It is known that cobordism provides a complete classification of surfaces as: a surface is cobordant to either $S^2$ or $\mathbb{R}P^2$. I am looking for a reference with contains a proof of this fact.



Reference request: full classification of surfaces as being cobordant to $S^2$ or $\mathbb{R}P^2$
The orientable closed surface with genus $g$ (denoted as $F\_g$) can be realized as the boundary of a central-symmetric closed body $K$ in $\Bbb{R}^3$ (for example, $F\_1 \cong \Bbb{T}^2$ can be realized as the boundary of a solid torus centered at the origin). Choose a large closed ball $B$ centered at the origin with its interior containing $K$, then $M=B \backslash (K^\circ)$ is the coboundary of $F\_g$ and $\Bbb{S}^2$.


Quotient $M$ by the group action generated from $x \mapsto -x$, the resulting manifold is the coboundary of the non-orientable closed surface with genus $[\dfrac{g+3}{2}]$ and $\Bbb{RP}^2$.


