In this article we study the Poisson algebra structure on the homology
of the totalization of a fibrant cosimplicial space associated with an
operad with multiplication. This structure is given as the Browder
operation induced by the action of little disks operad, which was
found by McClure and Smith. We show that the Browder operation
coincides with the Gerstenhaber bracket on the Hochschild homology,
which appears as the E2-term of the homology spectral sequence
constructed by Bousfield. In particular we consider a variant of the
space of long knots in higher dimensional Euclidean space, and show
that Sinha's homology spectral sequence computes the Poisson algebra
structure of the homology of the space. The Browder operation
produces a homology class which does not directly correspond to chord
diagrams.
Keywords
space of long knots, little disks operad,
Browder operation, Gerstenhaber bracket, McClure-Smith
machinery
