Algebraic equivalence of cycles and algebraic models of smooth manifolds

W. Kucharz^{∗}

1 Introduction and results

Throughout this paper the term real algebraic variety designates a locally ringed space isomorphic to an algebraic subset of ℝ^{n}, for some n, endowed with the Zariski topology and the sheaf of ℝ-valued regular functions. Morphisms between real algebraic varieties will be called regular maps. Basic facts on real algebraic varieties and regular maps can be found in [7]. Every real algebraic variety carries also the Euclidean topology, which is determined by the usual metric topology on ℝ. Unless explicitly stated otherwise, all topological notions related to real algebraic varieties will refer to the Euclidean topology.

Let X be a reduced quasiprojective scheme over ℝ. The set X (ℝ) of ℝ-rational points of X is contained in an affine open subset of X. Thus if X (ℝ) is dense in X, we can regard X (ℝ) as a real algebraic variety whose structure sheaf is the restriction of the structure sheaf of X; up to isomorphism, each real algebraic variety is of this form.

Given a compact nonsingular real algebraic variety X (as in [3, 7], nonsingular means that the irreducible components of X are pairwise disjoint, nonsingular, and of the same dimension), we can find a nonsingular quasiprojective scheme X over ℝ with X (ℝ) = X dense in X. Then we have the cycle homomorphism

cℓ_{ℝ} : Z^{k}(X) → H^{k}(X, ℤ/2)

∗The paper was written at the Max-Planck-Institut für Mathematik in Bonn, whose support and hospitality are gratefully acknowledged.

defined on the group Z^{k}(X) of algebraic cycles on X of codimension k : for any integral subscheme V of X of codimension k, the cohomology class cℓ_{ℝ}(V) is Poincaré dual to the homology class represented by the subvariety V(ℝ) of X assuming V(ℝ) has codimension k in X, and otherwise cℓ_{ℝ}(V) = 0 [10]. The subgroup

H_{alg}^{k}(X, ℤ/2) = cℓ_{ℝ}(Z^{k}(X))

of H^{k}(X, ℤ/2) plays a fundamental role in real algebraic geometry (cf. [9] for a short survey of its properties and applications). We define

Alg^{k}(X),

the main object of our investigation here, to be the image under cℓ_{ℝ} of the subgroup of Z^{k}(X) consisting of the cycles algebraically equivalent to 0 (we refer to [13, Chapter 10] for the theory of algebraic equivalence). Thus, by definition, Alg^{k}(X) is a subgroup of H_{alg}^{k}(X, ℤ/2). It readily follows that H_{alg}^{k}(X, ℤ/2) and Alg^{k}(X) do not depend on the choice of X. Note that Alg^{k}(X) can also be described as follows. An element u of H_{alg}^{k}(X, ℤ/2) belongs to Alg^{k}(X) if and only if there exist a compact nonsingular irreducible real algebraic variety T, two points t_{0} and t_{1} in T, and a cohomology class z in H_{alg}^{k}(X × T, ℤ/2) such that u = i_{t_{1}}^{∗}(z) − i_{t_{0}}^{∗}(z), where given t in T, we let i_{t} : X → X × T denote the map defined by i_{t}(x) = (x, t) for all x in X, while

i_{t}^{∗} : H^{∗}(X × T, ℤ/2) → H^{∗}(X, ℤ/2)

is the induced homomorphism (this does not force u = 0, the parameter space T being possibly disconnected).

Why is the group Alg^{k}(X) of interest? It was R. Silhol who first demonstrated that Alg^{1}(X) is important for understanding of H_{alg}^{1}(X, ℤ/2) [19]. In [17] it is proved, among other things, that Alg^{1}(−) is a birational invariant. The group Alg^{k}(X) strongly influences the behavior of H_{alg}^{n−k}(X, ℤ/2), where n = dim X [16, 17]. Substantial constructions of [18], at the borderline between real algebraic geometry and differential topology, depend on Alg^{k}(−). For some remarkable properties of Alg^{k}(X) contained in [1, 16] see also Theorem 2.1 in Section 2. It is in general very difficult to compute Alg^{k}(X), except the cases k = 0 or k = dim X (cf. for example [1] to see how these trivial cases are settled). In this paper we investigate the groups Alg^{k}(X) as X runs through the class of varieties diffeomorphic to a fixed variety. Below we make this precise.

