Differentiable monotone maps on manifolds. II

Abstract

Listen

Translate article

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