Let X the complement of the zaxis in R3 Let S1 be the unit

Let X the complement of the z-axis in R^3 . Let S^1 be the unit circle in the xy-plane. Give an explicit deformation retract (i.e. a formula) of X to S^1.

Solution

Given that X be a complement of Z-axis in R³

let x in X and z in Z

and given that S¹  be a unit circle i.e

S¹ = {a / |a|=1}

Given x = z^c

let z = (p,q,r) then

   x = (r,q,p)

let consider a function

f : X => S¹

then f(x)= {a /|a| =1}

f(r,q,p)= {a/ |a|=1}