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?
Best Answer
The quick answer to my question is: Yes, the space described in the question is a real counterexamle. It can even be "homogenized" to $W = \{\, (v,w,w') \in \mathbb{R}^3 : w' > 0, \, v(vw - w') = 0 \,\}$. This primary basic semialgebraic set is also disconnected but its stable(!) projection onto the $v$ coordinate produces the entire real line. In this case, the fiber-defining equations are homogeneous linear functionals in $(w,w')$ whose coefficients are polynomials in $v$. As such, it is a counterexample to Lemma 1 (i) in [2] which uses a (still) slightly more restrictive notion of stable projection than the one in the question.
I am in contact with the author, who has confirmed the problem, and we are working on a correction note. However, since the problem is with a definition (i.e., it fails to make a lemma true) which is, in one way or another, used throughout the book, understanding the implications might take some time. At this point, we are both convinced that this is a minor technical issue which does not impact the main theorems.
[2] Jürgen Richter-Gebert and Günter Ziegler: Realization spaces of 4-polytopes are universal, Bull. Amer. Math. Soc. 32 (1995), pp. 403-412.
Let me give a few details about the failure of this lemma in the homotopy equivalence case. There is a more complete write-up on my blog about everything I have gathered about this topic.
The most direct way to repair the definition of stable projection given in the question is to require in addition that the fibers $F_v = \pi^{-1}(v)$, for $v \in V$, each of which is a polyhedron, locally have the same dimension everywhere. To say this more precisely, let $\Psi_v w = b_v$ denote the linear system $\psi_j(v,w) = 0$ — the entries of matrix $\Psi_v$ and vector $b_v$ are polynomials in $v$. We require that for every $v \in V$ there exists an open neighborhood $U_v$ around $v$ in $\mathbb{R}^n$ and a square $r \times r$ submatrix $R_v$ of $\Psi_v$ such that everywhere on $U_v$: (a) $\operatorname{rank} \Psi_v = r$ and (b) $\det R_v \not= 0$. Proof of homotopy equivalence:
Thus, the assumption that the fibers, as polyhedra, locally have the same dimension everywhere leads to homotopy equivalence. Sure enough, the hyperbola counterexample fails the local equidimensionality condition at $v=0$.
The proof implies that some variations of the definition of stable projection lead to homotopy-equivalent semialgebraic sets, too. The loose criteria are: the fibers need to be convex and local sections must be obtainable. This can be achieved through different assumptions about the linear equations defining the fibers. In addition, the strict inequalities need not be linear as long as the fibers remain convex.