ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS WITH ...

ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS WITH ONE ISOTROPY TYPE KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN Abstract. Suppose G is a cyclic group and M a closed smooth G– manifold with exactly one isotropy type. We will show that there is a nonsingular real algebraic G–variety X which is equivariantly diffeomorphic to M and all G–vector bundles over X are strongly algebraic.

1. Introduction A vector bundle over a variety is said to be strongly algebraic if it is classified, up to homotopy, by an entire rational map to a Grassmannian with its canonical algebraic structure. We will catch up with definitions and background material in Section 2. In this paper we will show: Theorem 1.1. Suppose G is a cyclic group and M a closed smooth G– manifold with exactly one isotropy type. Then there is a nonsingular real algebraic G–variety X which is equivariantly diffeomorphic to M and all G–vector bundles over X are strongly algebraic. This theorem is a key ingredient in the proof of Theorem 1.2. [16] Suppose G is a cyclic group and M a closed smooth G–manifold. Then there is a nonsingular real algebraic G–variety X which is equivariantly diffeomorphic to M and all G–vector bundles over X are strongly algebraic. Our work is motivated by the Nash conjecture [24], which was shown to hold in [30], [1], and [2]. Early results on algebraic realization are due to Seifert [25] and Nash [24]. One may attempt to algebraically realize a closed manifold M simulaneously with some additional structure, such as a group action on M and vector bundles over M . The expectation is: Conjecture 1.3. (see [13] and [16]) Let G be a compact Lie group and M a closed smooth G manifold. Then there is a nonsingular real algebraic G–variety X which is equivariantly diffeomorphic to M and all G–vector bundles over X are strongly algebraic. Previously, this conjecture has been proven in the following special cases: Date: October 27, 2009. 1991 Mathematics Subject Classification. Primary 14P25, 57S15; Secondary 57S25. Key words and phrases. Algebraic Models, Equivariant Bordism. 1

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KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN

(1) (2) (3) (4)

G is the trivial group, see [5]. G is the product of an odd order group and a 2–torus, see [13]. The action of G on M is semifree, see [13]. All points of M have orbit type H, and N H/H is of odd order, see [29].

J. Hanson showed in [18] that for a closed smooth Z4 –manifold and one Z4 –bundle ξ over M one can find a non-singular real algebraic Z4 –variety X together with an equivariant diffeomorphism X → M that pulls back ξ to an equivariant strongly algebraic vector bundle over X. 1.1. Context. Let us sketch the bigger picture. Tognoli’s important breakthrough in the discussion of the algebraic realization problem was its reduction to a bordism problem [30]. The following meta-theorem is well supported: General Principle. [20, p. 154] If a topological situation is cobordant to an algebraic situation, then it is isomorphic to an algebraic situation (if you have the right notion of bordism and do a lot of work). Following the approach taken by Akbulut and King [1], the question of M n being diffeomorphic to a nonsingular real algebraic variety can be reformulated as asking whether the classifying map χM : M → BO(n) of its tangent bundle is cobordant to a algebraic map χX : X → BO(n). Strictly speaking, we need to (and we will) replace BO(n) by a finite approximation so that it becomes a variety. For the map to be algebraic we are requiring that X is a nonsingular real algebraic variety and χX is entire rational. Put this way, the questions is whether classes in the bordism group Nn (BO(n)) have algebraic maps as representatives. Once we apply the K¨ unneth formula (1.1)

N∗ (BO(n)) ∼ = N∗ ⊗ H∗ (BO(n), Z2 )

the question is whether classes in H∗ (BO(n), Z2 ) have algebraic Steenrod representatives, or as Akbulut and King put it [1, p. 436], whether BO(n) has totally algebraic homology1. In fact, Schubert cycles are algebraic representatives of the classes in H∗ (BO(n), Z2 ). It follows that a finite approximation of a product of BO(a)’s has totally algebraic homology as well. Such a space can be used as classifying space for a finite collection of vector bundles. The proof of Conjecture 1.3 in the non–equivariant case (G = 1) follows from generalities about strongly algebraic vector bundles and the fact that K–theory is finitely generated. Ivanov explains this program well in [19]. Little is know about other varieties having totally algebraic Z2 –homology. Here are two striking results: 1This only constitutes a shift in emphasis, because one still needs to represent classes

in N∗ algebraically.