All smooth (of class C^{∞}) manifolds that appear here are paracompact and without boundary. By Tognoli’s theorem [21, 7], any compact smooth manifold M has an algebraic model, that is, there exists a nonsingular real algebraic variety X diffeomorphic to M. We study how the groups Alg^{k}(X) vary as X run through the class of algebraic models of M. The kth Stiefel-Whitney class of M will be denoted by w_{k}(M), while [M] will stand for the fundamental class of M in H_{m}(M, ℤ/2), m = dim M. As usual, we use ∪ and <, > to denote the cup product and scalar (Kronecker) product.

Theorem 1.1 Let M be a compact smooth manifold of dimension m with m ≥ 2. Given a subgroup G of H^{1}(M, ℤ/2), the following conditions are equivalent:

(a) There exist an algebraic model X of M and a smooth diffeomorphism φ : M → X such that φ^{∗}(Alg^{1}(X)) = G,

(b) G is contained in the image of the reduction modulo 2 homomorphism H^{1}(M, ℤ) → H^{1}(M, ℤ/2) and for each integer ℓ, 1 ≤ ℓ ≤ m, and all u_{1}, …, u_{ℓ} in G, one has < u_{1} ∪ … ∪ u_{ℓ} ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] >= 0 for all nonnegative integers i_{1}, …, i_{r} with i_{1} + ⋯ + i_{r} = m − ℓ.

Furthermore, if (b) holds, then X in (a) can be chosen irreducible.

Our second result is the following.

Theorem 1.2 Let M be a compact connected smooth manifold of dimension m with m ≥ 3. Given a subgroup G of H^{m−1}(M, ℤ/2), the following conditions are equivalent:

(a) There exist an algebraic model X of M and a smooth diffeomorphism φ : M → X such that φ^{∗}(Alg^{m−1}(X)) = G, (b) < u ∪ w_{1}(M), [M] > 0 for all u in G.

Theorems 1.1 and 1.2 are proved in Section 2. We also have another result of the same type, Theorem 2.5 in Section 2, dealing with certain subgroups G of H^{k}(M, ℤ/2) for other values of k. Example 2.7 at the end of the paper shows how Theorems 1.1, 1.2, and 2.5 work in a special case.

2 Proofs

The groups H_{alg}^{k}(−, ℤ/2) and Alg^{k}(−) have the expected functorial properties. If f : X → Y is a regular map between compact nonsingular real algebraic varieties, then the induced homomorphism

f^{∗} : H^{∗}(Y, ℤ/2) → H^{∗}(X, ℤ/2)

satisfies

f^{∗}(H_{alg}^{∗}(Y, ℤ/2)) ⊆ H_{alg}^{∗}(X, ℤ/2) and f^{∗}(Alg^{∗}(Y)) ⊆ Alg^{k}(X).

Furthermore,

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

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

Alg^{∗}(X) = ⊕_{q≥0}Alg^{q}(X)

is an ideal of H_{alg}^{∗}(X, ℤ/2). These assertions concerning H_{alg}^{k}(−, ℤ/2) are proved in [10, 6] and they immediately imply the corresponding assertions about Alg^{k}(−).

Recall that if M is a smooth manifold, then a cohomology class u in H^{k}(M, ℤ/2), k ≥ 1, is said to be spherical, provided that u = f^{∗}(c), where f : M → S^{k} is a continuous (or equivalently smooth) map from M into the unit k-sphere S^{k} and c is the unique generator of the group H^{k}(S^{k}, ℤ/2) ≃ ℤ/2.

We shall make use of the following result.

Theorem 2.1 Let X be a compact nonsingular real algebraic variety. Then:

(i) < u ∪ v, [X] >= 0 for all u in Alg^{k}(X) and v in H_{alg}^{ℓ}(X, ℤ/2), where k + ℓ = dim X,

(ii) < u∪w_{i_{1}}(X)∪…∪w_{i_{r}}(X), [X] >= 0 for all u in Alg^{k}(X) and all nonnegative integers i_{1}, …, i_{r} with i_{1} + ⋯ + i_{r} = dim X − k,

(iii) If k = 1 or if k = dim X − 1 and X is connected, then every cohomology class in Alg^{k}(X) is spherical.

Reference for the proof. [16, Theorem 2.1] and [1, Theorem 1.1].

The next fact will also be very useful. Let B^{k} be a nonsingular irreducible real algebraic variety with precisely two connected components B_{0}^{k} and B_{1}^{k}, each diffeomorphic to S^{k}, k ≥ 1. For example, one can take

B^{k} = {(x_{0}, …, x_{k}) ∈ ℝ^{k+1} | x_{0}^{4} − 4x_{0}^{2} + 1 + x_{1}^{2} + ⋯ + x_{k}^{2} = 0}.

