On the same n-types for the wedges of the Eilenberg-Maclane spaces

Abstract

Given a connected CW-space X, SNT(X) denotes the set of all homotopy types [X′] such that the Postnikov approximations X ⁽ⁿ⁾ and X′⁽ⁿ⁾ are homotopy equivalent for all n. The main purpose of this paper is to show that the set of all the same homotopy ntypes of the suspension of the wedges of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type of itself, i.e., SNT(Σ(K(ℤ, 2a ₁) ∨ K(ℤ, 2a ₂) ∨∙∙∙∨ K(ℤ, 2a _k))) = * for a ₁ < a ₂ < ∙∙∙ < a _k, as a far more general conjecture than the original one of the same n-type posed by McGibbon and Møller (in [McGibbon, C. A. and Møller, J. M., On infinite dimensional spaces that are rationally equivalent to a bouquet of spheres, Proceedings of the 1990 Barcelona Conference on Algebraic Topology, Lecture Notes in Math., 1509, 1992, 285–293].)

Keywords

Same n-type / Aut / Basic Whitehead product / Samelson product / Bott-Samelson theorem / Tensor algebra / Cartan-Serre theorem / Hopf-Thom theorem

References

[1] Adams J. F.. An example in homotopy theory. Proc. Camb. Phil. Soc., 1957, 53: 922-923

[2] Arkowitz M., Curjel C. R.. Homotopy commutators of finite order (I). Quart. J. Math. Oxford, Ser. 2, 1963, 14: 213-219

[3] Bott R., Samelson H.. On the Pontryagin product in spaces of paths. Comment. Math. Helv., 1953, 27: 320-337

[4] Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations, Lecture Notes Math., 304, 1972.

[5] Félix Y., Halperin S., Thomas J. C.. Rational Homotopy Theory, GTM, 2001, New York: Springer-Verlag

[6] Gray B. I.. Spaces on the same n-type for all n. Topology, 1966, 5: 241-243

[7] Hilton P. J.. On the homotopy groups of the union of spheres. J. Lond. Math. Soc., 1955, 30(2): 154-172

[8] Lee D.-W.. On the same n-type conjecture for the suspension of the infinite complex projective space. Proc. Amer. Math. Soc., 2009, 137(3): 1161-1168

[9] Lee D.-W.. On the same n-type structure for the suspension of the Eilenberg-MacLane spaces. J. Pure Appl. Algebra, 2010, 214: 2027-2032

[10] Lee D.-W.. On the same n-type of the suspension of the infinite quaternionic projective space. J. Pure Appl. Algebra, 2013, 217: 1325-1334

[11] McGibbon C. A., Møller J. M.. On infinite dimensional spaces that are rationally equivalent to a bouquet of spheres, Proceedings of the 1990 Barcelona Conference on Algebraic Topology. Lecture Notes Math., 1992, 1509: 285-293

[12] McGibbon C. A., Steiner R.. Some questions about the first derived functor of the inverse limit. J. Pure Appl. Algebra, 1995, 103(3): 325-340

[13] Morisugi K.. Projective elements in K-theory and self-maps of SCP8. J. Math. Kyoto Univ., 1998, 38: 151-165

[14] Moriya S.. The de Rham homotopy theory and differential graded category. Math. Z., 2012, 271: 961-1010

[15] Ravenel D. C.. Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, 1992, Princeton: Princeton University Press

[16] Ruiz, F. G., A note on residue formulas for the Euler class of sphere fibrations, Chin. Ann. Math. Ser. B., 32(4), 2011, 615–618.

[17] Scheerer H.. On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces. Manuscripta Math., 1984, 51: 63-87

[18] Scoville N. A.. Lusternik-Schnirelmann category and the connectivity of X. Algebr. Geom. Topol., 2012, 12: 435-448

[19] Spanier E.. Algebraic Topology, 1966, New York: McGraw-Hill

[20] Whitehead G. W.. Elements of Homotopy Theory, 1978, New York: Springer-Verlag

[21] Wilkerson, C. W., Classification of spaces of the same n-type for all n, Proc. Amer. Math. Soc., 60, 1976, 279–285.