On RO(G)–graded equivariant “ordinary” cohomology where G is a power of ℤ∕2

Holler, John¹ ; Kriz, Igor¹

¹Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, United States

Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 741-763

Résumé

We compute the complete RO(G)–graded coefficients of “ordinary” cohomology with coefficients in ℤ∕2 for G = (ℤ∕2)n. As an important intermediate step, we identify the ring of coefficients of the corresponding geometric fixed point spectrum, revealing some interesting algebra. This is a first computation of its kind for groups which are not cyclic p–groups.

Détail

DOI : 10.2140/agt.2017.17.741

Classification : 55N91
Keywords: equivariant cohomology, geometric fixed points, isotropy separation

Bibliographie

[1] W C Abram, I Kriz, The equivariant complex cobordism ring of a finite abelian group, Math. Res. Lett. 22 (2015) 1573 | DOI

[2] S Araki, M Murayama, τ–cohomology theories, Japan. J. Math. 4 (1978) 363

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

[4] M F Atiyah, I M Singer, The index of elliptic operators, I, Ann. of Math. 87 (1968) 484 | DOI

[5] G E Bredon, Equivariant cohomology theories, 34, Springer (1967) | DOI

[6] T Tom Dieck, Bordism of G–manifolds and integrality theorems, Topology 9 (1970) 345 | DOI

[7] P Donovan, M Karoubi, Graded Brauer groups and K–theory with local coefficients, Inst. Hautes Études Sci. Publ. Math. 38 (1970) 5 | DOI

[8] A W M Dress, Notes on the theory of representations of finite groups, I : The Burnside ring of a finite group and some AGN-applications, Universität Bielefeld, Fakultät für Mathematik (1971)

[9] D S Freed, M J Hopkins, C Teleman, Loop groups and twisted K–theory, I, J. Topol. 4 (2011) 737 | DOI

[10] J P C Greenlees, Adams spectral sequences in equivariant topology, PhD thesis, University of Cambridge (1985)

[11] J P C Greenlees, J P May, Equivariant stable homotopy theory, from: "Handbook of algebraic topology" (editor I M James), North-Holland (1995) 277 | DOI

[12] M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. 184 (2016) 1 | DOI

[13] P Hu, I Kriz, Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence, Topology 40 (2001) 317 | DOI

[14] P Hu, I Kriz, Coefficients of the constant Mackey functor over cyclic p–groups, preprint (2010)

[15] P Hu, I Kriz, Topological Hermitian cobordism, J. Homotopy Relat. Struct. 11 (2016) 173 | DOI

[16] 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 | DOI

[17] I Kriz, The Z∕p–equivariant complex cobordism ring, from: "Homotopy invariant algebraic structures" (editors J P Meyer, J Morava, W S Wilson), Contemp. Math. 239, Amer. Math. Soc. (1999) 217 | DOI

[18] S Kriz, Equivariant cohomology and the super reciprocal plane of a hyperplane arrangement, preprint (2015)

[19] P S Landweber, Conjugations on complex manifolds and equivariant homotopy of MU, Bull. Amer. Math. Soc. 74 (1968) 271 | DOI

[20] G Lewis, J P May, J Mcclure, Ordinary RO(G)–graded cohomology, Bull. Amer. Math. Soc. 4 (1981) 208 | DOI

[21] L G Lewis Jr., The RO(G)–graded equivariant ordinary cohomology of complex projective spaces with linear Z∕p actions, from: "Algebraic topology and transformation groups" (editor T tom Dieck), Lecture Notes in Math. 1361, Springer (1988) 53 | DOI

[22] L G Lewis Jr., J P May, M Steinberger, J E Mcclure, Equivariant stable homotopy theory, 1213, Springer (1986) | DOI

[23] N Proudfoot, D Speyer, A broken circuit ring, Beiträge Algebra Geom. 47 (2006) 161

[24] H Terao, Algebras generated by reciprocals of linear forms, J. Algebra 250 (2002) 549 | DOI