Uniqueness of submanifold representing second cohomology class

Question

Setup

Let $M$ be a closed, orientable, smooth manifold, and $\alpha\in H^2(M)$. Further, let $N\subset M$ be a closed, orientable submanifold that is Poincare dual to $\alpha$, and let $E\rightarrow M$ be the complex line bundle whose Euler class is $\alpha$. Finally, let $s:M\rightarrow E$ be a section transverse to the zero section, and $Z=\{m\in M\mid s(m)=0\}$.

Question

Are $N$ and $Z$ cobordant?

Context

I know that $Z$ is also Poincare dual to $\alpha$, but I am wondering if $Z$ is "unique up to cobordism".

Answer 1

In a sense, it all boils down to the fact that the same space ($\mathbb CP^\infty$) is both $K(\mathbb Z,2)$ and $BSO(2)$:

1. Since $\mathbb CP^\infty$ is $K(\mathbb Z,2)$, we have $H^2(X)=[X,\mathbb CP^\infty]$. To get a realization of a class in $H^2(M)$ by a submanifold take the preimage of $\mathbb CP^{N-1}$ under a map $M\to\mathbb CP^N$ corresponding to the class [and transversal to $\mathbb CP^{N-1}$).

2. Moreover, any codimension 2 cooriented submanifold $Z\subset M$ is a preimage of $\mathbb CP^{N-1}\subset\mathbb CP^N$. Indeed:

   • since $\xi\to\mathbb CP^\infty$ is the universal $SO(2)$-bundle, the normal bundle of $Z$ in $M$ is $f^*\xi$ for some map $f\colon Z\to\mathbb CP^{N-1}$ (for some $N$);
   • choose a small open neighbourhood $U$ of $Z$ isomorphic to the normal bundle of $Z$ and let $Mf$ be the map from $M$ to the 'Thom space' (i.e. 1-point compactification) $M\xi$ of $\xi$ (sending everything outside of $U$ to the 'infinity' and using the natural map $f^*\xi\to\xi$ on $U$);
   • $M\xi=CP^N$ so we have a map $M\to\mathbb CP^N$, and the preimage of $\mathbb CP^{N-1}$ is exactly $Z$.

3. If we have two such realizations of the same class, we get a homotopy $M\times[0;1]\to CP^N$ — again make this map transversal and take the preimage of $\mathbb CP^{N-1}$ — that's a cobordism you're looking for.

Answer 2

Yes, they are equal. Let $M$ be a manifold, We say that two submanifolds $U$ and $V$ of $M$ of dimension $p$ are equivalent if they are cobordant. Denote by $L_p(M)$ the set of equivalent classes. The canonical map $L_p(M)\rightarrow H_p(M)$ is injective for $p\leq 3$, see the reference p.4 section 3.

http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/thom_-_cobordant_differentiable_manifolds.pdf