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Theorem 1.4. [6, Theorem 2] In every dimension ≥ 11 there are closed smooth manifolds, so that none of their algebraic realizations has totally algebraic homology. Theorem 1.5. [4] Every closed smooth manifold is homeomorphic to a real algebraic set with totally algebraic homology. The first result is in the smooth category, the second one in the topological one. See [3] for another result similar to Theorem 1.4. Add to this Theorem 1.6. [9] For every closed smooth manifold M of positive dimension there are uncountably many birationally inequivalent algebraic varieties that are diffeomorphic to M . It is tempting to pose Question 1.7. Let Y be a non–singular real algebraic variety. Which classes in N∗ (Y ) have algebraic representatives? Above results suggest that any answer depends in a subtle way not only on a differentiable structure of Y , but also on its algebraic structure. Bochnak and Kucharz studied such problems if Y is a sphere [8] and the results are delicate. The questions become less tractable if one imposes the action of a group G. One may use a partial ordering on the sets of subgroups of G to analyse N∗G (Y ) inductively. In the inductive step one uses a long exact ConnerFloyd sequence. Even if we inductively know that the classes in any two of three terms are algebraically represented, we have no general result that allows us to conclude that the remaining terms are algebraically represented. The relative terms in the Conner–Floyd sequence are typically rewritten as non–equivariant bordism groups. Still, their calculation remains delicate. Even for a simple group like Z4 it involves the calculation of a spectral sequence which does not collapse at the E2 –level. For G a cyclic group and Y of a very specific form, we were able to calculate enough about the relative terms in the Conner–Floyd sequence to deduce the results in [16]. The equivariant equivalent to (1.1) in the one–isotropy group case is (5.1). It involves equivariant (Borel) homology groups instead of regular homology. We can calculate these homology groups in only very few cases. In brief, as tempting as it is to ask Question 1.7, great care needs to be excercised so that the question yields an interesting answer. 1.2. Structure of the paper. In Section 2 we will provide basic definitions and background material. If the only isotropy group in Theorem 1.1 is called H, then its proof is known if H or G/H is of odd order. In Section 3 we will sketch the proofs. We also show an attempt at a proof in case H and G/H are both of even order. One encounters a bundle extension problem. Its vanishing would make the proof complete and our paper really short. Calculations in the later sections of the paper indcate that the extension problem does not vanish.

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In Section 4 we formulate a bordism theoretic result, Theorem 4.1, and deduce Theorem 1.1 from it. Theorem 4.1 asserts that the generators of a certain bordism group are algebraically represented. In Sections 5 we find generators for this bordism group, and in Section 7 we find algebraic representatives for these generators. Acknowledgements. The first named author would like to thank Osaka City University and Michigan State University for their hospitality. 2. Basic definitions and background material 2.1. Real algebraic varieties and entire rational maps. Let G be a compact Lie group and Ω an orthogonal representation of G. We think of an orthogonal representation as an underlying Euclidean space Rn together with an action of G via orthogonal maps. Definition 2.1. If V = {x ∈ Ω | p1 (x) = · · · = pm (x) = 0} is the common set of polynomials p1 ,. . . , pm : Ω → R and V is G–invariant, then we call it a real algebraic G–variety. We use the Euclidean topology on varieties and use the term ‘nonsingular’ with its standard meaning [32]. Let V ⊆ Rn and W ⊆ Rm be real algebraic varieties. is said to be regular if it extends to a map F : Rn → of its coordinates Fi (i.e., Fi = δi ◦ F : Rn → R where projection on the i–th coordinate) is a polynomial. We rational if there are regular maps p : Rn → Rm and q : f = p/q on V and q does not vanish anywhere on V . We will simplify our language by saying:

A map f : V → W Rm such that each δi : Rm → R is the say that f is entire Rn → R, such that

Terminology 2.2. A map to a nonsingular real algebraic variety is said to be algebraic, if its domain is a nonsingular real algebraic variety and the map is entire rational. An algebraic model for a map f : M → X from a smooth manifold to a variety is an algebraic map g : X → Y together with a diffeomorphism Φ : X → M , so that g ≃ f ◦ Φ. 2.2. Grassmannians and classification of vector bundles. Let Λ stand for R or C. Let Ω be a representation of G over Λ. Its underlying space is Λn for some n. We assume that the action of G preserves the standard bilinear form on Λn . Let EndΛ (Ω) denote the set of endomorphisms of Ω over Λ. It is a representation of G with the action given by G × EndΛ (Ω) → EndΛ (Ω) with

(g, L) 7→ gLg−1 .

Let d be a natural number. We set (2.1) GΛ (Ω, d) = {L ∈ EndΛ (Ω) | L2 = L, L∗ = L, trace L = d} (2.2) EΛ (Ω, d) = {(L, u) ∈ EndΛ (Ω) × Ω | L ∈ GΛ (Ω, d), Lu = u} Here L∗ denotes the adjoint of L. If one chooses an orthogonal (resp., unitary) basis of Ω, then EndΛ (Ω) is canonically identified with the set of n × n matrices, and L∗ is obtained by transposing L and conjugating its

