Toward the group completion of the Burau representation

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets the plane, interpret Burau representation knot theory as quantum field defined on it. Its determinant writhe is an invertible braided TQFT which group completes to define Hopkins-Mahowald model for integral homology E 2 E_2 Thom spectrum. We use these ideas infinite cyclic (Alexander) cover alttext="double-struck upper C"> C encoding="application/x-tex">\mathbb {C} , argue that by closely related SU(2)-valued Wess-Zumino-Witten string R Subscript plus Superscript 3"> mathvariant="double-struck">R + 3 {R}^3_+ .