[Paper Review] First cohomology of pure mapping class groups of big genus one and zero surfaces

George Domat, Paul Plummer|arXiv (Cornell University)|Apr 23, 2019

Geometric and Algebraic Topology9 references4 citations

TL;DR

This paper proves that the first integral cohomology of the pure mapping class group of an infinite-type genus one surface is trivial, while for genus zero surfaces, it constructs an uncountable family of homomorphisms to ℤ that do not factor through finite-type punctured spheres—demonstrating that not all such cohomology classes arise from finite-type subsurfaces.

ABSTRACT

We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.

Motivation & Objective

• To determine the structure of the first integral cohomology of pure mapping class groups of infinite-type surfaces of genus one and zero.
• To investigate whether all homomorphisms from these groups to ℤ factor through finite-type subsurfaces, particularly spheres with finitely many punctures.
• To construct explicit examples of cohomology classes in the genus zero case that do not arise from finite-type structures.
• To complete the picture of first cohomology for big pure mapping class groups by resolving the missing case of genus zero surfaces.

Proposed method

• Uses the Gervais star presentation to analyze the homology of finite-type exhaustion sequences of genus one surfaces.
• Applies continuity results from [APV17] and [D61] to show that non-trivial homomorphisms to ℤ must be continuous, leading to a contradiction via divergent image sequences.
• Constructs a specific homomorphism φ on the flute surface by assigning non-zero values to Dehn twists about specific curves γ_ai and γ_bi, with opposite signs.
• Shows that the homomorphism extends continuously to the full pure mapping class group using Lemma 4.1, which characterizes elements in the closure as infinite products of Dehn twists.
• Demonstrates that the constructed homomorphism does not factor through any forgetful map to a finite-type surface, since removing any puncture would trivialize the image of key Dehn twists.
• Generalizes the construction to uncountably many such homomorphisms by varying the set of curves assigned non-zero values, indexed by infinite-codimensional subsets of ℕ.

Experimental results

Research questions

• RQ1 Is the first integral cohomology of the pure mapping class group of an infinite-type genus one surface trivial?
• RQ2 Do all homomorphisms from the pure mapping class group of an infinite-type genus zero surface to ℤ factor through a finite-type punctured sphere?
• RQ3 Can one construct explicit, non-trivial cohomology classes in the genus zero case that are not supported on finite-type subsurfaces?
• RQ4 What is the structure of the first cohomology group of big pure mapping class groups when the surface has genus zero and infinitely many punctures?
• RQ5 How do the cohomology classes of infinite-type surfaces relate to those of their finite-type subsurfaces and quotients?

Key findings

• The first integral cohomology of the pure mapping class group of any infinite-type genus one surface is trivial: H¹(PMCG(S);ℤ) = 0.
• For genus zero surfaces, there exists a homomorphism from PMCG(S) to ℤ that does not factor through any forgetful map to a finite-type sphere.
• An uncountable family of such non-factoring homomorphisms exists, indexed by infinite-codimensional subsets of the positive integers.
• The constructed homomorphism is continuous and well-defined on the full pure mapping class group, extending from compactly supported elements via infinite products of Dehn twists.
• The key obstruction to factoring through finite-type surfaces is that removing any puncture would make certain Dehn twists trivial, but their images under the homomorphism are non-zero.
• The results complete the classification of first cohomology for big pure mapping class groups, distinguishing genus zero as the only case with non-trivial, non-finite-type-supported cohomology.