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