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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

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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.
2) A uniqueness theorem for axiomatic homology and cohomology of Polish spaces (=separable complete metric spaces), which is a common generalization of two old uniqueness theorems by Milnor (for axiomatic homology and cohomology of infinite simplicial complexes and of compact metric spaces). The proof of the theorem provides a combinatorial description of homology and cohomology of Polish spaces, in terms of simplicial chain complexes which satisfy lim colim = colim lim.
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).
4) A common correction of strong shape and compactly generated strong shape (which differ from each other essentially by permuting a limit with a colimit) of Polish spaces, which takes into account the topology on the indexing sets.
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