Anomaly cancellation in M-theory on orbifolds

We present calculation of the anomaly cancellation in M-theory on orbifolds $S^1\mathrm{/}{Z}_2$ and $T^5\mathrm{/}{Z}_2$ in the upstairs approach. The main requirement that allows one to uniquely define solutions to the modified Bianchi identities in this case is that the field strength $G$ be globally defined on $S^1$ or $T^5$ and properly transforming under $Z_2$. We solve for general $G$ that satisfies these requirements and explicitly construct anomaly-free theories in the upstairs approach. We also obtain the solutions in the presence of five-branes. All these constructions show equivalence of the downstairs and upstairs approaches. For example in the $S^1\mathrm{/}{Z}_2$ case the ten-dimensional gauge coupling and the anomaly cancellation at each wall are the same as in the downstairs approach.