Retraction by deformation

Question

in Theorem 3.1 in the book Morse theory by Milnor , in the end of the proof they say that :

$r_t$ is a deformation retract ?

how to prove this ?

please

thank you

Answer 1

A deformation retract is, by definition, a homotopy between a retraction and the identity map. This is exactly what we have here: the homotopy is given by

$$H:M^b \times [0,1] \rightarrow M^b, H(q,t) = r_t(q).$$