Timeline for How to show that the "bing's house with two rooms" is contractible?
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Mar 10, 2010 at 6:28 | comment | added | Ryan Budney | You might want to take a look at Marshall Cohen's book "A Course in Simple Homotopy Theory". He's quite explicit about these sorts of details. | |
| Mar 10, 2010 at 6:02 | comment | added | gylns | Ok,Thanks for your advice, maybe this is not a "problem" for you. | |
| Mar 10, 2010 at 5:57 | comment | added | Ryan Budney | If you remove the support walls I think this new space has to have a fundamental group. Think about a loop that runs up the "external" wall and down the tunnel. | |
| Mar 10, 2010 at 5:50 | comment | added | Ryan Budney | The main issue you seem to be having is that Hatcher is assuming a certain level of comfort with linear-algebraic constructions. A good way to achieve this level of maturity would be to work through much of the point-set and fundamental-group problems in a book like Munkres. | |
| Mar 10, 2010 at 5:48 | comment | added | gylns | Well, can you tell me if remove the two vertical rectangles (‘support walls’ for the two tunnels),it remains contractible? | |
| Mar 10, 2010 at 5:43 | comment | added | Ryan Budney | You can realize the deformation-retraction as a sequence (concatenation) of "elementary collapse" operations. In particular you can write the map as a piecewise construction, made of composites of rational polynomial functions. These elementary collapses appear in many places in Hatcher's book -- the main construction in Proposition 0.16 of Chapter 0 (page 15) is the first such explicit construction, I think. | |
| Mar 9, 2010 at 6:06 | comment | added | Harry Gindi | Parameterizing those maps is going to be a painful endeavor. | |
| Mar 9, 2010 at 5:39 | comment | added | gylns | But,I want a proof in mathematical style!!! | |
| Mar 9, 2010 at 1:04 | comment | added | Ryan Budney | It sounds like this is a language issue. Imagine a drinking glass full of wax. It's a solid object. By melting the wax and draining the liquid wax, you in effect "hollow out the chamber" -- the chamber being the glass full of wax. The hollow chamber is the empty glass. | |
| Mar 9, 2010 at 0:46 | comment | added | gylns | Sorry, my imagination is really poor,"hollowing out the chamber" is not clear. | |
| Mar 9, 2010 at 0:39 | comment | added | Petya | May be a good exercise is to take a loop which is looking non-trivial and try to see that it is contractible. | |
| Mar 9, 2010 at 0:35 | comment | added | Steven Gubkin | Hatcher gives a pretty lucid description. What part of it is not clear? He suggests visualizing a thickening of the space as made out of clay: have you tried using playdough? I have resorted to playdough many times when my visual imagination failed me. | |
| Mar 9, 2010 at 0:28 | comment | added | gylns | Yes, but is not clearly for me. | |
| Mar 9, 2010 at 0:26 | history | answered | Harry Gindi | CC BY-SA 2.5 |