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We construct a new minimal complex surface of general type with $p_g=0$,$K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact$\pi_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existencequestion for numerical Campedelli surfaces with all possible algebraicfundamental groups. The main techniques involved in the construction are arational blow-down surgery and a $\mathbb{Q}$-Gorenstein smoothing theory.

A complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$