On the Betti number of the image of a generic map (Q1359436)

Let \(f:M\to N\) be a proper map of class \(C^2\) of a closed \(m\)-dimensional manifold into an \(n\)-dimensional manifold with \(k=n -m>0\). The authors have already been studying the question how one can recognize that \(f\) is an embedding assuming that \(f\) belongs to a certain class of differentiable maps. The class of maps treated here are the generic maps for double points which has been introduced by \textit{F. Ronga} [Compositio Math. 27, 223-232 (1973; Zbl 0271.57009)]. An \(f:M \to N\) as above is generic for the double points if the 1-jet extension \(j^1f: M\to J^1 (M,N)\) of \(f\) is transverse to the submanifolds \(\Sigma^i =\{\alpha \in J^1(M,N) \mid \dim \ker \alpha =i\}\) for all \(i\) and if the \(l\)-fold product map \(f^l:M^l \to N^l\) is transverse to the diagonal \(\delta^l_N\) of \(N^l\) off the super diagonal \(\Delta^l_M =\{(x_1, \dots, x_l) \in M^l \mid x_i= x_j\) for some \(i\neq j\}\) of \(M^l\) for all \(l=2,3,4, \dots\). The main result says: Let \(f:M\to N\) be as above and generic for double points, \(k= n- m>0\). Then \(f\) is a differentiable embedding if and only if \(\breve\beta_{m-k+1} (M)= \breve \beta_{m-k+1} (f(M))\) and \(w_k(f) =v(f)\). The \(\breve\beta\)'s are Betti numbers with respect to Čech homology, \(w_k(f)\) is the \(k\)-th Stiefel-Whitney class, and \(v(f)\) is the image of the fundamental class \([M] \in H^c_m(M)\) by the composite \[ H^c_m(M) \to H^c_m (N) \to H^k(N) \to H^k(N), \] where the middle homomorphism is Poincaré duality isomorphism, while the other two are induced by \(f\) (the homology and cohomology groups have \(\mathbb{Z}_2\) coefficients). In order to show a possible behaviour of generic maps for double points, an example is presented which shows that the image of such a map need not be an ANR. This is the reason why the Čech homology and cohomology is used.