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

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.