Algebra Structures on the Twisted Eilenberg–Zilber Theorem

Abstract

Let G, G′, and G ×τ G′ be three simplicial groups (not necessarily abelian) and C N (G) ⊗ t C N (G′) be the “twisted” tensor product associated to C N (G ×τ G′) by the twisted Eilenberg–Zilber theorem. Here we prove that the pair (C N (G) ⊗ t C N (G′), μ) is a DGA-algebra where μ is the standard product of C N (G) ⊗ C N (G′). Furthermore, the injection of the twisted Eilenberg–Zilber contraction is a DGA-algebra morphism and the projection and the homotopy operator satisfy other weaker multiplicative properties.