Forms and chain complexes II (Ex)

Definition 0.1 [Ranicki1980, Proposition 4.6] . An n-dimensional $<wikitex>; {{beginthm|Definition|{{cite|Ranicki1980|Proposition 4.6}} }} An n-dimensional $\epsilon$-symmetric/$\epsilon$-quadratic) complex $(C, \phi)$/$(C, \psi)$ is called ''well-connected'' if $H_0(C)= 0$. {{endthm}} {{beginthm|Exercise}} Show that the boundary of a well-connected \epsilon$-symmetric/$\epsilon$-quadratic complex $(C, \phi)$/$(C, \psi)$ is called well-connected if $H_0(C)= 0$.

Exercise 0.2. Show that the boundary of a well-connected $1$-dimensional symmetric/quadratic complex gives a hyperbolic form.