Continuous homology of topological periodic homology of complex cobordism

We determine the continuous mod $p$ homology of the topological periodic homology $TP(MU)$ of the complex cobordism spectrum, as a graded algebra with Steenrod operations. The answer is given in terms of an explicit and purely algebraic construction $C_{+}$, analogous to Singer’s construction $R_{+}$. Its $Ext$-algebra provides the $E_2$-term for a multiplicative Adams-type spectral sequence converging strongly to the homotopy of $p$-completed $TP(MU)$.