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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Some mapping theorems

Author: R. C. Lacher
Journal: Trans. Amer. Math. Soc. 195 (1974), 291-303
MSC: Primary 57A15
MathSciNet review: 0350743
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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 ``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)$.

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PII: S 0002-9947(1974)0350743-2
Keywords: Mapping, acyclic, finiteness, cellularity
Article copyright: © Copyright 1974 American Mathematical Society