Timeline for Covering maps in real life that can be demonstrated to students

Current License: CC BY-SA 3.0

31 events

when toggle format	what		by	license	comment
Dec 10, 2017 at 9:55	history	edited	Martin Sleziak		removed (tag-removed) tag
Mar 14, 2013 at 14:57	comment	added	Lee Mosher		Will Sawin's example can be realized in a very pleasing manner which I learned recently at MoMath. Modify the usual construction of the Mobius band, using three strips of paper stacked on top of each other. Do a single half-twist of this triple stack, and then carefully stick the required pieces of tape into this contraption. You will get a paper model of the double covering of the Mobius band (the glued inner strip) by the annulus (the glued pair of outer strips).
Mar 14, 2013 at 9:17	comment	added	Sergey Melikhov		G.C.: The question is not equivalent to asking whether the mapping cylinder embeds in $\Bbb R^3$. Also, there are no problems with point-set topology because quotient topology wasn't mentioned. In fact, the question is equivalent to asking whether the mapping cylinder has a level-preserving embedding in $\Bbb R^3\times [0,1]$, where the mapping cylinder is endowed with the (metrizable) topology of quotient uniformity. If the domain is compact, this is the same as quotient topology, but if it's not, quotient topology is non-metrizable.
Mar 14, 2013 at 8:52	answer	added	Sergey Melikhov		timeline score: 1
Mar 14, 2013 at 1:27	history	edited	Brian Rushton	CC BY-SA 3.0	Removed unnecessary tag and edited typos
Dec 9, 2012 at 13:41	answer	added	Dror Bar-Natan		timeline score: 22
Dec 9, 2012 at 3:31	history	edited	Brian Rushton	CC BY-SA 3.0	Made CW ; Post Made Community Wiki
Dec 8, 2012 at 21:12	comment	added	Alison Miller		@Ketil Tveiten: Not quite. You actually get a paper-strip-glued-with-two-full-twists that way. The construction you want is to make a Mobius strip, but with two strips of paper stacked on top of each other (so that the gluing glues one end of each strip to the opposite end of the other).
Dec 8, 2012 at 9:27	comment	added	G.C.		Regarding the question of embedding the mapping cylinder, it is easy to do for a the covering of a circle given by a projection $\mathbf Z\to\mathbf Z/n$, but even for the simple case of the triple covering given by the map $f : S^1\sqcup S^1 \to S^1$ defined by $f(z)=z$ on the first circle and $f(z)=z^2$ on the second, it won't be possible. Similarly, you cannot embed the triple trivial covering of the tree with 4 nodes, one of which is 3-valent. Thus 'imbeddable coverings' are quite rare.
Dec 8, 2012 at 1:15	history	edited	Brian Rushton	CC BY-SA 3.0	Edited per G.C.'s comment.
Dec 7, 2012 at 13:56	comment	added	G.C.		(Pernickety comment) About your more precise question (embedding the mapping cylinder), there will be a problem of point set topology if you don't assume your covering is finite (since $\mathbf R^3$ is locally compact).
Dec 7, 2012 at 2:30	comment	added	Will Jagy		I like the one where the unit quaternions double cover $SO_3.$ The other idea is to roll a student in carpet. When the carpet just begins to touch, that it a single cover. Roll him over again and that is a double cover. Very real life.
Dec 7, 2012 at 2:05	history	edited	Brian Rushton	CC BY-SA 3.0	Minor edit
Dec 6, 2012 at 23:57	history	edited	Brian Rushton	CC BY-SA 3.0	Refocused question
Dec 6, 2012 at 20:42	history	edited	Brian Rushton	CC BY-SA 3.0	Refocused question
Dec 6, 2012 at 19:56	history	edited	Brian Rushton		edited tags
Dec 6, 2012 at 15:51	answer	added	G.C.		timeline score: 14
Dec 6, 2012 at 15:15	comment	added	Pablo Lessa		You should definitely look at the pictures on page 58 of Hatcher's book (math.cornell.edu/~hatcher/AT/ATch1.pdf). He gives several non-obvious coverings of the figure 8.
Dec 6, 2012 at 14:17	answer	added	Hiro Lee Tanaka		timeline score: 18
Dec 6, 2012 at 13:02	comment	added	Ketil Tveiten		@Brian Rushton: Think of cutting a paper Möbius strip (aka. paper-strip-glued-with-a-half-twist) along the midline. You get the paper-strip-glued-with-a-twist, which is the required nonstandard cylinder embedding.
Dec 6, 2012 at 12:57	comment	added	j.c.		Here are a few examples that I can't find homotopies for but are examples that fit the title. First, some obvious ones: tilings / wallpaper patterns are covering maps of the torus (certain orbifolds if you want); In nature, crystal structures give many nice examples as well e.g. en.wikipedia.org/wiki/Diamond_lattice . A nice set of examples are triply periodic minimal surfaces en.wikipedia.org/wiki/Triply_periodic_minimal_surface , which amazingly are realized in certain soft materials e.g. jstor.org/stable/54307 .
Dec 6, 2012 at 12:22	answer	added	Ronnie Brown		timeline score: 9
Dec 6, 2012 at 9:02	answer	added	ACL		timeline score: 18
Dec 6, 2012 at 6:23	answer	added	Guntram		timeline score: 19
Dec 6, 2012 at 6:01	answer	added	S. Carnahan♦		timeline score: 10
Dec 6, 2012 at 3:09	answer	added	Rodrigo A. Pérez		timeline score: 4
Dec 6, 2012 at 3:07	comment	added	Steve Huntsman		The covers of a Tanner graph (a bipartite graph with nodes given by symbols and checks) associated to a low-density parity check code are readily exhibited and applied: since the primary desideratum for efficient decoding is that the Tanner graph not have any short cycles, coverings naturally come to the fore. MacKay's book at inference.phy.cam.ac.uk/itprnn/book.html prominently features a universal cover of a Tanner graph (at the beginning of Part VI and) on page 566.
Dec 6, 2012 at 2:56	comment	added	Brian Rushton		Cool! Do you have a reference or image I could check out?
Dec 6, 2012 at 2:25	comment	added	Will Sawin		The orientable double cover of the mobius strip can be realized in $3$-space using a non-standard embedding of the cylinder.
Dec 6, 2012 at 2:22	history	edited	Brian Rushton	CC BY-SA 3.0	Changed title, removed "ambient"
Dec 6, 2012 at 1:57	history	asked	Brian Rushton	CC BY-SA 3.0

toggle format