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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!

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    I am really sorry I should have mentioned that it is reduced homology but I think now it is fine.2012-12-11

2 Answers 2

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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

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    well,I appreciate that you think it is a homework problem but in actual it is not.Thanks for the hint.2012-12-13
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Yo can find an idea for the solution here: http://www.math.ku.dk/~moller/blok1_05/AT-ex.pdf?q=allen-hatcher, page 26.