Title: Operads and knot spaces
Author: Dev Sinha
E-mail: dps@math.uoregon.edu
Abstract: Let F_m be the space of knotted intervals in I^m equipped with a
trivialization through immersions. We show that the totalization
of the Kontsevich operad provides a model for the embedding calculus
tower for F_m. Combined with results of Goodwillie-Klein-Weiss
and Volic, this resolves Kontsevich's conjecture of existence of such
a model which captures the homotopy type of F_m when m>3 and which
classifies finite-type framed knot invariants when m=3.
We carefully develop the Kontsevich operad, which is closely related
to the Fulton-MacPherson operad and weakly equivalent to the little
cubes operad. In doing so we show that the standard simplicial model
for the two-sphere carries an operad structure in the opposite category
of pointed sets. We apply the well-developed machinery of McClure and
Smith on operads with multiplication to deduce that our model has a
little two-cubes action.
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