Equivariant semi-topological invariants, Atiyah’s 𝐾𝑅-theory, and real algebraic cycles

Heller, Jeremiah; Voineagu, Mircea

doi:10.1090/S0002-9947-2012-05603-0

Equivariant semi-topological invariants, Atiyah’s $KR$-theory, and real algebraic cycles
HTML articles powered by AMS MathViewer

by Jeremiah Heller and Mircea Voineagu PDF

Trans. Amer. Math. Soc. 364 (2012), 6565-6603 Request permission

Abstract:

We establish an Atiyah-Hirzebruch type spectral sequence relating real morphic cohomology and real semi-topological $K$-theory and prove it to be compatible with the Atiyah-Hirzebruch spectral sequence relating Bredon cohomology and Atiyah’s $KR$-theory constructed by Dugger. An equivariant and a real version of Suslin’s conjecture on morphic cohomology are formulated, proved to come from the complex version of Suslin conjecture and verified for certain real varieties. In conjunction with the spectral sequences constructed here, this allows the computation of the real semi-topological $K$-theory of some real varieties. As another application of this spectral sequence we give an alternate proof of the Lichtenbaum-Quillen conjecture over $\mathbb {R}$, extending an earlier proof of Karoubi and Weibel.

References

M. F. Atiyah, $K$-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966), 367–386. MR 206940, DOI 10.1093/qmath/17.1.367
S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), no. 5, 1235–1253. MR 714776, DOI 10.2307/2374341
Spencer Bloch and Arthur Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181–201 (1975). MR 412191
J. Michael Boardman, Conditionally convergent spectral sequences, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 49–84. MR 1718076, DOI 10.1090/conm/239/03597
Kenneth S. Brown and Stephen M. Gersten, Algebraic $K$-theory as generalized sheaf cohomology, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 266–292. MR 0347943
Jean-Louis Colliot-Thélène, Raymond T. Hoobler, and Bruno Kahn, The Bloch-Ogus-Gabber theorem, Algebraic $K$-theory (Toronto, ON, 1996) Fields Inst. Commun., vol. 16, Amer. Math. Soc., Providence, RI, 1997, pp. 31–94. MR 1466971
David A. Cox, The étale homotopy type of varieties over $\textbf {R}$, Proc. Amer. Math. Soc. 76 (1979), no. 1, 17–22. MR 534381, DOI 10.1090/S0002-9939-1979-0534381-4
Pedro F. dos Santos, Algebraic cycles on real varieties and ${\Bbb Z}/2$-equivariant homotopy theory, Proc. London Math. Soc. (3) 86 (2003), no. 2, 513–544. MR 1971161, DOI 10.1112/S002461150201376X
Pedro F. dos Santos, A note on the equivariant Dold-Thom theorem, J. Pure Appl. Algebra 183 (2003), no. 1-3, 299–312. MR 1992051, DOI 10.1016/S0022-4049(03)00029-X
Pedro F. dos Santos and Paulo Lima-Filho, Quaternionic algebraic cycles and reality, Trans. Amer. Math. Soc. 356 (2004), no. 12, 4701–4736. MR 2084395, DOI 10.1090/S0002-9947-04-03663-3
Daniel Dugger, An Atiyah-Hirzebruch spectral sequence for $KR$-theory, $K$-Theory 35 (2005), no. 3-4, 213–256 (2006). MR 2240234, DOI 10.1007/s10977-005-1552-9
Daniel Dugger and Daniel C. Isaksen, Topological hypercovers and $\Bbb A^1$-realizations, Math. Z. 246 (2004), no. 4, 667–689. MR 2045835, DOI 10.1007/s00209-003-0607-y
Eric M. Friedlander, Christian Haesemeyer, and Mark E. Walker, Techniques, computations, and conjectures for semi-topological $K$-theory, Math. Ann. 330 (2004), no. 4, 759–807. MR 2102312, DOI 10.1007/s00208-004-0569-3
Eric M. Friedlander and H. Blaine Lawson, Duality relating spaces of algebraic cocycles and cycles, Topology 36 (1997), no. 2, 533–565. MR 1415605, DOI 10.1016/0040-9383(96)00011-0
Eric M. Friedlander and H. Blaine Lawson Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), no. 2, 361–428. MR 1185123, DOI 10.2307/2946609
Eric M. Friedlander and H. Blaine Lawson Jr., A theory of algebraic cocycles, Ann. of Math. (2) 136 (1992), no. 2, 361–428. MR 1185123, DOI 10.2307/2946609
Eric M. Friedlander and Barry Mazur, Filtrations on the homology of algebraic varieties, Mem. Amer. Math. Soc. 110 (1994), no. 529, x+110. With an appendix by Daniel Quillen. MR 1211371, DOI 10.1090/memo/0529
Eric M. Friedlander and Andrei Suslin, The spectral sequence relating algebraic $K$-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 6, 773–875 (English, with English and French summaries). MR 1949356, DOI 10.1016/S0012-9593(02)01109-6
Eric M. Friedlander and Vladimir Voevodsky, Bivariant cycle cohomology, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 138–187. MR 1764201
Eric M. Friedlander and Mark E. Walker, Comparing $K$-theories for complex varieties, Amer. J. Math. 123 (2001), no. 5, 779–810. MR 1854111
Eric M. Friedlander and Mark E. Walker, Function spaces and continuous algebraic pairings for varieties, Compositio Math. 125 (2001), no. 1, 69–110. MR 1818058, DOI 10.1023/A:1002464407035
