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The homotopy limit problem for Hermitian \(K\)-theory, equivariant motivic homotopy theory and motivic real cobordism. (English) Zbl 1247.14020
The homotopy limit problem originally posed by Thomason and studied in this paper, asks how close the canonical map from Hermitian \(K\)-theory to the \(\mathbb Z/2\)-homotopy fixed points of algebraic \(K\)-theory is to being an isomorphism, at least after \(2\)-completion. The main results solve this problem for all fields of characteristic \(0\) which satisfy a certain cohomological condition.
The approach taken involves applying a significant amount of equivariant motivic stable homotopy theory, which should be of wider interest.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19G38 Hermitian \(K\)-theory, relations with \(K\)-theory of rings
55P91 Equivariant homotopy theory in algebraic topology
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[1] Atiyah, M.F., K-theory and reality, Quart. J. math. Oxford ser. (2), 17, 367-386, (1966) · Zbl 0146.19101
[2] Balmer, P., Triangular Witt groups II, Math. Z., 236, 351-382, (2001) · Zbl 1004.18010
[3] Behrens, M.; Hopkins, M.J., Higher real K-theories and topological automorphic forms · Zbl 1254.55001
[4] Behrens, M.; Lawson, T., Topological automorphic forms · Zbl 1210.55005
[5] Berrick, A.J.; Karoubi, M., Hermitian K-theory of the integers, Amer. J. math., 127, 4, 785-823, (2005) · Zbl 1080.19006
[6] A.J. Berrick, M. Karoubi, P.A. Østvær, M. Schlichting, in preparation.
[7] A.J. Berrick, M. Karoubi, P.A. Østvær, Hermitian K-theory of totally Real 2-regular number fields, UIUC K-theory archive, No. 927, 2009.
[8] Bousfield, A.K., The localization of spectra with respect to homology, Topology, 18, 257-281, (1979) · Zbl 0417.55007
[9] Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander, The algebraic and geometric theory of quadratic forms, Amer. math. soc. colloq. publ., vol. 56, (2008), Amer. Math. Soc. Providence, RI · Zbl 1165.11042
[10] Elmendorf, A.D.; Kriz, I.; Mandell, M.A.; May, J.P., Rings, modules and algebras in stable homotopy theory, Math. surveys monogr., vol. 47, (1997), Amer. Math. Soc. Providence, RI · Zbl 0894.55001
[11] Gille, S., On Witt groups with support, Math. ann., 322, 103-137, (2002) · Zbl 1010.19003
[12] Greenlees, J.P.C.; May, J.P., Generalized Tate cohomology, Mem. amer. math. soc., 113, 543, (1995), viii+178 · Zbl 0876.55003
[13] Hill, M.A.; Hopkins, M.J.; Ravenel, D.C., On the non-existence of Kervaire invariant one · Zbl 1366.55007
[14] Hirschhorn, Philip S., Model categories and their localizations, Math. surveys monogr., vol. 99, (2003), Amer. Math. Soc. Providence, RI · Zbl 1017.55001
[15] Hornbostel, Jens, \(A^1\)-representability of Hermitian K-theory and Witt groups, Topology, 44, 3, 661-687, (2005) · Zbl 1078.19004
[16] Hornbostel, Jens; Schlichting, Marco, Localization in Hermitian K-theory of rings, J. lond. math. soc. (2), 70, 1, 77-124, (2004) · Zbl 1061.19003
[17] Hu, P.; Kriz, I., Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology, 40, 317-399, (2001) · Zbl 0967.55010
[18] Hu, P.; Kriz, I., Some remarks on real and algebraic cobordism, K-theory, 22, 335-366, (2001) · Zbl 1032.55003
[19] P. Hu, I. Kriz, K. Ormsby, Remarks on motivic homotopy theory over algebraically closed fields, J. K-Theory, doi:10.1017/is010001012jkt098. · Zbl 1248.14026
