An algebraic model for chains on Ω𝐵𝐺^{^{∧}}_{𝑝}

Benson, Dave

doi:10.1090/S0002-9947-08-04728-4

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ISSN 1088-6850(online) ISSN 0002-9947(print)

An algebraic model for chains on $\Omega BG{}^{^\wedge}_p$

Author: Dave Benson
Journal: Trans. Amer. Math. Soc. 361 (2009), 2225-2242
MSC (2000): Primary 55P35, 55R35, 20C20; Secondary 55P60, 20J06, 13C40, 14M10
Posted: November 19, 2008
MathSciNet review: 2465835
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Abstract: We provide an interpretation of the homology of the loop space on the -completion of the classifying space of a finite group in terms of representation theory, and demonstrate how to compute it. We then give the following reformulation. If is an idempotent in such that is the projective cover of the trivial module , and , then we exhibit isomorphisms for $n\ge 2$ :

$\displaystyle H_n(\Omega BG {}^{^\wedge}_p;k)$	$\displaystyle \cong \mathrm{Tor}_{n-1}^{e.kG.e}(kG.e,e.kG),$
$\displaystyle H^n(\Omega BG{}^{^\wedge}_p;k)$	$\displaystyle \cong \mathrm{Ext}^{n-1}_{e.kG.e}(e.kG,e.kG).$

Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations.

1. D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302 (93g:20099)
2. David John Benson and Henning Krause, Complexes of injective 𝑘𝐺-modules, Algebra Number Theory 2 (2008), no. 1, 1–30. MR 2377361 (2009d:20010), http://dx.doi.org/10.2140/ant.2008.2.1
3. Enrico Bombieri, A. Odlyzko, and D. Hunt, Thompson’s problem (𝜎²=3), Invent. Math. 58 (1980), no. 1, 77–100. Appendices by A. Odlyzko and D. Hunt. MR 570875 (81f:20019), http://dx.doi.org/10.1007/BF01402275
4. W. Bosma and J. Cannon, Handbook of Magma Functions, Magma Computer Algebra, Sydney, 1996.
5. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin, 1972. MR 0365573 (51 #1825)
6. Frederick R. Cohen and Ran Levi, On the homotopy theory of 𝑝-completed classifying spaces, Group representations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 157–182. MR 1603147 (99e:55026)
7. W. G. Dwyer and J. P. C. Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002), no. 1, 199–220. MR 1879003 (2003g:16010)
8. W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006), no. 2, 357–402. MR 2200850 (2006k:55017), http://dx.doi.org/10.1016/j.aim.2005.11.004
9. W. Dwyer, J. P. C. Greenlees, and S. Iyengar, Finiteness in derived categories of local rings, Comment. Math. Helv. 81 (2006), no. 2, 383–432. MR 2225632 (2007b:13021), http://dx.doi.org/10.4171/CMH/56
10. Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Hopf algebras of polynomial growth, J. Algebra 125 (1989), no. 2, 408–417. MR 1018954 (90j:16021), http://dx.doi.org/10.1016/0021-8693(89)90173-7
11. Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Elliptic Hopf algebras, J. London Math. Soc. (2) 43 (1991), no. 3, 545–555. MR 1113392 (92i:57033), http://dx.doi.org/10.1112/jlms/s2-43.3.545
12. Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Elliptic spaces. II, Enseign. Math. (2) 39 (1993), no. 1-2, 25–32. MR 1225255 (94f:55008)
13. N. L. Gordeev, Finite linear groups whose algebra of invariants is a complete intersection, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 2, 343–392 (Russian). MR 842586 (88a:13020)
14. T. H. Gulliksen, A homological characterization of local complete intersections, Compositio Math. 23 (1971), 251–255. MR 0301008 (46 #168)
15. T. H. Gulliksen, On the deviations of a local ring, Math. Scand. 47 (1980), no. 1, 5–20. MR 600076 (82c:13022)
16. Victor Kac and Keiichi Watanabe, Finite linear groups whose ring of invariants is a complete intersection, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 221–223. MR 640951 (83h:14042), http://dx.doi.org/10.1090/S0273-0979-1982-14989-8
17. Ran Levi, On finite groups and homotopy theory, Mem. Amer. Math. Soc. 118 (1995), no. 567, xiv+100. MR 1308466 (96c:55019)
18. Ran Levi, A counter-example to a conjecture of Cohen, Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994), Progr. Math., vol. 136, Birkhäuser, Basel, 1996, pp. 261–269. MR 1397737 (97d:55027)
19. Ran Levi, On homological rate of growth and the homotopy type of Ω𝐵𝐺^{∧}_{𝑝}, Math. Z. 226 (1997), no. 3, 429–444. MR 1483541 (98k:55016), http://dx.doi.org/10.1007/PL00004349
20. Ran Levi, On 𝑝-completed classifying spaces of discrete groups and finite complexes, J. London Math. Soc. (2) 59 (1999), no. 3, 1064–1080. MR 1709097 (2000m:55019), http://dx.doi.org/10.1112/S0024610799007279
21. Haruhisa Nakajima, Quotient singularities which are complete intersections, Manuscripta Math. 48 (1984), no. 1-3, 163–187. MR 753729 (86h:14039), http://dx.doi.org/10.1007/BF01169006
22. Haruhisa Nakajima, Quotient complete intersections of affine spaces by finite linear groups, Nagoya Math. J. 98 (1985), 1–36. MR 792768 (87c:14056)
23. Haruhisa Nakajima and Keiichi Watanabe, The classification of quotient singularities which are complete intersections, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 102–120. MR 775879 (86c:14002), http://dx.doi.org/10.1007/BFb0099359
24. John H. Walter, The characterization of finite groups with abelian Sylow 2-subgroups., Ann. of Math. (2) 89 (1969), 405–514. MR 0249504 (40 #2749)

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