I am reading Liviu I. Nicolaescu's book "Notes on Seiberg-Witten Theory". I am confused that in Corollary 4.3.22(b) (and Remark 4.4.2), it appears that the boundary map $\partial_\infty:\widehat{\mathfrak{M}_\mu}\to\mathfrak{M}_\sigma$ is a submersion , about an irreducible and strongly regular monopole.
Here, $\widehat{\mathfrak{M}_\mu}$ is the SW moduli space of a four-manifold with cylindrical end $\hat{N}$, and $\mathfrak{M}_\sigma$ is the SW moduli space of $N$ (which is the end of $\hat{N}$), $\sigma$ is the spin structure specified, and $\mu$ is a suitable exponential decay rate.
However, in the middle-bottom part of page 433 of the same book, the author sets $L_i^+:=Im(\partial_{\infty}^c:ker_{ex}\hat{\mathfrak{T}}_{\hat{C_i}}\to T_{C_{\infty}}\mathfrak{M}_{\sigma})$ and implicitly asserts that $L_i^+$ is a Lagrangian subspace, for example in the next page's equation (L).
Here, $ker_{ex}\hat{\mathfrak{T}}_{\hat{C_i}}$ denotes the kernel of the linearzied elliptic operator (of the Seiberg-Witten equation together with the gauge condition) in the extended-$L^2$ configuration space at the monopole $\hat{C_i}$.
From Proposition 4.3.28 of the same book, it appears to me that if $\hat{C_i}$ is irreducible and strongly regular, then $ker_{ex}\hat{\mathfrak{T}}_{\hat{C}_0}\cong H^1_{\hat{C}_0}\cong T_{\hat{C}_0}\widehat{\mathfrak{M}_\mu}$, and $\dim Im (\partial_{\infty}^c)=\dim Im (D\partial_{\infty}|_{\hat{C_i}})$. Here $ H^1_{\hat{C}_0}$ is the first homology of the corresponding elliptic complex. If $\partial_{\infty}$ is a submersion, we must have $\dim Im(D\partial_{\infty}|_{\hat{C_i}})\geq\dim T_{C_{\infty}}\mathfrak{M}_{\sigma}$, on the other hand $Im(\partial^c_{\infty})$ is a Lagrangian subspace of $T_{C_{\infty}}\mathfrak{M}_{\sigma}$, which is half dimensional. How is it possible?
Another question is that the proof in the construction of the submersion $\partial_{\infty}$ looks too general. It implies the bound on local dimension $\dim \widehat{\mathfrak{M}_\mu}\geq \dim \mathfrak{M}_\sigma$. It seems giving a positive answer to an extension problem, namely three dimensional monopole $C_i$ can be extended to the whole 4-manifold with cylindrical end as a 4-monopole, if $C_i$ is near enough to the image of a strongly regular, irreducible 4-monopole $\hat{C_i}$. I cannot find such kind of assertion in other reference or similar case. Do we have this kind of results on say the 2+1 SW setting (three dimensional Seiberg Witten moduli space asymptotic to vortex solution), or the Yang Mills setting? Thank you.
After some thoughts, I think the map $\partial_{\infty}$ is a submersion only when the codomain is restricted to its image, so we actually don't have the inequality $\dim \widehat{\mathfrak{M}_{\mu}}\geq\dim \mathfrak{M}_{\sigma}$, and $Im(D\partial_{\infty})$ remains a Lagrangian subspace of $\mathfrak{M}_{\sigma}$.