1
Deformations
Our first look at topology will involve some problems about the deforma -
tion of elastic bodies and surfaces. W e shall assume that the objects consid -
ered ar e mad e fro m a very elasti c material: thei r shap e ma y be change d a t
will, you can bend, distort, stretch , and compres s them a s much a s you like,
but o f course you may not tear them or glue parts of them together .
The deformation s tha t yo u wil l be aske d t o find will see m impossibl e a t
first glance . But actually they are not difficult t o visualize, as you can verify by
reading their descriptio n i n the solutio n section . However , w e emphaticall y
suggest tha t yo u tr y t o find the solutio n o n you r ow n befor e lookin g a t ou r
answers.
Problem 1.1 . Sho w that the elastic body represented i n Figure 1.1(a ) can
be deforme d s o a s t o becom e th e on e show n i n Figur e 1.1(b) . I n othe r
words, were the huma n bod y elastic enough, afte r makin g linke d ring s with
your inde x fingers an d thumbs , yo u coul d mov e you r hand s apar t withou t
separating the joined fingertips.
(a) (b )
FIGURE 1.1
I
http://dx.doi.org/10.1090/mawrld/004/01
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