Let B = B^{k} × ⋯ × B^{k} and B_{0} = B_{0}^{k} × ⋯ × B_{0}^{k} be the d-fold products, and let δ : B_{0} ↪ B be the inclusion map. It is known [18, Example 4.5] that

(2.2) H^{q}(B_{0}, ℤ/2) = δ^{∗}(H^{q}(B, ℤ/2)) = δ^{∗}(Alg^{q}(B)) for all q ≥ 0.

We now recall an important result from differential topology.

Theorem 2.3 Let P be a smooth manifold. Two smooth maps f : M → P and g : M → P, where M and N are compact smooth manifolds of dimension d, represent the same bordism class in the unoriented bordism group N_{∗}(P) if and only if for every nonnegative integer q and every cohomology class v in H^{q}(P, ℤ/2), one has

< f^{∗}(v) ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] >=< g^{∗}(v) ∪ w_{i_{1}}(N) ∪ … ∪ w_{i_{r}}(N), [N] >

for all nonnegative integers i_{1}, …, i_{r} with i_{1} + ⋯ + i_{r} = d − q.

Reference for the proof. [11, (17.3)].

If Y is a nonsingular real algebraic variety, then a bordism class in N_{∗}(Y) is said to be algebraic provided that it can be represented by a regular map f : X → Y of a compact nonsingular real algebraic variety X into Y, cf. [2, 3, 4, 5, 6].

A topological real vector bundle on a real algebraic variety Y is said to admit an algebraic structure if it is topologically isomorphic to an algebraic subbundle of the trivial vector bundle with total space Y ×ℝ^{p} for some p (cf. [7] for various characterizations of such vector bundles and for their basic properties).

Given smooth manifolds N and P, we endow the set C^{∞}(N, P) of all smooth maps from N into P with the C^{∞} topology [14] (in our applications N is always compact so it does not matter whether we take the weak C^{∞} topology or the strong one).

Our basic tools include the following approximation theorem.

Theorem 2.4 Let M be a compact smooth submanifold of ℝ^{n} and let W be a nonsingular real algebraic variety. Let f : M → W be a smooth map whose bordism class in N_{∗}(W) is algebraic. Suppose that M contains a (possibly empty) Zariski closed nonsingular subvariety L of ℝ^{n}, the restriction f|L : L → W is a regular map, and the restriction to L of the tangent bundle of M admits an algebraic structure. If 2 dim M + 1 ≤ n, then there exist a smooth embedding e : M → ℝ^{n}, a Zariski closed nonsingular subvariety X of ℝ^{n}, and a regular map g : X → W such that L ⊆ X = e(M), e|L : L → ℝ^{n} is the inclusion map, g|L = f|L, and g ∘ ē (where ē: M → e(M) is the smooth diffeomorphism defined by ē(x) = e(x) for all x in M) is homotopic to f. Furthermore, given a neighborhood U in C^{∞}(M, ℝ^{n}) of the inclusion map M ↪ ℝ^{n} and a neighborhood V of f in C^{∞}(M, W), the objects e, X, and g can be chosen in such a way that e is in U and g ∘ ē is in V.

Reference for the proof. Precisely this formulation is in [18, Theorem 4.2]. It is based on very similar results of [2, 3, 4, 5].

After these preparations we return to the main topic of our paper.

Theorem 2.5 Let M be a compact smooth manifold of dimension m. Let G be a subgroup of H^{k}(M, ℤ/2), where k ≥ 1. Assume that G is generated by spherical cohomology classes. If 2k + 1 ≤ m, then the following conditions are equivalent:

(a) There exist an algebraic model X of M and a smooth diffeomorphism φ : M → X such that φ^{∗}(Alg^{k}(X)) = G,

(b) For every integer ℓ satisfying ℓ ≥ 1 and ℓk ≤ m, one has

< u_{1} ∪ … ∪ u_{ℓ} ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] >= 0

for all u_{1}, …, u_{ℓ} in G and all nonnegative integers i_{1}, …, i_{r} with i_{1} +⋯+i_{r} = m−ℓk.

Furthermore, if (b) holds, then X in (a) can be chosen irreducible.

Proof. It follows from Theorem 2.1(ii) that (a) implies (b). Suppose then that (b) holds. We prove below that (a), with X irreducible, is satisfied. We assume that M is a smooth submanifold of ℝ^{2m+1}.

Let us set

Γ = {v ∈ H^{m−k}(M, ℤ/2)| < u ∪ v, [M] >= 0 for every u in G},
Γ = {v_{1}, …, v_{s}}.

