Algebras and Representation Theory

, Volume 16, Issue 1, pp 129–153 | Cite as

Upper and Lower Bounds of the (co)chain Type Level of a Space

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Abstract

We establish an upper bound for the cochain type level of the total space of a pull-back fibration. It explains to us why the numerical invariants for principal bundles over the sphere are less than or equal to two. Moreover computational examples of the levels of path spaces and Borel constructions, including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We also show that the chain type level of the homotopy fibre of a map is greater than the E-category in the sense of Kahl, which is an algebraic approximation of the Lusternik-Schnirelmann category of the map. The inequality fits between the grade and the projective dimension of the cohomology of the homotopy fibre.

Keywords

Level Semi-free module Triangulated category Formal space Borel construction L.-S. category 

Mathematics Subject Classifications (2010)

16E45 18E30 55R20 13D07 
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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Faculty of ScienceShinshu UniversityMatsumotoJapan

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