Quaternionic algebraic cycles and reality

dos Santos, Pedro; Lima-Filho, Paulo

doi:10.1090/S0002-9947-04-03663-3

Quaternionic algebraic cycles and reality
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by Pedro F. dos Santos and Paulo Lima-Filho PDF

Trans. Amer. Math. Soc. 356 (2004), 4701-4736 Request permission

Abstract:

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group $Gal({\mathbb C} / {\mathbb R})$. Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour’s quaternionic $K$-theory, and the other one classifies an equivariant cohomology theory ${\mathfrak Z}^*(-)$ which is a natural recipient of characteristic classes $KH^*(X) \to {\mathfrak Z}^*(X)$ for quaternionic bundles over Real spaces $X$.

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
Charles P. Boyer, H. Blaine Lawson Jr., Paulo Lima-Filho, Benjamin M. Mann, and Marie-Louise Michelsohn, Algebraic cycles and infinite loop spaces, Invent. Math. 113 (1993), no. 2, 373–388. MR 1228130, DOI 10.1007/BF01244311
Raoul Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. MR 110104, DOI 10.2307/1970106
Steven R. Costenoble and Stefan Waner, Fixed set systems of equivariant infinite loop spaces, Trans. Amer. Math. Soc. 326 (1991), no. 2, 485–505. MR 1012523, DOI 10.1090/S0002-9947-1991-1012523-4
S. R. Costenoble and S. Waner, Equivariant Poincaré duality, Michigan Math. J. 39 (1992), no. 2, 325–351. MR 1162040, DOI 10.1307/mmj/1029004526
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
D. Dugger, An Atiyah-Hirzebruch spectral sequence for $KR$-theory, Preprint, 2001.
Johan L. Dupont, Symplectic bundles and $KR$-theory, Math. Scand. 24 (1969), 27–30. MR 254839, DOI 10.7146/math.scand.a-10918
—, A note on characteristic classes for Real vector bundles, Preprint, 1999.
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
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Bruno Kahn, Construction de classes de Chern équivariantes pour un fibré vectoriel réel, Comm. Algebra 15 (1987), no. 4, 695–711 (French, with English summary). MR 877194, DOI 10.1080/00927872.1987.12088443
T. K. Lam, Spaces of real algebraic cycles and homotopy theory, Ph.D. thesis, SUNY at Stony Brook, 1990.
H. Blaine Lawson Jr., Algebraic cycles and homotopy theory, Ann. of Math. (2) 129 (1989), no. 2, 253–291. MR 986794, DOI 10.2307/1971448
Paulo Lima-Filho, The topological group structure of algebraic cycles, Duke Math. J. 75 (1994), no. 2, 467–491. MR 1290199, DOI 10.1215/S0012-7094-94-07513-3
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
H. Blaine Lawson Jr., Paulo Lima-Filho, and Marie-Louise Michelsohn, Algebraic cycles and equivariant cohomology theories, Proc. London Math. Soc. (3) 73 (1996), no. 3, 679–720. MR 1407465, DOI 10.1112/plms/s3-73.3.679
H. Blaine Lawson Jr., P. Lima-Filho, and M.-L. Michelsohn, On equivariant algebraic suspension, J. Algebraic Geom. 7 (1998), no. 4, 627–650. MR 1642736
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
—, Spaces of algebraic cycles and classical groups, Part II: Quaternionic cycles, Preprint, 1998.
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Algebraic cycles, Bott periodicity, and the Chern characteristic map, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 241–263. MR 974339, DOI 10.1090/pspum/048/974339
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Algebraic cycles and group actions, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 261–277. MR 1173046
J. Peter May, $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. MR 0494077
J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609, DOI 10.1007/BFb0068547
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
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
Jacob Mostovoy, Algebraic cycles and antiholomorphic involutions on projective spaces, Bol. Soc. Mat. Mexicana (3) 6 (2000), no. 2, 151–170. MR 1810845
Daniel Quillen, Higher algebraic $K$-theory. I, 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. 85–147. MR 0338129
Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452, DOI 10.1007/BF02684593
R. M. Seymour, The real $K$-theory of Lie groups and homogeneous spaces, Quart. J. Math. Oxford Ser. (2) 24 (1973), 7–30. MR 319193, DOI 10.1093/qmath/24.1.7