Geometry & Topology Volume 20, issue 2 (2016)

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Ethics Statement

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author Index

To Appear

Other MSP Journals

Slices of hermitian $K$–theory and Milnor's conjecture on quadratic forms

Oliver Röndigs and Paul Arne Østvær

Geometry & Topology 20 (2016) 1157–1212

DOI: 10.2140/gt.2016.20.1157

Bibliography

1	J K Arason, A Pfister, Beweis des Krullschen Durchschnittsatzes für den Wittring, Invent. Math. 12 (1971) 173 MR0294251
2	R Baeza, Über die Torsion der Witt-Gruppe W_q(A) eines semi-lokalen Ringes, Math. Ann. 207 (1974) 121 MR0327759
3	P Balmer, Witt groups, from: "Handbook of K–theory, Vol. 2" (editors E M Friedlander, D R Grayson), Springer (2005) 539 MR2181829
4	H Bass, John Milnor, the algebraist, from: "Topological methods in modern mathematics" (editors L R Goldberg, A V Phillips), Publish or Perish (1993) 45 MR1215959
5	A J Berrick, M Karoubi, Hermitian K–theory of the integers, Amer. J. Math. 127 (2005) 785 MR2154370
6	A J Berrick, M Karoubi, M Schlichting, P A Østvær, The homotopy fixed point theorem and the Quillen–Lichtenbaum conjecture in Hermitian K–theory, Adv. Math. 278 (2015) 34 MR3341783
7	S Bloch, S Lichtenbaum, A spectral sequence for motivic cohomology, (1995)
8	D Dugger, D C Isaksen, Motivic cell structures, Algebr. Geom. Topol. 5 (2005) 615 MR2153114
9	B I Dundas, O Röndigs, P A Østvær, Motivic functors, Doc. Math. 8 (2003) 489 MR2029171
10	E M Friedlander, Computations of K–theories of finite fields, Topology 15 (1976) 87 MR0394660
11	E M Friedlander, Motivic complexes of Suslin and Voevodsky, from: "Séminaire Bourbaki 1996/97", Astérisque 245, Soc. Math. France (1997) 355 MR1627118
12	D R Grayson, The motivic spectral sequence, from: "Handbook of K–theory, Volume 1" (editors E M Friedlander, D R Grayson), Springer (2005) 39 MR2181820
13	J J Gutiérrez, O Röndigs, M Spitzweck, P A Østvær, Motivic slices and coloured operads, J. Topol. 5 (2012) 727 MR2971612
14	R T Hoobler, The Merkuriev–Suslin theorem for any semi-local ring, J. Pure Appl. Algebra 207 (2006) 537 MR2265538
15	J Hornbostel, A¹–representability of Hermitian K–theory and Witt groups, Topology 44 (2005) 661 MR2122220
16	J Hornbostel, Some comments on motivic nilpotence, preprint (2015) arXiv:1511.07292
17	J Hornbostel, M Schlichting, Localization in Hermitian K–theory of rings, J. London Math. Soc. 70 (2004) 77 MR2064753
18	M Hoyois, S Kelly, P A Østvær, The motivic Steenrod algebra in positive characteristic, preprint (2013) arXiv:1305.5690
19	P Hu, I Kriz, K Ormsby, The homotopy limit problem for Hermitian K–theory, equivariant motivic homotopy theory and motivic Real cobordism, Adv. Math. 228 (2011) 434 MR2822236
20	D C Isaksen, A Shkembi, Motivic connective K–theories and the cohomology of A(1), J. K–Theory 7 (2011) 619 MR2811718
21	B Kahn, La conjecture de Milnor (d’après V Voevodsky), 245, Soc. Math. France (1997) 379 MR1627119
22	M Karoubi, Le théorème fondamental de la K–théorie hermitienne, Ann. of Math. 112 (1980) 259 MR592292
23	M Kerz, The Gersten conjecture for Milnor K–theory, Invent. Math. 175 (2009) 1 MR2461425
24	D Kobal, K–theory, Hermitian K–theory and the Karoubi tower, K–Theory 17 (1999) 113 MR1696428
25	M Levine, Relative Milnor K–theory, K–Theory 6 (1992) 113 MR1187704
26	M Levine, Homology of algebraic varieties : an introduction to the works of Suslin and Voevodsky, Bull. Amer. Math. Soc. 34 (1997) 293 MR1432056
27	M Levine, The homotopy coniveau tower, J. Topol. 1 (2008) 217 MR2365658
28	M Levine, Convergence of Voevodsky’s slice tower, Doc. Math. 18 (2013) 907 MR3084567
29	M Levine, A comparison of motivic and classical stable homotopy theories, J. Topol. 7 (2014) 327 MR3217623
30	S Lichtenbaum, The construction of weight-two arithmetic cohomology, Invent. Math. 88 (1987) 183 MR877012
31	C Mazza, V Voevodsky, C Weibel, Lecture notes on motivic cohomology, 2, Amer. Math. Soc. (2006) MR2242284
32	J Milnor, Algebraic K–theory and quadratic forms, Invent. Math. 9 (1969/1970) 318 MR0260844
