Cfrgtkky

This question is in relation with the definition of Euler-Integral transforms, page 147 of Michael Robinson book "Topological Signal Processing".

Definition: Euler characteristic and Constructible function

The euler characteristic $chi$ is a valuation from a cell complex $X_f$ to $mathbb{Z}$ defined by:
$$ chi(X_f) = sum_{c in X_f}^{}{-1^{dim(c)}}$$

${bf Remark:} $The definition of $chi$ makes it possible to define it for an arbitraty collection of cell. For instance, a sub-collection $X_f'$ of cells in $X_f$ is not always a cell complex, but we will admit that $chi(X_f')$ is well-defined. Even though the author doesn't make it clear in the book, this assumption seems to be largely used.

An integered valued constructible function is a function on a topological space $f:X mapsto mathbb{Z}$ such that there exists a non-unique cell-complex $X_f$ for which:

a) There is an homeomorphism $h : X mapsto X_f$

b) The function $f circ h$ is constant on each cell of $X_f$

This can be seen as a generalization of piecewise constant function on topological spaces having a cell-complex skeleton.

Problem statement

Given a topological space $X$ and a smooth map $P:X mapsto mathbb{R}$. We assume that $X$ is such that there is a homeomorphism $h$ between $X$ and a cell complex $X_f$ . We define the level sets of $P$:
$$ X_c ={ x in X, P_x(x,bullet) = s }$$

We define a constructible function f with the same underlying cell-structure (i.e, $fcirc h$ is piecewise constant on the cells of $X_f$).

The question is: Is $f_{vert X_c}$ still constructible for the homeomorphism $h$?

The author seems to consider so. But I am disturbed by the fact that the level set themselves can very well fail to be sub-cell complex, or even collections of cells.

For instance, if $Y$ is the ball centered in O in $mathbb{R}^2$ with the cell-complex structure made of one $0$-cell, one $1$-cell and one $2$-cell. Let's have $P : y mapsto | y |$, and $f$ is the indicatrice function of the ball.

The level set of $P$ for $s$ are the intersection between a circle of radius s and $Y$. Three cases can occur:

1. For $s=0$, the level set is the single point (0,0).


2. For $0<s<{max}$ Y_s is a closed circle in the interior of $Y$.


3. For $s = s_{max}$ the level set is the frontier made of the $1$-cell attached to the $0$-cell.

It seems that apart for the third case, the level set are neither cells or a collection of cells. Which results in my inability to understand how can the author use the function $f_{vert Y_s}$ as a constructible function (computing its euler integral for instance).

I am wondering if I missed something involved by $h$ being a homeomorphism, or by $P$ being smooth, as I don't seem to use any of these assumptions. I would appreciate some clues about what the author is really trying to do (for those who read the book) or about the constructability of $f_{vert X_c}$.