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establish it) and is known due to previous work of the author and Franz, and also, as it turns out, to Huebschmann's unpublished 1983 habilitation work.
Cite
@article{arxiv.2306.04860,
title = {A ring structure on Tor},
author = {Jeffrey D. Carlson},
journal= {arXiv preprint arXiv:2306.04860},
year = {2026}
}
Comments
Substantial corrections to the preceding draft are enumerated: briefly, the central claimed topological result is false without a modification which renders it already known; other results in the paper are true but do not accomplish what was desired. All this is elaborated upon in detail in a new preface