I am trying to reproduce some of the details in how the Mayer-Vietoris sequence for bordism should go, especially in showing exactness using this definition of the boundary operator. I have tried to work out some of the details, I'd be grateful to anyone who can take a look at them and perhaps fill in some of the gaps.
First recall that we want an exact sequence
$$\dots \rightarrow \Omega_{n+1}(X)\xrightarrow{\delta_{n+1}} \Omega_{n}(U\cap V)\rightarrow \Omega_{n}(U)\oplus \Omega_{n}(V)\rightarrow \Omega_{n}(X)\rightarrow \dots$$
Here's how I guess the $\delta_{n+1}$ should be defined. One takes a map $f:W\rightarrow X$ representing an element of $\Omega_{n+1}(X)$ and chooses a function $\phi:X\rightarrow [-1,1]$ such that $\phi|_{X\setminus U}=-1$ and $\phi_{X\setminus V}=1$. Then, using Thom transversality, one homotopes the composition $\phi\circ f$ to a smooth map $\tilde{\phi}$ that is tranverse onto the point 0. That way one can take the inverse image $M:=\tilde{\phi}^{-1}(0)$ and this will be a n-dimensional submanifold of $W$ with a map $f|_{M}:M\rightarrow U\cap V$. We get that it indeed maps to $U\cap V$ because we could choose the map $\tilde{\phi}$ to be arbitrarily close to $\phi\circ f$. The choice of $\delta_{n+1}$ is independant of the choice of $\tilde{\phi}$, as any two choices of $\tilde{\phi}$ will be homotopic and hence give bordant classes.
Now to show exactness at $\Omega_{n}(U\cap V)$ we want to show that the compositions $M\rightarrow U\cap V \rightarrow U$ and $M\rightarrow U\cap V\rightarrow V$ are null-bordant. For this we can simply take the submanifold $\tilde{\phi}^{-1}([0,1])$. This is then a manifold with boundary $\tilde{\phi}^{-1}(0)=M$ and hence this gives us our null-bordism of $M\rightarrow U$ and similarly for $M\rightarrow V$ we can take the null-bordism $\tilde{\phi}^{-1}([-1,0])$.
The main problem for me begins with the reverse direction of exactness. Starting with a representative $M\rightarrow U\cap V$ such that $M\rightarrow U\cap V\rightarrow U$ and $M\rightarrow U\cap V\rightarrow V$ are null-bordant, say by null-bordisms $W_1\rightarrow U$ and $W_2\rightarrow V$, we can paste together these manifolds along the common boundary $M$ to get a representative $[W_1\cup_{M} W_2\rightarrow X]\in \Omega_{n+1}(X)$. What I can't figure out is then how to show that this represents maps back to $M\rightarrow U\cap V$ under the map $\delta_{n+1}$.
Anyone have any ideas/references for this? I'd be very grateful for any help!
