Let
be the space of smooth embeddings from the circle to a closed manifold
.
We introduce a new spectral sequence converging to
for a simply connected
closed manifold
of
dimension
or more, which
has an explicit
–page
and a computable
–page.
As applications, we compute some part of the cohomology for
with some conditions
on the dimensions
and
, and prove
that the inclusion
to the immersions induces an isomorphism on
for some simply
connected
–manifolds.
This gives a restriction on a question posed by Arone and Szymik. The idea to construct
the spectral sequence is to combine a version of Sinha’s cosimplicial model for the knot
space and a spectral sequence for a configuration space by Bendersky and Gitler. The
cosimplicial model consists of configuration spaces of points (with a tangent vector)
in
.
We use Atiyah duality to transfer the structure maps on the configuration
spaces to maps on Thom spectra of the quotient of a direct product of
by
the fat diagonal. This transferred structure is the key to defining our spectral
sequence, and is also used to show that Sinha’s model can be resolved into simpler
pieces in a stable category.