RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 Forthcoming seminars Seminar calendar List of seminars Archive by years Register a seminar Search RSS Forthcoming seminars

You may need the following programs to see the files

Geometric Topology Seminar
April 5, 2018 14:00–16:00, Moscow, Math Department of the Higher School of Economics, Room 108

Limit of colimits versus colimit of limits

S. A. Melikhov

Abstract: The cohomology of an infinite simplicial complex is generally different from the limit of the cohomology of its finite subcomplexes. As shown by Milnor, the difference is measured by lim$^1$ – the derived functor of the (inverse) limit functor. While the use of lim$^1$ (and in extreme cases also lim$^2$, lim$^3$, etc.) suffices to deal with passages to the limit in homology and cohomology of compact spaces and of simplicial complexes (and more generally ANRs), homology and cohomology of non-compact non-ANRs has been very poorly understood until recently, due to the lack of understanding of the interaction between limits and colimits (=inverse and direct limits). I will talk about a recent progress in this area. All necessary notions will be defined (apart from homology and cohomology of simplicial complexes).
Here is the plan (probably, for more than one talk): 0) Some basics (lim, colim, lim$^1$, etc.)
1) A new functor lim$^1_fg$ and how it applies to describe the natural homomorphism colim$_i$lim$_k G_{ik}\to$lim$_k$colim$_i G_{ik}$ in the context of cohomology of locally compact separable metric spaces.
3) A correction of the functors lim$^p$ taking into account the topology of the indexing set (when it is uncountable), and a similar correction of “strong” homology and cohomology of Polish spaces (which differ from the usual ones essentially by permuting a limit with a colimit), which removes the notorious dependence of simplest assertions on additional axioms of set theory (and instead provides new spectral sequences for computing the usual homology and cohomology of Polish spaces).