The $\eta$-periodic motivic sphere over fields has attracted considerable interest in the past: Ananyevskiy–Levine–Panin have proved that its rationalization is Eilenberg–MacLane, and various authors have studied its homotopy groups over specific fields. In this talk I will explain recent results obtained in joint work with Mike Hopkins, in which we establish structural properties of the $\eta$-periodic category, over in principle general bases (containing $\frac{1}{2}$). In particular we compute over $\mathbb{Z}[\frac{1}{2}]$ the $\eta$-periodic stable stems, as well as the $\eta$-periodic algebraic $SL$-cobordism groups.