  A Kosterlitz-Thouless phase transition in the ground state of an
antiferromagnetic spin-$\frac{1}{2}$ Heisenberg chain with nearest and
next-nearest-neighbor interactions is re-investigated from a different
perspective: An unequivocal correspondence is found between components of the
scalar product $\langle \Psi_0| \Psi_0 \rangle$ and geometrical objects. One
can classify these objects according to whether any two of them can be
transformed into each other in a continuous way (belong to the same homotopy
class). A finite size scaling of the "connection term`` $\langle
\Psi_0|\partial \Psi_0 \rangle$ with respect to chain length (16, 18, 20, 22,
24 spins) for each homotopy class of above mentioned objects leads to the
critical value of $\lambda$ with rather high accuracy.
