In his book [1], Richter-Gebert introduces a notion of *stable
equivalence* for primary basic semialgebraic sets (subsets of
$\mathbb{R}^n$ defined by a conjunction of polynomial equations
and strict inequalities with rational coefficients) and claims
that stably equivalent sets have the same homotopy type.
From the rest of the book, it then follows that (among other interesting
things) **all** the homotopy types of primary basic semialgebraic sets
occur as realization spaces of 4-polytopes.

Apparently, there have been many different notions of stable equivalence in the literature. He writes

The common idea is that semialgebraic sets that only differ by a "trivial fibration" and a rational change of coordinates should be considered as stably equivalent, while semialgebraic sets that differ in certain "characteristic properties" should not turn out to be stably equivalent. In particular, stable equivalence should preserve the thomotopy type, and respect the algebraic complexity and singularity structure.

My question concerns one kind of map which is used in the definition
of stable equivalence, namely *stable projection* which relates to the
"trivial fibration" aspect in the above quote. Richter-Gebert claims
in [1, Lemma 2.5.2], that if $W$ stably projects onto $V$, then $W$
and $V$ have the same homotopy type but omits the proof. I would like
to know why this is true.

Let me recite the definition. Let $V \subseteq \mathbb{R}^n$ and
$W \subseteq \mathbb{R}^{n+m}$ be basic semialgebraic sets with
$V = \pi(W)$ where $\pi$ the coordinate projection which deletes
the last $m$ coordinates. This projection is *stable* according to
[1, Section 2.5] if there exist polynomials $\phi_i, \psi_j \in
\mathbb{Q}[v_1, \dots, v_n, w_1, \dots, w_m]$ (where $i$ and $j$ run
over finite, possibly empty index sets) such that (1) the degree of
all $\phi_i$ and $\psi_j$ in the variables $w_1, \dots, w_m$ is 1,
and (2)
$$
W = \{\, (v,w) \in \mathbb{R}^{n+m} :
\text{$v \in V$ and $\phi_i(v,w) > 0$ and $\psi_j(v,w) = 0$}
\,\}.
$$
Thus, all fibers of $\pi$ are relative interiors of rational polyhedra
whose defining affine functionals vary polynomially in the image point $v$.
In particular they are convex sets and it seems "intuitively clear" that
these convex sets may be (uniformly!) deformation-retracted to yield a
copy of $V$ inside of $W$, thus proving the homotopy equivalence.

After many discussions about the technicalities behind this idea, my colleague came up with the following example: $$ W = \{\, (v,w) \in \mathbb{R}^2 : v(vw - 1) = 0 \,\}. $$ $W$ is the union of a hyperbola with the $w$-axis. This set projects to the entire $v$-axis and its fibers are all given by a single rational affine-linear equation in $w$ whose parameters polynomially depend on $v$. This is a stable projection per definition, but $W$ has three path-connected components whereas $V$ only has one, which contradicts the homotopy equivalence.

I wonder if I overlooked something. Is this really a counterexample to Richter-Gebert's lemma? If yes, can someone who is more versed in these topological notions (or alternative definitions of stable equivalence) think of a way to repair the definition in a way that preserves the intention of the book: showing homotopy type universality of realization spaces of polytopes?

More concretely, I wonder if there are **any additional topological assumptions
to make about $\pi$ to ensure that it has local sections around every point in
$V$** (that is, for every point $v$ an open neighborhood and a continuous
right-inverse for $\pi$ defined on that neighborhood). Since the fibers are
convex, I believe that a partition of unity can be used to convex-combine
the locally finitely many local sections to obtain a continuous global one,
which $W$ then may be contracted to. But finding local sections seems hard
especially on points where the rank of the linear system $\Psi_v \cdot w = b_v$
occuring in the definition of the fiber $\pi^{-1}(v)$ is locally non-constant
(the rank may increase in every neighborhood of $v$, which is what happens
in the above example at $v=0$).

These are observations made by a non-topologist who has run out of
colleagues to pester. I would also **greatly** appreciate pointers to
the proper jargon and any literature dealing with problems of this kind.

If I **did** overlook something and this is not a counterexample, how to
prove [1, Lemma 2.5.2] (the homotopy part) properly?