Bigraded cohomology of ℤ∕2–equivariant Grassmannians

Dugger, Daniel¹

¹ Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

Geometry & topology, Tome 19 (2015) no. 1, pp. 113-170.

Résumé

We determine the $RO\left(G\right)$–graded Eilenberg–MacLane cohomology (with mod $2$ coefficients) of the real, infinite Grassmannians in the case $G=\mathbb{Z}∕2$. Possible connections to motivic characteristic classes of quadratic bundles are briefly discussed.

Détail

DOI : 10.2140/gt.2015.19.113

Classification : 55N91, 55R40
Keywords: equivariant Grassmannian, equivariant characteristic classes

Bibliographie

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

[2] M F Atiyah, K–theory, W. A. Benjamin (1967)

[3] R Carter, G Segal, I Macdonald, Lectures on Lie groups and Lie algebras, 32, Cambridge Univ. Press (1995)

[4] J L Caruso, Operations in equivariant Z∕p–cohomology, Math. Proc. Cambridge Philos. Soc. 126 (1999) 521

[5] A Delzant, Définition des classes de Stiefel–Whitney d’un module quadratique sur un corps de caractéristique différente de 2, C. R. Acad. Sci. Paris 255 (1962) 1366

[6] D Dugger, An Atiyah–Hirzebruch spectral sequence for KR–theory, K–Theory 35 (2005) 213

[7] K K Ferland, L G Lewis Jr., The RO(G)–graded equivariant ordinary homology of G–cell complexes with even-dimensional cells for G = Z∕p, 794, Amer. Math. Soc. (2004)

[8] O Foundation, Online encyclopedia of integer sequences (2012)

[9] C Giusti, P Salvatore, D Sinha, The mod-2 cohomology rings of symmetric groups, J. Topol. 5 (2012) 169

[10] G Hochschild, The structure of Lie groups, Holden-Day (1965)

[11] W C Kronholm, A freeness theorem for RO(Z∕2)–graded cohomology, Topology Appl. 157 (2010) 902

[12] W C Kronholm, The RO(G)–graded Serre spectral sequence, Homology, Homotopy Appl. 12 (2010) 75

[13] J P May, Equivariant homotopy and cohomology theory, 91, Amer. Math. Soc. (1996)

[14] J Milnor, Algebraic K–theory and quadratic forms, Invent. Math. 9 (1969/1970) 318

[15] F Morel, V Voevodsky, A1–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. (1999) 45

[16] D Rydh, A minimal set of generators for the ring of multisymmetric functions, Ann. Inst. Fourier (Grenoble) 57 (2007) 1741

[17] M E Shulman, Equivariant local coefficients and the RO(G)–graded cohomology of classifying spaces, PhD thesis, The University of Chicago (2010)

[18] D P Sullivan, Geometric topology : Localization, periodicity and Galois symmetry, 8, Springer (2005)

[19] G S Tripathi, Geometric models for higher Grothendieck–Witt groups in A1–homotopy theory,