Some mapping theorems

Various mapping theorems are proved, culminating in the following result for mappings f from a closed $(2k + 1)$-manifold M to another, N: If ``almost all'' point-inverses of f are strongly acyclic in dimensions less than k and if ``almost all'' point-inverses of f have Euler characteristic equal to one, then all but finitely many point-inverses are totally acyclic. (Here ``almost all'' means ``except on a zero-dimensional set in N".) More can be said when $k = 1$: If f is a monotone map between closed 3-manifolds and if the Euler characteristic of almost-all point-inverses is one, then all but finitely many point-inverses of f are cellular in M; consequently M is the connected sum of N and some other closed 3-manifold and f is homotopic to a spine map. Other results include an acyclicity criterion using the idea of ``nonalternating'' mapping and the following result for PL maps $\phi$ between finite polyhedra X and Y: If the Euler characteristic of each point-inverse of $\phi$ is the integer c then $\chi (X) = c\chi (Y)$.