We study here an alternative technique to probe the Dark Ages (DA) and the Epoch of Reonization (EoR) that makes use of the Comptonization of the CMB spectrum modified by physical effects occurring during this epoch related to the emergence of the 21-cm radiation background. Inverse Compton scattering of 21-cm photon background by thermal and non-thermal electrons residing in the atmospheres of cosmic structures like galaxy clusters, radiogalaxy lobes and galaxy halos, produces a specific form of Sunyaev-Zeldovich effect (SZE) that we refer to as SZE-21cm. We derive the SZE-21cm in a general relativistic approach which is required to describe the correct spectral features of this astrophysical effect. We calculate the spectral features of the thermal and non-thermal SZE-21cm in galaxy clusters and in radiogalaxy lobes, and their dependence on the history of physical mechanisms occurring during the DA and EoR. We study how the spectral shape of the SZE-21cm can be used to establish the global features in the mean 21-cm spectrum generated during and prior to the EoR, and how it depends on the properties of the (thermal and non-thermal) plasma in cosmic structures. We find that the thermal and non-thermal SZE-21cm have peculiar spectral shapes that allow to investigate the physics and history of the EoR and DA. Its spectrum depends on the gas temperature (for the thermal SZE-21cm) and on the electrons minimum momentum (for the non-thermal SZE-21cm). The global SZE-21cm signal can be detected (in $sim 1000$ hrs) by SKA1-low in the frequency range $ u simgt 75-90$ MHz, for clusters in the temperature range 5 to 20 keV, and the difference between the SZE-21cm and the standard SZE can be detected by SKA1 or SKA2 at frequencies depending on the background model and the cluster temperature. [abridged]
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