is* indeed somewhat like "half" a handle! Of course being non-orientable it can't be embedded in 3-space, only immersed, i.e. with self-intersection, like the Klein bottle. But any extra handles on an otherwise non-orientable surface can be turned inside out to become Klein-like handles (and vice versa). So handle ==> 1 onto the genus = 2 off the E_C <== inverted-handle Incidentally, another way to think of a crosscap is as a Mobius strip. A Mobius strip is just a sphere with crosscap, which has had a very small circle cut out of it. The self-intersecting bit can then be moved to go through the hole, leaving a proper embedding in 3-space, to wit, the Mobius strip. Or, you can visualize this big sphere, with a smallish circle cut out of it, and a narrow strip re-inserted across this gap, but twisted once... the whole thing is actually a Mobius strip, but now it looks more like a sphere with something *on* it! It's a sphere with a crosscap, with the circular hole just standing by for engineering convenience. If you have a cross-CAP on your sphere, And you give it a circle-shaped tear, Then just shake it about And untangle it out And a Moebius strip will appear! .... or dually ... The Moebius strip is a pain, When you cut it again and again, But if you should wedge A large disk round the edge Then you just get a PROjective plane. You can also see the same thing by cutting a Klein bottle into two, lengthways, through the plane of symmetry. The two left-over halves turn out to be just two Mobius strips! Again, 2 crosscaps =^= 1 handle. A mathematician named Klein Said, "The Mobius strip is divine." He observed, "If you glue Up the edges of two," Then you get a strange bottle like mine. Fun stuff! ------------------------------------------------------------------------------- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------------------------------- There was a young man from Lahore || There was a young woman from Crewe Whose limericks stopped at line four || Whose limericks stopped at line two, Things seemed to be fine || But there'd be no punch line ||--------------------------------------- || There was an old chap from Verdun ------------------------------------------------------------------------------- (I once knew an even better one about a man called Nero - now how did it go...?) -------------------------------------------------------------------------------