Elimination of \(B_2\) singularities I

For a \(C^\infty \) $C^{\infty }$ stable map \(f:M \rightarrow Q\) $f:M\to Q$ of a compact 3-manifold with boundary into a surface without boundary, we show that f is homotopic to a stable map which has no \(B_2\) $B_2$ points if the Euler characteristic of each connected component of the boundary of M is even.