×

zbMATH — the first resource for mathematics

Noetherian loop spaces. (English) Zbl 1229.55011
This paper studies properties of loop spaces having Noetherian mod \(p\) cohomology, paying attention to differences between such loop spaces and those with finite mod \(p\) cohomology; here \(p\) is any prime number.
More precisely, a \(p\)-Noetherian group is a triple \((X,BX,e)\) such that the cohomology \(H^*(X; \mathbb F_p)\) is a Noetherian algebra (it is finitely generated as an algebra), \(BX\) is a \(p\)-complete space, and \(e: X\rightarrow \Omega BX\) (where \(\Omega BX\) is the space of loops on \(BX\)) is a weak homotopy equivalence. In this setting, \(X\) is called the Noetherian loop space, and \(BX\) is referred to as the classifying space of \(X\). In particular, \(p\)-compact groups in the sense of W. Dwyer and C. Wilkerson [Ann. Math. (2) 139, No. 2, 395–442 (1994; Zbl 0801.55007)] are \(p\)-Noetherian groups.
The authors study the cohomology of the classifying space \(BX\) and show that it is as small as expected. More precisely, they prove that for any \(p\)-Noetherian group \((X,BX,e)\), \(H^*(BX; \mathbb F_p)\) is finitely generated as an algebra over the Steenrod algebra and, in addition, the module of indecomposable elements, \(QH^*(BX; \mathbb F_p)=\tilde H^*(BX; \mathbb F_p)/\tilde H^*(BX; \mathbb F_p)\cdot \tilde H^*(BX; \mathbb F_p)\), belongs to \(\mathcal U_1\) of the Krull filtration, \(\mathcal U_1\subset \mathcal U_2\subset \cdots\), of the category \(\mathcal U\) of unstable modules over the Steenrod algebra (\(\mathcal U_0\) consists of the locally finite unstable modules and contains \(QH^*(X; \mathbb F_p)\); that is, \(QH^*(BX; \mathbb F_p)\) lies only one stage higher than where \(QH^*(X; \mathbb F_p)\) lies).
MSC:
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
13A02 Graded rings
13E05 Commutative Noetherian rings and modules
55P20 Eilenberg-Mac Lane spaces
55P60 Localization and completion in homotopy theory
55S10 Steenrod algebra
55T10 Serre spectral sequences
57T10 Homology and cohomology of Lie groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aguadé, J., Smith, L.: On the mod p torus theorem of John Hubbuck. Math. Z. 191, 325-326 (1986) · Zbl 0591.55003 · doi:10.1007/BF01164037 · eudml:173678
[2] Andersen, K., Grodal, J.: The classification of 2-compact groups. J. Amer. Math. Soc. 22, 387-436 (2009) · Zbl 1360.55014
[3] Andersen, K., Grodal, J., Møller, J. M., Viruel, A.: The classification of p-compact groups for p odd. Ann. of Math. (2) 167, 95-210 (2008) · Zbl 1149.55011 · doi:10.4007/annals.2008.167.95 · annals.math.princeton.edu
[4] Bousfield, A. K.: Localization and periodicity in unstable homotopy theory. J. Amer. Math. Soc. 7, 831-873 (1994) · Zbl 0839.55008 · doi:10.2307/2152734
[5] Bousfield, A. K., Kan, D. M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Math. 304, Springer, Berlin (1972) · Zbl 0259.55004
[6] Browder, W.: Torsion in H -spaces. Ann. of Math. (2) 74, 24-51 (1961) · Zbl 0112.14501 · doi:10.2307/1970305
[7] Cartan, H., Moore, J. C., Thom, R., Serre, J.-P.: Séminaire Henri Cartan de l’ École Nor- male Supérieure, 1954/1955. Alg‘ebres d’Eilenberg-MacLane et homotopie, exposés 2-16. Secrétariat mathématique, Paris (1955) · Zbl 0067.15601 · www.numdam.org · www.numdam.org
