On the number of singularities of a generic surface with boundary in a 3-manifold (Q1277023)

The authors generalize a theorem of \textit{A. Szücs} [Bull. Lond. Math. Soc. 18, 60-66 (1986; Zbl 0563.57015)], who found a congruence relating the number of triple points, the number of cross caps of a generic smooth map \(f:M\to\mathbb{R}^3\) of a closed smooth surface \(M\) into \(\mathbb{R}^3\) with the Euler characteristic of \(M\). They prove that for the generic smooth map \(f:M\to N\) of a compact surface \(M\) with boundary into a connected 3-manifold \(N\) with boundary \((f^{-1}(\partial N)=\partial M)\), such that one of the conditions: (1) \(M\) is orientable and \(H_1(N)=0\), (2) \(M\) is orientable and \(f_* [M,\partial M]=0\) in \(H_2(N,\partial N)\), (3) \(\partial N\) is orientable and \(f_* [M,\partial M]=0\) in \(H_2(N,\partial N)\) is fulfilled, the following congruence is true \[ T(f)= \sum^k_{i=1} n(x_i,f)+ \sum^m_{j=1} n'(y_j,f) +\chi (M) +\beta_0 (\partial M)\bmod 2 \] where \(T(f)\) is the number of triple points of \(f\), \(\chi(M)\) is the Euler characteristic of \(M\), \(x_1,\dots,x_k\) are the cross caps, \(y_1,\dots,y_m\in f(\partial M)\) are the boundary double points, \(\beta_0\) denotes the number of connected components, and \(n(x_i,f)\), \(n'(y_j,f)\) are the corresponding indices of characteristic points.