Since 2k + 1 ≤ m, the homology class in H_{k}(M, ℤ/2) Poincaré dual to v_{i} can be represented by a compact smooth submanifold N_{i} of M [20, Théorème II.26]. Thus we have

e_{i_{∗}}([N_{i}]) = v_{i} ∩ [M],

where e_{i} : N_{i} ↪ M is the inclusion map and ∩ stands for the cap product. We may assume that N_{1}, …, N_{s} are pairwise disjoint. Note that

(1) < e_{i}^{∗}(u), [N_{i}] >=< u ∪ v_{i}, [M] > for all u in H^{k}(M, ℤ/2).

Indeed standard properties of ∪, ∩, <, > (cf. for example [12]) yield

< e_{i}^{∗}(u), [N_{i}] > =< u, e_{i_{∗}}([N_{i}]) >
=< u, v_{i} ∩ [M] >
=< u ∪ v_{i}, [M] > .

The bilinear map

H^{k}(M, ℤ/2) × H^{m−k}(M, ℤ/2) → ℤ/2, (u, v) →< u ∪ v, [M] >

is a dual pairing [12, p. 300, Proposition 8.13] and hence

G = {u ∈ H^{k}(M, ℤ/2)| < u ∪ v, [M] >= 0 for every v in Γ}.

By applying (1), we obtain

(2) G = {u ∈ H^{k}(M, ℤ/2)| < e_{i}^{∗}(u), [N_{i}] >= 0 for 1 ≤ i ≤ s}.

We shall now successively modify M and N_{1}, …, N_{s} to ensure that they satisfy some additional desirable conditions.

Let γ_{n,m} denote the universal vector bundle on the Grassmannian 𝔾_{n,m} of m-dimensional vector subspaces of ℝ^{n}. Assuming that n is large enough, we can find a smooth classifying map h_{i} : N_{i} → 𝔾_{n,m} for the restriction τ(M)|N_{i} of the tangent bundle τ(M) of M (this means that the vector bundles τ(M)|N_{i} and h_{i}^{∗}γ_{n,m} are isomorphic). Recall that 𝔾_{n,m} is endowed with a canonical structure sheaf which makes it into a real algebraic variety in the sense of this paper [7, Theorem 3.4.4] (𝔾_{n,m} is an affine real algebraic variety according to the terminology used in [7]). Moreover, 𝔾_{n,m} is nonsingular [7, Proposition 3.4.3] and every bordism class in N_{∗}(𝔾_{n,m}) is algebraic [7, Proposition 11.3.3] and [3, Lemma 2.7.1]. It follows that Theorem 2.4 can be applied to h_{i} : N_{i} → 𝔾_{n,m} (with L empty) and hence modifying M, we may assume that N_{i} is a Zariski closed nonsingular subvariety of ℝ^{2m+1} and h_{i} : N_{i} → 𝔾_{n,m} is a regular map for 1 ≤ i ≤ s.

Let u_{1}, …, u_{d} be spherical cohomology classes generating G. Using the same notation as in (2.2), choose a smooth map f_{j} : M → B^{k} such that f_{j}(M) ⊆ B_{0}^{k} and f_{j}^{∗}(H^{1}(B^{k}, ℤ/2)) is the subgroup of G generated by u_{j}. By (2.2), we have

(3) G = f^{∗}(H^{k}(B, ℤ/2)) = f^{∗}(Alg^{k}(B)),

where f = (f_{1}, …, f_{d}) : M → B = B^{k} × ⋯ × B^{k}.

We assert that the maps (f|N_{i}, h_{i}) : N_{i} → B × 𝔾_{n,m} and (c_{i}, h_{i}) : N_{i} → B × 𝔾_{n,m}, where c_{i} : N_{i} → B is a constant map sending N_{i} to a point in B_{0}, represent the same class in the bordism group N_{∗}(B × 𝔾_{n,m}). By Theorem 2.3 and Künneth’s theorem in cohomology, in order to prove the assertion it suffices to show that given cohomology classes ξ in H^{p}(B, ℤ/2) and η in H^{q}(𝔾_{n,m}, ℤ/2), we have

\begin{matrix}< (f|N_{i}, h_{i})^{∗}(ξ × η)∪ & w_{j_{1}}(N_{i}) ∪ … ∪ w_{j_{r}}(N_{i}), [N_{i}] >= \\ < (c_{i}, h_{i})^{∗}(ξ × η) ∪ w_{j_{1}}(N_{i}) ∪ … ∪ w_{j_{r}}(N_{i}), [N_{i}] > \end{matrix}

for all nonnegative integers j_{1}, …, j_{r} satisfying j_{1} + ⋯ + j_{r} = k − (p + q). Since (f|N_{i}, h_{i})^{∗} (ξ × η) = (f|N_{i})^{∗}(ξ) ∪ h_{i}^{∗}(η) and (c_{i}, h_{i})^{∗}(ξ × η) = c_{i}^{∗}(ξ) ∪ h_{i}^{∗}(η), the last displayed equality is equivalent to

