Is there any chain complex $A_\bullet$ such that $H_n(A_\bullet)$ is $\mathbb{Z}/2\mathbb{Z}$ for all $n\in\mathbb{Z}$?

Question

Is there any chain complex $A_\bullet$ such that $H_n(A_\bullet)$ is $\mathbb{Z}/2\mathbb{Z}$ for all $n\in\mathbb{Z}$ and $H_n(A_\bullet/2A_\bullet)=\mathbb{Z}/2\mathbb{Z}$?

score 4 · Answer 1 · answered Sep 13 '15 at 20:18

4

What about $A_n=\mathbb Z/2\mathbb Z$ and $d_n=0$?

answered Sep 13 '15 at 20:18

Hagen von Eitzen

374,180

(More generally, the homology of any complex with zero differential is itself.) – Pedro Sep 14 '15 at 00:13

Is there any chain complex $A_\bullet$ such that $H_n(A_\bullet)$ is $\mathbb{Z}/2\mathbb{Z}$ for all $n\in\mathbb{Z}$?

1 Answers1