Upper and lower bounds of the (co)chain type level of a space

Details Upper and lower bounds of the (co)chain type level of a space Upper and lower bounds of the (co)chain type level of a space

2010-06-14

arxiv.org/abs/1006.2669v3

Mathematics

Algebraic Topology

22 pages. Minor corrections

Scientific paper

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 invariant for a principal bundle 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.

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