  A topological approach to the theory of equilibrium phase transitions in
statistical physics is based on the Topological Hypothesis (TH), which claims
that phase transitions are due to changes of the topology of suitable
submanifolds in the configuration space. In this paper we examine in detail the
anti-ferromagnetic mean-field XY model and study topology of the sub-energy
manifolds. The latter can be interpreted mechanically as configuration space of
a linkage with one telescopic leg. We use methods of Morse theory to describe
explicitly the Betti numbers of this configuration space. We apply these
results to the anti-ferromagnetic mean-field XY model and compute the
exponential growth rate of the total Betti number. The previous authors instead
of the total Betti number studied the Euler characteristic. We show that in the
presence of an external magnetic field the model undergoes a single "total
Betti number phase transition".
