Infinite-dimensional manifold triples of homeomorphism groups

In this paper we give some characterizations of (s, Σ, σ)-, (s2, s × σ, σ2)- and (s∞, σ∞, σf∞)-manifold triples under the stability condition. As an application we show that if M is a compact PL n-manifold (n ⩾ 1, n ≠ 4 and ∂M = θ for n = 5) and H(M) is an ANR, then (H(M)∗, HLIP(M)∗, HPL(M)) is an (s, Σ, σ)-manifold triple, where H(M) (HLIP(M) or HPL(M)) is the group of (Lipschitz or PL) homeomorphisms of M with the compact-open topology and H(M)∗ (H(LIP)(M)∗) is the subgroup consisting of (Lipschitz) homeomorphisms approximated by PL-homeomorphisms. We also show that a triple of homeomorphism groups of the real line is an (s∞, σ∞, σf∞)-manifold triple.