Let $X$ be a space and $x_0\in X$ a base point. The Hurewicz map $\pi_k(X,x_0)\longrightarrow H_k(X)$ factors through oriented bordism $\pi_k(X,x_0)\longrightarrow MO_k(X)\longrightarrow H_k(X).$ Ordinary homology $H_k(X)$ can be defined as $MO_k(X)$, but taking oriented simplicial complexes with maximal faces of dimension $k$ instead of manifolds of dimension $k$: Define $S_k(X)=\{{\rm singular~oriented~simplicial~ complexes~in~}X~{\rm with~ maximal~ faces~ of ~dimension} ~k\}$ and $\partial_k\colon S_k(X)\rightarrow S_{k-1}(X)$ sends a simplicial complex $Y$ to its boundary (the singular sub-simplicial complex of $Y$ generated by those faces of $Y$ that are faces of only one (maximal) face). Then $H_k(X)\cong H_k(S_*(X))$. If we take the subcomplex $SM_*(X)$ generated by those oriented simplicial complexes that are oriented manifolds, then $H_k(SM_*(X))\cong MO_*(X)$.
Say that a homology theory $E_*$ is ''bordism-like'' if 1) $E_k(X) = H_k(SQ_*(X))$, for some subchain complex $SQ_*(X)\subset S_*(X)$ generated by the simplicial complexes that have some property $Q$ and 2) the Hurewicz map to ordinary homology factors throgh $E_*$.
Question: Is there a initial bordism-like homology theory? (might be $MO_*$).
Remarks: 1. I've tried restricting to the subcomplex generated by coproducts of spheres ${\mathbb S^k}$, discs $D^k$ and cylinders ${\mathbb S^{k-1}}\times I$, but it does not satisfy excision. 2. Probably I should be writing PL-bordism instead of $MO_*$.