I am reading [these notes](http://www.massey.math.neu.edu/Massey/Massey_preprints/perverse_milnor.pdf) on nearby and vanishing cycles, where an initial assumption is made: the author talks about a complex analytic function $f:X\to \mathbb C$, assuming $X$ is closed in an open subset $U\subset \mathbb C^N$ and $f$ it is the restriction of a function $\tilde f:U\to \mathbb C$. Then some statements are made regarding the cohomology of the Milnor fiber $F\_{f,p}$ for $p\in f^{-1}(0)$. The function $\tilde f$ is never mentioned again. So I was wondering: 



> 
> What is the relation between the sheaf of vanishing cycles $\phi\_f$ (living over the singular locus
>  of $X\_0=f^{-1}(0)$) and $\phi\_{\tilde f}$ (living over the singular
>  locus of $U\_0=\tilde f^{-1}(0)$)?
> 
> 
> 


In particular I would like to understand under which assumptions one can conclude that $$\chi\_{top}(F\_{f,p})=\chi\_{top}(F\_{\tilde f,p}).$$


Even "stronger" question: Can we say that the reduced cohomologies 
$$
\mathcal H^i(\phi\_f \underline{\mathbb Q}\_X)\_p\cong\tilde H^i(F\_{f,p},\mathbb Q),\qquad \tilde H^i(F\_{\tilde f,p},\mathbb Q)\cong\mathcal H^i(\phi\_{\tilde f} \underline{\mathbb Q}\_U)\_p
$$
agree?


Thank you for reading!



Relation between Milnor fiber and its restriction via vanishing cycles
There is no general relationship between the cohomology of the Milnor fibers of $\tilde f$ and $f$, in this setting. At a point $p\in X\_0$, the Milnor fiber of $f$ is the intersection of $X$ and the Milnor fiber of $\tilde f$. Hence, the topologies of the two fibers may be very different.


If $X$ is a hyperplane or hypersurface, the situation is manageable and one can analyze how the topology of the Milnor fibers of $f$ and $\tilde f$ compare, but the equality of the Euler characteristics of the Milnor fibers would be very special. For instance, if $\tilde f$ has a 1-dimensional critical locus $\Sigma\tilde f$ at the origin, and $X$ is a hyperplane $H$ such that $f$ has an isolated critical point at the origin, the the Euler characteristics of the two Milnor fibers would be the same if and only if the relative polar curve of $\tilde f$ with respect to $H$ is empty; this is equivalent to saying that $\Sigma\tilde f$ is smooth at the origin and defines a $\mu$-constant (Milnor number constant) family of isolated critical points.


The standard reason for starting with $\tilde f$, even though one is interested in $f$, is so that one can use conormal techniques and the differential $d\tilde f$ when calculating.


