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Consider a surface $S_g$ of genus $g$. The first homology group $H_1(S, \mathbb{Z})$ is the abelianization of $\pi_1(S)$, and is easy to vizualize (we get two generators for each handle).

How is $H_1(S, \mathbb{R})$ defined? Is it the same as $H_1(S, \mathbb{Z})$ but with real coefficients? How does one interpret a 1-cycle with, say a $\sqrt{2}$ coefficient?

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    $\begingroup$ You interpret it exactly as with integer coefficients! $\endgroup$ – Mariano Suárez-Álvarez Nov 15 '16 at 5:17
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    $\begingroup$ Your question leaves me wondering how do you interpret cycles with integer coefficients. it would probably be useful if you told us what your interpretation is, so that we can see what changes (or does not) with you allow for real coefficients. $\endgroup$ – Mariano Suárez-Álvarez Nov 15 '16 at 5:19
  • $\begingroup$ Hmmm. So lets say $S$ is the torus. The usual homology group is generated by two elements, say $a, b$. Then $2a$ is a cycle that goes around $a$ twice. That makes sense. But what would $\sqrt{2} a$ mean? $\endgroup$ – user98246 Nov 15 '16 at 5:19
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    $\begingroup$ Well, no. if $a$ is a $1$-simplex which is a closed curve, then $2a$ is certainly not a curve that does anything two times. It is homologous to one, at the very most. $2a$ is simply a sum of simplices with integer coefficients. $\endgroup$ – Mariano Suárez-Álvarez Nov 15 '16 at 5:20
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    $\begingroup$ But that interpretation is wrong, in so far as twice a $1$-cycle is not a curve that goes around anything twice: it is simply that $1$-cocycle taken «with multiplicity two». Yes, if $a$ is a close curve, then the $1$-cycle $2a$ is cohomologous to the $1$-cycle $a^2$ corresponding to the concatenation of $a$ with itself. But that is a rather completely different thing. $\endgroup$ – Mariano Suárez-Álvarez Nov 15 '16 at 5:25
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If $X$ is a space and $S(X)$ is the chain complex of its singular simplices, then the homology with integer coefficients is simply the homology of $S(X)$.

Now if $A$ is an abelian group, the homology of $X$ with coefficients in $A$ is simply the homology of the complex $A\otimes S(X)$, which is exactly the chain complex with elements the formal linear combinations of singular simplices with coefficients in $A$.

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