I am calculating the torus cohomology groups with $\mathbb{Z}/2$ coefficients and I get to that $$H^1(T, \mathbb{Z}/2)\cong Hom(H_1(T), \mathbb{Z}/2)=Hom(\mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}/2)$$ but I don't know what P is, could someone please tell me what $Hom(\mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}/2)$ is? Thank you!
2026-03-26 07:58:22.1774511902
what $Hom(\mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}/2)$ is?
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Hint For any $f \in Hom(\mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}/2)$ you have $$f(m,n)=mf(1,0)+nf(0,1) \pmod{2}$$
Now, there are four choices for $f(1,0), f(0,1)$ thus this leads to a group of 4 elements. Note also that $f+f=0$ for each $f$.