Special consequences of o-minimality

When giving bounds on the topology of sets defined using Pfaffian functions, one invokes constantly the o-minimality of the structure generated by those functions. In this section are gathered a few minor results that will be often used in the next chapters.

Lemma 1.76 (Existence of limits) Let f : (0, ε)→R be definable. Then, the function f has a well-defined limit in R∪ {±∞}.

Proof: This is a simple consequence from the monotonicity theorem (Theorem 1.46).

There exists δ >0 such that the restriction off to (0, δ) is continuous, and one of strictly increasing, constant, or strictly decreasing. The case wheref is constant on (0, δ) is trivial.

If f is strictly increasing on that interval, then either it is bounded from above, and then f must have a finite limit at 0, or it is not bounded and the limit of f is +∞. The case

where f is decreasing is similar. ✷

Lemma 1.77 (Critical values) Let q :U ⊆Rⁿ →R be a C^∞ definable function. Then, q has finitely many critical values.

Proof: The set of critical values ofqis a definable subset ofR.(Ifqis a Pfaffian function, this set is actually sub-Pfaffian, see Example 1.17.) By Sard’s lemma, it must be of measure zero, and a definable subset of R of measure zero can only be finite. ✷

Recall that we denote by bi(X) the i-th Betti number of X (see Notation 1.28) and b(X) =P

ibi(X).

Lemma 1.78 (Deformation of basic sets) Let U be a domain of bounded complexity, g an exhausting function for U and let X = {x ∈ U | q1(x) = · · · = qr(x) = 0, p1(x) >

0, . . . , p_s(x) > 0} be a basic semi-Pfaffian set, and for ε > 0 and t ∈ (R₊)^s, define X_ε = {x ∈ U | q1(x) = · · · = qr(x) = 0, p1(x) ≥ εt1, . . . , ps(x) ≥ εts, g(x) ≥ ε}. Then for all ε≪1, b(Xε) =b(X).

Proof: The groups H∗(X) are the direct limit of the singular homology groups of the compact subsets of X ([Spa, Theorem 4.4.5].) Thus, b(X) = lim_ε→0b(X_ε), and by the generic triviality theorem (Theorem 1.58), this sequence is eventually stationary. ✷ Lemma 1.79 (topology of compact limits) Let Kε be a decreasing sequence of com-pact definable sets defined for ε > 0, and let K be their intersection. Then, for all ε ≪1, and all 0≤i≤n, we have

bi(Kε) =bi(K)

Proof: Since all the sets considered are triangulable, their homological type is that of a polyhedron, and the ˇCech homology ˇH∗ and the singular homology H∗ are isomorphic.

Since the sequenceKnis compact and decreasing and the limit is compact too, we have [ES]

H∗(K) = lim

←−H∗(Kε). (1.21)

But by generic triviality, the setsKεare homeomorphic forε ≪1,hence the limit in (1.21)

becomes eventually stationary. ✷

Chapter 2

Betti numbers of semi-Pfaffian sets

This chapter is devoted to the study of the possible complexity bounds that can be proved on the Betti numbers of semi-Pfaffian sets defined on a domain of bounded complexity.

These results include bounds for the sum of Betti numbers of compact and non-compact Pfaffian varieties (Theorem 2.8 and Theorem 2.10), bounds for the sum of Betti num-bers of basic semi-Pfaffian sets (Lemma 2.14) and semi-Pfaffian sets given by P-closed formulas (the main result of this chapter, Theorem 2.17). Theorem 2.25 gives a bound on C(V;P), the number of connected sign cells of the family P on V that was introduced in Definition 1.29, and this allows to establish in Theorem 2.32 a bound on the Borel-Moore homology of arbitrary (locally closed) semi-Pfaffian sets. In particular, this last result provides an upper-bound for the sum of Betti numbers of any compact semi-Pfaffian set, without requiring the defining formula to beP-closed.

Recently, Gabrielov and Vorobjov [GV4] generalized the results of the present chapter:

they established a general, single-exponential bound for the sum of the Betti numbers of any semi-Pfaffian set, without any assumption on its topology or defining formula. Such a result was not known even in the algebraic case, and the precise statement was added at the end of this chapter (Theorem 2.34) for reference purposes.

The setting for the present chapter will be the following: we will consider a fixed Pfaffian chain f of length ℓ and degreeα in a domainU ⊆Rⁿ of bounded complexity for the chain f. We will let g be an exhausting function for U,and γ = deg_fg.

Throughout this chapter,p1, . . . , ps,andq1, . . . , qr,will be Pfaffian functions in the fixed chain f, and we’ll write P for {p₁, . . . , p_s} and deg_fP for max{deg_fp_i | 1≤ i≤ s}. The numberβ will be a common upper-bound for deg_fpi and deg_fqj.We’ll let q=q²₁+· · ·+q²_r and V =Z(q1, . . . , qr) =Z(q). The dimension of V will be denoted byd. We’ll let

V(n, ℓ, α, β, γ) = 2^{ℓ(ℓ−1)/2}β(α+β−1)ⁿ⁻¹γ

2[n(α+β−1) +γ+ min(n, ℓ)α]^ℓ. (2.1) The chapter is organized as follows.

• In the first section, we show that b(V) can be bounded in terms ofV(n, ℓ, α, β, γ).

47

• In section 2.2, we show that if X is a semi-Pfaffian subset of a compact variety V given by a P-closed formula, b(X)≤(5s)^dV(n, ℓ, α,2β, γ).

• In section 2.3, we show thatC(V;P)≤Σ(s, d)V(n, ℓ, α, β^⋆, γ),whereβ^⋆ = max(β, γ) and

Σ(s, d) = X

0≤i≤d

4s+ 1 i

.

• The last section is devoted to proving that the rank of the Borel-Moore homology groups of a locally closed X with the above format is bounded by an expression of the form

b^BM(X)≤s^2d2^ℓ(ℓ−1)O(nβ+ min(n, ℓ)α)^2(n+ℓ); for some constant depending onU.

The main results of this chapters are inspired by similar results in the semi-algebraic case by Basu, Pollack and Roy [B2, BPR1, BPR3]. Note that more analogues could also be formulated for more recent results in the same vein [B1, B4, B5]. Indeed, the o-minimality of the structure generated by Pfaffian functions ensures that most arguments can still be used. The use of infinitesimals in those papers can be avoided most of the time by placing oneself in a compact setting and replacing the infinitesimals in small real numbers. (The proof of Theorem 2.17 is an example of how one can compute a real number r such that the condition thatε is an infinitesimal can be replaced by ε < r.)

As in the work of Basu, Pollack and Roy, one of the ideas behind the bounds is the notion of combinatorial level of a family of functions P.

Definition 2.1 (Combinatorial level) LetX ⊆ U be a semi-Pfaffian set andP a family of functions on U. The combinatorial level of the couple (X,P) is the largest integer m such that there exists x in X and m functions in P vanishing at x.

This leads to a combinatorial definition of the idea of general position.

Definition 2.2 (General position) Let V be a Pfaffian variety. The set P is said to be in general positionwith V if the combinatorial level of (V,P) is bounded by dim(V).