zbMATH — the first resource for mathematics

Motivic cell structures. (English) Zbl 1086.55013
A fundamental problem in motivic homotopy theory is understanding how spaces can be “built” from each other. In particular, how much can be built by spheres only? The cellular spaces in the paper under review are similar to schemes with algebraic cell decompositions or linear varieties, but are better suited for homotopical arguments. More precisely, a motivic space is stably cellular if its associated motivic suspension spectrum is cellular, i.e. up to stable equivalence can be built from motivic spheres by homotopy colimit constructions.
Cellular spectra are very well behaved; for instance, a map of cellular spectra is a stable equivalence if it induces an isomorphism of homotopy groups.
The authors establish that the general linear, Grassmannian and Stiefel varieties are stably cellular, and likewise the algebraic K-theory and cobordism spectra are cellular. They remark that claims by Hopkins and Morel will then imply that the motivic cohomology spectrum itself is cellular.
The paper also contains further examples (showing e.g. that all toric varieties and some quadrics are stably cellular) and Künneth formulas for cellular spectra, as well as a final chapter on the compact generators of the stable motivic homotopy category.

55U35 Abstract and axiomatic homotopy theory in algebraic topology
14F42 Motivic cohomology; motivic homotopy theory
Full Text: DOI EMIS EuDML arXiv
[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, Springer (1969) 1 · Zbl 0193.51702
[3] B A Blander, Local projective model structures on simplicial presheaves, \(K\)-Theory 24 (2001) 283 · Zbl 1073.14517 · doi:10.1023/A:1013302313123
[4] J M Boardman, Conditionally convergent spectral sequences, Contemp. Math. 239, Amer. Math. Soc. (1999) 49 · Zbl 0947.55020
[5] A K Bousfield, E M Friedlander, Homotopy theory of \(\Gamma\)-spaces, spectra, and bisimplicial sets, Lecture Notes in Math. 658, Springer (1978) 80 · Zbl 0405.55021
[6] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972) · Zbl 0259.55004
[7] M Bökstedt, A Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993) 209 · Zbl 0802.18008 · numdam:CM_1993__86_2_209_0 · eudml:90218
[8] P Deligne, Poids dans la cohomologie des variétés algébriques, Canad. Math. Congress, Montreal, Que. (1975) 79 · Zbl 0334.14011
[9] E D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer (1996) · Zbl 0842.55001 · doi:10.1007/BFb0094429
[10] D Dugger, Universal homotopy theories, Adv. Math. 164 (2001) 144 · Zbl 1009.55011 · doi:10.1006/aima.2001.2014
[11] D Dugger, S Hollander, D C Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004) 9 · Zbl 1045.55007 · doi:10.1017/S0305004103007175
[12] W G Dwyer, J P C Greenlees, S Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357 · Zbl 1155.55302 · doi:10.1016/j.aim.2005.11.004
[13] W G Dwyer, J Spaliński, Homotopy theories and model categories, North-Holland (1995) 73 · Zbl 0869.55018
[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) · Zbl 0894.55001
[15] W Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press (1993) · Zbl 0813.14039
[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) · Zbl 0885.14002
[17] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003) · Zbl 1017.55001
[18] M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999) · Zbl 0909.55001
[19] M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63 · Zbl 1008.55006 · doi:10.1016/S0022-4049(00)00172-9
[20] P Hu, I Kriz, Some remarks on Real and algebraic cobordism, \(K\)-Theory 22 (2001) 335 · Zbl 1032.55003 · doi:10.1023/A:1011196901303
[21] D C Isaksen, Flasque model structures for simplicial presheaves, \(K\)-Theory 36 (2005) · Zbl 1116.18008 · doi:10.1007/s10977-006-7113-z
[22] J F Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000) 445 · Zbl 0969.19004 · emis:journals/DMJDMV/vol-05/15.html · eudml:121235
[23] R Joshua, Algebraic \(K\)-theory and higher Chow groups of linear varieties, Math. Proc. Cambridge Philos. Soc. 130 (2001) 37 · Zbl 0984.19002 · doi:10.1017/S030500410000476X
[24] B Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. \((4)\) 27 (1994) 63 · Zbl 0799.18007 · numdam:ASENS_1994_4_27_1_63_0 · eudml:82359
[25] F Morel, V Voevodsky, \(\mathbbA^1\)-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) · Zbl 0983.14007 · doi:10.1007/BF02698831 · numdam:PMIHES_1999__90__45_0 · eudml:104163
[26] S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. \((3)\) 80 (2000) 491 · Zbl 1026.18004 · doi:10.1112/S002461150001220X
[27] B Totaro, Chow groups, Chow cohomology and linear varieties, preprint (1995) · Zbl 1329.14018
[28] V Voevodsky, Voevodsky’s Seattle lectures: \(K\)-theory and motivic cohomology, Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 283 · Zbl 0941.19001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.