On genera of polyhedra

Yuriy Drozd; Petro Kolesnyk

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 401-410
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Yuriy Drozd, and Petro Kolesnyk. "On genera of polyhedra." Open Mathematics 10.2 (2012): 401-410. <http://eudml.org/doc/269672>.

@article{YuriyDrozd2012,
abstract = {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.},
author = {Yuriy Drozd, Petro Kolesnyk},
journal = {Open Mathematics},
keywords = {Stable homotopy category; Polyhedron; Genus; Cancellation; Orders in semisimple algebras; genera of polyhedra; polyhedron; genus; cancellation; orders in semisimple algebras},
language = {eng},
number = {2},
pages = {401-410},
title = {On genera of polyhedra},
url = {http://eudml.org/doc/269672},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Yuriy Drozd
AU - Petro Kolesnyk
TI - On genera of polyhedra
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 401
EP - 410
AB - 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.
LA - eng
KW - Stable homotopy category; Polyhedron; Genus; Cancellation; Orders in semisimple algebras; genera of polyhedra; polyhedron; genus; cancellation; orders in semisimple algebras
UR - http://eudml.org/doc/269672
ER -

References

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  1. [1] Cohen J.M., Stable Homotopy, Lecture Notes in Math., 165, Springer, Berlin-New York, 1970 
  2. [2] Curtis C.W., Reiner I., Methods of Representation Theory I, John Wiley & Sons, New York, 1981 
  3. [3] Curtis C.W., Reiner I., Methods of Representation Theory II, John Wiley & Sons, New York, 1987 
  4. [4] Drozd Y.A., Adèles and integral representations, Izv. Akad. Nauk SSSR Ser. Mat., 1969, 33, 1080–1088 (in Russian) 
  5. [5] Drozd Y.A., Matrix problems, triangulated categories and stable homotopy classes, preprint available at http://arxiv.org/abs/0903.5185 Zbl1259.55004
  6. [6] Hu S.-T., Homotopy Theory, Pure Appl. Math., 8, Academic Press, New York-London, 1959 
  7. [7] Jacobson N., Structure of Rings, Amer. Math. Soc. Colloq. Publ., 37, American Mathematical Society, Providence, 1956 
  8. [8] Sullivan D.P., Geometric Topology: Localization, Periodicity and Galois Symmetry, K-Monogr. Math., 8, Springer, Dordrecht, 2005 
  9. [9] Switzer R.W., Algebraic Topology - Homotopy and Homology, Grundlehren Math. Wiss., 212, Springer, Berlin-Heidelberg-New York, 1975 Zbl0305.55001

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