We consider the stable homotopy category S of polyhedra (finite cell complexes). We say that two polyhedra X,Y are in the same genus and write X ∼ Y if X p ≅ Y p for all prime p, where X p denotes the image of Xin the localized category S p. We prove that it is equivalent to the stable isomorphism X∨B 0 ≅Y∨B 0, where B 0 is the wedge of all spheres S n such that π nS(X) is infinite. We also prove that a stable isomorphism X ∨ X ≅ Y ∨ X implies a stable isomorphism X ≅ Y.
@article{bwmeta1.element.doi-10_2478_s11533-011-0123-y,
author = {Yuriy Drozd and Petro Kolesnyk},
title = {On genera of polyhedra},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {401-410},
zbl = {1244.55006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0123-y}
}
Yuriy Drozd; Petro Kolesnyk. On genera of polyhedra. Open Mathematics, Tome 10 (2012) pp. 401-410. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0123-y/
[1] Cohen J.M., Stable Homotopy, Lecture Notes in Math., 165, Springer, Berlin-New York, 1970
[2] Curtis C.W., Reiner I., Methods of Representation Theory I, John Wiley & Sons, New York, 1981
[3] Curtis C.W., Reiner I., Methods of Representation Theory II, John Wiley & Sons, New York, 1987
[4] Drozd Y.A., Adèles and integral representations, Izv. Akad. Nauk SSSR Ser. Mat., 1969, 33, 1080–1088 (in Russian)
[5] Drozd Y.A., Matrix problems, triangulated categories and stable homotopy classes, preprint available at http://arxiv.org/abs/0903.5185 | Zbl 1259.55004
[6] Hu S.-T., Homotopy Theory, Pure Appl. Math., 8, Academic Press, New York-London, 1959
[7] Jacobson N., Structure of Rings, Amer. Math. Soc. Colloq. Publ., 37, American Mathematical Society, Providence, 1956
[8] Sullivan D.P., Geometric Topology: Localization, Periodicity and Galois Symmetry, K-Monogr. Math., 8, Springer, Dordrecht, 2005
[9] Switzer R.W., Algebraic Topology - Homotopy and Homology, Grundlehren Math. Wiss., 212, Springer, Berlin-Heidelberg-New York, 1975 | Zbl 0305.55001