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entries. This description specifies GΛ (Ω, d) and EΛ (Ω, d) as real algebraic G–varieties. Define p : EΛ (Ω, d) → GΛ (Ω, d) as projection on the first factor. This defines the universal G–vector bundle over GΛ (Ω, d). If Ω is a universe (Ω contains each irreducible representation of G an infinite number of times) then GR (Ω, d) serves as classifying space for G vector bundles of dimension d over nice spaces. In other words, there is a 1 − 1 correspondence between isomorphism classes of G–vector bundles over M and equivariant homotopy classes from M to GR (Ω, V ). As long as we work with varieties we suppose that Ω is sufficiently large, but of finite dimension. Being sufficiently large is a stable condition. The following remark tells us how we may trucate Ω. Suppose for all points in M that their isotropy group is contained in H. Let V be a representation of H. Denote the multiplicity of an irreducible representation α in W by χ(W, α). Then GR (Ω, V ) classifies G–vector bundles over M if χ(ResH Ω, αǫ ) > χ(V, αǫ ) + dim M + 1 for all irreducible representations αǫ of H so that χ(V, αǫ ) 6= 0. For details on this discussion we refer the reader to [31, Section 3]. In our setting the preferred concept of a vector bundle is the one of a strongly algebraic vector bundle. One has this notion with real and complex coefficients, Λ = R or Λ = C. Definition 2.3. A strongly algebraic G–vector bundle over a real algebraic G–variety is a bundle whose classifying map to GΛ (Ξ, k) is equivariantly homotopic to an entire rational map. Occasionally, we think of G–vector bundles as equivariant maps to a Grassmannian GΛ (Ξ, k). Then we need to allow stabilization of Ξ. 2.3. Fixed point sets in Grassmannians. The results in this subsection are taken from [13, §10]. Let H be a subgroup of G. We would like to describe the H–fixed point set GR (Ω, d). For a real representation V of H of dimension d we set: (2.3)

GR (Ω, V ) = {L ∈ GR (Ω, d) | L(Ω) ∼ = V }.

So, we are requiring that L(Ω) is invariant under the action of H and isomorphic to V as a representation of H. Let α be an irreducible representation of H. We let Ωα and Vα be the summands of Ω and V that restrict to multiples of α. There are three levels of analysis. We decompose GR (Ω, d)H into components. Then we decompose each component as a product, and finally we describe each factor in classical terms. In the third step we assume that the group G is cyclic.

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Proposition 2.4. Let V be the set of all real representations of H of dimension d. Then GR (Ω, d)H = {L ∈ GR (Ω, d) | hLh−1 = L ∀ h ∈ H} G (2.4) = G (Ω, V ) R

V ∈V

and GR (Ω, d)H and its components GR (Ω, V ) are G–invariant subvarieties of GR (Ω, d).

Proposition 2.5. Let GR (Ω, V ) be as in Proposition 2.4. Then Y (2.5) GR (Ω, V ) = GR (Ωα , Vα ). α

The product ranges over the irreducible representations of H, and each factor in (2.5) is invariant under the action of G. The remainder of this subsection is specific to cyclic groups. Definition 2.6. Suppose G is a cylic group and H is a subgroup. An irreducible representation of H is said to be good if it is the restriction of an irreducible representation of G. Otherwise it is said to be bad. Proposition 2.7. If H and G/H are of even order, then there exists eactly one bad irreducible representation of H. We denote it by αb . Its underlying space is R and a generator of H acts by multiplication with −1. There is an irreducible representation α eb of G (and of S 1 ) that restricts to twice αb . For all other irreducible representations of H there exists a representation α e of G that restricts to α.

Finally, we study the factors in (2.5). The action of G on Ωα induces one on GR (Ωα , Vα ), and this action is trivial if restricted to H. Thus C = G/H acts on GR (Ωα , Vα ). The next proposition describes GR (Ωα , Vα ) as a space together with its action of C.

Proposition 2.8. If Ωα is of countably infinite dimension and α is good, then GR (Ωα , aα) is either BO(a) or BU (a), depending on whether α has R or C as underlying space, and the action of C on GR (Ωα , aα) is trivial. If α is bad, then GR (Ωα , aα) = BO(a). If a is odd, then GR (Ωα , aα)C = ∅. If a is even, then (2.6)

i

GR (Ωα , aα)C = GC (C∞ , a/2) = BU (a/2) ֒→ GR (R∞ , a) = BO(a).

If Ωα is of finite dimension, then the BO’s and BU ’s are trucated. To understand the embedding BU (a/2) ⊂ BO(a) in (2.6), we identify Cn and R2n , and view a complex subspace of Cn of complex dimension a/2 as a real subspace of R2n of dimension a. For future reference we mention the relation between mod 2 Chern classes and Stiefel–Whitney classes: (2.7)

i∗ (w2s ) = cs .

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This follows because the real universal bundle over BO(a) restricts to the realification of the complex universal bundle over BU (a/2). We will also apply Proposition 2.9. Let GR (Ωα , aα) be as above. If α is the bad irreducible representation of H, then the action of C on GR (Ωα , aα) extends to an action of S 1 . In general, the induced action of C on H∗ (GR (Ωα , aα), Z2 ) and H ∗ (GR (Ωα , aα), Z2 ) is trivial. Proof. The first assertion follows because the action of C on α eb , hence on 1 Ωα , extends to an action of S , see Proposition 2.7. If α is good, then C acts trivially on GR (Ωα , aα) and our claim is trivial. If α is bad, then the action of C on GR (Ωα , aα) is the restriction of an action of S 1 , and the claim follows from continuity.  2.4. Results from the literature. Previously we showed: Proposition 2.10. [15, Proposition 2.13] Let G be a compact Lie group and M a closed smooth G–manifold. Suppose that for every finite collection of G– vector bundles over M there is an algebraic model X, such that each bundle in this collection, pulled back over X, is strongly algebraic. Then M has an algebraic model over which all G–vector bundles are strongly algebraic. Theorem 2.11. [15, Theorem C] Let G be a compact Lie group. An equivariant map from a closed smooth G–manifold to a nonsingular real algebraic G–variety has an algebraic model if and only if its cobordism class has an algebraic representative. 3. Proof of Theorem 1.1 in two special cases The proof of Theorem 1.1 is known in case H or C = G/H is of odd order. We will review these proofs below. If H and C are of even order one may attempt the same proof as in case H is of odd order and C of even order, but one encounters a bundle extension problem if the fibre of the bundle has a bad representation (see Definition 2.6) as a summand. Sketch of proof of Theorem 1.1 if |C| is odd. For a detailed proof see [29]. Let M be a closed smooth G–manifold, all of whose points have isotropy group H. Let ξ1 , . . . , ξk be G–vector bundles over M . Because C is of odd order, induction is onto in bordism. Specifically, see [15, Proposition 5.2], (3.1)

