Homotopy classes that are trivial mod ℱ

Arkowitz, Martin¹ ; Strom, Jeffrey¹

¹Dartmouth College, Hanover NH 03755, USA

Algebraic and Geometric Topology, Tome 1 (2001) no. 1, pp. 381-409

Résumé

If ℱ is a collection of topological spaces, then a homotopy class α in [X,Y ] is called ℱ–trivial if

for all A ∈ℱ. In this paper we study the collection Zℱ(X,Y ) of all ℱ–trivial homotopy classes in [X,Y ] when ℱ = S, the collection of spheres, ℱ = ℳ, the collection of Moore spaces, and F = Σ, the collection of suspensions. Clearly

and we find examples of finite complexes X and Y for which these inclusions are strict. We are also interested in Zℱ(X) = Zℱ(X,X), which under composition has the structure of a semigroup with zero. We show that if X is a finite dimensional complex and ℱ = S, ℳ or Σ, then the semigroup Zℱ(X) is nilpotent. More precisely, the nilpotency of Zℱ(X) is bounded above by the ℱ–killing length of X, a new numerical invariant which equals the number of steps it takes to make X contractible by successively attaching cones on wedges of spaces in ℱ, and this in turn is bounded above by the ℱ–cone length of X. We then calculate or estimate the nilpotency of Zℱ(X) when ℱ = S, ℳ or Σ for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems.

Détail

DOI : 10.2140/agt.2001.1.381

Keywords: cone length, trivial homotopy

Bibliographie

[1] J F Adams, The sphere, considered as an $H$–space $\mathrm{mod} p$, Quart. J. Math. Oxford. Ser. $(2)$ 12 (1961) 52

[2] M Arkowitz, K I Maruyama, D Stanley, The semigroup of self-homotopy classes which induce zero on homotopy groups, Kyushu J. Math. 56 (2002) 89

[3] H J Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press (1989)

[4] R Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960) 249

[5] J D Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998) 284

[6] F R Cohen, J C Moore, J A Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. $(2)$ 110 (1979) 549

[7] O Cornea, There is just one rational cone-length, Trans. Amer. Math. Soc. 344 (1994) 835

[8] B Eckmann, P J Hilton, Operators and cooperators in homotopy theory, Math. Ann. 141 (1960)

[9] S Eilenberg, S Mac Lane, On the groups $H(\Pi,n)$ II: Methods of computation, Ann. of Math. $(2)$ 60 (1954) 49

[10] Y Félix, A Murillo, A bound for the nilpotency of a group of self homotopy equivalences, Proc. Amer. Math. Soc. 126 (1998) 625

[11] P Freyd, Stable homotopy, from: "Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965)", Springer (1966) 121

[12] P Hilton, Homotopy theory and duality, Gordon and Breach Science Publishers (1965)

[13] P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Mathematics Studies 15, North-Holland Publishing Co. (1975)

[14] I M James, Reduced product spaces, Ann. of Math. $(2)$ 62 (1955) 170

[15] I M James, Spaces associated with Stiefel manifolds, Proc. London Math. Soc. $(3)$ 9 (1959) 115

[16] I M James, On category, in the sense of Lusternik–Schnirelmann, Topology 17 (1978) 331

[17] R M Kane, The homology of Hopf spaces, North-Holland Mathematical Library 40, North-Holland Publishing Co. (1988)

[18] H J Marcum, D Randall, A note on self-mappings of quaternionic projective spaces, An. Acad. Brasil. Ci. 48 (1976) 7

[19] C A Mcgibbon, Phantom maps, from: "Handbook of algebraic topology", North-Holland (1995) 1209

[20] C A Mcgibbon, Self-maps of projective spaces, Trans. Amer. Math. Soc. 271 (1982) 325

[21] M Mimura, H Toda, Cohomology operations and homotopy of compact Lie groups. I, Topology 9 (1970) 317

[22] H Oshima, Self-homotopy set of a Hopf space, Quart. J. Math. Oxford Ser. $(2)$ 50 (1999) 483

[23] P Pavešić, On the group $\mathrm{Aut}_\Omega(X)$, Proc. Edinb. Math. Soc. $(2)$ 45 (2002) 673

[24] G J Porter, The homotopy groups of wedges of suspensions, Amer. J. Math. 88 (1966) 655

[25] Y B Rudyak, On category weight and its applications, Topology 38 (1999) 37

[26] H Scheerer, D Tanré, Variation zum Konzept der Lusternik–Schnirelmann–Kategorie, Math. Nachr. 207 (1999) 183

[27] P A Schweitzer, Secondary cohomology operations induced by the diagonal mapping, Topology 3 (1965) 337

[28] W Sierpiński, Elementary theory of numbers, North-Holland Mathematical Library 31, North-Holland Publishing Co. (1988)

[29] W Singhof, On the Lusternik–Schnirelmann category of Lie groups, Math. Z. 145 (1975) 111

[30] J Strom, Essential category weight and phantom maps, from: "Cohomological methods in homotopy theory (Bellaterra, 1998)", Progr. Math. 196, Birkhäuser (2001) 409

[31] H Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton University Press (1962)

[32] G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer (1978)

[33] K Yamaguchi, Self-homotopy equivalences of $\mathrm SO(4)$, Hiroshima Math. J. 30 (2000) 129