The Adams–Novikov spectral sequence and Voevodsky’s slice tower

Levine, Marc¹

¹ Fakultät Mathematik, Campus Essen, Universität Duisburg-Essen, 45117 Essen, Germany

Geometry & topology, Tome 19 (2015) no. 5, pp. 2691-2740.

Résumé

We show that the spectral sequence induced by the Betti realization of the slice tower for the motivic sphere spectrum agrees with the Adams–Novikov spectral sequence, after a suitable reindexing. The proof relies on a partial extension of Deligne’s décalage construction to the Tot–tower of a cosimplicial spectrum.

Détail

DOI : 10.2140/gt.2015.19.2691

Classification : 14F42, 55T15, 55P42
Keywords: Morel–Voevodsky stable homotopy category, slice tower, Adams–Novikov spectral sequence

Bibliographie

[1] J F Adams, Stable homotopy and generalised homology, University of Chicago Press (1974)

[2] J Ayoub, Note sur les opérations de Grothendieck et la réalisation de Betti, J. Inst. Math. Jussieu 9 (2010) 225 | DOI

[3] C Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010) 245

[4] A K Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geom. Topol. 7 (2003) 1001 | DOI

[5] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972)

[6] P Deligne, Théorie de Hodge, II, Inst. Hautes Études Sci. Publ. Math. (1971) 5

[7] D Dugger, D C Isaksen, Motivic cell structures, Algebr. Geom. Topol. 5 (2005) 615 | DOI

[8] J Heller, K Ormsby, Galois equivariance and stable motivic homotopy theory,

[9] P S Hirschhorn, Model categories and their localizations, 99, Amer. Math. Soc. (2003)

[10] J Hornbostel, Localizations in motivic homotopy theory, Math. Proc. Cambridge Philos. Soc. 140 (2006) 95 | DOI

[11] M Hovey, Model categories, 63, Amer. Math. Soc. (1999)

[12] M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63 | DOI

[13] M Hoyois, From algebraic cobordism to motivic cohomology, J. Reine Angew. Math. 702 (2015) 173 | DOI

[14] D C Isaksen, Etale realization on the A1–homotopy theory of schemes, Adv. Math. 184 (2004) 37 | DOI

[15] J F Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987) 35 | DOI

[16] J F Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445

[17] T Lawson, Notes on motivic homotopy theory, Lecture notes from a seminar given by Mike Hopkins at Harvard University (2004)

[18] M Levine, The slice filtration and Grothendieck–Witt groups, Pure Appl. Math. Q. 7 (2011) 1543 | DOI

[19] M Levine, Convergence of Voevodsky’s slice tower, Doc. Math. 18 (2013) 907

[20] M Levine, A comparison of motivic and classical stable homotopy theories, J. Topol. 7 (2014) 327 | DOI

[21] F Morel, An introduction to A1–homotopy theory, from: "Contemporary developments in algebraic –theory" (editors M Karoubi, A O Kuku, C Pedrini), ICTP Lect. Notes 15, Abdus Salam Int. Cent. Theoret. Phys. (2004) 357

[22] F Morel, The stable A1–connectivity theorems, –Th. 35 (2005) 1 | DOI

[23] N Naumann, M Spitzweck, P A Østvær, Motivic Landweber exactness, Doc. Math. 14 (2009) 551

[24] I Panin, K Pimenov, O Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology, Homotopy Appl. 10 (2008) 211

[25] I Panin, K Pimenov, O Röndigs, On Voevodsky’s algebraic K–theory spectrum, from: "Algebraic topology" (editors N A Baas, E M Friedlander, B Jahren, P A Østvær), Abel Symp. 4, Springer (2009) 279 | DOI

[26] P Pelaez, Multiplicative properties of the slice filtration, 335, Soc. Math. France (2011)

[27] P Pelaez, The unstable slice filtration, Trans. Amer. Math. Soc. 366 (2014) 5991 | DOI

[28] J Riou, Catégorie homotopique stable d’un site suspendu avec intervalle, Bull. Soc. Math. France 135 (2007) 495

[29] D P Sinha, Operads and knot spaces, J. Amer. Math. Soc. 19 (2006) 461 | DOI

[30] D P Sinha, The topology of spaces of knots: Cosimplicial models, Amer. J. Math. 131 (2009) 945 | DOI

[31] M Spitzweck, Slices of motivic Landweber spectra, J. –Theory 9 (2012) 103 | DOI

[32] G Vezzosi, Brown–Peterson spectra in stable A1–homotopy theory, Rend. Sem. Mat. Univ. Padova 106 (2001) 47

[33] V Voevodsky, A1–homotopy theory, from: "Proceedings of the International Congress of Mathematicians, Vol. I", Geronimo (1998) 579

[34] V Voevodsky, A possible new approach to the motivic spectral sequence for algebraic K–theory, from: "Recent progress in homotopy theory" (editors D M Davis, J Morava, G Nishida, W S Wilson, N Yagita), Contemp. Math. 293, Amer. Math. Soc. (2002) 371 | DOI

[35] V Voevodsky, Motivic Eilenberg–Maclane spaces, Publ. Math. Inst. Hautes Études Sci. (2010) 1 | DOI

[36] C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994) | DOI