On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups
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On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups Solomon Jekel1 · Rita Jiménez Rolland2 Received: 21 February 2020 / Revised: 20 July 2020 / Accepted: 24 August 2020 © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
Abstract The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this largely expository note, we determine the non-vanishing behavior of the powers of this class. Our argument relies on restricting the cohomology classes to torsion subgroups of the mapping class group. Keywords Mapping class group · Cohomology · Euler class · Torsion
1 Introduction Let gk denote the pure mapping class group of a closed orientable surface g of genus g ≥ 1 with k ≥ 0 marked points. Homological properties of the mapping class group of surfaces of finite type have been studied for the last 40 years. For instance, cohomology classes of mapping class groups correspond to characteristic classes of surface bundles. Furthermore, for surfaces of genus g ≥ 2, the rational cohomology of gk coincides with the cohomology of the moduli space Mg,k of Riemann surfaces of genus g with k marked points.
Rita Jiménez Rolland is grateful for the financial support from PAPIIT DGAPA-UNAM grant IA104010 and from CONACYT grant CB-2017-2018-A1-S-30345-F-3125.
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Rita Jiménez Rolland [email protected] Solomon Jekel [email protected]
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Mathematics Department, Northeastern University, Boston, MA 02115, USA
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Instituto de Matemáticas, Universidad Nacional Autónoma de México, Oaxaca de Juárez, Oaxaca 68000, Mexico
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S. Jekel, R. Jiménez Rolland
Some of the first homological calculations for mapping class groups are due to Harer. He computed H2 (gk ; Z) for genus g ≥ 5 in [9] and proved a remarkable homological stability theorem in [10], which was a key result in the proof of Mumford’s conjecture for H ∗ (Mg ; Q) by Madsen and Weiss [19]. Miller [21] and Morita [23] constructed non-trivial cohomology classes in H ∗ (gk ; Q), while Glover and Mislin [6] used torsion subgroups of the mapping class groups to detect torsion in their cohomology. In the same spirit, we use torsion elements in the mapping class group g1 of a surface of genus g ≥ 1 with one marked point to show the non-vanishing of some classes in H ∗ (g1 ; Z). For g ≥ 2, Nielsen defined a faithful action of g1 on the circle S1 which identifies g1 with a subgroup of the group Homeo+ (S1 ) of orientation-preserving homeomorphisms of the circle (see, for example, [8] and Sect. 4). This monomorphism ρ : g1 → Homeo+ (S1 ) pulls back the discrete universal Euler class E and its powers En to g1 producing classes ρ ∗ (En ) =: En ∈ H 2n (g1 ; Z) for each n ≥ 1. As we review in Sect. 2, the nth cup product powers En
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