Analytic cycles on real analytic manifolds

J. Bochnak and W. Kucharz^{∗}

1 Introduction

Let X be a compact real analytic manifold of dimension n. In this paper we study homology classes in H_{∗}(X, ℤ/2) represented by analytic subsets of X. Here a subset V of X is called analytic if V = f^{−1}(0), where f : X → ℝ is an analytic function. Thus analytic sets are zeros of global analytic functions.

Each analytic set V ⊆ X of dimension k represents a homology class in H_{k}(X, ℤ/2), written [V]_{X}, that corresponds to the fundamental class of V (cf. [7]), and the set of all classes of this type forms a subgroup of H_{k}(X, ℤ/2), denoted H_{k}^{an}(X, ℤ/2). It is convenient to define and work with

H_{an}^{ℓ}(X, ℤ/2) = D_{X}^{−1}(H_{k}^{an}(X, ℤ/2)),

where D_{X} : H^{ℓ}(X, ℤ/2) → H_{k}(X, ℤ/2) is the Poincaré duality isomorphism and k + ℓ = n. Any cohomology class in H_{an}^{∗} is said to be analytic or represented by an analytic set.

It is well known that each homology class in H_{n−1}(X, ℤ/2) is represented by an analytic submanifold of X of codimension 1. In particular, one has always H^{1}(X, ℤ/2) = H_{an}^{1}(X, ℤ/2). The situation is quite different for H^{2}(X, ℤ/2), where the analogous equality is not always satisfied.

∗The paper was written at the Max-Planck-Institut für Mathematik in Bonn, whose support and hospitality are gratefully acknowledged. Both authors were partially supported by European grant RAAG 2002-2006.

Surprisingly, the subgroup H_{an}^{2}(X, ℤ/2) admits a rather elegant and purely topological description. Let Vect(X) denote the set of isomorphism classes of topological real vector bundles over X and define

W (X) = {v ∈ H^{2}(X, ℤ/2)| v = w_{2}(ξ) for some ξ in Vect(X)},

where w_{2}(ξ) is the second Stiefel-Whitney class of ξ. One easily sees that W (X) is a subgroup of H^{2}(X, ℤ/2), cf. for example [5, Proposition 1.4]. For X of dimension less than 6, one has H^{2}(X, ℤ/2) = W (X). However, for each n ≥ 6, there exists a compact n-dimensional real analytic manifold X with H^{2}(X, ℤ/2) ≠ W (X), cf. [19].

Theorem 1.1 Let X be a compact real analytic manifold and let v be a cohomology class in H^{2}(X, ℤ/2). Then v is analytic if and only if v is in W (X). Briefly,

H_{an}^{2}(X, ℤ/2) = W (X).

Corollary 1.2 For each n ≥ 6 there exists a compact n-dimensional real analytic manifold X with H^{2}(X, ℤ/2) ≠ H_{an}^{2}(X, ℤ/2).

Similar problems for complex analytic manifolds and cohomology classes with coefficients in ℤ were investigated by Atiyah and Hirzebruch [2] by completely different methods.

The proof of Theorem 1.1 is given in Section 3.

2 The Grothendieck formula in real analytic setting

Let X be a paracompact real analytic manifold. By an analytic subset of X we mean a subset of the form V = f^{−1}(0), where f : X → ℝ is an analytic function (the reader may refer to Whitney and Bruhat [21] for fundamental properties of such globally defined real analytic sets). Let O(X) (resp. O_{x}) denote the ring of real analytic functions on X (resp. the ring of germs of real analytic functions at x in X) and let

I(V) = {f ∈ O(X)| f|V = 0}
J_{V,x} = {f ∈ O_{x}| f|V = 0}

be the ideals of functions or germs of functions vanishing on V. We define the local dimension dim V_{x} of V at x to be the Krull dimension of the ring O_{x}/J_{V,x}, and dimension dim V of V as

dim V = max{dim V_{x}| x ∈ V }.

