The Eilenberg-Moore spectral sequence converges from the classical Tor of a span of cohomology rings to the differential Tor of a span of cochain algebras (which is the cohomology of the homotopy pullback). These are both rings, the first classically and the second as a corollary of the Eilenberg-Zilber theorem.

One might well ask when a more general differential Tor of DGAs admits a ring structure, though apparently no one did. We will show that when the DGAs in question admit a certain sort of \(E_3\)-algebra structure, Tor is a commutative graded algebra.

We have not done this out of an innocent interest in homotopy-commutative algebras. In 1960s and ’70s there was a flurry of activity developing A-infinity-algebraic techniques with an aim toward computing the Eilenberg–Moore spectral sequence (for example, of a loop space or homogeneous space). Arguably the most powerful result this program produced was the 1974 theorem of Munkholm that the sequence collapses when the three input spaces have polynomial cohomology over a given principal ideal domain, which however only gives the story on cohomology groups. Our result shows that Munkholm’s map is in fact an isomorphism of rings.

The proof hinges on homotopy properties of the (1-)category of augmented DGAs. This work is all joint with several large commutative diagrams, who should be considered the true authors.

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