(4) \begin{matrix}< (f|N_{i})^{∗}(ξ) ∪ h_{i}^{∗}(η)∪ & w_{j_{1}}(N_{i}) ∪ … ∪ w_{j_{r}}(N_{i}), [N_{i}] >= \\ < c_{i}^{∗}(ξ) ∪ h_{i}^{∗}(η) ∪ w_{j_{1}}(N_{i}) ∪ … ∪ w_{j_{r}}(N_{i}), [N_{i}] > . \end{matrix}

We now justify (4). If p is not a multiple of k, then ξ = 0 and hence (4) holds. It remains to consider two cases: (p, q) = (0, k) and (p, q) = (k, 0). If (p, q) = (0, k), then (f|N_{i})^{∗}(ξ) = c_{i}^{∗}(ξ), which implies (4). If (p, q) = (k, 0), then c_{i}^{∗}(ξ) = 0 and (4) is reduced to

(5) < (f|N_{i})^{∗}(ξ) ∪ h_{i}^{∗}(η), [N_{i}] >= 0.

Since f|N_{i} = f ∘ e_{i}, we have

(f|N_{i})^{∗}(ξ) ∪ h_{i}^{∗}(η) = e_{i}^{∗}(f^{∗}(ξ)) ∪ h_{i}^{∗}(η) = λe_{i}^{∗}(f^{∗}(ξ)),

where λ = 0 or λ = 1. Hence (5) follows from (2) and (3). This means that (4) always holds and therefore the proof of the assertion is complete.

Since (c_{i}, h_{i}) : N_{i} → B × 𝔾_{n,m} is a regular map, the assertion implies that the bordism class of (f|N_{i}, h_{i}) : N_{i} → B × 𝔾_{n,m} in N_{∗}(B × 𝔾_{n,m}) is algebraic. Theorem 2.4 can be applied to (f|N_{i}, h_{i}) : N_{i} → B × 𝔾_{n,m} (with L empty) and therefore modifying M and f, we may assume that N_{i} is still a Zariski closed nonsingular subvariety of ℝ^{2m+1} and (f|N_{i}, h_{i}) : N_{i} → B × 𝔾_{n,m} is a regular map for 1 ≤ i ≤ s. By construction, τ(M)|N_{i} admits an algebraic structure (being isomorphic to h_{i}^{∗}γ_{n,m}). Note that N = N_{1} ∪…∪N_{s} is a Zariski closed nonsingular subvariety of ℝ^{2m+1} and

(6)

f|N : N → B is a regular map,

(7)

τ(M)|N admits an algebraic structure.

We can further modify f so that it is constant on some open subset U of M which is disjoint from N and has a nonempty intersection with each connected component of M. Let P be a compact k-dimensional smooth submanifold of U such that each connected component of M contains a connected component of P, each connected component of P is diffeomorphic to S^{k}, and the restriction τ(M)|P is a trivial vector bundle. There is a smooth diffeomorphism σ of ℝ^{2m+1} such that σ(x) = x for x in N and σ(P) is a Zariski closed nonsingular irreducible subvariety of ℝ^{2m+1}. Replacing M by σ(M), we may assume that P itself is a Zariski closed nonsingular irreducible subvariety of ℝ^{2m+1}.

Note that N ∪ P is a Zariski closed nonsingular subvariety of ℝ^{2m+1}. Since f is constant on P, it follows from (6) that

(8)

f|(N ∪ P) : N ∪ P → B is a regular map.

Furthermore, in view of (7), we get

(9)

τ(M)|(N ∪ P) admits an algebraic structure.

We claim that f : M → B and a constant map M → B sending M to a point in B_{0} represent the same class in the bordism group N_{∗}(B). We verify the claim via Theorem 2.3. It suffices to show that given a positive integer q and a cohomology class ξ in H^{q}(B, ℤ/2), we have

(10) < f^{∗}(ξ) ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] >= 0

for all nonnegative integers i_{1}, …, i_{r} with i_{1} + ⋯ + i_{r} = m − q. If q is not a multiple of k, then ξ = 0 and (10) holds. If q = ℓk ≤ m, then ξ is a linear combination of cohomology classes of the form ξ_{1} ∪ … ∪ ξ_{ℓ}, where ξ_{1}, …, ξ_{ℓ} are in H^{k}(B, ℤ/2). By (3), the cohomology classes f^{∗}(ξ_{1}), …, f^{∗}(ξ_{ℓ}) are in G and hence (10) follows from (b). Thus the claim is proved.