The set RegV of globally regular points of V is, by definition, the collection of all points x in V for which the local ring O_{x}/I(V)O_{x} is regular of Krull dimension dim V. We write “globally regular” to emphasize that the ideal I(V)O_{x}, instead of J_{V,x}, is used in our definition. It follows from [21, Proposition 16] that V \RegV is an analytic subset of X (defined by global analytic functions on X) of dimension less than dim V ([21, Proposition 16] is applicable since the sheaf of ideals on X with stalk I(V)O_{x} at each point x in X is coherent, cf. [9, Corollaire (I,8)], and therefore by virtue of Cartan’s Theorem A it is the largest coherent sheaf of ideals on X with zero locus containing V).

We shall use the following facts:

(i) If x is in RegV, then I(V)O_{x} = J_{V,x} (cf. [1, p. 225, Proposition 4.4d]); in particular, there is a neighborhood U of x in X such that V ∩U is an analytic submanifold of U of dimension dim V (however, a point of V having such a neighborhood is not necessarily globally regular).

(ii) For each x in X, the ring O_{x} is flat over O(X)_{m_{x}}, the localization of the ring O(X) at the maximal ideal

m_{x} = {f ∈ O(X)| f(x) = 0} (cf. [1, p. 226]).

(iii) If X is compact, the ring O(X) is regular (that is, Noetherian whose localization at each maximal ideal, which is always of the form m_{x} for some x in X, is a regular local ring) of finite Krull dimension (cf. [9] and [1, p. 225, Proposition 4.4d]).

The following result is a real analytic version of the Grothendieck formula (for its real algebraic version, not needed here, the reader may consult [6]).

Theorem 2.1 Let X be a compact real analytic manifold and let v be a cohomology class in H_{an}^{2}(X, ℤ/2). Then there exists an analytic vector bundle ξ over X such that w_{1}(ξ) = 0 and w_{2}(ξ) = v.

Proof. Let V be an analytic subset of X of codimension 2 representing v, that is, D_{X}(v) = [V]_{X}. In the proof it is convenient to assume, which we can do without loss of generality, that X is connected and V is irreducible (in the class of analytic subsets of X defined by global real analytic function on X). Let n = dim X.

Step 1. First we shall construct an auxiliary projective module of rank 2 over an appropriate localization of the ring O(X).

Let x_{0} be a point in RegV. Since O_{x_{0}} is a regular local ring, there exist (see [14, p. 168, Proposition 1.10]) two elements a_{1} and a_{2} in I(V) such that

I(V)O_{x_{0}} = (a_{1}, a_{2})O_{x_{0}}.

Hence, using flatness (property (ii) above), one gets

I(V)O(X)_{m_{x_{0}}} = (a_{1}, a_{2})O(X)_{m_{x_{0}}}.

Since the ideal I(V) is finitely generated and V \RegV is an analytic subset of V, we can choose λ in O(X) satisfying

(1) λI(V) ⊆ (a_{1}, a_{2})O(X) and λ(x_{0}) ≠ 0, V \RegV ⊆ λ^{−1}(0).

Define

Y = X\(V ∩ λ^{−1}(0)) and W = V \(V ∩ λ^{−1}(0)).

Irreducibility of V implies

dim(X\Y) = dim(V ∩ λ^{−1}(0)) < n − 2.

Clearly, W is an analytic subset of Y of codimension 2 and W ⊂ RegV.

Consider the map F : X × ℝ^{2} → ℝ^{2} defined by

F (x, t) = F_{t}(x) = (a_{1}(x) + t_{1}b(x)^{2}, a_{2}(x) + t_{2}b(x)^{2}),

where x is in X, t = (t_{1}, t_{2}) is in ℝ^{2}, and b is an element of O(X) with b^{−1}(0) = V. Observe that F |Y ×ℝ^{2} is transverse to 0 in ℝ^{2}. Indeed, let (x, t) be a point in Y ×ℝ^{2} and F (x, t) = 0. If x is not in W, then transversality at (x, t) is obvious. If x is in W, then λ(x) ≠ 0 and hence (1) implies I(V)O_{x} = (a_{1}, a_{2})O_{x}. Since x belongs to RegV, it follows from (i) that also J_{V,x} = (a_{1}, a_{2})O_{x}, which implies transversality of (a_{1}, a_{2}) : X → ℝ^{2} at x to 0 in ℝ^{2}, and therefore also transversality of F_{t} at x and F at (x, t) to 0 in ℝ^{2}.

