## Some mapping theorems

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- by R. C. Lacher
- Trans. Amer. Math. Soc.
**195**(1974), 291-303 - DOI: https://doi.org/10.1090/S0002-9947-1974-0350743-2
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## Abstract:

Various mapping theorems are proved, culminating in the following result for mappings*f*from a closed $(2k + 1)$-manifold

*M*to another,

*N*: If âalmost allâ point-inverses of

*f*are strongly acyclic in dimensions less than

*k*and if âalmost allâ point-inverses of

*f*have Euler characteristic equal to one, then all but finitely many point-inverses are totally acyclic. (Here âalmost allâ means âexcept 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 ânonalternatingâ 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 $\chi (X) = c\chi (Y)$.

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## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**195**(1974), 291-303 - MSC: Primary 57A15
- DOI: https://doi.org/10.1090/S0002-9947-1974-0350743-2
- MathSciNet review: 0350743