Cobordism of Morse–Bott Functions on Surfaces

Abstract We study Morse–Bott functions on surfaces and compute their cobordism group by defining Morse–Bott maps, which yield an equivalence relation between functions. The generators of the cobordism group correspond to the number of Morse singularities mod two or endowed with signs and to the number of components of Morse–Bott singular loci endowed with signs. We give geometric cobordism invariants and also prove a formula about the relation of the numbers of different singularity types of a Morse–Bott function on a surface. We also get a similar formula about the Euler characteristic of the surface. We show that for $n \ge 3$ the rank of the cobordism group of Morse–Bott functions on $n$-dimensional manifolds is equal to infinity.