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Journal of Topology Advance Access originally published online on October 28, 2007
Journal of Topology 2008 1(1):217-267; doi:10.1112/jtopol/jtm004
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© 2007 London Mathematical Society

The homotopy coniveau tower

Marc Levine

Department of Mathematics, Northeastern University, Boston, MA 02115, USA marc{at}neu.edu


  Abstract

We examine the ‘homotopy coniveau tower’ for a general cohomology theory on smooth k-schemes and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. In addition, we show that the homotopy coniveau tower agrees with Voevodsky's slice tower for S1-spectra, giving a proof of a connectedness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a tower of functors on the Morel–Voevodsky stable homotopy category, and we identify this P1-stable homotopy coniveau tower with Voevodsky's slice tower for P1-spectra. We also show that the zeroth layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general P1-spectrum the structure of a module over motivic cohomology. This recovers and extends results of Voevodsky on the zeroth layer of the slice filtration, and yields a spectral sequence that is reminiscent of the classical Atiyah–Hirzebruch spectral sequence.

Received January 23, 2007.

2000 Mathematics Subject Classification 14C25, 19E15 (primary), 19E08, 14F42, 55P42 (secondary)

The author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and support of the NSF via grants DMS 0140445 and DMS-0457195.


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