A note on the codimension of a map (Q914207)

The main result in the paper is the following Reduction Theorem: Let f: \(N\to P\) be a smooth proper map between finite dimensional smooth manifolds. Then \[ co\dim (f)=\sum_{q\in P}co\dim_{C_ q}(f). \] In this theorem \(C_ q=C\cap f^{-1}(q)\) and \(C=critical\) set of f. Moreover codim(f) is the right-left codimension as used by Mather, i.e. the codimension of the tangent space to the right-left orbit of f in the space of smooth maps from N to P. For \(S\subset f^{-1}(q)\) the \(co\dim_ S(f)\) is defined similarly as a multigerm version, taken over all points in S. This theorem generalizes results of Mather, more precisely that locally infinitesimal stability implies (global) infinitesimal stability. Moreover the formula implies that mappings of finite codimension have only a finite number of special critical values and gives specific bounds for the codimension of the germs of mappings over these critical values.