A survey of homotopy nilpotency and co-nilpotency. (English) Zbl 1469.55005

Recall the concept of the homotopy nilpotency and co-nilpotency of spaces as follows. First, let \(X\) be an \(H\)-space, \(\varphi_{X,1}=\mbox{id}_X:X\to X\) and let \(\varphi_{X,2}:X^2\to X\) the commutator map. Then define the family of maps \(\{\varphi_{X,n}:X^n\to X\}_{n\geq 1}\) inductively by the equality \(\varphi_{X,n+1}=\varphi_{X,2}\circ (\mbox{id}_X\times \varphi_{X,n}).\) An \(H\)-space \(X\) is called homotopy nilpotent of class \(n\) if \(\varphi_{X,n+1}\) is null-homotopic but \(\varphi_{X,n}\) is not. In this case, we write \(\mbox{nil }X=n\). Dually we can define the concept of co-nilpotency of a co-\(H\)-space. Next, define the concept of the homotopy nilpotency (resp. homotopy co-nilpotency) of a pointed space \(X\) by means of its loop space \(\Omega X\) (resp. its suspension space \(\Sigma X\)).
This paper is concerned with reviewing known results and stating new results on the homotopy nilpotency and co-nilpotency of spaces. Especially, the author considers the systematic study of the homotopy nilpotency of a homogenous space \(G/K\) for a Lie group \(G\) with its closed subgroup \(K<G\). In particular, he studies the homotopy nilpotency of the loop spaces \(\Omega G_{n,m}(\mathbb{K})\), \(\Omega F_{n_1,n_2,\dots ,n_k}(\mathbb{K})\) and \(\Omega V_{n,m}(\mathbb{K})\) for \(\mathbb{K}=\mathbb{R}\), \(\mathbb{C}\) or \(\mathbb{H}\), where \(G_{n,m}(\mathbb{K})\), \(F_{n_1,n_2,\dots ,n_k}(\mathbb{K})\) and \(V_{n,m}(\mathbb{K})\)) denote the Grassmann, flag and the Stiefel manifold of the field of real and complex numbers or the skew \(\mathbb{R}\)-algebra of quaternions.


55P15 Classification of homotopy type
20F18 Nilpotent groups
55P45 \(H\)-spaces and duals
55P60 Localization and completion in homotopy theory
Full Text: DOI


