Decompositions of tame parametrised chain complexes

Abstract: We show a classification result for tame filtered chain complexes with indexing posets of dimension 1. Filtered chain complexes, on the one hand, arise from filtrations of finite point clouds. On the other hand, they are the cofibrant replacements of any tame parametrised chain complex, once an appropriate model category structure is defined. Posets of dimension 1 form the other fundamental piece of this presentation. Examples of these are given by natural, integer, real numbers with the usual order, trees and zigzags.

Our classification result states that there are only two types of cofibrant (filtered) indecomposables in the category tame(Q, ch(C)), where Q is a poset of dimension 1, and C is an appropriate category. They are either disks (trivial homology) of indecomposable projectives in tame(Q, C) or spheres (non-trivial homology) on the minimal projective resolution of the homology of the chain complex. Both of them are nonzero in only two consecutive degrees. If time allows, we will also show a technique, based on “glueing”, to construct indecomposables in a functor category by “smaller” indecomposables. Examples obtained in this way will also show that the results presented above, for functors indexed by posets of dimension 1, are not immediately generalisable.

This presentation is based on joint work with Wojciech Chachólski, Barbara Giunti and Claudia Landi.

algebraic topologycategory theory

Audience: researchers in the topic

Bilkent Topology Seminar

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