Let X be the 2 complex obtained from $S^{1}$ with its usual cell structure by attaching
two 2 cells by maps of degrees 2 and 3 , respectively.
(a) Compute the homology groups of all the subcomplexes $A ⊂ X $ and the corre-
sponding quotient complexes X/A .
(b) Show that X is homotopy equivalent to
$S^{2}$ and that the only subcomplex $A ⊂ X$ for which the quotient
map X →X/A is a homotopy equivalence is the trivial subcomplex, the 0 cell.
I have calculated the homologies and these are:
Case 1 : A is 1-skeleton ,$H_0(X/A)= Z $, $H_2(X/A)= Z\bigoplus Z$ and $0$ otherwise.
Case 2: For other non-trivial proper subcomplexes ,$H_i(X/A)= Z$ for $i=0,2$ and $0$
otherwise.
But I need some help for the second part of question.
Thanks!
Best Answer
I know this a Hatcher HW problem, so I won't give away the answer. But Allen Hatcher himself thought this problem was too hard, and gives an extra hint on this page (you have to scroll down to see it):Hatcher Additional Exercises