Boundedness of bundle diffeomorphism groups over a circle

Abstract

In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle 𝜋 : 𝑀 → 𝑆¹ with fiber 𝑁 and structure group 𝛤 and 𝑟 ∈ ℤ _{≥ 0} ∪ { ∞ } we distinguish an integer 𝑘 = 𝑘(𝜋, 𝑟) ∈ ℤ _{≥ 0} and construct a function 𝜈^∧ : Diff^𝑟_𝜋(𝑀)₀ → ℝ_𝑘. When 𝑘 ≥ 1, it is shown that the bundle diffeomorphism group Diff^𝑟_𝜋(𝑀)₀ is bounded and 𝑐𝑙𝑏_𝜋𝑑 Diff^𝑟_𝜋(𝑀)₀ ≤ 𝑘 + 3, if Diff^𝑟_𝜚,𝑐(𝐸)₀ is perfect for the trivial fiber bundle 𝜚 : 𝐸 → ℝ with fiber 𝑁 and structure group 𝛤. On the other hand, when 𝑘 = 0, it is shown that 𝜈^∧ is a unbounded quasimorphism, so that Diff^𝑟_𝜋(𝑀)₀ is unbounded and not uniformly perfect. We also describe the integer 𝑘 in term of the attaching map 𝜑 for a mapping torus 𝜋 : 𝑀_𝜑 → 𝑆¹ and give some explicit examples of (un)bounded groups.