We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$.

$$\Omega_0^{SO}=\mathbb{Z}$$ $$\Omega_1^{SO}=0$$ $$\Omega_2^{SO}=0$$ $$\Omega_3^{SO}=0$$ $$\Omega_4^{SO}=\mathbb{Z}$$ $$\Omega_5^{SO}=\mathbb{Z}/2$$ $$\Omega_6^{SO}=0$$ $$\Omega_7^{SO}=0$$ $$\Omega_8^{SO}=\mathbb{Z} \oplus \mathbb{Z}$$ $$\Omega_9^{SO}=\mathbb{Z}/2 \oplus \mathbb{Z}/2$$ $$\Omega_{10}^{SO}=\mathbb{Z}/2$$ $$\Omega_{11}^{SO}=\mathbb{Z}/2$$

Some of the cobordism invariants for these bordism groups only require the Stiefel-Whitney class $w_j(TM)$. But $w_j(TM)$ can be defined on oriented triangulable manifolds without smooth differentiable structures. So those cobordism invariants may still survive when we loosen the condition to study the oriented triangulable manifolds *without* smooth differentiable structures.

Do we know any bordism theory for **oriented triangulable manifolds without smooth differentiable structures** that produces the similar bordism groups as above, but only maintain those cobordism invariants that only require oriented triangulable manifolds (such as those only require Stiefel-Whitney class $w_j(TM)$)? That is, could we remove the $SO$ structure above, and only keep the triangulation, say $trg$, and study its bordism group? I expect something like:

$$\Omega_5^{triag}=\mathbb{Z}/2, \text{: cobordism invariant: $w_2w_3$}$$

$$\Omega_9^{triag}=\mathbb{Z}/2 \oplus \dots \text{: cobordism invariant: $w_4w_5$}$$

$$\Omega_{13}^{triag}=\mathbb{Z}/2 \oplus \dots \text{: cobordism invariant: $w_6w_7$}$$

$$\Omega_{4k+1}^{triag}=\mathbb{Z}/2 \oplus \dots \text{: cobordism invariant: $w_{2k}w_{2k+1}$, for $k\geq1$}$$

I suppose the Piecewise Linear PL is similar to what I want. I do not think I am considering the topological manifold Top. But I do not require the smooth structure, nor the $SO$ or $O$ tangential structure. But I am not sure that the Stiefel-Whitney class $w_j(TM)$ requires the PL structure. The Stiefel-Whitney class $w_j(TM)$ may require only the triangulation and branching rule on manifold $M$.