33	J Milnor, Collected papers of John Milnor, V : Algebra, Amer. Math. Soc. (2010) MR2841244
34	F Morel, Voevodsky’s proof of Milnor’s conjecture, Bull. Amer. Math. Soc. 35 (1998) 123 MR1600334
35	F Morel, Suite spectrale d’Adams et invariants cohomologiques des formes quadratiques, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 963 MR1696188
36	F Morel, Milnor’s conjecture on quadratic forms and mod 2 motivic complexes, Rend. Sem. Mat. Univ. Padova 114 (2005) 63 MR2207862
37	F Morel, 𝔸¹–algebraic topology over a field, 2052, Springer (2012) MR2934577
38	D Orlov, A Vishik, V Voevodsky, An exact sequence for K_∗^M∕2 with applications to quadratic forms, Ann. of Math. 165 (2007) 1 MR2276765
39	I Panin, C Walter, On the motivic commutative ring spectrum BO, preprint (2010) arXiv:1011.0650
40	P Pelaez, Multiplicative properties of the slice filtration, 335, Soc. Math. France (2011) MR2807904
41	P Pelaez, On the functoriality of the slice filtration, J. K–Theory 11 (2013) 55 MR3034283
42	A Pfister, On the Milnor conjectures : history, influence, applications, Jahresber. Deutsch. Math.-Verein. 102 (2000) 15 MR1769021
43	D Popescu, General Néron desingularization, Nagoya Math. J. 100 (1985) 97 MR818160
44	O Röndigs, On the η–inverted sphere, preprint (2016) arXiv:1602.08798
45	O Röndigs, P A Østvær, Motives and modules over motivic cohomology, C. R. Math. Acad. Sci. Paris 342 (2006) 751 MR2227753
46	O Röndigs, P A Østvær, Modules over motivic cohomology, Adv. Math. 219 (2008) 689 MR2435654
47	O Röndigs, M Spitzweck, P A Østvær, Cellularity of hermitian K–theory and Witt theory, preprint (2015) arXiv:1603.05139
48	M Schlichting, Hermitian K–theory, derived equivalences and Karoubi’s fundamental theorem, preprint (2012) arXiv:1209.0848
49	M Spitzweck, Relations between slices and quotients of the algebraic cobordism spectrum, Homology, Homotopy Appl. 12 (2010) 335 MR2771593
50	M Spitzweck, A commutative ℙ¹–spectrum representing motivic cohomology over Dedekind domains, preprint (2013) arXiv:1207.4078
51	A A Suslin, On the K–theory of local fields, from: "Proceedings of the Luminy conference on algebraic K–theory" (editors E M Friedlander, M Karoubi), Elsevier (1984) 301 MR772065
52	A A Suslin, Torsion in K₂ of fields, K–Theory 1 (1987) 5 MR899915
53	A Suslin, Voevodsky’s proof of the Milnor conjecture, from: "Current developments in mathematics, 1997" (editors R Bott, A Jaffe, D Jerison, G Lusztig, I Singer, S T Yau), Int. Press (1999) 173 MR1700298
54	A Suslin, V Voevodsky, Bloch–Kato conjecture and motivic cohomology with finite coefficients, from: "The arithmetic and geometry of algebraic cycles" (editors B B Gordon, J D Lewis, S Müller-Stach, S Saito, N Yui), NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ. (2000) 117 MR1744945
55	R W Thomason, The homotopy limit problem, from: "Proceedings of the Northwestern Homotopy Theory Conference" (editors H R Miller, S B Priddy), Contemp. Math. 19, Amer. Math. Soc. (1983) 407 MR711065
56	V Voevodsky, A¹–homotopy theory, from: "Proceedings of the International Congress of Mathematicians, Volume I" (1998) 579 MR1648048
57	V Voevodsky, Voevodsky’s Seattle lectures : K–theory and motivic cohomology, from: "Algebraic K–theory" (editors W Raskind, C Weibel), Proc. Sympos. Pure Math. 67, Amer. Math. Soc. (1999) 283 MR1743245
58	V Voevodsky, Triangulated categories of motives over a field, from: "Cycles, transfers, and motivic homology theories", Ann. of Math. Stud. 143, Princeton Univ. Press (2000) 188 MR1764202
59	V Voevodsky, Open problems in the motivic stable homotopy theory, I, from: "Motives, polylogarithms and Hodge theory, Part I" (editors F Bogomolov, L Katzarkov), Int. Press Lect. Ser. 3, Int. Press (2002) 3 MR1977582
60	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 MR1890744
61	V Voevodsky, Motivic cohomology with Z∕2–coefficients, Publ. Math. Inst. Hautes Études Sci. (2003) 59 MR2031199
62	V Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. (2003) 1 MR2031198
63	V Voevodsky, On the zero slice of the sphere spectrum, Tr. Mat. Inst. Steklova 246 (2004) 106 MR2101286