[8] Castellana, N., Crespo, J. A., Scherer, J.: On the homotopy groups of p-completed classifying spaces. Manuscripta Math. 118, 399-409 (2005) · Zbl 1086.55007 · doi:10.1007/s00229-005-0587-9
[9] Castellana, N., Crespo, J. A., Scherer, J.: Deconstructing Hopf spaces. Invent. Math. 167, 1-18 (2007) · Zbl 1109.55005 · doi:10.1007/s00222-006-0518-8
[10] Castellana, N., Crespo, J. A., Scherer, J.: On the cohomology of highly connected covers of Hopf spaces. Adv. Math. 215, 250-262 (2007) · Zbl 1126.57016 · doi:10.1016/j.aim.2007.03.011
[11] Chachólski, W., Pitsch, W., Scherer, J., Stanley, D.: Homotopy exponents for large H -spaces. Int. Math. Res. Notices 2008, no. 16, art. ID rnn061, 5 pp. · Zbl 1163.55005 · doi:10.1093/imrn/rnn061
[12] Crespo, J. A.: Structure of mod p H -spaces with finiteness conditions. In: Cohomological Methods in Homotopy Theory (Bellaterra, 1998), Progr. Math. 196, Birkhäuser, Basel, 103- 130 (2001) · Zbl 0990.55004
[13] Dehon, F.-X., Gaudens, G.: Espaces profinis et probl‘emes de réalisabilité. Algebr. Geom. Topol. 3, 399-433 (2003) · Zbl 1022.55012 · doi:10.2140/agt.2003.3.399 · emis:journals/UW/agt/AGTVol3/agt-3-13.abs.html · eudml:122950 · arxiv:math/0306271
[14] Dwyer, W. G., Mislin, G.: On the homotopy type of the components of map\ast (BS3, BS3). In: Algebraic Topology (Barcelona, 1986), Lecture Notes in Math. 1298, Springer, Berlin, 82-89 (2001) · Zbl 0654.55014
[15] Dwyer, W. G., Wilkerson, C. W.: Spaces of null homotopic maps. In: International Con- ference on Homotopy Theory (Marseille-Luminy, 1988), Astérisque 191, 6, 97-108 (1990) · Zbl 0731.55009
[16] Dwyer, W. G., Wilkerson, C. W.: A new finite loop space at the prime two. J. Amer. Math. Soc. 6, 37-64 (1993) · Zbl 0769.55007 · doi:10.2307/2152794
[17] Dwyer, W. G., Wilkerson, C. W.: Homotopy fixed-point methods for Lie groups and finite loop spaces. Ann. of Math. (2) 139, 395-442 (1994) · Zbl 0801.55007 · doi:10.2307/2946585
[18] Evens, L.: The cohomology ring of a finite group. Trans. Amer. Math. Soc. 101, 224-239 (1961) · Zbl 0104.25101 · doi:10.2307/1993372
[19] Evens, L.: The spectral sequence of a finite group extension stops. Trans. Amer. Math. Soc. 212, 269-277 (1975) · Zbl 0331.18022 · doi:10.2307/1998624
[20] Farjoun, E. D.: Cellular Spaces, Null Spaces and Homotopy Localization. Lecture Notes in Math. 1622, Springer, Berlin (1996) · Zbl 0842.55001 · doi:10.1007/BFb0094429
[21] Harada, M., Kono, A.: Cohomology mod p of the 4-connective fibre space of the classi- fying space of classical Lie groups. Proc. Japan Acad. Ser. A Math. Sci. 60, 63-65 (1984) · Zbl 0559.55021 · doi:10.3792/pjaa.60.63
[22] Harada, M., Kono, A.: Cohomology mod p of the 4-connected cover of the classifying space of simple Lie groups. In: Homotopy Theory and Related Topics (Kyoto, 1984), Adv. Stud. Pure Math. 9, North-Holland, Amsterdam, 109-122 (1987) · Zbl 0656.55013
[23] Hubbuck, J. R.: On homotopy commutative H -spaces. Topology 8, 119-126 (1969) · Zbl 0176.21301 · doi:10.1016/0040-9383(69)90004-4
[24] Jeanneret, A., Osse, A.: The K-theory of p-compact groups. Comment. Math. Helv. 72, 556- 581 (1997) · Zbl 0895.55001 · doi:10.1007/s000140050034
[25] Kuhn, N. J.: On topologically realizing modules over the Steenrod algebra. Ann. of Math. (2) 141, 321-347 (1995) · Zbl 0849.55022 · doi:10.2307/2118523
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.