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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



General position maps for topological manifolds in the $ {2\over 3}$rds range

Author: Jerome Dancis
Journal: Trans. Amer. Math. Soc. 216 (1976), 249-266
MSC: Primary 57A15
MathSciNet review: 0391098
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Abstract: For each proper map f of a topological m-manifold M into a topological q-manifold Q, $ m \leqslant (2/3)q - 1/3$, we build an approximating map g such that the set of singularities S of g is a locally finite simplicial $ (2m - q)$-complex locally tamely embedded in M, $ g(S)$ is another locally finite complex $ g\vert:S \twoheadrightarrow g(S)$ is a piecewise linear map and g is a locally flat embedding on the complement of S.

Furthermore if $ f\vert\partial M$ is a locally flat embedding then we construct g so that it agrees with f on $ \partial M$ even when $ f(\partial M)$ meets $ \operatorname{Int} Q \cap f({\operatorname{Int}}\;M)$.

In addition we present two other general position lemmas. Also, we show that given two codimension $ \geqslant 3$ locally flat topological submanifolds M and V of a topological manifold Q, $ \dim \;M + \dim \;V - \dim \;Q \leqslant 3$, then we can move M so that M and V are transverse in Q.

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Keywords: General position, topological manifolds, piecewise linear maps, transversality
Article copyright: © Copyright 1976 American Mathematical Society

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