Subject: Re: many responses Date: Fri, 16 Jan 2004 21:17:50 -0000 From: "Ronald Brown" the fundamental group of SO(3). You can see stills from a movie on Pivoted Lines and the Projective Plane on http://www.cpm.informatics.bangor.ac.uk/centre/ (do a search on Mobius or rotations or projective). Search also on Dirac. For the movie, you have to buy the CDRom advertised there on the RPAMath section. Lou Kauffman has a great graphics on this. There is a nice group theory question hidden here. MHA Newman showed that the Dirac trick on n strings is related to a quotient H of the braid group on n strings (that is how you can prove that the strings do not untwist after a rotation through 360 degrees). Question: Is the quaternion group of order 8 embedded in this quotient? I am not a good enough group theorist to solve this, but a Bangor undergraduate student Susan Hale in a project a few years ago verified this experimentally, I think! Ronnie Brown http://www.bangor.ac.uk/~mas010 ________________________________________________________________ Subject: \pi_1 SO_3 Date: Fri, 16 Jan 2004 21:25:34 -0000 From: "Neil Strickland" > The fact that the fundamental group of O(3) is Z/2 > can be illustrated (literally) by the contortion with a water glass > rotating wrist and arm through 720 degress > I once saw a series of still illustrating this > perhaps in Scientific American This is in Feynman and Weinberg "Elementary Particles and the Laws of Physics" (the 1986 Dirac Memorial Lectures) (Cambridge University Press) on page 30. There are only four photos, so it isn't too clear. There are probably better versions elsewhere. Neil _______________________________________________________________ Subject: [Fwd: balinese water glass dance?] Date: Fri, 16 Jan 2004 17:33:22 -0500 From: jim stasheff Thanks to all those who have written me directly but not to Don about the pcitures in Bredon In addition, Jack Morava wrote: > > Hi Jim, > > That Scientific American article is by our mutual buddy Herb > Bernstein; I'll try to dig up a reference. > ____________________________________________________________ Subject: Spinner spanner -> board and strings!! Date: Fri, 16 Jan 2004 16:40:32 -0700 (MST) From: David Pengelley Jim Stasheff wrote: > Subject: spinner spanner > Date: Thu, 15 Jan 2004 08:51:15 -0500 (EST) > From: James Stasheff > > The fact that the fundamental group of O(3) is Z/2 > can be illustrated (literally) by the contortion with a water glass > rotating wrist and arm through 720 degress > I once saw a series of still illustrating this > perhaps in Scientific American > > anyone have any reference? This motion is performed repeatedly, with both hands simultaneously, and burning candles replacing the water glass, as part of the "Phillipine Candle Dance", which I once saw performed by a female dance troupe from the Philippines. It is indeed a wonderful illustration that the fundamental group of SO(3) has an "apparently nontrivial" element of order two. It's a great hit in any classroom. (No, Jim, sorry, I don't know a published reference.) Now much, much better is the following related object (also related to, but better than, the Dirac scissors and belt), which I learned from Arkady Vaintrob; I think it's well known in Russia. I call it "board and strings". Take a rectangular piece of cardboard, label one side near one edge with an arrow for reference, attach three (preferably distinctly colored) strings to either surface of the cardboard in a row in the center in the long direction (just punch holes in the cardboard and tie the strings with knots on the other side). Now attach each of the other string ends to a rod (e.g., a pencil), keeping the strings parallel. Voila, you have a "board and strings". Now here are the "rules for using it". The rod is to stay fixed, and permanently anchored to something "effectively" very big (like a student's hand, for instance). The cardboard starts out in a fixed position ("home"), and is allowed to be twisted and turned in any way you wish, coming back to its original location in space (although not necessarily with the arrow in the same place!). This is called a "move". The strings have gone along for the ride, and thus may be rather twisted up. The strings are allowed, however, to be deformed (isotopy) as much as you wish (including passing around the "back" of the board, for which the person holding the board will have to "switch hands"; but NOT allowing the strings around the rod, which remember is "permanently anchored to something "effectively" very big), so that up to string isotopy there may be vastly fewer moves than expected. Under composition of moves this is of course is a group. Which "familiar" one is it isomorphic to? (I'll put the answer one screen down, at the very end of this message, in case you want to figure it out yourself first.) Of course a move in which the arrow comes back