Generic fibre product of one-dimensional manifolds (Q1873308)

Let \(M\) and \(N\) be connected, compact, one-dimensional manifolds of infinite class and let \(G(M,N)\) be the set of pairs of mappings \((f,g)\) such that \(f: M\to\mathbb R\) and \(g: N\to\mathbb R\) are good Morse functions (the critical values are distinct) of infinite class with \(f(x)\neq g(y)\) for any critical point \(x\) of \(f\) and any critical point \(y\) of \(g\). Then by well known results from differential topology \(G(M,N)\) is open and dense in the product space of functional spaces equipped with the strong topology that coincides with the weak topology in this case. Moreover for any \((f,g)\) in \(G(M,N)\) the fibred product \(f_\top g= \{(x,y)\mid f(x)= g(y)\}\) is a compact one-dimensional submanifold of the torus \(M\times N\) and by the classification theorem of one-dimensional manifolds is characterized by the finite number of its connected components that are diffeomorphic to the circle. Since \(M\times N\) is a torus the components of \(f_\top g\) can be contractible. The author gives a simple efficient algorithm, in terms of the critical points of \(f\) and \(g\), to determine the number of contractible components of \(f_\top g\) and the number of non-contractible components.