A Cubical Approach to Homotopy Orbits of Circle Actions

For a space X belonging to a class including all double suspensions of finite type, we construct for any field k a cochain complex hos(X) such that H∗ ( hos(X) ) ∼= H∗(LXhS1 ; k), the cohomology of the homotopy orbit space of the natural S-action on the free loop space LX. The model hos∗(X) is “small” and thus lends itself relatively easily to explicit computation. Furthermore hos∗(X) fits into an extension of cochain complexes that models the fibration LX −→ LXhS1 −→ BS . This concrete link with topology proves essential for the construction of the cochain complex for computing spectrum cohomology of topological cyclic homology in [HR]. Introduction For a space X belonging to a class including all double suspensions of finite type, we construct here for any field k a cochain complex hos(X) such that H ( hos(X) ) ∼= H(LXhS1 ;k), the cohomology of the homotopy orbit space of the natural S-action on the free loop space LX, also known as the equivariant or Borel cohomology of the free loop space. The model hos(X) is “small” and thus lends itself relatively easily to explicit computation, as we illustrate in section 2.4. Furthermore, hos(X) fits into a commuting diagram