When computing special bordism groups, I often need to determine existence of (singular) smooth $4$-manifolds with fixed fundamental group and certain properties like the spin behaviour (i.e. being totally non-spin, spin or almost spin).

Sometimes I'm able to do it algebraically by means of the edge homomorphism of certain Atyiah-Hirzebruch spectral sequences but if I'm not able to compute the stable page of it I don't know how to build the examples.

Let me be more clear with an example: In the AHSS for computing the bordism group $\Omega_4(\xi)$ for a stable bundle $\xi \colon K(D_{2n};1)\times BSpin\to BSO$, where $D_{2n}$ is the Dihedral group, for other reasons I know that signature and the edge homomorphism induce an isomorphism: $$\mathfrak{sg}\oplus \mathfrak{e} \colon \Omega_4(\xi)\cong 8\cdot \mathbb{Z} \oplus E_{4,0}^{\infty}$$ $$ M \mapsto (\mathfrak{sg}(M), c_*[M])$$ where $c \colon M\to K(D_{2n};1)$ is the classifying map of the universal cover.

I know that $E_{4,0}^{\infty}=0$ or $\mathbb{Z}/2$ (I'm unable to determine what is $d_3 \colon E_{4,0}^3\cong \mathbb{Z}/2 \to E_{1,2}^3\cong \mathbb{Z}/2$)

In order to see which case can occur, I currently want to establish existence of a singular $4$ manifold $(M,c)$, with the following properties:

• It's orientable
• $\pi_1M \cong D_{2n}$
• It's an element of $\Omega_4(\xi)$
• The element $c_*[M]$ in $H_4(D_{2n};\mathbb{Z})$ is non zero,

I think surgery might be a way to deal with this problem, but since I've never learned it, I would like to have some guidance about what to learn in order to solve my problem. Any suggestions or ideas are welcome.

If needed I can add some explanation about any of the notion I introduced here, I tried to keep explanation at minimum in order to avoid lengthy and boring questions. Let me know if something need more explanation.