By a standard transversality theorem [12, p. 79, Theorem 2.7], we can choose t in ℝ^{2} such that for

F_{t} = (f_{1}, f_{2}) : X → ℝ^{2},

the restriction F_{t}|Y : Y → ℝ^{2} is transverse to 0 in ℝ^{2}. It follows that for each point x in W, one has J_{V,x} = (f_{1}, f_{2})O_{x} and hence in view of (i),

(2) I(V)O_{x} = (f_{1}, f_{2})O_{x}.

Using again property (ii) (flatness), one deduces

(3) I(V)O(X)_{m_{x}} = (f_{1}, f_{2})O(X)_{m_{x}}.

for all x in W. Condition (2) implies

(4) f_{1}^{−1}(0) ∩ f_{2}^{−1}(0) ∩ Y = W ∪ W′ with W ∩ W′ = ∅.

The intersection V′ of all analytic subsets of X containing W′ is an analytic subset of X (cf. [21, p. 154]) satisfying

(5) V′ ⊆ f_{1}^{−1}(0) ∩ f_{2}^{−1}(0).

Claim. W′ = V′ ∩ Y. The Claim is proved as follows. By (3), for each point x in W, there exists an analytic function μ_{x} in O(X) such that

μ_{x}I(V) ⊆ (f_{1}, f_{2})O(X) and μ_{x}(x) ≠ 0.

Let μ_{1}, …, μ_{p} be a finite system of generators of the ideal of O(X) generated by all μ_{x} with x in W, and let μ = μ_{1}^{2} + ⋯ + μ_{p}^{2}. Then

(6) W ⊂ ⋃_{x∈W}(X\μ_{x}^{−1}(0)) = X\ ⋂_{x∈W} μ_{x}^{−1}(0) = X\μ^{−1}(0),

and μI(V) ⊆ (f_{1}, f_{2})O(X). The last inclusion implies

(7) f_{1}^{−1}(0) ∩ f_{2}^{−1}(0) ⊆ μ^{−1}(0) ∪ V.

By (4) and (5),

(8) V′ ∩ Y ⊆ W ∪ W′.

Since in view of (4) and (7),

W ∪ W′ ⊆ (μ^{−1}(0) ∪ V) ∩ Y = (μ^{−1}(0) ∩ Y) ∪ (V ∩ Y) = (μ^{−1}(0) ∩ Y) ∪ W,

one gets W′ ⊆ μ^{−1}(0) ∩ Y ⊆ μ^{−1}(0), the sets W and W′ being disjoint. Hence also

(9) V′ ⊆ μ^{−1}(0).

Conditions (8), (9), and (6) imply

V′ ∩ Y ⊆ (W ∪ W′) ∩ μ^{−1}(0) = W′ ∩ μ^{−1}(0) ⊆ W′

and the Claim follows, the inclusion W′ ⊆ V′ ∩ Y being trivial.

Choose ψ_{1} and ψ_{2} in O(X) with ψ_{1}^{−1}(0) = V and ψ_{2}^{−1}(0) = V′. Then ψ_{1}^{−1}(0) ∩ Y = W and, by the Claim, ψ_{2}^{−1}(0) ∩ Y = W′. In view of (4) and the fact that (f_{1}, f_{2})|Y : Y → ℝ^{2} is transverse to 0 in ℝ^{2}, for each point x in Y, the germ of ψ_{1}ψ_{2} at x belongs to the ideal (f_{1}, f_{2})O_{x}. Hence making use of (ii), one gets

(10) ψ_{1}ψ_{2} ∈ (f_{1}, f_{2})O(X)_{m_{x}} for all x in Y = X\(V ∩ λ^{−1}(0)).

