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 prove existence of such manifolds "algebraically", i.e. after studying the edge homomorphism of certain Atyiah-Hirzebruch spectral sequences. But if I'm not able to compute the stable page of the AHSS I don't know how to proceed. This is why I need to study a new, more topological, approach to this problem.

Let me be more clear with an example: Let $$\xi:= \eta_w \oplus Bp \colon K(D_{2n};1)\times BSpin\to BSO$$ be a stable bundle, where $D_{2n}$ is the Dihedral group and $\eta_w$ is a vector bundle such that $w_1(\eta_w)=0$ and $w_2(\eta_w)=w \in H^2(D_{2n};\mathbb{Z}_2)$. The Bordism group $$\Omega_4(\xi)$$ contains as elements almost spin orientable $4$-manifolds with fundamental group $D_{2n}$ and such that $c^*(w)=w_2(M)$ where $c \colon M\to K(D_{2n};1)$ is the classifying map of the universal cover (this is a fancy way to say that $M$ has normal $1$-type $\xi$). You can think of it as a special $B$-bordism group.

An application of the AHSS for computing the bordism group $\Omega_4(\xi)$ gives that the signature and the edge homomorphism induce an isomorphism: $$\mathfrak{sg}\oplus \mathfrak{e} \colon \Omega_4(\xi)\cong 8\cdot \mathbb{Z} \oplus E_{4,0}^{\infty}$$ $$ M \mapsto (\mathfrak{sg}(M), c_*[M])$$

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 would like to establish existence (or not) of a closed $4$ manifold $M$, with the following properties:

• It's orientable
• It's an element of $\Omega_4(\xi)$ (i.e. has normal $1$-type $\xi$)
• The element $c_*[M]$ in $H_4(D_{2n};\mathbb{Z})$ is non zero

I think surgery theory might be a way to deal with this problem, but since I've never studied it, I need some guidance in understanding HOW surgery could help me and WHERE can I learn the tools I need.

PS: If needed I can add more explanations about the background of the problem I tried to describe here. I tried to keep explanations at minimum in order to avoid lengthy and boring questions. Let me know if something need more explanation.