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 they are denoted as odd or even manifolds).
Usually I'm able to prove existence of such manifolds "algebraically", i.e. after studying the edge homomorphism of certain Atyiah-Hirzebruch spectral sequences and then infer the existence (or not) if such manifolds. This method as a drawback though: if I'm not able to compute the stable page of the AHSS I can't proceed. This is the reason why I'm interested in 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 with $n=0 \pmod{4}$ 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.
In the comments I was linked this paper. I have started reading it, but I'd like to know which chapter should I focus on: a quick search of "dihedral" and a glance at the index don't seem to show where surgery is used in a way I might be interested in. After a more careful read, it seems that lots of the results require properties on $\pi_1$ that I clearly don't have: torsion-free, virtually $\mathbb{Z}^n$, $PD_r$-group... Moreover the section about stable classification (the reason why I'm computing such bordism groups) doesn't tell us much, withouth proving anything. It's possible that due to my inexperience in this field, I'm unable to find the right chapters, but if someone could point me out where I should focus it would be very helpful.
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.
