The weak \(b\)-principle: Mumford conjecture (Q2798995)

The author has proposed a stable homotopy version of the \(h\)-principle and the \(b\)-principle for differential relations, which he has described in short form in the review paper [Contemp. Math. 621, 101--112 (2014; Zbl 1346.55007)].NEWLINENEWLINEIn the present paper, the author introduces and studies a new class of maps, which he terms oriented, colored, broken submersions. This class of maps satisfies a version of the \(b\)-principle, and for oriented submersions from a smooth manifold \(M^{n+2}\) to a closed, simply connected, smooth \(n\)-dimensional manifold \(N^n\), the class provides good approximations to the class of all oriented submersions, in the sense that every oriented submersion is bordant to an oriented, colored, broken submersion.NEWLINENEWLINEIn the main result, it is proved that the standard Mumford conjecture proved by \textit{I. Madsen} and \textit{M. Weiss} [Ann. Math. (2) 165, No. 3, 843--941 (2007; Zbl 1156.14021)] fits very well into a general setting of the \(b\)-principle for oriented, colored, broken submersions.