Locally flat imbedding

An imbedding (cf. Immersion) $q$ of one topological manifold $M=M^m$ into another $N=N^n$ such that for any point $x\in M$ there are charts in a neighbourhood $U$ of $x$ and in a neighbourhood $V$ of the point $qx$ in $N$ in which the restriction of $q$ to $U$ linearly maps $U$ to $V$. In other words, $q$ is locally linear in suitable coordinate systems. Equivalently: There are neighbourhoods $U$ of a point $x\in M$ and $V$ of the point $qx\in N$ such that the pair $(V,qU)$ can be mapped homeomorphically onto a standard pair $(D^n,D^m)$ or $(D^n,D_+^m)$, where $D^k$ is the unit ball of the space $\mathbf R^k$ with centre at the origin and $D_+^k$ is the intersection of this ball with the half-space $x_k\geq0$.

Any imbedding of a circle and an arc into a plane is locally flat; however, a circle or an arc can be imbedded in $\mathbf R^k$ with $k\geq3$ in a manner that is not locally flat (see Wild imbedding; Wild sphere). Any smooth imbedding is locally flat in the smooth sense (that is, in the definition the coordinates can be chosen to be smooth). A piecewise-linear imbedding need not be locally flat, not only in the piecewise-linear sense, but even not in the topological sense; for example, a cone with vertex in $\mathbf R_+^4$ over a closed polygon knotted in the bounding plane $\mathbf R^3$. For $n\neq4$ and $m\neq n-2$ there is a homotopy criterion for an imbedding to be locally flat: For every point $x\in M$ and neighbourhood $U$ of the point $qx$ there is a neighbourhood $V\subset U$ such that any loop in $V\setminus qM$ is homotopic to zero in $U\setminus qM$ (local simple connectedness). If $m=n-2$, then such a criterion holds for $n\neq4$, but is essentially more complicated. For $m=4$ the question remains unsettled (1989). For $m=n-1$ and $m=n-2$ a locally flat imbedding has a topological normal bundle.

References

[a1]	J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , Topology of manifolds , Markham (1970)