Define a multiplicatively closed subset S of the ring O(X) by

S = {f ∈ O(X)| f^{−1}(0) ⊂ V ∩ λ^{−1}(0)}

and consider the localization A = S^{−1}O(X) of O(X) at S. Each maximal ideal of A is of the form m_{x}A, where m_{x} is the maximal ideal of O(X) corresponding to a point x in Y. Hence (10) implies that ψ_{1}ψ_{2} belongs to (f_{1}, f_{2})A, that is,

ψ_{1}ψ_{2} = (h_{1}f_{1} + h_{2}f_{2})/c,

where h_{1}, h_{2} are in O(X) and c is in S. It follows from the last equation that the matrices

g_{12} = ( \begin{matrix}h_{1}/cψ_{2}^{2} & h_{2}/cψ_{2}^{2} \\ −f_{2}ψ_{1}/ψ_{2} & f_{1}ψ_{1}/ψ_{2} \end{matrix} ) , g_{21} = ( \begin{matrix}f_{1}ψ_{2}/ψ_{1} & −h_{2}/cψ_{1}^{2} \\ f_{2}ψ_{2}/ψ_{1} & h_{1}/cψ_{1}^{2} \end{matrix} ) ,

with entries in the field O(X)_{(0)} of fractions of O(X), are invertible and satisfy g_{21} = g_{12}^{−1}. The entries of g_{21} and g_{12} are analytic on U_{1} = Y \W and U_{2} = Y \W′, respectively, and {U_{1}, U_{2}} is an open cover of Y. Regarding A as a subring of O(X)_{(0)}, set

P = {(α, β) ∈ A^{2} × A^{2}| β = g_{21}α},

where elements of A^{2} are viewed as column vectors. Since the ring O(X) is Noetherian (recall (iii)), so is the localization A of O(X), and hence P is a finitely generated submodule of the A-module A^{2} × A^{2}.

We assert that P is a projective A-module of rank 2. To establish this we just have to prove that the localization P_{m} of P at each maximal ideal m of A is a free A_{m}-module of rank 2 [14, p. 113, Corollary 3.6]. We proceed as follows. Define

\begin{matrix}γ_{1} = & cψ_{1}^{2}(( \begin{matrix}1 \\ 0 \end{matrix} ) , g_{21} ( \begin{matrix}1 \\ 0 \end{matrix} )) = (( \begin{matrix}cψ_{1}^{2} \\ 0 \end{matrix} ) , ( \begin{matrix}cf_{1}ψ_{1}ψ_{2} \\ f_{2}ψ_{2} \end{matrix} )), \\ γ_{2} = & cψ_{1}^{2}(( \begin{matrix}0 \\ 1 \end{matrix} ) , g_{21} ( \begin{matrix}0 \\ 1 \end{matrix} )) = (( \begin{matrix}0 \\ cψ_{1}^{2} \end{matrix} ) , ( \begin{matrix}−h_{2} \\ h_{1} \end{matrix} )), \\ η_{1} = & cψ_{2}^{2}(g_{12} ( \begin{matrix}1 \\ 0 \end{matrix} ) , ( \begin{matrix}1 \\ 0 \end{matrix} )) = (( \begin{matrix}h_{1} \\ −cf_{2}ψ_{1}ψ_{2} \end{matrix} ) , ( \begin{matrix}cψ_{2}^{2} \\ 0 \end{matrix} )), \\ η_{2} = & cψ_{2}^{2}(g_{12} ( \begin{matrix}0 \\ 1 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \end{matrix} )) = (( \begin{matrix}h_{2} \\ cf_{1}ψ_{1}ψ_{2} \end{matrix} ) , ( \begin{matrix}0 \\ cψ_{2}^{2} \end{matrix} )). \end{matrix}

Each of these elements is in P. Given any maximal ideal m = m_{x}A of A, with x in U_{1} (resp. x in U_{2}), the element cψ_{1}^{2} (resp. cψ_{2}^{2}) is a unit in A_{m}, and hence γ_{1}, γ_{2} (resp. η_{1}, η_{2}) generate freely P_{m} over A_{m}. The assertion is proved.

