On the Lê-Milnor fibration for real analytic maps

Manifolds Which Admit Maps with Finitely Many Critical Points Into Spheres of Small Dimensions

“…When singular points arise from the fibered links above, each critical point s contributes with χ(F s ) − 1 to χ(M ), where F s is the local fiber around s. This holds more generally for all fibered links. On the other hand if dimensions were of the form (2n + 1, k), then local fibers should verify χ(F s ) = 1, according to [2,14,23]. This shows that the contribution of every critical point is trivial in odd dimensions and hence the previous arguments cannot work.…”