[20] P. Hu, I. Kriz, K. Ormbsy, Convergence of the motivic Adams spectral sequence, J. K-Theory, doi:10.1017/is011003012jkt150, in press. · Zbl 1309.14018
[21] Jardine, J.F., Simplicial presheaves, J. pure appl. algebra, 47, 35-87, (1987) · Zbl 0624.18007
[22] Jardine, J.F., Motivic symmetric spectra, Doc. math., 5, (2000) · Zbl 0969.19004
[23] Jouanolou, J.-P., Une suite de Mayer-Vietoris in K-théorie algébrique, (), 293-316 · Zbl 0291.14006
[24] Karoubi, M., Periodicite de la K-theorie hermitienne, (), 301-411 · Zbl 0274.18016
[25] Karoubi, Max, Localisation de formes quadratiques II, Ann. sci. ec. norm. super. (4), 8, 99-155, (1975) · Zbl 0345.18007
[26] Karoubi, Max, Le théorème fondamental de la K-théorie hermitienne, Ann. of math. (2), 112, 2, 259-282, (1980) · Zbl 0483.18008
[27] Kobal, Damjan, K-theory, Hermitian K-theory and the karoubi tower, K-theory, 17, 2, 113-140, (1999) · Zbl 0931.19003
[28] Lam, T.Y., Introduction to quadratic forms over fields, Grad. stud. math., vol. 67, (2005), Amer. Math. Soc. Providence, RI · Zbl 1068.11023
[29] Landweber, P.S., Conjugations on complex manifolds and equivariant homotopy of MU, Bull. amer. math. soc., 74, 271-274, (1968) · Zbl 0181.26801
[30] Levine, M., Chowʼs moving lemma and the homotopy coniveau tower, K-theory, 37, 129-209, (2006) · Zbl 1117.19003
[31] Levine, M., The homotopy coniveau tower, J. topol., 1, 217-267, (2008) · Zbl 1154.14005
[32] Lewis, L.G.; May, J.P.; Steinberger, M., Equivariant stable homotopy theory, Lecture notes in math., vol. 1213, (1986), Springer-Verlag Berlin · Zbl 0611.55001
[33] Luna, Domingo, Slices étales, (), 81-105 · Zbl 0286.14014
[34] Mandell, M., Equivariant symmetric spectra, (), 399-452 · Zbl 1074.55003
[35] May, J.P., The geometry of iterated loop spaces, Lecture notes in math., vol. 271, (1972), Springer-Verlag · Zbl 0244.55009
[36] May, J.P., Multiplicative infinite loop space theory, J. pure appl. algebra, 26, 1-69, (1982) · Zbl 0532.55013
[37] May, J.P.; Thomason, R., The uniqueness of infinite loop space machines, Topology, 17, 205-224, (1978) · Zbl 0391.55007
[38] Morel, Fabien; Voevodsky, Vladimir, \(\mathbf{A}^1\)-homotopy theory of schemes, Publ. math. inst. hautes études sci., 90, 45-143, (1999) · Zbl 0983.14007
[39] Orlov, D.; Vishik, A.; Voevodsky, V., An exact sequence for \(K_\ast^M / 2\) with applications to quadratic forms, Ann. of math. (2), 165, 1, 113, (2007)
[40] Quebbemann, H.G.; Scharlau, W.; Schulte, M., Quadratic and Hermitian forms in additive and abelian categories, J. algebra, 59, 264-289, (1979) · Zbl 0412.18016
[41] Schlichting, M., Hermitian K-theory of exact categories, J. K-theory, 5, 105-165, (2010) · Zbl 1328.19009
[42] Schlichting, M., The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. math., 179, 349-433, (2010) · Zbl 1193.19005
[43] Schlichting, M., The homotopy limit problem and (etale) Hermitian K-theory
[44] Thomason, R.W., The homotopy limit problem, () · Zbl 0528.55008
[45] Vladimir Voevodsky, The Milnor conjecture, UIUC K-theory archive, no. 170, 1996.
[46] Vladimir Voevodsky, Lectures on motivic cohomology 2000/2001 (written by Pierre Deligne), UIUC K-theory archive, No. 527, 2001. · Zbl 1005.19001
[47] Voevodsky, V., Open problems in the motivic stable homotopy theory I, (), 3-34 · Zbl 1047.14012
[48] Voevodsky, V., Motivic cohomology with \(\mathbf{Z} / 2\)-coefficients, Publ. math. inst. hautes études sci., 98, (2003)
[49] Weibel, C., Homotopy algebraic K-theory, Contemp. math., 83, 461-488, (1989)
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