The fixed point set of a piecewise linear $(PL)$ map $h:P \times I \rightarrow P$ is the set of points where $h$ coincides with the projection $\pi :P\times I \rightarrow P$; it is denoted by Fix$(h)$ and is a subpolyhedron of $P\times I$. When $P$ is a compact polyhedron, we show how to deform $h$ (with appropriate control) to a new $PL$ map $h'$ so that Fix$(h')$ is as nice as possible. Indeed it is not hard to arrange that Fix$(h')$ have dimension $\leq 1$ (Theorem A), but one would wish for a map $h'$ such that Fix$(h')$ is a manifold of dimension $\leq 1$. This is achieved in Theorem B. If $P$ is a $PL$ manifold, Theorem B reduces to a standard $PL$ transversality theorem (Theorem C).