Hatcher-Algebraic-Topology · Problem 4.I.3
Homotopy Theory
Mathematics — Graph Theory · stable-splittings-of-spaces
Problem
Extending Proposition 4I.3, show that the skeleton of the suspension of a high-dimensional lens space with fundamental group of order is homotopy equivalent to the wedge sum of the skeleta of the spaces , if these 's are chosen to have the minimum number of cells in each dimension, as described in the remarks following the proof.Intuitive explanation
The key insight is that suspension preserves the skeletal structure while amplifying homological information: suspending a space adds a dimension uniformly, so the skeleton of the suspension captures how the original space's lower skeleta interact through the lens of its fundamental group's -torsion. By decomposing the lens space into minimal-cell building blocks (each encoding a single torsion generator), you can show that suspension respects this additive structure—the skeleton becomes a wedge sum because the cells in complementary dimensions don't interact within that range. This reduces a complex global problem to understanding how suspension acts on simpler pieces.
Intuitive explanation
The key insight is that suspension preserves the skeletal structure while amplifying homological information: suspending a space adds a dimension uniformly, so the skeleton of the suspension captures how the original space's lower skeleta interact through the lens of its fundamental group's -torsion. By decomposing the lens space into minimal-cell building blocks (each encoding a single torsion generator), you can show that suspension respects this additive structure—the skeleton becomes a wedge sum because the cells in complementary dimensions don't interact within that range. This reduces a complex global problem to understanding how suspension acts on simpler pieces.