  Consider a local chain Differential Graded algebra, such as the singular
chain complex of a pathwise connected topological group.
  In two previous papers, a number of homological results were proved for such
an algebra: An Amplitude Inequality, an Auslander-Buchsbaum Equality, and a Gap
Theorem. These were inspired by homological ring theory.
  By the so-called looking glass principle, one would expect that analogous
results exist for simply connected cochain Differential Graded algebras, such
as the singular cochain complex of a simply connected topological space.
  Indeed, this paper establishes such analogous results.

Rings and Algebras;K-Theory and Homology