[Paper Review] Surface Homeomorphisms That Do Not Extend to Any Handlebody and the Johnson Filtration

Jamie Bradley Jorgensen|ArXiv.org|May 28, 2008

Geometric and Algebraic Topology18 references3 citations

TL;DR

This paper proves the existence of surface homeomorphisms on genus 3 or higher surfaces that do not extend to any handlebody, even when lying arbitrarily deep in the Johnson filtration of the mapping class group. Using Lie ring theory and the Johnson homomorphisms, the author constructs robust homeomorphisms in the k-th term of the filtration that fail to extend to handlebodies, demonstrating that algebraic constraints (deep filtration) do not guarantee geometric extendability.

ABSTRACT

We prove the existence of homeomorphisms of a closed, orientable surface of genus 3 or greater that do not extend to any handlebody bounded by the surface. We show that such homeomorphisms exist arbitrarily deep in the Johnson filtration of the mapping class group. The second and third terms of the Johnson filtration are the well-known Torelli group and Johnson subgroup, respectively. Richard Hain has obtained very similar results by different methods.

Motivation & Objective

• To resolve the geometric obstruction of mapping class group elements not extending to handlebodies despite strong algebraic constraints.
• To investigate whether homeomorphisms lying deep in the Johnson filtration can fail to extend to any handlebody.
• To construct explicit examples of such non-extendable homeomorphisms using Lie ring and homomorphism theory.
• To determine the precise genus and filtration depth conditions under which such non-extendable homeomorphisms exist.
• To show that such homeomorphisms exist arbitrarily deep in the Johnson filtration for genus ≥3 surfaces.

Proposed method

• Utilizes the Johnson filtration of the mapping class group, focusing on the k-th term 𝒥(k) and its associated Johnson homomorphisms τ_k.
• Applies the Morita-Heap homomorphisms and Lie ring techniques to analyze the image of τ_k and its real representation.
• Constructs a vector bundle E over a surface to reduce the problem to a dimensional comparison between the image of τ_k and the space of symmetric tensors.
• Employs Levine’s estimate for dim(im(j_* ∘ τ_k)^ℝ) to compare dimensions and establish non-extendability.
• Uses the Witt formula to compute the dimension of free Lie algebras, particularly C(k, m−1), to bound the image size.
• Applies inductive arguments and inequality analysis (e.g., g(g+1)/2 < ∑C(3,m−1)) to verify conditions for non-extendability across genus and filtration depth.

Experimental results

Research questions

• RQ1 Do there exist surface homeomorphisms in the Johnson filtration that do not extend to any handlebody?
• RQ2 Can such non-extendable homeomorphisms be found arbitrarily deep in the Johnson filtration for surfaces of genus ≥3?
• RQ3 What is the relationship between the algebraic depth in the Johnson filtration and the geometric property of handlebody extension?
• RQ4 For which pairs (g,k) does the dimension of the image of τ_k fail to support extension to a handlebody?
• RQ5 Are there robust homeomorphisms in 𝒥(k) − 𝒥(k+1) that do not extend to any handlebody?

Key findings

• There exist homeomorphisms in 𝒥(k) − 𝒥(k+1) that do not extend to any handlebody for genus g ≥ 3 and sufficiently large k, as confirmed by dimension comparisons.
• For k=2 (Torelli group), non-extendable homeomorphisms exist when g ≥ 7, as g(g+1)/2 < (g choose 3).
• For k=3, non-extendable homeomorphisms exist when g ≥ 5, based on the inequality g(g+1)/2 < ∑_{m=3}^g C(3,m−1).
• The construction is robust: if f is a non-extendable homeomorphism in 𝒥(k) − 𝒥(k+1), then so is f^n for all n ≠ 0.
• The result holds in the shaded region of Figure 5.10, which specifies the (g,k) pairs where the dimension inequality ensures non-extendability.
• The paper confirms that such non-extendable homeomorphisms exist arbitrarily deep in the Johnson filtration for genus ≥3 surfaces, resolving a key geometric-algebraic dichotomy.