- #1

Lonewolf

- 336

- 1

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?