General position maps for topological manifolds in the ${2\over 3}$rds range
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- by Jerome Dancis
- Trans. Amer. Math. Soc. 216 (1976), 249-266
- DOI: https://doi.org/10.1090/S0002-9947-1976-0391098-9
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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 $g|:S \twoheadrightarrow g(S)$ is a piecewise linear map and g is a locally flat embedding on the complement of S. Furthermore if $f|\partial M$ 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.References
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Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 216 (1976), 249-266
- MSC: Primary 57A15
- DOI: https://doi.org/10.1090/S0002-9947-1976-0391098-9
- MathSciNet review: 0391098