Step 2. Construction of a topological vector bundle ζ over X with w_{2}(ζ) = v.

Consider the finitely generated projective A-module P of rank 2 constructed in Step 1. Since the ring O(X) is regular of finite Krull dimension (recall (iii)), its projective dimension is also finite [14, p. 208, Proposition 2.4]. Hence, by [3, p. 453, Proposition 2.1, p. 492, Proposition 6.1], there exists a finitely generated projective O(X)-module Q such that the A-modules A ⊗_{O(X)} Q and P are stably isomorphic.

Each element of A determines an analytic function in O(Y). This gives rise to a canonical homomorphism A → C(Y) from A into the ring C(Y) of all continuous functions from Y into ℝ. The C(Y)-modules C(Y) ⊗_{A} (A ⊗_{O(X)} Q) and C(Y) ⊗_{A} P are stably isomorphic. Making use of the restriction homomorphisms O(X) → C(Y) and C(X) → C(Y), we obtain the following isomorphisms of C(Y)-modules:

C(Y) ⊗_{A} (A ⊗_{O(X)} Q) ≅ C(Y) ⊗_{O(X)} Q ≅ C(Y) ⊗_{C(X)} (C(X) ⊗_{O(X)} Q).

Hence the finitely generated projective C(Y)-modules C(Y) ⊗_{C(X)} (C(X) ⊗_{O(X)} Q) and C(Y) ⊗_{A} P are stably isomorphic.

Let ζ (resp. η) be the topological real vector bundle over X (resp. Y) corresponding to C(X) ⊗_{O(X)} Q (resp. C(Y) ⊗_{A} P), cf. [18]. It follows that ζ|Y and η are stably isomorphic.

For our purposes we need to describe the vector bundle η more explicitly. Define

E = {(y, u_{1}, u_{2}) ∈ Y × ℝ^{2} × ℝ^{2}| u_{2} = g_{21}(y)u_{1} for y in U_{1} and
u_{1} = g_{12}(y)u_{2} for y in U_{2}},
π : E → Y, π(y, u_{1}, u_{2}) = y.

Since {U_{1}, U_{2}} is an open cover of Y, the set E is a closed analytic submanifold of Y ×ℝ^{2}×ℝ^{2} and π is an analytic map. For each point y in Y, the fiber E_{y} = π^{−1}(y) is a vector subspace of {y} × ℝ^{2} × ℝ^{2}. Furthermore, the map

U_{k} × ℝ^{2} → π^{−1}(U_{k}), (y, u) → (y, g_{1k}(y)u, g_{2k}(y)u)

is an analytic diffeomorphism for k = 1, 2, where g_{kk}(y) denotes the identity matrix. Thus η′ = (E, π, Y) is an analytic real vector bundle of rank 2 over Y.

We claim that η and η′ are isomorphic as topological vector bundles. In order to demonstrate this claim it suffices to show that the C(Y)-module Γ of all continuous sections of η′ is isomorphic to C(Y) ⊗_{A} P (cf. [18]). Note that

Γ = {(φ, ψ) ∈ C(Y)^{2} × C(Y)^{2}| ψ(y) = g_{21}(y)φ(y) for y in U_{1} and
φ(y) = g_{12}(y)ψ(y) for y in U_{2}}.

On the other hand, since P is a projective submodule of the A-module A^{2} × A^{2}, we obtain that C(Y) ⊗_{A} P is isomorphic to the submodule C(Y)P of the C(Y)-module C(Y)^{2} × C(Y)^{2}, where C(Y)P is the submodule generated by P over C(Y) (this follows from the fact that any projective module is flat). One readily check C(Y)P = Γ, and hence the vector bundles η and η′ are topologically isomorphic, as claimed.

