The barcode of persistence is thus for K/Z 3 in Fig. : Consequenctly, the birthdeath time coincides for F(K) and F(K/Z 3 )which verifies our result in Theorem 5.2. One can also justify results from Theorem 4.3. Clearly, for K 8 , its quotient space is a cone with one sigular point (nontrivial stabilizer group). Example 2 (Lattice group action on simple topological crystal).