Diagrams up to cohomology

Dwyer, W.; Wilkerson, C.

doi:10.1090/S0002-9947-96-01550-4

Diagrams up to cohomology
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by W. G. Dwyer and C. W. Wilkerson PDF

Trans. Amer. Math. Soc. 348 (1996), 1863-1883 Request permission

Abstract:

We compute (under suitable assumptions) how many ways there are to take a diagram in the homotopy category of spaces and perturb it to get another diagram which looks the same up to cohomology. Sometimes there are no perturbations. This can shed light on the question of whether the $p$-completion of the classifying space of a particular connected compact Lie group is determined up to homotopy by cohomological data.

References

A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
W. G. Dwyer, The centralizer decomposition of $BG$, Proceedings 1994 Barcelona Conference on Algebraic Topology (to appear).
W. G. Dwyer and D. M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23 (1984), no. 2, 139–155. MR 744846, DOI 10.1016/0040-9383(84)90035-1
W. G. Dwyer and D. M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23 (1984), no. 2, 139–155. MR 744846, DOI 10.1016/0040-9383(84)90035-1
W. G. Dwyer and D. M. Kan, Equivariant homotopy classification, J. Pure Appl. Algebra 35 (1985), no. 3, 269–285. MR 777259, DOI 10.1016/0022-4049(85)90045-3
W. G. Dwyer, D. M. Kan, and J. H. Smith, Homotopy commutative diagrams and their realizations, J. Pure Appl. Algebra 57 (1989), no. 1, 5–24. MR 984042, DOI 10.1016/0022-4049(89)90023-6
W. G. Dwyer and D. M. Kan, Centric maps and realization of diagrams in the homotopy category, Proc. Amer. Math. Soc. 114 (1992), no. 2, 575–584. MR 1070515, DOI 10.1090/S0002-9939-1992-1070515-X
W. G. Dwyer, H. R. Miller, and C. W. Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), no. 1, 29–45. MR 1153237, DOI 10.1016/0040-9383(92)90062-M
W. G. Dwyer and C. W. Wilkerson, A cohomology decomposition theorem, Topology 31 (1992), no. 2, 433–443. MR 1167181, DOI 10.1016/0040-9383(92)90032-D
W. G. Dwyer and C. W. Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), no. 1, 37–64. MR 1161306, DOI 10.1090/S0894-0347-1993-1161306-9
W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395–442. MR 1274096, DOI 10.2307/2946585
W. G. Dwyer and C. W. Wilkerson, The center of a $p$-compact group, Contemporary Mathematics 181, Proceedings of the 1993 Čech Conference (Northeastern Univ.), Amer. Math. Soc., Providence, RI, 1995, pp. 119–157.
Stefan Jackowski and James McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992), no. 1, 113–132. MR 1153240, DOI 10.1016/0040-9383(92)90065-P
S. Jackowski, J. McClure and R. Oliver, Maps between classifying spaces revisited.
Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un $p$-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135–244 (French). With an appendix by Michel Zisman. MR 1179079
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
D. Notbohm, Homotopy uniqueness of classifying spaces of compact connected Lie groups at primes dividing the order of the Weyl group, Habilitationsschrift, Göttingen, 1992.
Bob Oliver, Higher limits via Steinberg representations, Comm. Algebra 22 (1994), no. 4, 1381–1393. MR 1261265, DOI 10.1080/00927879408824911
K. Premadasa, Thesis (Purdue University) 1994.