to where it started means that the board has followed a based loop in SO(3); I'll leave it to you to see how the fundamental group of SO(3) is thus reflected in this group, and how it generalizes the Philippine Candle Dance phenomenon beautifully. A classroom unit in which groups of students figure this all out from scratch, leading to group theory, subgroups, quotient groups, isomorphism, etc., is the most wildly(!) successful topic I have ever taught in a required Mathematics Appreciation class, for students with poor background who don't want to be taking another mathematics class in their lives. It's also a big winner in a beginning abstract algebra course. Probably this is the best "hands-on" activity I have ever come across to do with students; topology wins! I usually spend some weeks on it in class; they learn so much, and love it. I'd be happy to share my written assignments with anyone who asks, although most of the "instruction" happens in the classroom, not in writing. David Pengelley (davidp@nmsu.edu) Mathematics, New Mexico State University, Las Cruces, NM 88003 USA Tel: 505-646-3901=dept., 505-646-2723=my office; Fax: 505-646-1064 http://math.nmsu.edu/~davidp Answer: The eight element group of unit quaternions! _____________________________________________________________________________ Subject: More on Spinner Spanner re SO(3) Date: Fri, 16 Jan 2004 17:13:47 -0700 (MST) From: David Pengelley More on the fundamental group of SO(3): Maybe everyone already knows this, but I also have what I find a more satisfying intuitive geometric way of seeing that this element has order two (I set aside being convinced that the element is nontrivial, which seems intuitive because you feel your arm may break, but I think can only be proven with some serious mathematics). The fact that your arm twists up with one full rotation around the vertical, and then somehow untwists the second time, instead of breaking, has always seemed more like magic than explanation to me. Using the fact that "we know" that based paths up to homotopy is forming a group, I am therefore "theoretically comfortable" that seeing that this element has order two is equivalent to seeing that it is its own inverse; and this is easy! The element consists of one full rotation (imagine rotating a sphere) around its vertical axis, say, clockwise, looking from above. This element is a closed path in SO(3). But now let us deform it via a homotopy of closed paths, as follows. Just slowly tilt the vertical axis (in any direction), always still performing one full rotation as the closed path. Slowly tilt the axis further in that direction, and eventually all the way over until it is again vertical, all the while imagining the "full turn" which is the closed path in SO(3) at each stage of the homotopy of closed paths. When you're done turning the axis over, the axis has flipped upside down, so the closed path is now a full counterclockwise turn, which is obviously the inverse closed path in SO(3). Thus the original closed path is homotopic to its inverse. Easy! (Notice that because I had to choose a horizontal direction to effect my flip, there is a nontrivial higher homotopy lurking here; what does it "mean"?) Now, it would be nice to see visually, using my sphere image above, say, an explicit homotopy between the composition of the original loop followed by itself and the original loop followed by its inverse, since we know that the latter is homotopic to the identity in a standard, textbook, way. Then the two homotopies concatenated would provide a visual nullhomotopy for the 720 degree twist. (One of my dreams is to make art based on this!) As it is, our arms are doing something quite curious to achieve the nullhomotopy; in particular, your hand has to go once under the elbow, and then once over. Best, David David Pengelley (davidp@nmsu.edu) Mathematics, New Mexico State University, Las Cruces, NM 88003 USA Tel: 505-646-3901=dept., 505-646-2723=my office; Fax: 505-646-1064 http://math.nmsu.edu/~davidp ___________________________________________________________ Subject: Re: two postings Date: Fri, 16 Jan 2004 13:50:16 -0700 (MST) From: "Peter D. Zvengrowski" Dear Jim, Here are two replies, both to today's question and to one from a week or two ago. 1) For the generator of the fundamental group of $SO(3)$ try the book by Montesinos, Classical Tesselations and 3-Manifolds, p.99. 2) For the cohomology ring of Seifert manifolds, see a couple of papers on exactly this topic (at least for those with infinite fundamental group, which is most Seifert manifolds) by Bryden, Hayat-Legrand, Zieschang, and myself in recent Topology and its Appl. We did not consider the question you asked of filling in the generic fibres with discs, but clearly it will be closely related to the results for the Seifert manifold itself. Hope this helps a little. Regards, Peter (Zvengrowski)