Henceforth we shall identify η with η′ and regard η as an analytic vector bundle over Y. The analytic map

s : Y → E, s(y) = (y, ( \begin{matrix}ψ_{1}(y) \\ 0 \end{matrix} ) , ( \begin{matrix}f_{1}(y)ψ_{2}(y) \\ f_{2}(y)ψ_{2}(y) \end{matrix} ))

is a section of η = η′ = (E, π, Y) with s^{−1}(0_{E}) = W, where 0_{E} is the image of the zero section of η. On U_{2} the section s is represented by (f_{1}, f_{2}) : U_{2} → ℝ^{2}, and therefore s is transverse to 0_{E}.

Let [W]_{Y} be the Borel-Moore homology class in H_{n−2}^{BM}(Y, ℤ/2) represented by W (since Y and possibly W are not compact, the use of the Borel-Moore homology is necessary [7]). Since W = s^{−1}(0_{E}) and s is transverse to 0_{E}, it follows that

(11) D_{Y} (w_{2}(η)) = [W]_{Y} ,

where D_{Y} : H^{∗}(Y, ℤ/2) → H_{∗}^{BM}(Y, ℤ/2) is the Poincaré duality isomorphism.

As ζ|Y and η are stably isomorphic, one has

(12) j^{∗}(w_{2}(ζ)) = w_{2}(ζ|Y) = w_{2}(η),

where j : Y ↪ X is the inclusion map and j^{∗} : H^{∗}(X, ℤ/2) → H^{∗}(Y, ℤ/2) the induced homomorphism.

On the other hand, commutativity of the diagram

H^{2}(X, ℤ/2) ⟶^{j^{∗}} H^{2}(Y, ℤ/2)
D_{X} ↓ ↓^{D_{Y}}
H_{n−2}(X, ℤ/2) ⟶^{ρ} H_{n−2}^{BM}(Y, ℤ/2),

where ρ is the restriction homomorphism (which is defined for each open subset Y of X, cf. [7]), implies

j^{∗}(v) = j^{∗}(D_{X}^{−1}([V]_{X})) = D_{Y}^{−1}(ρ([V]_{X})) = D_{Y}^{−1}([W]_{Y}).

Hence, according to (11),

(13) j^{∗}(v) = w_{2}(η).

The homomorphism j^{∗} is injective, as we can see by examining the exact sequence

H^{2}(X, Y, ℤ/2) → H^{2}(X, ℤ/2) ⟶^{j^{∗}} H^{2}(Y, ℤ/2).

Indeed, since H^{2}(X, Y, ℤ/2) is isomorphic to H_{n−2}(X\Y, ℤ/2) (cf. [17, p. 351, Lemma 14]) and dim(X\Y) < n − 2, we have H^{2}(X, Y, ℤ/2) = 0 and j^{∗} is injective. Hence comparing (12) and (13), one gets w_{2}(ζ) = v. The construction in Step 2 is accomplished.

Step 3. Construction of an analytic real vector bundle ξ over X with w_{1}(ξ) = 0 and w_{2}(ξ) = v.

Let r be the rank of ζ and Θ = Λ^{r}ζ. Since w_{1}(Θ) = w_{1}(ζ) (cf. [13, p. 246]), the topological vector bundle ξ = ζ ⊕ Θ ⊕ Θ ⊕ Θ over X satisfies w_{1}(ξ) = 0 and w_{2}(ξ) = v.

In order to complete the proof it remains to recall that each topological vector bundle over X is isomorphic to an analytic one. This is true since any vector bundle over X is a pullback of the universal vector bundle over an appropriate Grassmannian and the following approximation theorem holds: For any compact real analytic manifolds Z and Z′, the set of analytic maps from Z into Z′ is dense in the space of all smooth (of class C^{∞}) maps from Z into Z′, cf. [16, Theorem 1.6.5] and [11, Theorem 3]. □

