Hatcher-Algebraic-Topology · Problem 2.2.28
Homology
Mathematics — Graph Theory · mayer-vietoris-sequence
Problem
(a) Use the Mayer--Vietoris sequence to compute the homology groups of the space obtained from a torus by attaching a M\"obius band via a homeomorphism from the boundary circle of the M\"obius band to the circle in the torus. (b) Do the same for the space obtained by attaching a M\"obius band to via a homeomorphism of its boundary circle to the standard .Intuitive explanation
The Mayer–Vietoris sequence is a tool for computing homology by decomposing a space into two overlapping pieces whose homology you already know. The key insight is to identify your two open sets (the torus/projective plane and a neighborhood of the Möbius band), find their intersection (the boundary circle), then use the long exact sequence to track how homology classes either vanish, persist, or merge when you glue. Pay careful attention to how the boundary map relates homology in the intersection to homology in each piece—this is where the topology of the attaching map reveals itself.
Intuitive explanation
The Mayer–Vietoris sequence is a tool for computing homology by decomposing a space into two overlapping pieces whose homology you already know. The key insight is to identify your two open sets (the torus/projective plane and a neighborhood of the Möbius band), find their intersection (the boundary circle), then use the long exact sequence to track how homology classes either vanish, persist, or merge when you glue. Pay careful attention to how the boundary map relates homology in the intersection to homology in each piece—this is where the topology of the attaching map reveals itself.