Some mapping theorems

Author:
R. C. Lacher

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

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Abstract: Various mapping theorems are proved, culminating in the following result for mappings *f* from a closed -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 : 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 between finite polyhedra *X* and *Y*: If the Euler characteristic of each point-inverse of is the integer *c* then .

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0350743-2

Keywords:
Mapping,
acyclic,
finiteness,
cellularity

Article copyright:
© Copyright 1974
American Mathematical Society