Hatcher Exercise 2.1.4

Form the triangular parachute from a regular -simplex with all vertices identified to a single vertex. Label the vertex , the edges , and the face . We wish to determine the simplicial homology groups. We have and (of course, all brackets in this context mean the free abelian group on the enclosed generators).

For , we have and . For , we have . Thus, and .

For , we have . Then, , and is precisely . and higher are all the zero map.

It’s clear that we have .

Next, . Consider the homomorphism given by . Observe that is precisely , which is exactly an equivalent description of . We conclude .

is because is injective, and all higher homology groups are zero as well.