I was going over Example 2.10 of Botnan, Magnus Bakke. "Topological Data Analysis Spring 2020.".
I thought the number of holes is 3 which includes the hole from the linear combination of c1+c2.. but seems not. Could someone tell me why not?
Example 2.10
Today, I presented this exercise to my colleagues and After a discussion w/h them, we realised that $Z_1(K)$ actually contains linear combinations of cycles as they are mapped to zero, i.e., they form a kernel!, so the 3 coming from dim$Z_1(K)$ was covering my case!