I can't image this, Someone can give a clear illustration?
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2$\begingroup$ This question seems too localized. $\endgroup$– Harry GindiCommented Mar 9, 2010 at 0:29
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5$\begingroup$ Ken Baker has put up some beautiful images here: sketchesoftopology.wordpress.com/2010/03/25/bings-house $\endgroup$– j.c.Commented Mar 31, 2010 at 3:03
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1$\begingroup$ An update to jc's comment above: Ken Baker made a subsequent post that describes a deformation retraction: sketchesoftopology.wordpress.com/2010/06/23/… $\endgroup$– RamsayCommented Apr 9, 2012 at 11:13
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$\begingroup$ I am confused by the very first step in Ken's illustrations. I don't see how this sequence flickr.com/photos/sketchesoftopology/4644072542 is a continuous deformation that stays within the original complex. Could someone comment on that? (The remaining steps are then straightforward.) $\endgroup$– Tomáš BzdušekCommented Apr 5, 2018 at 23:20
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1 Answer
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This is covered in Chapter 0 (the introductory chapter) of Algebraic Topology by Allen Hatcher.
answered Mar 9, 2010 at 0:26
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3$\begingroup$ 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. $\endgroup$ Commented Mar 9, 2010 at 0:35
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4$\begingroup$ 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. $\endgroup$ Commented Mar 9, 2010 at 1:04
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2$\begingroup$ 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. $\endgroup$ Commented Mar 10, 2010 at 5:43
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4$\begingroup$ 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. $\endgroup$ Commented Mar 10, 2010 at 6:28