The Braid Skein Algebra of the Heisenberg Manifold

Abstract

We define and study the space C_n(ℍ) of configurations of n points in the Heiseberg manifold ℍ, and other closely related spaces: SC_n(ℍ) (special configurations), C^g_n (ℍ) and SC^g _n (ℍ) (under gauge equivalence). These spaces are defined by removing a codimension 2 (singular) locus from the relevant n-fold Cartesian product. We give explicit presentations of their fundamental groups (Theorem 6.2.5, Theorem 7.1.14, Theorem 6.3.5, Theorem 7.1.7), showing in particular that the fundamental group of C^g_n (ℍ) is isomorphic to 𝔸 ⁿ , the double affine Artin group of type GL_n (Corollary 6.3.6), and the fundamental group of SC^g _n (ℍ) is isomorphic to An, the double affine Artin group of type PSL_n (Theorem 7.1.7).

The n-braid skein algebras of the Heisenberg manifold are obtained by imposing the HOMFLY skein relations on the relevant fundamental group. We show that BSk^g _n (ℍ) (for configurations under gauge equivalence) is isomorphic to ℍⁿ _q,t, the double affine Hecke algebra of type GL_n, and SBSk^g_n (ℍ) (for special configurations under gauge equivalence) is isomorphic to ℍ ⁿ _q,t, the double affine Hecke algebra of type PSL_n (Corollary 7.3.1). Furthermore, BSk_n(ℍ) is isomorphic to BSk^g _n (ℍ), and SBSk_n(ℍ) is isomorphic to SBSk^g _n (ℍ) (Corollary 7.3.2). We show that the Morton-Samuelson braid skein algebra is isomorphic to BSk^g _n (ℍ₂) and this elucidates the topological meaning of the MS-skein relation (Figure 26).

One result that emerges from our considerations is the proof of the K(π, 1)-conjecture (Arnol’d, Brieskorn, Pham, Thom, and Van der Lek [34, Chapter IV, Conjecture 1.3] for extended Coxeter arrangements) for the double affine hyperplane arrangement of type An. This is the first confirmation of K(π, 1)-conjecture for a non-central arrangement which is not of Coxeter type. We also establish that all of the considered configuration spaces for the Heisenberg manifold are K(π, 1).

We construct configuration spaces of ℍ_r the r-Heisenberg manifold. We determine each of the configuration spaces C_n(ℍ_r), C^g_n (ℍ_r), SC_n(ℍ_r), and SC^g _n (ℍ_r) are K(π, 1). We show SC^g ₂ (ℍ_r) is homeomorphic to the Lens space L(r, 1) minus four fibers (Theorem 9.2.1).