Algebraic Topology · Problem 2.1.19
Homology — Allen Hatcher
Mathematics — Topology · singular-homology-computations
Problem
Compute the homology groups of the subspace of consisting of the four boundary edges plus all points in the interior whose first coordinate is rational.Intuitive explanation
This problem asks you to find the homology of a wildly disconnected space—think of it as the boundary of a square plus a dense, scattered set of vertical line segments filling the interior. The key insight is recognizing that the interior points form a totally disconnected set (no path connects two points with rational first coordinates unless they're the same), so they contribute nothing to higher homology. Focus on how the four boundary edges connect and what "holes" or independent cycles actually survive in this fragmented space; use the fact that a dense subset of isolated vertical fibers can't create genuine 2-dimensional voids.
Intuitive explanation
This problem asks you to find the homology of a wildly disconnected space—think of it as the boundary of a square plus a dense, scattered set of vertical line segments filling the interior. The key insight is recognizing that the interior points form a totally disconnected set (no path connects two points with rational first coordinates unless they're the same), so they contribute nothing to higher homology. Focus on how the four boundary edges connect and what "holes" or independent cycles actually survive in this fragmented space; use the fact that a dense subset of isolated vertical fibers can't create genuine 2-dimensional voids.