# Question 13 in section 2.2 Hatcher Homology

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 $A\subset X$ and the corresponding quotient complexes $X/A$.

I'm having trouble with this problem, the subcomplexes should just be the skeleta right? I'm very confused could anyone get me started here?

• Check the definition of subcomplex. There are five here; $A$ just the 0-cell, $A= S^1$, $A = S^1$ with the 2-cell attached, $A = S^1$ with the 3-cell attached, and $A=X$.
– user98602
Commented Oct 27, 2014 at 18:46
• @MikeMiller Calling them "2-cell" and "3-cell" are bad ideas IMO... They are both 2-cells, of degrees 2 and 3. Commented Oct 27, 2014 at 19:07
• @NajibIdrissi It's a terrible idea! Thanks for correcting me, my head must have been elsewhere when writing that.
– user98602
Commented Oct 27, 2014 at 19:09
• Can someone walk me through how to compute the homology of one of the S1 with an attached 2 cell subcomplexes and then I can figure out how to do the other one? Commented Oct 27, 2014 at 21:03
• What kind of homology is this? Singular, simplicial, cellular...? Do you know that they are all equivalent? Because if so, computing the cellular homology of this space should be simple. Commented Oct 28, 2014 at 9:57

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