In this article we investigate a pair of surjective local ring maps
S1ββRβS2β and their relation to the canonical projection RβS1ββRβS2β, where S1β,S2β are Tor-independent over R. Our main
result asserts a structural connection between the homotopy Lie algebra of
S:=S1ββRβS2β, denoted Ο(S), in terms of those of R,S1β and
S2β. Namely, Ο(S) is the pullback of (adjusted) Lie algebras along the
maps Ο(Siβ)βΟ(R) in various cases, including when the maps above have
residual characteristic zero. Consequences to the main theorem include
structural results on Andr\'{e}-Quillen cohomology, stable cohomology, and Tor
algebras, as well as an equality relating the Poincar\'{e} series of the common
residue field of R,S1β,S2β and S.Comment: 20 pages. Corrected a mistake in 1.7; simplified and reorganized
Sections 4 and