′ ′ ′ (M ; ξ1 , . . . , ξk ) ∼ IndG H (M ; ξ1 , . . . , ξk )

where M ′ is a closed H–manifold with trivial action and ξ1′ , . . . , ξk′ are H–vector bundles over M ′ . Here ∼ indicates a G cobordism. There is a nonsingular real algebraic H–variety X ′ together with a diffeomorphism Φ′ : X ′ → M ′ , so that ξ1′ , . . . , ξk′ pull back to strongly algebraic bundles over X ′ . This is due to the classical theory, enhenced so that we may impose actions on the fibres of the bundles. We apply induction and

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turn our algebraic H–date into algebraic G–data. We obtain a nonsingu′ lar real algebraic G–variety IndG H X and a G–equivariant diffeomorphism G ′ G G G ′ ′ ′ ′ Φ = IndG H Φ : IndH X → IndH M . The bundles IndH ξ1 , . . . , IndH ξk will G ′ pull back to strongly algebraic bundles over IndH X . In conclusion, the right hand side in (3.1) consists of algebraic data. Theorem 2.11 tells us that (M ; ξ1 , . . . , ξk ), the data on the left hand side in (3.1), is diffeomorphic to an algebraic situation. As (M ; ξ1 , . . . , ξk ) is equivariantly diffeomorphic to an algebraic situation for any finite number of G–vector bundles over M , it follows from [15, Proposition 2.13], quoted as Proposition 2.10, that there is a real algebraic G–variety X, equivariantly diffeomorphic to M , so that all G–vector bundles over X are strongly algebraic. 

Proof of Theorem 1.1 if |H| is odd and |C| is even. The ideas in this proof are based on the techniques employed in the proof of Lemma 4.2 in [15]. Let τ be an element of G that maps to the unique element of order 2 in C. Let M be a closed smooth G–manifold, all of whose points have isotropy group H, with G–vector bundles ξ1 , . . . , ξk over it. Let W be the mapping cylinder of the orbit map M → M/τ . It is a smooth G manifold with boundary M . A generator h of H acts on the fibres of the bundles ξj , and this action is not multiplication with −1. Thus we can take a unique, continuous square root of the action, and conclude that the bundles ξj descent to G–vector bundles ξ j over M/τ . The bundle over M/τ is a quotient. If we denote 1

the square root of the action by x 7→ h 2 x, then we obtain ξ j by identifying 1

τ (x) with h 2 x. Pulling back the ξ i along the retraction W → M/τ , we obtain G–vector bundle extension of the ξj over W . Thus (M ; ξ1 , . . . , ξk ) is a boundary. Being cobordant to the empty set, (M ; ξ1 , . . . , ξk ) is cobordant to an algebraic situation. According to the Theorem 2.11, there is a nonsingular real algebraic variety X and an equivariant diffeomorphism Φ : X → M , so that the bundles ξ1 , . . . , ξk pull back to strongly algebraic bundles over X. The final agument in the proof, going from the realization of finite collections of bundles to the realization of all bundles is as in the previous proof. 

Attempted proof of Theorem 1.1 if |H| and |C| are even. If H and C are of even order, we may still attempt to proceed as in the previous proof. We can still construct the mapping cylinder W . Lacking the square root, we would not know how to form a quotient bundle and extend the G–vector bundles ξj over W . The problem arises if the fibre of the bundle ξj has the bad irreducible representation αb as a summand. 

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4. Reduced problem We will set up a bordism theoretic result and deduce Theorem 1.1 from it. Throughout this section G denotes a cyclic group, H a subgroup of G and C = G/H. Let αb denote the bad irreducible representation of H as in Definition 2.6, and let α eb be the representation of G that restricts to twice αb , see Proposition 2.7. For a given sequence of natural numbers A = (a1 , . . . , ak ), we define (4.1)

Fb = GR (Ωb , a1 αb ) × · · · × GR (Ωb , ak αb ).

