Acyclicity of complements of σ-compact weakly infinite-dimensional subsets in Hilbert cube

Abstract

Acyclicity properties of subsets in the Hilbert cube are a subject of intensive study in infinitedimensional topology. The complement of a compact, finite-dimensional subset in the Hilbert cube is acyclic. This result was generalised to compact C-spaces, and compact trt-dimensional spaces. The final generalisation of the above result was proved for weakly infinite-dimensional compact subspaces of the Hilbert cube. It is a well-known fact that the Hilbert cube cannot be made disconnected by removing a weakly infinite-dimensional (not necessarily closed) subspace. Some strongly infinite-dimensional compacta can separate the Hilbert cube. Further, a previous study proved that the complement of a weakly infinite-dimensional subset of the Hilbert cube is continuum connected. However, path-connectivity of the complement of a noncompact weakly infinite-dimensional subspace in the Hilbert cube is not established. In 2017, a result on the homology of complements of compact weakly infinite-dimensional spaces was proved. In particular, this showed that the complement of a weakly infinite-dimensional compact subspace of the Hilbert cube is path-connected since it has trivial 0-dimensional homology. In this research, we have generalised this result to complements of 𝜎-compact weakly infinite-dimensional spaces. We prove that if 𝑋 = ⋃𝑖=1 ∞ 𝑋𝑖 is a 𝜎-compact weakly infinite-dimensional subspace of 𝑄, where each 𝑋𝑖 is compact, then the complement of 𝑋 has trivial 0-dimensional Steenrod homology. This argument is based on using Milnor’s short exact sequence for Steenrod homology to the sequence of spaces 𝐾𝑖 , which are closed tubular neighbourhoods of paths in the complement of 𝑋𝑖 . Since there are continuum connected spaces with non-trivial 0-dimensional Steenrod homology, the above theorem is a generalisation of the continuum connectedness theorem proved in literature.