[1] Martin Arkowitz. Induced mappings of homology decompositions. InHomotopy and geometry (Warsaw, 1997), volume 45 ofBanach Center Publ., pages 225-233. Polish Acad. Sci. Inst. Math., Warsaw, 1998, https://eudml.org/doc/208905. · Zbl 0941.55008
[2] Martin Arkowitz.Introduction to homotopy theory. Universitext. Springer, New York, 2011, doi: 10.1007/978-1-4419-7329-0. · Zbl 1232.55001
[3] Martin Arkowitz, Marek Golasiński. Co-H-structures on Moore spaces of type(G,2). Canad. J. Math., 46(4):673-686, 1994, doi: 10.4153/CJM-1994-037-0. · Zbl 0829.55006
[4] Clemens Berger, Dominique Bourn. Central reflections and nilpotency in exact Mal’tsev categories.J. Homotopy Relat. Struct., 12(4):765-835, 2017, doi: 10.1007/s40062-0160165-8. · Zbl 1397.18021
[5] I. Berstein, T. Ganea. Homotopical nilpotency.Illinois J. Math., 5:99-130, 1961, doi: 10.1215/ijm/1255629648. · Zbl 0096.17602
[6] Israel Berstein, Emmanuel Dror. On the homotopy type of non-simply-connected coH-spaces.Illinois J. Math., 20(3):528-534, 1976, doi: 10.1215/ijm/1256049794. · Zbl 0323.55018
[7] Georg Biedermann, William G. Dwyer. Homotopy nilpotent groups.Algebr. Geom. Topol., 10(1):33-61, 2010, doi: 10.2140/agt.2010.10.33. · Zbl 1329.55008
[8] A. K. Bousfield, D. M. Kan.Homotopy limits, completions and localizations. Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972, doi: 10.1007/978-3-540-38117-4. · Zbl 0259.55004
[9] Cristina Costoya, Jérôme Scherer, Antonio Viruel. A torus theorem for homotopy nilpotent loop spaces.Ark. Mat., 56(1):53-71, 2018, doi: 10.4310/ARKIV.2018.v56.n1.a5. · Zbl 1396.55006
[10] M. C. Crabb, W. A. Sutherland, P. Zhang. Homotopy nilpotency.Quart. J. Math. Oxford Ser. (2), 50(198):179-196, 1999, doi: 10.1093/qjmath/50.198.179. · Zbl 0932.55007
[11] Ethan S. Devinatz, Michael J. Hopkins, Jeffrey H. Smith. Nilpotence and stable homotopy theory. I.Ann. of Math. (2), 128(2):207-241, 1988, doi: 10.2307/1971440. · Zbl 0673.55008
[12] Rosona Eldred. Goodwillie calculus via adjunction and LS cocategory.Homology Homotopy Appl., 18(2):31-58, 2016, doi: 10.4310/HHA.2016.v18.n2.a2. · Zbl 1360.55013
[13] T. Ganea. On the loop spaces of projective spaces.J. Math. Mech., 16:853-858, 1967, doi: 10.2307/45277192. · Zbl 0148.17104
[14] Tudor Ganea. Lusternik-Schnirelmann category and cocategory.Proc. London Math. Soc. (3), 10:623-639, 1960, doi: 10.1112/plms/s3-10.1.623. · Zbl 0101.15802
[15] Tudor Ganea. Cogroups and suspensions.Invent. Math., 9:185-197, 1969/70, doi: 10.1007/BF01404323. · Zbl 0194.55103
[16] M. Golasiński. Homotopy nilpotency of some homogeneous spaces. Submitted, 2020.
[17] M. Golasiński. On two equivalent notions of the homotopy nilpotency. Submitted, 2020.
[18] MarekGolasiński,DacibergLimaGonçalves,PeterWong.Exponentsof [Ω(Sr+1),Ω(Y)]. InAlgebraic topology and related topics, Trends Math., pages 103-122. Birkhäuser/Springer, Singapore, 2019, doi: 10.1007/978-981-13-5742-8_7.
[19] Marek Golasiński, John R. Klein. On maps into a co-H-space.Hiroshima Math. J., 28(2):321-327, 1998, doi: 10.32917/hmj/1206126763. · Zbl 0921.55009
[20] Peter Hilton.Homotopy theory and duality. Gordon and Breach Science Publishers, New York-London-Paris, 1965. · Zbl 0155.50801
[21] M. J. Hopkins. Formulations of cocategory and the iterated suspension. 113:212-226, 1984, http://www.numdam.org/item/AST_1984__113-114__212_0. · Zbl 0559.55013
[22] M. J. Hopkins. Nilpotence and finiteH-spaces.Israel J. Math., 66(1-3):238-246, 1989, doi: 10.1007/BF02765895. · Zbl 0684.55007
[23] M. Hovey. Lusternik-Schnirelmann cocategory.Illinois J. Math., 37(2):224-239, 1993, doi: 10.1215/ijm/1255987145. · Zbl 0802.55003
[24] I. M. James. On fibre spaces and nilpotency. II.Math. Proc. Cambridge Philos. Soc., 86(2):215-217, 1979, doi: 10.1017/S0305004100056024. · Zbl 0411.55014
[25] Donald W. Kahn. A note onH-spaces and Postnikov systems of spheres.Proc. Amer. Math. Soc., 15:300-307, 1964, doi: 10.2307/2034058. · Zbl 0127.38904
[26] Shizuo Kaji, Daisuke Kishimoto. Homotopy nilpotency inp-regular loop spaces.Math. Z., 264(1):209-224, 2010, doi: 10.1007/s00209-008-0459-6. · Zbl 1185.55007
[27] Daisuke Kishimoto. Homotopy nilpotency in localized SU(n).Homology Homotopy Appl., 11(1):61-79, 2009, http://projecteuclid.org/euclid.hha/1251832560. · Zbl 1181.55011
[28] Willi Meier. Homotopy nilpotency and localization.Math. Z., 161(2):169-183, 1978, doi: 10.1007/BF01214929. · Zbl 0367.55009
[29] Aniceto Murillo, Antonio Viruel. Lusternik-Schnirelmann cocategory: a Whitehead dual approach. InCohomological methods in homotopy theory (Bellaterra, 1998), volume 196 ofProgr. Math., pages 323-347. Birkhäuser, Basel, 2001, doi: 10.1007/978-30348-8312-2_20. · Zbl 0986.55006
[30] G. J. Porter. Homotopical nilpotence ofS3.Proc. Amer. Math. Soc., 15:681-682, 1964, doi: 10.2307/2034577. · Zbl 0133.16501
[31] G. J. Porter. Higher-order Whitehead products.Topology,3:123-135,1965, doi: 10.1016/0040-9383(65)90039-X. · Zbl 0149.20204
[32] Dieter Puppe. Homotopiemengen und ihre induzierten Abbildungen. I.Math. Z., 69:299-344, 1958, doi: 10.1007/BF01187411. · Zbl 0092.39803
[33] Vidhyanath K. Rao. Spin(n)is not homotopy nilpotent forně7.Topology, 32(2):239- 249, 1993, doi: 10.1016/0040-9383(93)90017-P. · Zbl 0776.55006
[34] Vidhyānāth K. Rao. Homotopy nilpotent Lie groups have no torsion in homology. Manuscripta Math., 92(4):455-462, 1997, doi: 10.1007/BF02678205. · Zbl 0880.22005
[35] Victor P. Snaith. Some nilpotentH-spaces.Osaka Math. J., 13(1):145-156, 1976, http: //projecteuclid.org/euclid.ojm/1200769310. · Zbl 0329.55010
[36] James Stasheff.H-spaces from a homotopy point of view. Lecture Notes in Mathematics, Vol. 161. Springer-Verlag, Berlin-New York, 1970, doi: 10.1007/BFb0065896. · Zbl 0205.27701
[37] Stephen Theriault. The dual polyhedral product, cocategory and nilpotence.Adv. Math., 340:138-192, 2018, doi: 10.1016/j.aim.2018.09.037. · Zbl 1432.55003
[38] George W. Whitehead. On products in homotopy groups.Ann. of Math (2), 47:460- 475, 1946, doi: 10.2307/1969085. · Zbl 0060.41106
[39] George W. Whitehead. On mappings into group-like spaces.Comment. Math. Helv., 28:320-328, 1954, doi: 10.1007/BF02566938. · Zbl 0058.16801
[40] Nobuaki Yagita. Homotopy nilpotency for simply connected Lie groups.Bull. London Math. Soc., 25(5):481-486, 1993, doi: 10.1112/blms/25.5.481. · Zbl 0796.57015
[41] Donald Yau. Clapp-Puppe type Lusternik-Schnirelmann (co)category in a model category.J. Korean Math. Soc., 39(2):163-191, 2002, doi: 10.4134/JKMS.2002.39.2.163. · Zbl 0999.55002
[42] Alexander Zabrodsky.Hopf spaces. North-Holland Publishing Co., AmsterdamNew York-Oxford, 1976. North-Holland Mathematics Studies, Vol. 22, Notas de Matemática, No. 59 · Zbl 0339.55009
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.