Eric M. Friedlander and Mark E. Walker, Semi-topological $K$-theory of real varieties, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 219–326. MR 1940670
Eric M. Friedlander and Mark E. Walker, Semi-topological $K$-theory using function complexes, Topology 41 (2002), no. 3, 591–644. MR 1910042, DOI 10.1016/S0040-9383(01)00023-4
Eric M. Friedlander and Mark E. Walker, Rational isomorphisms between $K$-theories and cohomology theories, Invent. Math. 154 (2003), no. 1, 1–61. MR 2004456, DOI 10.1007/s00222-003-0300-0
Daniel R. Grayson, Weight filtrations via commuting automorphisms, $K$-Theory 9 (1995), no. 2, 139–172. MR 1340843, DOI 10.1007/BF00961457
Daniel R. Grayson and Mark E. Walker, Geometric models for algebraic $K$-theory, $K$-Theory 20 (2000), no. 4, 311–330. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part IV. MR 1803641, DOI 10.1023/A:1026506218989
J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178. MR 1230773, DOI 10.1090/memo/0543
Jeremiah Heller and Mircea Voineagu, Vanishing theorems for real algebraic cycles, Amer. J. Math. To appear.
J. F. Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35–87. MR 906403, DOI 10.1016/0022-4049(87)90100-9
J. F. Jardine, Generalized étale cohomology theories, Progress in Mathematics, vol. 146, Birkhäuser Verlag, Basel, 1997. MR 1437604, DOI 10.1007/978-3-0348-0066-2
Max Karoubi and Charles Weibel, Algebraic and Real $K$-theory of real varieties, Topology 42 (2003), no. 4, 715–742. MR 1958527, DOI 10.1016/S0040-9383(02)00069-1
Tsz Kin Lam, Spaces of real algebraic cycles and homotopy theory, ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Ph.D.)–State University of New York at Stony Brook. MR 2685475
H. Blaine Lawson, Paulo Lima-Filho, and Marie-Louise Michelsohn, Algebraic cycles and the classical groups. I. Real cycles, Topology 42 (2003), no. 2, 467–506. MR 1941445, DOI 10.1016/S0040-9383(02)00018-6
Marc Levine, The homotopy coniveau tower, J. Topol. 1 (2008), no. 1, 217–267. MR 2365658, DOI 10.1112/jtopol/jtm004
L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482, DOI 10.1007/BFb0075778
L. Gaunce Lewis Jr., Equivariant Eilenberg-Mac Lane spaces and the equivariant Seifert-van Kampen and suspension theorems, Topology Appl. 48 (1992), no. 1, 25–61. MR 1195124, DOI 10.1016/0166-8641(92)90120-O
P. Lima-Filho, On the equivariant homotopy of free abelian groups on $G$-spaces and $G$-spectra, Math. Z. 224 (1997), no. 4, 567–601. MR 1452050, DOI 10.1007/PL00004297
J. P. May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. MR 1413302, DOI 10.1090/cbms/091
Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284
Stephen A. Mitchell, Hypercohomology spectra and Thomason’s descent theorem, Algebraic $K$-theory (Toronto, ON, 1996) Fields Inst. Commun., vol. 16, Amer. Math. Soc., Providence, RI, 1997, pp. 221–277. MR 1466977
I. Panin, Oriented cohomology theories of algebraic varieties, $K$-Theory 30 (2003), no. 3, 265–314. Special issue in honor of Hyman Bass on his seventieth birthday. Part III. MR 2064242, DOI 10.1023/B:KTHE.0000019788.33790.cb
Andreas Rosenschon and Paul Arne Østvær, The homotopy limit problem for two-primary algebraic $K$-theory, Topology 44 (2005), no. 6, 1159–1179. MR 2168573, DOI 10.1016/j.top.2005.04.004
Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, DOI 10.1016/0040-9383(74)90022-6
A. Suslin, On the Grayson spectral sequence, Tr. Mat. Inst. Steklova 241 (2003), no. Teor. Chisel, Algebra i Algebr. Geom., 218–253; English transl., Proc. Steklov Inst. Math. 2(241) (2003), 202–237. MR 2024054
Andrei Suslin and Vladimir Voevodsky, Relative cycles and Chow sheaves, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 10–86. MR 1764199
Jyh-Haur Teh, Harnack-Thom theorem for higher cycle groups and Picard varieties, Trans. Amer. Math. Soc. 360 (2008), no. 6, 3263–3285. MR 2379796, DOI 10.1090/S0002-9947-07-04432-7
R. W. Thomason, Algebraic $K$-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552. MR 826102
Vladimir Voevodsky, Motivic cohomology with $\textbf {Z}/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104. MR 2031199, DOI 10.1007/s10240-003-0010-6
Mircea Voineagu, Semi-topological $K$-theory for certain projective varieties, J. Pure Appl. Algebra 212 (2008), no. 8, 1960–1983. MR 2414696, DOI 10.1016/j.jpaa.2008.01.004
—, Cylindrical Homomorphisms and Lawson Homology, To appear in Journal of $K$-theory (2009).
Mark Edward Walker, Motivic complexes and the K-theory of automorphisms, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 2694778
Mark E. Walker, Adams operations for bivariant $K$-theory and a filtration using projective lines, $K$-Theory 21 (2000), no. 2, 101–140. MR 1804538, DOI 10.1023/A:1007896730627
Mark E. Walker, Semi-topological $K$-homology and Thomason’s theorem, $K$-Theory 26 (2002), no. 3, 207–286. MR 1936355, DOI 10.1023/A:1020649830539
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136