From reading the Morishita article 0904.3399 (page 24), there is a following analogue of Poincare conjecture:

Suppose that k is a number field whose ring of integers \mathscr O_k http://latex.mathoverflow.net/png?%5Cmathscr%20O%5Fk is “cohomologically \mathbb Z http://latex.mathoverflow.net/png?%5Cmathbb%20Z”, namely formula edited out http://latex.mathoverflow.net/png?%7B%7D%5EcH%5Ei%20%28%5Cmathrm%7BSpec%7D%5C%2C%20%20%0A%5Cmathscr%20O%5Fk%20%2C%20%5Cmathbb%20Z%29%20%3D%20%7B%7D%5EcH%5E%20i%20%28%5Cmathrm%7BSpec%7D%5C%2C%20%5Cmathbb%20Z%2C%20%5Cmathbb%20Z%29 for i ≥ 0. Then formula edited out http://latex.mathoverflow.net/png?%5Cmathscr%20O%5Fk must be \mathbb Z http://latex.mathoverflow.net/png?%5Cmathbb%20Z.

From reading the Morishita article 0904.3399 (page 24), there is a following analogue of Poincare conjecture:

Suppose that k is a number field whose ring of integers $\mathscr O_k$ is “cohomologically $\mathbb Z$”, namely $${}^{c}H^{i}(\text{Spec}\, \mathscr O_k,\mathbb{Z}) = {}^{c}H^{i}(\text{Spec}\, \mathbb{Z},\mathbb{Z})$$ for $i ≥ 0$. Then $\mathscr O_k$ must be $\mathbb Z$.