On proper surjections with locally trivial Leray sheaves (Q1919876)

It is proven that the image of a map \(f : X \to Y\) of an orientable manifold \(X\) is a generalized manifold provided all preimages \(f^{-1} (y)\) have the shape of a closed, connected, orientable manifold and the dimension of \(Y\) is finite. This is a strong generalization of a well-known fact that the image of a cell-like map of a manifold is a generalized manifold if it is finite-dimensional. Also it is proven that if all preimages \(f^{-1} (y)\) have the shape of the same manifold \(N\), then the formula \(\chi (X) = \chi (N) \chi (Y)\) holds for the Euler characteristic over \(\mathbb{Q}\). This formula gives a restriction to the existence of certain decompositions of manifolds.