The factors GR (Ωb , aαb ) of F were defined and discussed in Section 2.3 and Proposition 2.8. In the upcoming sections we will show: Theorem 4.1. The classes in NnC [free](Fb ) have algebraic representatives for all n and sufficiently large Fb . In this theorem, Ωb is of large finite dimension so that Fb is a variety and we may talk about bordism classes that have algebraic maps (see Terminology 2.2) as representatives. Also, NnC [free](Fb ) is the bordism group of C–equivariant maps from closed smooth C–manifolds to Fb , compare [11]. As indicated, actions on domains, and on bordisms in between them, are assumed to have free actions. Deduce Theorem 1.1 from Theorem 4.1. Consider a manifold M as in Theorem 1.1. Let ξ1 , . . . , ξk be a collection of G vector bundles over M . Denote the classifying map by f : M → G = GR (Ω, d1 ) × · · · × GR (Ω, dk ). To prove Theorem 1.1, it suffices to show that, for every collection of bundles, the bordism class of f : M → G has an algebraic representative. This follows from Proposition 2.10 and Theorem 2.11. Let H denote the unique isotropy group of the action on M . We think of f : M → G as a representative of a class in N∗G [{H}](G). As H acts trivially on M , f factors though GH . The components of GR (Ω, d)H are G–invariant (see Proposition 2.4) and so are the ones of GH . The decomposition of GH into components leads to a direct sum decomposition of the bordism group. Hence, Theorem 1.1 follows from the Special Case 4.2. Let F be a component of GH and n a natural number. Then each class of NnG [{H}](F) has an algebraic representative. The notation indicates that we are considering G–equivariant maps to F, and on their domains (and the bordisms between them) H is the only isotropy group. We obtain isomorphsms N G [{H}](F) ∼ = N C [free](F] ∼ = N∗ (EC ×C F). ∗

∗

For the first isomorphism we divide out the kernel H of the actions, leading to C = G/H actions. The seond isomorphism is as in [11].

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If we apply Proposition 2.4 to a product of Grassmannians, then we see that F is of the form F = GR (Ω, V1 ) × · · · × GR (Ω, Vk ) for some sequence of representations Vj of H, 1 ≤ j ≤ k. We may factor this component of GH as (4.2)

b

g

F=F ×F =

k Y

j=1

GR (Ωb , aj,b αb ) ×

k Y Y

GR (Ωǫ , aj,ǫ αǫ ).

j=1 ǫ∈E g

Here αb is the unique bad irreducible representation of H, Ωb is a multiple of α eb , and aj,b is the multiplicity of αb in Vj . The second product of the second factor is similar, only E g is an index set for the good irreducible representations of H, αǫ is the representation indexed by ǫ ∈ E g . The action of C on Fg is trivial (see Proposition 2.8), so that EC ×C F = Fg × (EC ×C Fb ).

(4.3)

According to the K¨ unneth formula [11, Section 19] N∗ (EC ×C F) ∼ = N∗ (Fg ) ⊗N∗ N∗ (EC ×C Fb ) (4.4) ∼ = N∗ (Fg ) ⊗ N∗G [{H}](Fb ). Given a dimension m, we need to show that classes in Nm (EC ×C F) have algebraic representatives. They are represented by sums of products of the form f1 × f2 : M1 × M2 → Fg × Fb .

(4.5)

The first factor has as domain a closed smooth manifold M1 with trivial action of G. The range of f1 is a product of truncated BO(a)’s and BU (a)’s, and f1 classifies a collection of bundles. There is an algebraic model for such a map according to the classical theory [5]. The factors of Fg have encoded the action on the fibres of the individual bundles. The second factor in (4.5) has as domain a closed smooth manifold all of whose points have isotropy group H. We assumed (Theorem 4.1) that this map has an algebraic model. It follows that the map in (4.5) has an algebraic model. With this we have verified that our Special Case 4.2 follows from Theorem 4.1, and so does Theorem 1.1.  5. The computation of H∗ (EC ×C Fb , Z2 ) Standard techiques reduce the computation of N∗C [free](Fb ), the bordism group from Theorem 4.1, to a homology problem (5.1) N C [free](Fb ) ∼ = N∗ (EC ×C Fb ) ∼ = N∗ ⊗Z H∗ (EC ×C Fb , Z2 ). ∗

2

These bordism groups are not computable in the sense of Stong [28]. Still, to prove Theorem 4.1, we try to gain sufficient computational insight, identify generators, and show that these are algebraically represented. This program is carried out in Sections 5–7.

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We may suppose that C is of even order. Otherwise Theorem 1.1 is already proved. In addition, Fb would be a point, and the computation of N∗C [free](Fb ) would be well known. To calculate H∗ (EC ×C Fb , Z2 ), we use the Leray-Serre spectral sequence of the fibration Fb → EC ×C Fb → BC.

(5.2)

The local coefficient system H∗ (Fb , Z2 ) is simple because C acts trivially on H ∗ (Fb , Z2 ), see Proposition 2.9. We also exploit the fact that we are using coefficients in the field Z2 (use [26, p. 457] or [22, Proposition 5.4]). Then we find that (5.3) E p,q ∼ = H p (BC, H q (Fb , Z2 )) ∼ = H p (BC, Z2 ) ⊗ H q (Fb , Z2 ). 2

We study the factors of E2p,q in (5.3). By definition, see (4.1): (5.4)

Fb = GR (Ωb , a1 αb ) × · · · × GR (Ωb , ak αb ).

