Duality on the (co)chain type levels of maps
video: comming soon.
abstract:
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
EckmannHilton 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), 17311781.
[K1] K. Kuribayashi, Upper and lower bounds of the (co)chain
type level of a space, to appear in Algebras and Representation
Theory,
arXiv:math.AT/1006.2669.
[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,
arXiv:math.AT/1102.3271.
