Spectral sequences of fibre tolerance spaces

The paper presents the theoretical base for the construction of spectral sequences of tolerant exfoliations. Namely, the authors give a number of important qualities of singular cubes in tolerant exfoliations. The fundamental base group operation on the group of fiber homology of tolerant exfoliation theorem is proved. According to the general theory of spectral sequences the first and the second terms of spectral sequence of tolerant exfoliations are got.