To avoid truncation, we let Ωb be a countably infinite number of copies of α eb , the representation of G that restricts to twice αb . That makes (see Proposition 2.8) (5.5)

Fb = BO(a1 ) × · · · × BO(ak ).

The Z2 cohomology of BO(a) is a polynomial ring in the universal Stiefel– Whitney classes wi = wi (γ a ) of the canonical a–plane bundle over BO(a) (see [10] and [23]): (5.6)

H ∗ (BO(a), Z2 ) = Z2 [w1 , . . . , wa ].

We define a differential ∇ on H ∗ (BO(a), Z2 ). On generators we set ( wi−1 if i + a is even and i 6= 0 (5.7) ∇(wi ) = 0 if i + a is odd or i = 0. For convenience, we denoted the non-trivial constant polynomial 1 in (5.6) by w0 . With this differential (H ∗ (BO(a), Z2 ), ∇) is a DGA. The definition of this differential is geometrically motivated, as we will see in the proof of Proposition 5.2. In the upcoming computation we will use:


n 2m 2m+1 . (5.8) ∇ w2b+1 = w2b+1 w2b+2 = ∇ w∗2m = 0 and ∇ w2b+2 The differential extends to products:

(5.9)

H ∗ (Fb , Z2 ) = H ∗ (BO(a1 ), Z2 ) ⊗ · · · ⊗ H ∗ (BO(ak ), Z2 ).

Alluding to Borel’s analysis of H ∗ (BO(a), Z2 ) in terms of symmetric functions, we write (5.10) Sa ∼ = H ∗ (BO(a), Z2 ). For A = (a1 , . . . , ak ) and Fb as in (4.1) we abbreviate and compute that (5.11)

SA = Sa1 ⊗ · · · ⊗ Sak = H ∗ (Fb , Z2 ).

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KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN

We will need double–indices for the Stiefel–Whitney classes, so wj,i refers to the i–th Stiefel–Whitney class in the j–th factor in (5.9). We denote the homology and the cycles of the differential graded algebra (SA , ∇) by H∗ (SA ) and Z∗ (SA ). Proposition 5.1. [12] If at least one entry of A is odd, then (SA , ∇) is acyclic. If A = (a1 , . . . , ak ) = (2b1 , . . . , 2bk ) is a sequence of even nonnega2 tive integers, then H∗ (SA ) = Z2 [{wj,2s | 0 ≤ j ≤ k, 1 ≤ s ≤ bj }]. Proof. If a is odd and z is a cycle in SA , then z is a boundary: ∇(z ∪ wt,1 ) = ∇(z) ∪ wt,1 + z ∪ ∇(wt,1 ) = 0 + z ∪ 1 = z. Here wt,1 is the first Stiefel-Whitney class in H ∗ (BO(at )) for some t for which at is odd. For the second claim, according to the K¨ unneth formula, it suffices to prove the formula for (SA , ∇) with A = (2b) of length 1. We proceed by induction over b. If b = 0, then Fb is a point, and we read the formula as saying that H∗ (SA ) = Z2 , which is true. We assume that the assertion holds for A = (2b), and show that it holds A′ = (2b + 2). This follows once we show that any cycle f in SA′ can be expressed in the form 2N 2 , + · · · + c2N w2b+2 f = h + c0 + c2 w2b+2

(5.12)

where h is a boundary in SA′ , and c0 , . . . , c2N are cycles in SA . Express f ∈ SA′ in the form X 2m+1 n 2m n w2b+2 + gn,2m+1 w2b+1 w2b+2 f= gn,2m w2b+1 n,m≥0

(5.13)

=

X

n−1 2m+1 n 2m w2b+2 gn,2m w2b+1 w2b+2 + gn−1,2m+1 w2b+1

n>0,m≥0

+

X

2m g0,2m w2b+2

m≥0

where the g∗,∗ are in SA . We use the formulae in (5.8) to calculute ∇(f ): X n+1 2m 2m+1 n 2m n w2b+2 . + gn,2m+1 w2b+1 ∇(gn,2m )w2b+1 w2b+2 + ∇(gn,2m+1 )w2b+1 w2b+2 n,m≥0

As f is a cycle, we deduce that ∇g0,2m = 0 and ∇gn,2m + gn−1,2m+1 = 0 for n > 0 and m ≥ 0. We set: X 2m 2 2N g0,2m w2b+2 = c0 + c2 w2b+2 + · · · c2N w2b+2 . m≥0

The coefficients g0,2m are cycles in SA , and this sum, which equals the last sum in (5.13), accounts for all terms on the right hand side of (5.12), except for h. We also set X n−1 2m+1 n 2m h= gn,2m w2b+1 w2b+2 + gn−1,2m+1 w2b+1 w2b+2 . n>0,m≥0

ALGEBRAIC REALIZATION WITH ONE ISOTROPY TYPE

13

Each summand is a boundary because
n−1 2m+1 n−1 2m+1 n 2m = ∇(gn,2m )w2b+1 w2b+2 + gn,2m w2b+1 w2b+2 ∇ gn,2m w2b+1 w2b+2

2m n n−1 2m+1 . w2b+2 w2b+2 + gn,2m w2b+1 = gn−1,2m+1 w2b+1

This verifies that a cycle is of the form asserted in (5.12).



