Homological periods and higher cycles

For any scheme which is algebraic over a subfield of the complex numbers we here construct an homological regulator from Suslin homology to period homology and a higher cycle class map from Bloch’s higher Chow group to the period Borel-Moore homology. Over algebraic numbers, making use of the motivic Albanese, we provide a purely geometric description of these period homologies in degree 1 and we characterise the $\mathbb{Q}\mathrm{/}\mathbb{Z}$-cokernel of these regulators in terms of torsion zero-cycles, showing that Grothendieck period conjectures imply generalised Roĭtman theorems.