The relative join operad and polyhedral products

Abstract

An inclusion of nonvoid simplicial complexes induces an arrow of polyhedral products. Applying Ayzenberg's polyhedral join to both complexes gives a symmetric relative join operad. Its endpoint suboperads recover Abramyan--Panov substitution and Ayzenberg composition. Neither endpoint operad nor either relative join operad is finitely generated. The suboperad with nonvoid lower complex arises as the nonempty power-set quotient of the subset-inclusion operad. When tensoring preserves colimits of nonempty finite diagrams, the suboperad acts on arrows by polyhedral colimits up to coherent natural isomorphism. The action induces a set-operad algebra on isomorphism classes of arrows. The dual construction produces Stanley--Reisner quotient arrows, and the colimit action refines Eldridge's loop-space decomposition to arrows. PL ball--boundary pairs form a suboperad of the relative join operad, and minimal interior faces give an operad morphism. Principal pairs have odd-dimensional spherical moment-angle homotopy fibers and yield a closure result for Eldridge's loop-space class.

arXiv comment24 pages. Substantially revised and retitled. Develops the relative join operad and its action on polyhedral-product arrows, with applications to Stanley-Reisner quotient maps, PL ball-boundary pairs, and a relative refinement of Eldridge's loop decomposition