The claim implies that the class of f : M → B in the bordism group N_{∗}(B) is algebraic. In view of (8) and (9) we can apply Theorem 2.4 to f : M → B (with L = N ∪ P). Hence there exist a Zariski closed nonsingular subvariety X of ℝ^{2m+1}, a smooth diffeomorphism φ : M → X, and a regular map g : X → B such that X = φ(M), φ(x) = x for all x in N ∪P, and f is homotopic to g ∘φ. Clearly, X is irreducible, the variety P being irreducible. In order to complete the proof it remains to show φ^{∗}(Alg^{k}(X)) = G. We argue as follows. Since g : X → B is a regular map, we have g^{∗}(Alg^{k}(B)) ⊆ Alg^{k}(X). Hence using (3) and f^{∗} = (g ∘ φ)^{∗} = φ^{∗} ∘ g^{∗}, we obtain

G = f^{∗}(Alg^{k}(B)) = φ^{∗}(g^{∗}(Alg^{k}(B))) ⊆ φ^{∗}(Alg^{k}(X)).

Suppose there is an element w in Alg^{k}(X) such that φ^{∗}(w) is not in G. By (2), one can find i, 1 ≤ i ≤ s, for which

< e_{i}^{∗}(φ^{∗}(w)), [N_{i}] >≠ 0.

If ϵ_{i} : N_{i} ↪ X is the inclusion map, then ϵ_{i} = φ ∘ e_{i} and hence ϵ_{i}^{∗}(w) = e_{i}^{∗}(φ^{∗}(w)). It follows that

< ϵ_{i}^{∗}(w), [N_{i}] >≠ 0.

This contradicts Theorem 2.1(i) since ϵ_{i}^{∗}(w) is in Alg^{k}(N_{i}), the map ϵ_{i} being regular. Thus φ^{∗}(Alg^{k}(X)) = G and the proof is complete. □

Proposition 2.6 Let M be a compact smooth manifold of dimension m with m ≥ 2. Let G be a subgroup of H^{m−1}(M, ℤ/2). Assume that G is generated by spherical cohomology classes. Then the following conditions are equivalent:

(a) There exist an algebraic model X of M and a smooth diffeomorphism φ : M → X such that φ^{∗}(Alg^{m−1}(X)) = G,

(b) < u ∪ w_{1}(M), [M] >= 0 for all u in G, and when m = 2, then in addition < u_{1} ∪ u_{2}, [M] >= 0 for all u_{1} and u_{2} in G.

Futhermore, if (b) holds, then X in (a) can be chosen irreducible.

Proof. It follows from Theorem 2.1(ii) that (a) implies (b). Suppose then that (b) holds. We prove below that (a), with X irreducible, is satisfied. In the proof we assume that M is a smooth submanifold of ℝ^{2m+1}.

Let us set

Γ = {v ∈ H^{1}(M, ℤ/2)| < u ∪ v, [M] >= 0 for all u in G}, Γ = {u_{1}, …, u_{s}}.

If n is sufficiently large and A = ℙ^{n}(ℝ) × ⋯ × ℙ^{n}(ℝ) is the product of s copies of real projective n-space ℙ^{n}(ℝ), then there exists a smooth map f : M → A for which

(1) Γ = f^{∗}(H^{1}(A, ℤ/2)).

Let u_{1}, …, u_{d} be spherical cohomology classes generating G. Using the same notation as in (2.2), with k = m − 1, choose a smooth map g_{j} : M → B^{m−1} such that g_{j}(M) ⊂ B_{0}^{m−1} and g_{j}^{∗}(H^{m−1}(B^{m−1}, ℤ/2)) is the subgroup of G generated by u_{j}. Note that (2.2) implies

(2) G = g^{∗}(H^{m−1}(B, ℤ/2)) = g^{∗}(Alg^{m−1}(B)),

where g = (g_{1}, …, g_{d}) : M → B = B^{m−1} × ⋯ × B^{m−1}.

