### The homotopy coniveau filtration, by Marc Levine

Following the construction of Friedlander-Suslin, we define the "homotopy
coniveau tower" for a general cohomology theory on smooth k-schemes. Assuming
the cohomology theory satisfies some natural axioms, we show that the
homotopy coniveau tower is homotopy invariant, satisfes a localization
property, and can be made functorial. We also give an interpretation of the
layers in the tower in terms of a generalized version of Bloch's higher Chow
groups. In particular, this gives a new proof that the layers of this tower
for K-theory agree with motivic cohomology, independent of the results of
Bloch-Lichtenbaum or Friedlander-Suslin.
We show how these constructions lead to a tower of functors on the
Morel-Voevodsky motivic stable homotopy category, and identify this stable
homotopy coniveau tower with Voevodsky's slice filtration. We also show that
the 0th layer for the motivic sphere spectrum is the motivic cohomology
spectrum, which gives the layers for a general P^1-spectrum the structure of
a module over motivic cohomology. This recovers and extends recent results of
Voevodsky on the 0th layer of the slice filtration, and yields a spectral
sequence that is reminiscent of the classical Atiyah-Hirzebruch spectral
sequence. For the K-theory P^1-spectrum, one recovers the
Bloch-Lichtenbaum/Friedlander-Suslin spectral sequence.

Marc Levine <marc@neu.edu>