Katsuhiko Kuribayashi

Duality on the (co)chain type levels of maps

slide: PDF.
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 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.
[K3]K. Kuribayashi, Duality on the (co)chain type levels, preprint (2011), arXiv:math.AT/1108.3890.