Let
be an Alexandrov space collapsing to an Alexandrov space
of lower dimension.
Suppose
has no proper
extremal subsets and let
denote a regular fiber. We slightly improve the result of Perelman to construct an infinitely
long exact sequence of homotopy groups and a spectral sequence of cohomology groups for
the pair
.
The proof is an application of the good coverings of Alexandrov spaces introduced by
Mitsuishi and Yamaguchi. We also extend this result to each primitive extremal subset
of .
Keywords
Alexandrov spaces, collapse, good coverings, extremal
subsets