

Geometric Topology Seminar
September 24, 2018 15:30–17:30, Moscow, Math Department of the HSE (Usachyova, 6), room 212






Corrected derived limits, or How to expel set theory from algebraic topology
S. A. Melikhov^{} 
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Abstract:
We address the following problem
"Does direct limit ($colim$) commute with inverse limit ($\lim$) when higher derived limits ($\lim^p$) are taken into account?"
in a natural topological context: $\lim$ and $colim$ are applied to the homology (or cohomology) of finite simplicial complexes which approximate in two ways (as an inverse system of direct systems, or as a direct system of inverse systems) a given separable metrizable space $X$.
It is likely that this problem (which is stated precisely in terms of a BousfieldKan spectral sequence) cannot be solved in the usual set theory ZFC. At least, this is known to be so for some specific (very simple) spaces $X$. But we show that the problem has positive solution (for any $X$ in the case of cohomology, and for finitedimensional $X$ in the case of homology), if the derived functors $\lim^p$ are "corrected" so as to take into account a natural topology on the indexing poset. The correction involves sheaves (particularly, Leray sheaves), as well as probability measures and measurable functions. Further details can be found in https://arxiv.org/abs/1809.00022
The theory of the "corrected" $\lim^p$ is still in its infancy, and there are a lot of very basic problems about them (see the last page in the linked paper) which could well be the subject of a course/diploma/thesis work by any student who has some idea about sheaves and homology algebra.
Website:
https://arxiv.org/abs/1809.00022
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