We define the cobordism group of Morse functions on manifolds and show that it is an infinite cyclic group for dimension two in the oriented case. We also give an explicit Morse function which gives a generator of the group.
Kazuichi IKEGAMI. Osamu SAEKI. "Cobordism group of Morse functions on surfaces." J. Math. Soc. Japan 55 (4) 1081 - 1094, October, 2003. https://doi.org/10.2969/jmsj/1191418765