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Let $N_g$ be a closed nonorientable surface of genus $g$. I will try to compute the homology groups and I want you to help me with certain steps and correct my mistakes - I will use this as an outline.

(i) Cell decomposition:

  • one 2-cell ($e^2$)
  • g 1-cells ($a_1,...,a_g$)
  • one 0-cell ($v$)

(ii) Attaching maps:

  • Attaching map of $e^2$ is $f=a_1^2...a_g^2$
  • Attaching map of $a_i$ is $g=v$

(iii) The chain groups:

  • $C_0=\Bbb{Z}$
  • $C_1=\Bbb{Z}^g$
  • $C_2=\Bbb{Z}$ and the rest of them are zero

(iv) The boundary homomorphisms:

$\dots \xrightarrow{d_3} C_2(N_g) \xrightarrow{d_2} C_1(N_g) \xrightarrow{d_1} C_0(N_g) \xrightarrow{d_0} 0$

  • $d_0=0$
  • $d_1:\Bbb{Z}^g\to\Bbb{Z}$ and this should be the zero map but how do I show that?
  • $d_2:\Bbb{Z}\to\Bbb{Z}^g$ I'm totally clueless about this one, once I know $d_1$ and $d_2$ everything is easy.

Here is the definition of $d_n$:

$d_n(e_\alpha^n)=\sum_\beta d_{\alpha\beta}e_\beta^{n-1}$ where $d_{\alpha\beta}$ is the degree of the map $S_\alpha^{n-1}\to X^{n-1}\to S_\beta^{n-1}$ that is the composition of the attaching map $\phi$ of $e_\alpha^n$ with the quotient map $q$ collapsing $X^{n-1}-e_\beta^{n-1}$ to a point.

So to understand $d_1$ I need to compute the degree of the map $q\circ\phi:S^{n-1}\to S^{n-1}$. But $H_0(S^0)\to H_0(X^0)\to H_0(S^0)$ is the zero map because $H_0(X^0)=0$, which means that the degree is zero, so the coefficient is zero and $d_1$ is zero, is that true?

Now, what is $d_2(1)=d_1a_1+...+d_ga_g$?

Any help would be greatly appreciated.

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  • Hint: to compute $d_2$ use the formula for $f$, but convert multiplication to addition. – Moishe Kohan Oct 17 '13 at 13:54
  • Alright, so this time to compute the $d_{\alpha\beta}$'s we look at the map $H_1(S^1)\to H_1(X^1)\to H_1(S^1)$ that is $\Bbb{Z}\to H_1(X^1)\to \Bbb{Z}$. For instance, what is the image of $1$ under this map? Intuitively it is $(2,...,2)$; I replaced all the $a_i$'s with $1$ and used the formula $2a_1+...+2a_g$ but I don't know why this works, how can I replace $1$ with $a_i$? I don't really understand the transition between $\Bbb{Z}$ and $H_1(S^1)$, also what kind of map is $\Bbb{Z}\to H_1(X^1)\to \Bbb{Z}$? – Xena Oct 17 '13 at 14:04
  • Do not replace all $a_i$ with ones! They are independent generators of 1st homology of $X^1$. – Moishe Kohan Oct 17 '13 at 14:14
  • Ok, then I will replace only one $a_i$ with 1. The quotient map will take care of the rest - it collapses them to a point? I seriously don't know how to put it rigourously. – Xena Oct 17 '13 at 14:17
  • Do not replace anything! This gives you the required map $d_2$: Send the generator of $C_2$ to twice the sum of generators of $C_1$. – Moishe Kohan Oct 17 '13 at 14:30
  • Then why do we need to collapse anything? – Xena Oct 17 '13 at 14:36
  • I did not say you should collapse anything. – Moishe Kohan Oct 17 '13 at 14:38
  • But the quotient map does that – Xena Oct 17 '13 at 14:39

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