Braids, mapping class groups, and categorical delooping

Abstract

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map φ : β 2g → Γ g,1 from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of [25]. By extending the homomorphism to a functor of monoidal 2-categories, φ is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology.

Braid groups and discrete diffeomorphisms of the punctured disk

“…Remark 1.9. Unlike the main theorem in [ST07] and [ST08] which says that φ induces a trivial map in the stable homology, we show that Φ induces a nontrivial map…”

“…Birman and Hilden [BH73] proved that this map is injective. The main theorem in [ST07] and [ST08] is that ψ induces the trivial map on the stable homology.…”

Mapping class groups of covers with boundary and braid group embeddings

“…In the 1980s J. Harer conjectured that the induced map on stable homology H * (B ∞ ; Z/2Z) → H * (Γ ∞ ; Z/2Z) is trivial. The conjecture was proved by Song and Tillman in [24,Theorem 1.1]. A stronger version of Harer's conjecture was proved for a large family of non-geometric embeddings of the braid group in [5].…”

“…have since been constructed in the works of Bödigheimer-Tillman [5], Kim-Song [16], Song [23], Song-Tillman [24], and Szepietowski [27].…”

Braid groups, mapping class groups and their homology with twisted coefficients

“…It is natural to study the behaviour in homology of the Birman-Hilden inclusion ϕ : Br 2g+1 → Γ g,1 . Song and Tillmann [15], and later Segal and Tillmann [14], show that the map ϕ * is stably trivial in homology with constant coefficients. More precisely:…”