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 functorial commutative graded algebra structure generalizing the classical structure when the algebras are genuine commutative differential graded algebras. As a topological corollary, Munkholm's Eilenberg--Moore collapse result for pullbacks of spaces with polynomial cohomology is enhanced to a ring isomorphism. New applications compute the Borel cohomology rings of homogeneous spaces and singular cohomology rings of biquotients with minimal restrictions on coefficient rings; existing results on the singular cohomology rings of free loop spaces and (generalized) homogeneous spaces are respectively recovered and strengthened.