On page 228 of the paper "Constant terms in powers of a Laurent polynomial" (by J.J. Duistermaat and Wilberd van der Kallen) a 'chain' $\Delta_\tau$ is defined as $f^{-1}([0,\tau])\cap \Delta$ and I have two questions about this:
- How does the intersection of a chain with a set represent a chain? So can someone please explain the definition of $\Delta_\tau$?
- How does the equality $\partial \Delta_\tau = \Gamma_\tau - \Gamma_0$ follow? Why is $\partial \Delta_\tau$ a difference of two cycles with images in $M_\tau$ and $M_0$?
Thank you so much for any help or suggestions.
UPDATE: I was able to prove the existence of a diffeomorphism $f^{-1}(0,\tau_0) \cong M_{\tau_0} \times (0,\tau_0)$ such that $f$ becomes the projection on the second coordinate, and my intuition tells me that this has to be used to construct "an extra nice" chain $\Delta$ such that $\Delta_\tau$ behaves well. I am not sure about the claim of a homeomorphism $f^{-1}[\tau,\tau_0] \cong M_{\tau_0} \times [\tau,\tau_0]$ for $\tau=0$ in the article (they also write $[0,\tau_0] \subset U$, which is clearly false as $0$ is assumed critical).
