Motivic cell structures

Dugger, Daniel¹ ; Isaksen, Daniel C²

¹Department of Mathematics, University of Oregon, Eugene OR 97403, USA
²Department of Mathematics, Wayne State University, Detroit MI 48202, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 615-652

Résumé

An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K–theory and algebraic cobordism spectra are both cellular, and prove some Künneth theorems for cellular objects.

Détail

DOI : 10.2140/agt.2005.5.615

Keywords: motivic cell structure, homotopy theory, celllular object

Bibliographie

[1] , Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics 270, Springer (1972)

[2] J F Adams, Lectures on generalised cohomology, from: "Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three)", Springer (1969) 1

[3] B A Blander, Local projective model structures on simplicial presheaves, $K$–Theory 24 (2001) 283

[4] J M Boardman, Conditionally convergent spectral sequences, from: "Homotopy invariant algebraic structures (Baltimore, MD, 1998)", Contemp. Math. 239, Amer. Math. Soc. (1999) 49

[5] A K Bousfield, E M Friedlander, Homotopy theory of $\Gamma$–spaces, spectra, and bisimplicial sets, from: "Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II", Lecture Notes in Math. 658, Springer (1978) 80

[6] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972)

[7] M Bökstedt, A Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993) 209

[8] P Deligne, Poids dans la cohomologie des variétés algébriques, from: "Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1", Canad. Math. Congress, Montreal, Que. (1975) 79

[9] E D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer (1996)

[10] D Dugger, Universal homotopy theories, Adv. Math. 164 (2001) 144

[11] D Dugger, S Hollander, D C Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004) 9

[12] W G Dwyer, J P C Greenlees, S Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357

[13] W G Dwyer, J Spaliński, Homotopy theories and model categories, from: "Handbook of algebraic topology", North-Holland (1995) 73

[14] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, American Mathematical Society (1997)

[15] W Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press (1993)

[16] W Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Springer (1998)

[17] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)

[18] M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999)

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

[20] P Hu, I Kriz, Some remarks on Real and algebraic cobordism, $K$–Theory 22 (2001) 335

[21] D C Isaksen, Flasque model structures for simplicial presheaves, $K$–Theory 36 (2005)

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

[23] R Joshua, Algebraic $K$–theory and higher Chow groups of linear varieties, Math. Proc. Cambridge Philos. Soc. 130 (2001) 37

[24] B Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. $(4)$ 27 (1994) 63

[25] F Morel, V Voevodsky, $\mathbb{A}^1$–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999)

[26] S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. $(3)$ 80 (2000) 491

[27] B Totaro, Chow groups, Chow cohomology and linear varieties, preprint (1995)

[28] V Voevodsky, Voevodsky's Seattle lectures: $K$–theory and motivic cohomology, from: "Algebraic $K$–theory (Seattle, WA, 1997)", Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 283