3 Analytic cycles

Proof of Theorem 1.1. Theorem 2.1 implies H_{an}^{2}(X, ℤ/2) ⊆ W (X). To prove the inclusion in the opposite direction, let u be in W (X) and choose a topological real vector bundle ξ of rank p over X with w_{2}(ξ) = u. Let f : X → G_{m,p}(ℝ) be a continuous map into the Grassmannian G_{m,p}(ℝ) for some m, such that ξ is isomorphic to the pullback f^{∗}γ of the universal vector bundle γ over G_{m,p}(ℝ). Since H^{∗}(G_{m,p}(ℝ), ℤ/2) = H_{an}^{∗}(G_{m,p}(ℝ), ℤ/2) (in fact, each cohomology class in H^{∗}(G_{m,p}(ℝ), ℤ/2) is represented by an algebraic subset [4, Proposition 11.3.3]), Proposition 3.1 below implies that the cohomology class u = w_{2}(ξ) = w_{2}(f^{∗}γ) = f^{∗}(w_{2}(γ)) is in H_{an}^{2}(X, ℤ/2). This shows W (X) ⊆ H_{an}^{2}(X, ℤ/2), which completes the proof. □

Proposition 3.1 Let f : X → Y be a continuous map between compact real analytic manifolds. Then the induced homomorphism f^{∗} : H^{∗}(Y, ℤ/2) → H^{∗}(X, ℤ/2) satisfies

f^{∗}(H_{an}^{k}(Y, ℤ/2)) ⊆ H_{an}^{k}(X, ℤ/2)

for each k ≥ 0.

Proof. Let w be a cohomology class in H_{an}^{k}(Y, ℤ/2). We shall show that f^{∗}(w) is in H_{an}^{k}(X, ℤ/2).

Choose an analytic subset W of Y such that w = D_{Y}^{−1}([W]_{Y}). Let {W_{1}, …, W_{s}} be a stratification of W satisfying Whitney’s condition (a) (for the definition see [10] or [4, p. 242], for the existence see [15]). The set of smooth maps from X into Y transverse to each of the submanifolds W_{1}, …, W_{s} is open and dense in the space of all smooth maps [8, Proposition 3.6] (it is Whitney’s condition (a) that guarantees openness). Since the set of analytic maps from X into Y is dense in the space of all smooth maps, there exists an analytic map g : X → Y homotopic to f and transverse to W_{i} for 1 ≤ i ≤ s. The inverse image V = g^{−1}(W) is an analytic subset of X and, by [7, Proposition 2.15], the cohomology class v = D_{X}^{−1}([V]_{X}) satisfies v = g^{∗}(w). The proof is complete since v is in H_{an}^{k}(X, ℤ/2) and v = g^{∗}(w) = f^{∗}(w). □

As a corollary to Proposition 3.1 we immediately obtain the following result.

Proposition 3.2 For any compact real analytic manifold X, the group

H_{an}^{∗}(X, ℤ/2) = ⊕_{k≥0}H_{an}^{k}(X, ℤ/2)

is a subring of the cohomology ring H^{∗}(X, ℤ/2).

Proof. Let u and v be cohomology classes in H_{an}^{k}(X, ℤ/2) and H_{an}^{ℓ}(X, ℤ/2), respectively. It readily follows that the cross product u × v belong to H_{an}^{k+ℓ}(X × X, ℤ/2). Recall that for the cup product we have u ∪ v = δ^{∗}(u × v), where δ : X → X × X is defined by δ(x) = (x, x) for all x in X. Hence u ∪ v is in H_{an}^{k+ℓ}(X, ℤ/2) by virtue of Proposition 3.1. □

In contrast to Proposition 3.1, analytic homology classes to not behave in a functorial manner.

Remark 3.3. With notation as in Proposition 3.1, it is not true, in general, that

(∗) f_{∗}(H_{∗}^{an}(X, ℤ/2)) ⊆ H_{∗}^{an}(Y, ℤ/2).

Indeed, let Y be a compact real analytic manifold having a homology class z which is not in H_{∗}^{an}(Y, ℤ/2). By [20], there are a compact real analytic manifold X and an analytic map f : X → Y such that f_{∗}([X]) = z. Hence the validity of (∗) would lead to a contradiction.

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

J. Bochnak Department of Mathematics Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: bochnak@cs.vu.nl

W. Kucharz Department of Mathematics and Statistics University of New Mexico Albuquerque, New Mexico 87131-1141 U.S.A. e-mail: kucharz@math.unm.edu