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?

  • $\begingroup$ 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 $A=X$. $\endgroup$
    – user98602
    Commented Oct 27, 2014 at 18:46
  • 2
    $\begingroup$ @MikeMiller Calling them "2-cell" and "3-cell" are bad ideas IMO... They are both 2-cells, of degrees 2 and 3. $\endgroup$ Commented Oct 27, 2014 at 19:07
  • $\begingroup$ @NajibIdrissi It's a terrible idea! Thanks for correcting me, my head must have been elsewhere when writing that. $\endgroup$
    – user98602
    Commented Oct 27, 2014 at 19:09
  • $\begingroup$ 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? $\endgroup$
    – EgoKilla
    Commented Oct 27, 2014 at 21:03
  • $\begingroup$ 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. $\endgroup$ Commented Oct 28, 2014 at 9:57

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