Let $F$ be an orientable, compact, connected surface with $n$ boundary components.

We want to consider the first homology group with integer coefficients of $F$, denoted by $H_1(F;\mathbb{Z})$. My book claims that we have $$H_1(F;\mathbb{Z})=\oplus_{2g+n-1}\mathbb{Z}.\qquad (1)$$

I have two questions: What is an homology group with integer coefficients? I only know singular homology groups or relative homology groups.

How do we get the formula $(1)$? I know that an orientable closed genug $g$ surface has the first homology group $\oplus_{2g}\mathbb{Z}$. Does this help somehow?

  • $\begingroup$ The homology groups $H_i(F,\Bbb Z)$ are the plain vanilla singular homology groups that you may have seen written as $H_i(F)$. And if you start with an orientable closed surface then you can use a Mayer-Vietoris argument to prove that removing a disc adds a summand $\Bbb Z$ to the $H_1$. $\endgroup$ – Lord Shark the Unknown May 1 '17 at 14:47
  • $\begingroup$ So it is nothing else but the abiliazation of the fundamental group, isn't it? Can I get equation $(1)$ if I have already proven the equation for $n=1$? $\endgroup$ – user369147 May 1 '17 at 14:52

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