Nonexistence of locally flat approximations in codimension two

M. A. Shtan'ko

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part III, Tome 83 (1979), pp. 93-100

Résumé

In this paper we prove that for any $n\geqslant6$ there exists a closed, piecewise-linearly imbedded in $E^n$ manifold $M_{pL}^{n-2}$, not admitting locally flat approximations. This manifold can be assumed, here, to be homotopically not equivalent to a smooth one if $n\geqslant10$. We also prove that for any $n\geqslant7$ there exists a closed topological manifold $M^{n-2}_{\mathrm{TOP}}\subset E^n$ not admitting locally flat approximation. This manifold can be assumed to be homotopically not equivalent with a piecewise-linear one.