We can choose f and g so that the map (f, g) : M → A × B is constant on some open subset U of M which has a nonempty intersection with each connected component of M. Let C be a compact smooth curve in U such that each connected component of M contains a connected component of C and the restriction τ(M)|C is trivial. There is a smooth diffeomorphism σ of ℝ^{2m+1} such that σ(C) is a Zariski closed nonsingular irreducible curve in ℝ^{2m+1}. Replacing M by σ(M), we may assume that C itself is a Zariski closed nonsingular irreducible curve in ℝ^{2m+1}.

We assert that the maps (f, g) : M → A × B and (f, c) : M → A × B, where c : M → B is a constant map sending M to a point in B_{0}, represent the same class in the bordism group N_{∗}(A × B). By Theorem 2.3 and Künneth’s theorem in cohomology, in order to prove the assertion it suffices to show that given cohomology classes ξ in H^{p}(A, ℤ/2) and η in H^{q}(B, ℤ/2), we have

\begin{aligned}< (f, g)^{∗} & (ξ × η) ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] >= \\ & < (f, c)^{∗}(ξ × η) ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] > \end{aligned}

for all nonnegative integers i_{1}, …, i_{r} satisfying i_{1}+⋯+i_{r} = m−(p+q). Since (f, g)^{∗}(ξ×η) = f^{∗}(ξ) ∪ g^{∗}(η) and (f, c)^{∗}(ξ × η) = f^{∗}(ξ) ∪ c^{∗}(η), the last displayed equality is equivalent to

(3)

\begin{matrix}< f^{∗}(ξ)∪ & g^{∗}(η) ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] >= \\ < f^{∗}(ξ) ∪ c^{∗}(η) ∪ w_{i_{1}}(M) ∪ … ∪ w_{i_{r}}(M), [M] > . \end{matrix}

If q is not a multiple of m − 1, then η = 0 and hence (3) holds. If q = 0, then g^{∗}(η) = c^{∗}(η) and (3) is also satisfied. It remains to consider the following three cases: (p, q) = (1, m − 1), (p, q) = (0, m − 1), and (p, q) = (0, 2) with m = 2. In each of these cases c^{∗}(η) = 0. If (p, q) = (1, m − 1), then (3) is reduced to

< f^{∗}(ξ) ∪ g^{∗}(η), [M] >= 0,

which holds in view of (1), (2), and the definition of Γ. If (p, q) = (0, m − 1), then (3) is equivalent to

< f^{∗}(ξ) ∪ g^{∗}(η) ∪ w_{1}(M), [M] >= 0,

which follows from (2) and (b) (note that f^{∗}(ξ) ∪ g^{∗}(η) = λg^{∗}(η), where λ = 0 or λ = 1). If (p, q) = (0, 2), m = 2, then f^{∗}(ξ) ∪ g^{∗}(η) = λg^{∗}(η), where λ = 0 or λ = 1, and (3) is equivalent to

< λg^{∗}(η), [M] >= 0.

The last equality follows from (b) since η is a linear combination of cohomology classes of the form η_{1} ∪ η_{2}, where η_{1}, η_{2} are in H^{1}(B, ℤ/2), and in view of (2), g^{∗}(η_{1}), g^{∗}(η_{2}) are in G. This completes the proof of (3) and hence the assertion holds.

We shall now prove

(4)

the bordism class of (f, g) : M → A × B in N_{∗}(A × B) is algebraic.

Since every bordism class in N_{∗}(A) is algebraic [3, Lemma 2.7.1], in view of Theorem 2.4, there exist a Zariski closed nonsingular subvariety Y of ℝ^{2m+1}, a smooth diffeomorphism ψ : M → Y, and a regular map f̄ : Y → A such that f is homotopic to f̄ ∘ ψ. Clearly, (f, c) : M → A × B and (f̄, c ∘ ψ^{−1}) : Y → A × B represent the same bordism class in N_{∗}(A × B). Note that (f̄, c ∘ ψ^{−1}) : Y → A × B is a regular map, c ∘ ψ^{−1} : Y → B being constant. Hence (4) follows from the assertion proved above.

By construction, (f, g) : M → A × B is constant on C and τ(M)|C is a trivial vector bundle. Thus (4) allows us to apply Theorem 2.4 to (f, g) : M → A × B (with L = C). Therefore there exist a Zariski closed nonsingular subvariety X of ℝ^{2m+1}, a smooth diffeomorphism φ : M → X, and a regular map (α, β) : X → A × B such that X = φ(M), φ(x) = x for all x in C, and (f, g) is homotopic to (α, β) ∘ φ = (α ∘ φ, β ∘ φ). Obviously, X is irreducible, the curve C being irreducible.

