In Chapter 0 of Hatcher, there is the example of thickened letters (p.2). Now, I don't get, why one would expect that A, B etc. (letters with holes) would not be deformation retracts. What is the reason they are not?
thickened letters and deformation retracts
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$\begingroup$
algebraic-topology
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0In my edition the picture is on page 1 – 2017-01-20
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0The picture shows that the thin letters are defomation retracts of the thick letters. – 2017-01-20
1 Answers
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The the thin letters are deformation retracts of their corresponding thickened letters. This is, what the picture shows.
A and B are neither deformation retracts of each other, nor homotopy equivalent to each other. Intuitively they are not transformable into one another by bending, shrinking or expanding. Obviously a deformation retract of A is the $S^1$ (just shrink the two end pieces) and B is homotopy equivalent to the figure 8, which is a wedge of two circles $S^1\vee S^1$. Since their fundamental group $\pi_1(A)=\pi_1(S^1)=\mathbb{Z}$ and $\pi_1(B)=\pi_1(S^1\vee S^1)=\mathbb{Z}\ast \mathbb{Z}$ are different, they are not homotopy equivalent.
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0So, he meant that $A$, $B$ are not deformation retracts of each other. That makes a lot more sense. Thank you. – 2017-01-23