How is it that with taking coefficients over a "simpler" ring, we are able to recover more information about the homology of a space? For example, if we take the projective plane as the union of a mobius band and the 2-disk glued along their boundaries, with coefficients in R we only get something in [tex]H_0(P^2)[/tex]. Over R, we get something in [tex]H_0(P^2)[/tex] and [tex]H_1(P^2)[/tex], but it is only in [tex]\mathbb{Z}_2[/tex] do we realise that the projective plane is a "true" 2-dimensional object, not being homotopically equaivalent to an object of lower dimension.(adsbygoogle = window.adsbygoogle || []).push({});

We have in the Mayer-Vietoris sequence a section that looks like

[tex]0 \rightarrow H_2(P^2) \rightarrow H_1(S^1) \rightarrow H_1(D^2)\oplus H_1(M) \rightarrow ...[/tex]

And taking coefficients over [tex]\mathbb{Z}_2[/tex] we get

[tex]0 \rightarrow H_2(P^2) \rightarrow \mathbb{Z}_2 \rightarrow \mathbb{Z}_2 \rightarrow ... [/tex]

Where the homomorphism from [tex]\mathbb{Z}_2 \rightarrow \mathbb{Z}_2[/tex] defined by sending 1 to 2, which is due to the circle wrapping twice around the mobius band, which means that the map from [tex] H_2(P^2) \rightarrow \mathbb{Z}_2[/tex] is an isomorphism. I guess my question is that is there any other reason that [tex]\mathbb{Z}_2[/tex] is able to pick out this extra structure in homology than other rings than that there is a homomorphism sending an element d in the ring to 2d? A more geometric interpretation?

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# Is a homomorphism sending an element

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