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Geometric Topology Seminar
April 12, 2018 14:00–16:50, Moscow, Math Department of the Higher School of Economics, Room 108

Limit of colimits versus colimit of limits II

S. A. Melikhov

This time we will study a situation where limits do not commute with colimits (and we will discuss specific examples of non-commutativity), but their “commutator” can be computed in terms of lim$^1$ (which will be defined and explained) and a new functor lim$^1_{fg}$. Namely, there are two well-known approximations to the Steenrod–Sitnikov homology $H_n(X)$ of a Polish space $X$: “Čech homology” $qH_n(X)$, which is a limit of colimits of homology groups of finite simplicial complexes, and “Čech homology with compact supports” $pH_n(X)$, which is a colimit of limits of homology groups of the same finite simplicial complexes.
The natural map $pH_n(X)\to qH_n(X)$ is generally neither injective (P. S. Alexandrov, 1947) nor surjective (E. F. Mishchenko, 1953), but it remains an open problem whether it is surjective when $X$ is locally compact. We show that for a locally compact $X$ the dual map in cohomology $pH^n(X)\to qH^n(X)$ is surjective and compute its kernel. This computation has applications to embeddability of compacta in Euclidean spaces. In homology, we show that $pH_n(X)\to qH_n(X)$ is surjective and compute its kernel when $X$ is a “compactohedron”, i.e. contains a compactum whose complement is homeomorphic to a simplicial complex.