Universal meager $F_{\sigma }$-sets in locally compact manifolds

In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1{]}^n$ we construct a meager $F_{\sigma }$-subset $X\subset M$ which is universal meager in the sense that for each meager subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\subset X$. We also prove that any two universal meager $F_{\sigma }$-sets in $M$ are ambiently homeomorphic.