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We give a new CR invariant treatment of the bigraded Rumin complex andrelated cohomology groups via differential forms. A key benefit is theidentification of balanced $A_\infty$-structures on the Rumin and bigradedRumin complexes. We also prove related Hodge decomposition theorems. Among manyapplications, we give a sharp upper bound on the dimension of the Kohn--Rossigroups $H^{0,q}(M^{2n+1})$, $1\leq q\leq n-1$, of a closed strictlypseudoconvex manifold with a contact form of nonnegative pseudohermitian Riccicurvature; we prove a sharp CR analogue of the Fr\"olicher inequalities interms of the second page of a natural spectral sequence; we give new proofs ofselected topological properties of closed Sasakian manifolds; and we generalizethe Lee class $\mathcal{L}\in H^1(M;\mathscr{P})$ -- whose vanishing isnecessary and sufficient for the existence of a pseudo-Einstein contact form --to all nondegenerate orientable CR manifolds.

The bigraded Rumin complex via differential forms