Algebraic Topology · Problem 2.1.4

You're collapsing the boundary of a 2-simplex down to a point, which creates a space that "looks like" a sphere from a homological perspective. The key insight is to track how this identification affects the chain complex: the three vertices become one, so 0-chains collapse, but the three edges and the 2-face remain distinct and non-degenerate. Use the boundary maps carefully to see which cycles survive the quotient—you'll find that the parachute has a non-trivial 1-cycle (from the "rim" of edges) and a 2-cycle (the collapsed face), giving you the homology of $S^2$.

You're collapsing the boundary of a 2-simplex down to a point, which creates a space that "looks like" a sphere from a homological perspective. The key insight is to track how this identification affects the chain complex: the three vertices become one, so 0-chains collapse, but the three edges and the 2-face remain distinct and non-degenerate. Use the boundary maps carefully to see which cycles survive the quotient—you'll find that the parachute has a non-trivial 1-cycle (from the "rim" of edges) and a 2-cycle (the collapsed face), giving you the homology of $S^2$.