Hatcher, Algebraic Topology, Chapter 2, Section 1
4. Compute the simplicial homology groups of the triangular parachute obtained from by identifying its three vertices to a single point.
Proof. The space , as well as the space obtained by the identification, can be seen in this (crude) sketch.
For the homology calculations, note that has three vertices , , three edges, and one face (call it ). Under the given identification, the space will then have one vertex , three edges (all of the form ), and one face . In particular, then, and . Moreover, we have the following chain complex1:
To determine the simplicial homology of , it suffices to determine the actions of , . Note that maps the face to the linear combination of the edges and that maps any edge to the linear combination . Hence, , 2, , and , and so it follows that
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