It remains to prove φ^{∗}(Alg^{m−1}(X)) = G. Since β : X → B is a regular map, we have β^{∗}(Alg^{m−1}(B)) ⊆ Alg^{m−1}(X). Making use of g^{∗} = (β ∘ φ)^{∗} = φ^{∗} ∘ β^{∗} and (2), we get

G = g^{∗}(Alg^{m−1}(B)) = φ^{∗}(β^{∗}(Alg^{m−1}(B))) ⊆ φ^{∗}(Alg^{m−1}(X)).

Suppose there exists w in Alg^{m−1}(X) such that φ^{∗}(w) is not in G. We obtain a contradiction as follows. The bilinear map

H^{m−1}(M, ℤ/2) × H^{1}(M, ℤ/2) → ℤ/2, (u, v) →< u ∪ v, [M] >

is a dual pairing [12, p.300, Proposition 8.13] and hence one can find an element v in Γ with < φ^{∗}(w) ∪ v, [M] >≠ 0. By (1), we have v = f^{∗}(z) for some z in H^{1}(A, ℤ/2). Since f^{∗} = (α ∘ φ)^{∗} = φ^{∗} ∘ α^{∗}, we get v = φ^{∗}(α^{∗}(z)). Thus < φ^{∗}(w) ∪ φ^{∗}(α^{∗}(z)), [M] >≠ 0, which yields

(5) < w ∪ α^{∗}(z), [X] >≠ 0.

Note that α^{∗}(z) is in H_{alg}^{1}(X, ℤ/2), the map α : X → A being regular and H^{1}(A, ℤ/2) = H_{alg}^{1}(A, ℤ/2). Hence (5) contradicts Theorem 2.1(i). The proof is complete. □

Proof of Theorem 1.1. Obviously, every spherical cohomology class with coefficients in ℤ/2 is the reduction modulo 2 of a cohomology class with coefficients in ℤ. Therefore it follows from Theorem 2.1(ii), (iii) that (a) implies (b). Suppose (b) holds. The first part of (b) guarantees that every cohomology class in G is spherical [15, p. 49, Theorem 7.1]. Thus (a), with X irreducible, holds by virtue of Theorem 2.5 and Proposition 2.6 (Proposition 2.6 is required only when m = 2). □

Proof of Theorem 1.2. We already know that, by Theorem 2.1(ii), (a) implies (b). Suppose (b) is satisfied. Since M is connected, given u in H^{m−1}(M, ℤ/2) with < u∪w_{1}(M), [M] >= 0, we get u ∪ w_{1}(M) = 0. The last equality implies that the homology class in H_{1}(M, ℤ/2) Poincaré dual to u can be represented by a compact smooth curve in M with trivial normal vector bundle, cf. for example [8, p. 599]. This in turn implies that u is spherical [20, Théorème II.1]. Hence every cohomology class in G is spherical. In view of Proposition 2.6, condition (a), with X irreducible, holds. □

We conclude the paper with an example.

Example 2.7. Let T^{m} = S^{1} × ⋯ × S^{1} be the m-fold product with m ≥ 2. Clearly, H_{alg}^{ℓ}(T^{m}, ℤ/2) = H^{ℓ}(T^{m}, ℤ/2) for all ℓ ≥ 0 and hence, by Theorem 2.1(i),

Alg^{k}(T^{m}) = 0 for all k ≥ 0.

On the other hand, let G be a subgroup of H^{k}(T^{m}, ℤ/2) and suppose that one of the following conditions is satisfied:

(i) k = 1 and G ≠ H^{1}(T^{m}, ℤ/2),

(ii) k = m − 1 and m ≥ 3,

(iii) k ≥ 1, 2k+1 ≤ m, m is not divisible by k, and G is generated by spherical cohomology classes,

(iv) m = kℓ, where ℓ is an integer satisfying ℓ ≥ max{2m/(m − 1), dim_{ℤ/2} G + 1}, and G is generated by spherical cohomology classes.

Then there exist an algebraic model X of T^{m} and a smooth diffeomorphism φ : T^{m} → X such that

φ^{∗}(Alg^{k}(X)) = G.

Indeed, since the tangent bundle to T^{m} is trivial, we have w_{i}(T^{m}) = 0 for all i ≥ 1. Furthermore, if either (i) or (iv) is satisfied, m = kℓ, and u_{1}, …, u_{ℓ} are in G, then u_{1}, ∪ … ∪ u_{ℓ} = 0. Thus X and φ with the required property exist in view of Theorems 1.1, 1.2, and 2.5.

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

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