Differentiable monotone maps on manifolds. II

Abstract:

Let ${M^n}$ and ${N^n}$ be closed manifolds, and let $G$ be any (nonzero) module. (1) If $f:{M^3} \to {N^3}$ is ${C^3}$ $G$-acyclic, then there is a closed ${C^3}$ $3$-manifold ${K^3}$ such that ${N^3}\# {K^3}$ is diffeomorphic to ${M^3}$, and ${f^{ - 1}}(y)$ is cellular for all but at most $r$ points $y \in {N^3}$, where $r$ is the number of nontrivial $G$-cohomology $3$-spheres in the prime decomposition of ${K^3}$. (2) If $f:{M^3} \to {M^3}$ or $f:{S^3} \to {M^3}$ is $G$-acyclic, then $f$ is cellular. In case $G$ is $Z$ or ${Z_p}$ ($p$ prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If $f:{M^n} \to {M^n}$ or $f:{S^n} \to {M^n}$ is real analytic monotone onto, then $f$ is a homeomorphism.