Duality on the (co)chain type levels of maps
video: comming soon.
This talk surveys the main results in [K1], [K2] and [K3]. By
using the notion of the levels of objects in a triangulated
category introduced by Avramov et al. in [ABIM], we define the
(co)chain type level for a map, which is a numerical homotopy
invariant of a space over a given space. In this talk, we
investigate topological and homological interpretations of the
levels. Especially, the cochain type level for a map provides a
lower bound of the number of fibrations which construct the map
in an appropriate sense. Moreover the chain type level gives an
upper bound of L.-S. category in the rational case. One might
expect that the two types of levels are related by a sort of
Eckmann-Hilton duality. We then also discuss duality on the
(co)chain type levels with exact functors which connect certain
derived and coderived categories.
[ABIM] L. L. Avramov, R. -O. Buchweitz, S. B. Iyengar and
C. Miller, Homology of perfect complexes, Adv. Math.
223 (2010), 1731-1781.
[K1] K. Kuribayashi, Upper and lower bounds of the (co)chain
type level of a space, to appear in Algebras and Representation
[K2] K. Kuribayashi, On the levels of maps and topological
realization of objects in a triangulated category, to appear in
J. Pure and Appl. Algebra,