  In this paper we introduce and study a higher-dimensional analogue of the
giant component in continuum percolation. Using the language of algebraic
topology, we define the notion of giant k-dimensional cycles (with 0-cycles
being connected components). Considering a continuum percolation model in the
flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1)
appear in the regime known as the thermodynamic limit. We also prove that the
thresholds for the emergence of the giant k-cycles are increasing in k and are
tightly related to the critical values in continuum percolation. Finally, we
provide bounds for the exponential decay of the probabilities of giant cycles
appearing.