Serre [27] quotes older sources for the computation of H ∗ (BC, Z2 ):   if C is of odd order, Z 2 ∗ ∼ (5.14) H (BC, Z2 ) = Z2 [x] if |C| is twice an odd number,   2 Z2 [x, y]/hx = 0i if |C| is divisible by 4.

Here x denotes a class in grading 1 and y a class in grading 2. Set (5.15)

0 6= ζ (p) ∈ H p (BC, Z2 )

This makes sense because in this section we are assuming that C is of even order and H p (BC, Z2 ) ∼ = Z2 for all p ≥ 0. So ζ (p) is the unique nonzero class in grading p. A complete understanding of the spectral sequence is obtained from Proposition 5.2. Let ζ ⊗ u ∈ H p (BC, Z2 ) ⊗ H q (Fb , Z2 ) be a typical generator in the E2 –term of the spectral sequence of the fibration in (5.2), then d2 (ζ ⊗ u) = (ζ (2) ∪ ζ) ⊗ ∇u. The spectral sequence collapses at the E3 -level and ( {ζ (p) ⊗ u | u ∈ Z∗ (SA )} p,∗ p,∗ (5.16) E∞ = E3 = {ζ (p) ⊗ u | u ∈ H∗ (SA )}

if 0 ≤ p ≤ 1 if p ≥ 2.

In the limit we find

( Zq (SA ) H p (BC, Z2 ) ⊗ H ∗ (EC ×C Fb , Z2 ) ∼ = Hq (SA ) p+q=∗ M

if 0 ≤ p ≤ 1 if p ≥ 2.

Proof. Throughout the proof we will use Z2 –coefficients. Kosniowski shows that the transgression sends wi to ζ (2) ∪ ∇(wi ), see [21, p. 90]. The formula for the differential is an immediate consequence of the product structure. The expression for the E3 -term follows easily from the one for the E2 term, the formula for the differential, and the ring structure of H ∗ (BC, Z2 ). It remains to be shown that the spectral sequence collapses at E3 -level. If one of the entries in A is odd, then H∗ (SA ) = 0 and only the first two columns of the E3 -term of the spectral sequence are nonzero. It follows that the spectral sequence collapses at this level. If all entries in A are even, then we follow the ideas employed in Kosniowski’s proof, [21, 3.3.11 Lemma]. Let us set A = (a1 , . . . , ak ) = (2b1 , . . . , 2bk ).

14

KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN

In (2.6) we saw that GR (Ωb , aαb )G = GC (C∞ , a/2) = BU (a/2) ⊂ GR (R∞ , a) = BO(a). We have a diagram of fibrations:

(5.17)

(Fb )C −−−−→ EC ×C (Fb )C −−−−→ BC       y y y Fb

−−−−→

EC ×C Fb

−−−−→ BC.

e of the Then E(C) ×C (Fb )C = BC × (Fb )C , and the spectral sequence E e2 -level. Let top fibration in (5.17) collapses at the E e2 ∼ θ2 : E2 ∼ = H ∗ (BC) ⊗ H ∗ (Fb ) → E = H ∗ (BC) ⊗ H ∗ ((Fb )C )

be the induced map. As before, let ζ (p) ∈ H p (BC) be the unique nonzero class in the indicated group. Let wj,i be the i-th Stiefel-Whitney class of the canonical aj -plane bundle over the factor BO(aj ) of Fb . Let cj,i be the i-th Chern class (modulo 2) of the canonical bj -plane bundle over the factor BU (bj ) of (Fb )C . Then, because the vertical map in the center of the diagram in (5.17) is the identity on the first factor and forgets to complex structure on the second one, we find θ2 (ζ (p) ) = ζ (p) (5.18)

θ2 (wj,2i ) = cj,i θ2 (wj,2i−1 ) = 0.

We mentioned the second equality in (2.7). The map θ2 is multiplicative and commutes with the differentials in the spectral sequences. We have a diagram

(5.19)

E3p,q   d3 y

θ p,q

3 −−− −→

p+3,q−2

e p,q E 3  e yd3 =0

θ3 e p+3,q−2 E3p+3,q−2 −− −−−→ E 3

In Proposition 5.1 we saw that H∗ (SA ) has a basis consisting of classes 2 . If the first superscript is at least 2, then E ∗,∗ = H ∗ (BC) ⊗ H (S ). wj,2r ∗ A 3 Using the first two formulas in (5.18), we conclude that θ3p+3,q−2 is injective. This implies that d3 = 0, so that E3 = E4 . Proceeding inductively, we see that E3 = E4 = E5 = · · · . The spectral sequence for the fibration in (5.2) collapses at the E3 -level as asserted.  6. Totally Algebraic Z2 –Homology of a Cyclic Group. The Schubert cycles provide algebraic representatives of all Z2 –homology classes of BO(n). As Akbulut and King express this observation in [1], the

ALGEBRAIC REALIZATION WITH ONE ISOTROPY TYPE

15

Grassmannians have totally algebraic homology. We interpret some classical results as saying BC has totally algebraic homology. Consider the finite approximations Es C of EC and Bs C of BC: (6.1)

Es C = U (s)/ ({1} × U (s − 1)) & Bs C = U (s)/ (C × U (s − 1)) .

