If I understand correctly, I can obtain the $O$-cobordism group of $$ \Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4, $$ The 3d cobordism invariants have 4 generators of mod 2 classes, are generated by $$g^3,$$ $$g w_2'(V_{SO(3)}),$$ $$w_3'(V_{SO(3)}),$$ $$g w_1(T)^2.$$

Denote:

The $T$ for the spacetime tangent bundle.

The $O(3)=\mathbb{Z}_2 \times SO(3)$.

The $V_{SO(3)}$ for the vector bundle of $SO(3)$.

Here $g$ is related to the $\mathbb{Z}_2$ generator of $g=H^1(B\mathbb{Z}_2,\mathbb{Z}_2)$.

Questions: What are the corresponding 3-manifold generators of O-co/bordism invariants $\Omega_{O}^3(BO(3))$, for $g^3,$ $g w_2'(V_{SO(3)}),$ $w_3'(V_{SO(3)}),$ $g w_1(T)^2.$?

Hint: I think the manifold generator for $g^3$ is $\mathbb{RP}^3$.

The manifold generator for $g w_1(T)^2$ is $S^1 \times \mathbb{RP}^2$?

What are manifold generators of $g w_2'(V_{SO(3)}),$ and $w_3'(V_{SO(3)})$? Can each manifold generator of 4 generators of mod 2 classes be unique and distinct?

If I understand correctly, I can obtain the $O$-cobordism group of $$ \Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4, $$ The 3d cobordism invariants have 4 generators of mod 2 classes, are generated by $$g^3,$$ $$g w_2'(V_{SO(3)}),$$ $$w_3'(V_{SO(3)}),$$ $$g w_1(T)^2.$$

Denote:

The $T$ for the spacetime tangent bundle.

The $O(3)=\mathbb{Z}_2 \times SO(3)$.

The $V_{SO(3)}$ for the vector bundle of $SO(3)$.

Here $g$ is related to the $\mathbb{Z}_2$ generator of $g=H^1(B\mathbb{Z}_2,\mathbb{Z}_2)$.

Questions: What are the corresponding 3-manifold generators of O-co/bordism invariants $\Omega_{O}^3(BO(3))$, for $g^3,$ $g w_2'(V_{SO(3)}),$ $w_3'(V_{SO(3)}),$ $g w_1(T)^2.$?

Hint: I think the manifold generator for $g^3$ is $\mathbb{RP}^3$.

The manifold generator for $g w_1(T)^2$ is $S^1 \times \mathbb{RP}^2$?

What are manifold generators of $g w_2'(V_{SO(3)}),$ and $w_3'(V_{SO(3)})$? Can each manifold generator of 4 generators of mod 2 classes be unique and distinct?