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I have seen a rough statement in chapter 2 of Hatcher's Algebraic Toplogy book that states homology groups with coefficients in $\mathbb{Z}$ contain less information rather than homology groups with coefficients in $\mathbb{Z}_m$. Further, in a question from my professor, I asked why people use homology with integers coefficients and he said homology theory with coefficients in $\mathbb{R}$ has less information. To which extent this statement holds? Is it true homology with coefficients in a bigger group contains less information?

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homology groups with coefficients in $\mathbb{Z}$ contain less information rather than homology groups with coefficients in $\mathbb{Z}_m$

This is totally false. Homology with coefficients in $\mathbb{Z}$ contains more information than homology with coefficients in any other group. Indeed, by the universal coefficient theorem, homology with any other coefficients is determined by homology with coefficients in $\mathbb{Z}$.

The main reason we use other coefficients besides $\mathbb{Z}$ despite them having less information is that they can be easier to work with. In general, working with coefficients in a field is easier, since then your homology groups are vector spaces (and in particular, they always have a basis). Another reason is that sometimes other coefficients arise naturally for geometric reasons. For instance, using coefficients in $\mathbb{Z}_2$ gives us Poincaré duality even on nonorientable manifolds, and (co)homology of smooth manifolds with coefficients in $\mathbb{R}$ is naturally related to differential forms by the de Rham theorem.

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  • $\begingroup$ Thank you very much for your answer. $\endgroup$ – mathvc_ Jun 6 '17 at 7:19

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