Before revision:  Also, the \mathbb Z _2-reduction is computable and, if M is (k-1)-connected, it determines whether there is a map f\colon |K| \to M which has an even number of crossings of f(\sigma) and f(\tau) for each pair (\sigma, \tau) of disjoint k-simplices of K.
After revision:  Also, the \mathbb{Z _2-reduction is computable and, if M is (k-1)-connected, it determines whether there is a map f\colon |K| \to M which has an even number of crossings of f(\sigma) and f(\tau) for each pair (\sigma, \tau) of disjoint k-simplices of K.