A ring structure on Tor

Abstract

We prove that within a natural class of E_3-algebras, the graded Tor group induced by a span of E_3-algebra maps carries a graded algebra structure generalizing the classical structure when the algebras are genuine commutative differential graded algebras. We attempt to prove, as a topological corollary, that Munkholm's Eilenberg--Moore collapse result for pullbacks of spaces with polynomial cohomology can be enhanced to a ring isomorphism. This is not achieved, and in fact the claim as stated in the previous drafts is false. If additionally, 2 is assumed to be a unit of the base ring, then that claim is true (not that the results in this paper establish it) and is known due to previous work of the author and Franz, and also, as it turns out, to Huebschmann's unpublished 1983 habilitation work.