The local (co)homology theorems for equivariant bordism

La Vecchia, Marco¹

¹ Mathematics Institute, University of Warwick, Coventry, United Kingdom

Geometry & topology, Tome 28 (2024) no. 2, pp. 627-639.

Résumé

We generalize the completion theorem for equivariant ${MU}_G$–module spectra for finite extensions of a torus to compact Lie groups using the splitting of global functors proved by Schwede. This proves a conjecture of Greenlees and May.

Détail

DOI : 10.2140/gt.2024.28.627

Keywords: equivariant complex bordism, local (co)homology, completion, equivariant stable homotopy theory

Bibliographie

[1] J F Adams, J H Gunawardena, H Miller, The Segal conjecture for elementary abelian p–groups, Topology 24 (1985) 435 | DOI

[2] M F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961) 23 | DOI

[3] M F Atiyah, F Hirzebruch, Vector bundles and homogeneous spaces, from: "Differential geometry" (editor C B Allendoerfer), Proc. Sympos. Pure Math. 3, Amer. Math. Soc. (1961) 7

[4] M F Atiyah, G B Segal, Equivariant K–theory and completion, J. Differential Geometry 3 (1969) 1

[5] G Carlsson, G B Segal’s Burnside ring conjecture for (Z∕2)k, Topology 22 (1983) 83 | DOI

[6] G Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. 120 (1984) 189 | DOI

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

[8] J P C Greenlees, K–homology of universal spaces and local cohomology of the representation ring, Topology 32 (1993) 295 | DOI

[9] J P C Greenlees, J P May, Derived functors of I–adic completion and local homology, J. Algebra 149 (1992) 438 | DOI

[10] J P C Greenlees, J P May, Localization and completion theorems for MU–module spectra, Ann. of Math. 146 (1997) 509 | DOI

[11] A Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. 9 (1957) 119 | DOI

[12] R Hartshorne, Local cohomology, 41, Springer (1967) | DOI

[13] E Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Math. Scand. 44 (1979) 37 | DOI

[14] W H Lin, On conjectures of Mahowald, Segal and Sullivan, Math. Proc. Cambridge Philos. Soc. 87 (1980) 449 | DOI

[15] W H Lin, D M Davis, M E Mahowald, J F Adams, Calculation of Lin’s Ext groups, Math. Proc. Cambridge Philos. Soc. 87 (1980) 459 | DOI

[16] P Löffler, Bordismengruppen unitärer Torusmannigfaltigkeiten, Manuscripta Math. 12 (1974) 307 | DOI

[17] M A Mandell, J P May, Equivariant orthogonal spectra and S–modules, 755, Amer. Math. Soc. (2002) | DOI

[18] M C Mccord, Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969) 273 | DOI

[19] T Nikolaus, P Scholze, On topological cyclic homology, Acta Math. 221 (2018) 203 | DOI

[20] D C Ravenel, The Segal conjecture for cyclic groups, Bull. London Math. Soc. 13 (1981) 42 | DOI

[21] S Schwede, Global homotopy theory, 34, Cambridge Univ. Press (2018) | DOI

[22] S Schwede, Splittings of global Mackey functors and regularity of equivariant Euler classes, Proc. Lond. Math. Soc. 125 (2022) 258 | DOI

[23] S Schwede, Chern classes in equivariant bordism, preprint (2023)

[24] G B Segal, C T Stretch, Characteristic classes for permutation representations, Math. Proc. Cambridge Philos. Soc. 90 (1981) 265 | DOI

[25] C T Stretch, Stable cohomotopy and cobordism of abelian groups, Math. Proc. Cambridge Philos. Soc. 90 (1981) 273 | DOI