[Paper Review] Rigidity of Fibering

Igor Rivin|arXiv (Cornell University)|Jun 22, 2011

Geometric and Algebraic Topology30 references3 citations

TL;DR

This paper investigates the rigidity of fibrations on Kähler manifolds and 3-manifolds by analyzing group-theoretic obstructions via fundamental groups and monodromy representations. It establishes that under strong geometric and arithmetic conditions—such as finite outer automorphism groups or property T—extensions of groups corresponding to fibrations are virtually direct products, leading to strong finiteness and rigidity results for multiple fibrations.

ABSTRACT

Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -- this could be described as Seifert fibering)There are group-theoretic obstructions to the existence of even one fibering, and in some cases (such as Kahler manifolds or three-dimensional manifolds) the question reduces to a group-theoretic question. In this note we summarize the author's state of knowledge of the subject.

Motivation & Objective

• To understand the number of possible fibrations a Kähler manifold or 3-manifold can admit, framed as group extensions with prescribed kernel and base groups.
• To identify group-theoretic obstructions—such as property T or finite outer automorphism groups—that limit or prevent fibering.
• To analyze the role of monodromy representations in classifying fibrations and determining when extensions are virtually direct products.
• To establish finiteness and rigidity results for fibrations under geometric and arithmetic constraints on the fundamental group.

Proposed method

• Use of the Stallings fibration theorem and Beauville-Siu theorem as foundational analogs in the complex and geometric categories.
• Application of cohomological dimension arguments to rule out fibrations when the fundamental group has dimension ≤2.
• Employment of monodromy representations into Out(N) to classify extensions of the form 1→N→G→B→1.
• Leveraging results from geometric group theory—such as finite images of lattices in semisimple groups in Out(F_k) or Out(Surface group)—to deduce virtual direct product structure.
• Utilization of signature multiplicativity theorems (e.g., Chern-Hirzebruch-Serre) to constrain fibrations with trivial or amenable monodromy.
• Application of Mostow rigidity and finite Out(N) theorems to show that extensions with hyperbolic or arithmetic fundamental groups are virtually direct products.

Experimental results

Research questions

• RQ1 Under what group-theoretic conditions does a Kähler manifold admit multiple fibrations over orbifolds of negative Euler characteristic?
• RQ2 When is a group extension 1→N→G→B→1 with N a Kähler group and B a hyperbolic orbifold group necessarily a virtual direct product?
• RQ3 How do arithmetic and geometric constraints (e.g., finite Out(N), property T, or rank of semisimple Lie groups) affect the number of possible fibrations?
• RQ4 What role does monodromy play in determining the structure of fibrations, especially when it acts trivially or with finite image?
• RQ5 Can signature multiplicativity theorems be used to rule out certain fibrations, particularly when monodromy is amenable or finite?

Key findings

• If a Kähler manifold has fundamental group of cohomological dimension ≤2, it cannot fiber over a hyperbolic orbifold, due to group cohomology obstructions.
• Extensions with non-elementary Fuchsian kernel and base groups are subject to arithmetic obstructions that limit the number of possible fibrations based on rank constraints.
• When the fiber group N is solvable (e.g., abelian), strong rigidity results follow from recent advances in geometric group theory, implying that such fibrations are highly constrained.
• For lattices B in semisimple Lie groups of real rank >1, monodromy representations into Out(N) are finite when N is a free group, surface group, or centerless hyperbolic group, leading to virtual direct product structure.
• If N is a closed hyperbolic 3-manifold group or a centerless lattice in a semisimple group of rank ≥2, then Out(N) is finite, implying that any extension is a virtual direct product.
• When monodromy is finite or amenable, the signature of the total space of a surface bundle over a surface vanishes, providing a topological obstruction to certain fibrations.