Algebraic cycles and the classical groups II: Quaternionic cycles

Lawson, H Blaine¹ ; Lima-Filho, Paulo² ; Michelsohn, Marie-Louise¹

¹ Department of Mathematics, Stony Brook University, Stony Brook, New York 11794, USA
² Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA

Geometry & topology, Tome 9 (2005) no. 3, pp. 1187-1220.

Résumé

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space $\mathbb{P}\left(V\right)$, where $V$ carries a real structure, and completely determined their homotopy type. We also extended some functors in $K$–theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel–Whitney classes. In this sequel, we establish corresponding results in the case where $V$ has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a similarly simple description. The stabilized space of quaternionic algebraic cycles admits a nontrivial infinite loop space structure yielding, in particular, a delooping of the total Pontrjagin class map. This stabilized space is directly related to an extended notion of quaternionic spaces and bundles ($KH$–theory), in analogy with Atiyah’s real spaces and $KR$–theory, and the characteristic classes that we introduce for these objects are nontrivial. The paper ends with various examples and applications.

Détail

DOI : 10.2140/gt.2005.9.1187

Keywords: quaternionic algebraic cycles, characteristic classes, equivariant infinite loop spaces, quaternionic $K$–theory

Bibliographie

[1] M F Atiyah, $K$–theory and reality, Quart. J. Math. Oxford Ser. $(2)$ 17 (1966) 367

[2] C P Boyer, H B Lawson Jr, P Lima-Filho, B M Mann, M L Michelsohn, Algebraic cycles and infinite loop spaces, Invent. Math. 113 (1993) 373

[3] A Dold, R Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. $(2)$ 67 (1958) 239

[4] P F Dos Santos, Algebraic cycles on real varieties and $\mathbb{Z}/2$-equivariant homotopy theory, Proc. London Math. Soc. $(3)$ 86 (2003) 513

[5] P F Dos Santos, P Lima-Filho, Quaternionic algebraic cycles and reality, Trans. Amer. Math. Soc. 356 (2004) 4701

[6] P F Dos Santos, P Lima-Filho, Quaternionic $K$–theory for real varieties, in preparation

[7] J L Dupont, Symplectic bundles and $KR$–theory, Math. Scand. 24 (1969) 27

[8] J Dupont, A note on characteristic classes for real vector bundles, preprint (1999)

[9] E M Friedlander, H B Lawson Jr, A theory of algebraic cocycles, Ann. of Math. $(2)$ 136 (1992) 361

[10] W Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2, Springer (1984)

[11] F R Harvey, H B Lawson Jr, On boundaries of complex analytic varieties I, Ann. of Math. $(2)$ 102 (1975) 223

[12] T K Lam, Spaces of real algebraic cycles and homotopy theory, PhD thesis, SUNY, Stony Brook (1990)

[13] H B Lawson Jr, Algebraic cycles and homotopy theory, Ann. of Math. $(2)$ 129 (1989) 253

[14] H B Lawson Jr, P Lima-Filho, M L Michelsohn, On equivariant algebraic suspension, J. Algebraic Geom. 7 (1998) 627

[15] H B Lawson Jr, P Lima-Filho, M L Michelsohn, Algebraic cycles and the classical groups I: Real cycles, Topology 42 (2003) 467

[16] H B Lawson Jr, M L Michelsohn, Algebraic cycles, Bott periodicity, and the Chern characteristic map, from: "The mathematical heritage of Hermann Weyl (Durham, NC, 1987)", Proc. Sympos. Pure Math. 48, Amer. Math. Soc. (1988) 241

[17] P Lima-Filho, Lawson homology for quasiprojective varieties, Compositio Math. 84 (1992) 1

[18] P Lima-Filho, Completions and fibrations for topological monoids, Trans. Amer. Math. Soc. 340 (1993) 127

[19] P Lima-Filho, On the equivariant homotopy of free abelian groups on $G$–spaces and $G$–spectra, Math. Z. 224 (1997) 567

[20] J P May, $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra, Lecture Notes in Mathematics 577, Springer (1977) 268

[21] J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press (1974)

[22] J Mostovoy, Quaternion flavored cycle spaces, ICMS preprint, Edinburgh University (1975)