As homogeneous spaces they have unique real algebraic structures, so that U (s) acts on them real algebraically [14]. The orbit maps Es C → Bs C are regular, and we have a filtered system of fibrations with regular inclusions Es C → Es+1 C and Bs C → Bs+1 C. Having this system in mind, it makes sense to ask whether BC has totally algebraic homology. See (5.14) for the computation of H ∗ (BC, Z2 ). Proposition 6.1. For any ζ(r) in Hr (BC, Z2 ) and a sufficiently large s, there exists a nonsingular real algebraic variety Z of dimension r and an entire rational map κ : Z → Bs C, such that κ∗ [Z] = ζ(r) . Proof of Proposition 6.1. Express C as a product C2 ×Codd , where C2 stands for the Sylow 2–subgroup of C and Codd for the product of the other Sylow subgroups. Accordingly, BC = BCodd × BC2 . For each factor we have a filtered system, as described above, and the product provides one for BC. Homologically speaking, with Z2 coefficients, BCodd is a point, and classes are easily represented algebraically. Without loss of generality, we may thus assume for the remainder of the argument that C = C2 . Again, nothing needs to be said in case ζ(r) = 0. From now on we suppose that ζ(r) 6= 0. If 2 divides |C| and 4 does not, then one represents ζ(r) as the natural inclusion of RP r ֒→ RP s = Bs C. As an embedding of a subvariety, this inclusion is certainly algebraic. From now on we assume that 4 divides |C| and ζ(r) 6= 0. One reference for our discussion is 3.4.7 LEMMA in [21]. Assume that r is even. Then we consider pairs (Z, κ) = (RP r , ρr ), where ρr : RP r → Bs C is a finite approximation of the classifying map for the fibration C ×Z2 S r → RP r . Here Z2 acts on C as a subgroup and on S r antipodally. For sufficiently large s we have an expanded classifying diagram:

(6.2)

ρer

=

inc

π

C ×Z2 S r −−−−→ S 2s−1 = Es Z2 −−−−→ S 2s−1 = Es C     π ′  π2 y y y C RP r

−−−−→ RP 2s−1 = Bs Z2 −−−−→

Bs C

The algebraic structure on all spaces in this diagram was specified in (6.1). The fibration is classified by a map ρr : RP r → BC, which we may take as the composition of the natural, regular inclusion RP r ⊂ Bs Z2 and the regular orbit map π : Bs Z2 → Bs C. That means that ρr = π ◦ inc is regular as well.

16

KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN

As second case, suppose that r is odd. Consider the classifying diagram, finitely approximated, for the principal C–bundle (first vertical map)

(6.3)

S 1 ×Z2 S r−1   y

gr e

−−−−→ Es C   y gr

S 1 /C × RP r−1 −−−−→ Bs C The action of S 1 on EC induces an action of S 1 /C on Bs C, and the bundle is classified by (6.4)

gr : (S 1 /C) × RP r−1 → Bs C where (t, x) 7→ tρr−1 (x).

Earlier we saw that ρr−1 is regular. We compose ρr−1 with the regular action of (S 1 /C ′ ). The result is gr , and this map is regular. The map gr : (S 1 /C) × RP r−1 → BC is an algebraic representative of ζ(r) in this last case.  7. Algebraic representation of bordism generators (Proof of Theorem 4.1) In Theorem 4.1 we asserted that classes in NnC [free](Fb ) have algebraic representatives. If C is not of even order, then Fb is a point, and the theorem holds due to classical results. So, we suppose that 2 divides |C|. Standard techiques reduce the computation of N∗C [free](Fb ) to a homology problem (7.1)

N∗C [free](Fb ) ∼ = N∗ (EC ×C Fb ) ∼ = N∗ ⊗Z2 H∗ (EC ×C Fb , Z2 ).

The first of these isomorphisms is natural, but the second one is not. Steenrod representation provides a map Ψ : H∗ (EC ×C Fb , Z2 ) −→ N∗ (EC ×C Fb ). For a given homogeneous class y ∈ H∗ (EC ×C Fb , Z2 ) one needs to find a map f : M → EC ×C Fb so that f∗ [M ] = y. Then Ψ(y) is the bordism class of f : M → EC ×C Fb . One defines the map on a basis and extends it linearly to make sure that it is a homomorphism. After tensoring with N∗ the construction results in an isomorphism. The composition of the maps in the top row of the following pull-back diagram represents the corresponding class in N∗C [free](Fb ). e

(7.2)

f π f −−− −→ Es C × Fb −−−2−→ Fb M     y y f

M −−−−→ EC ×C Fb

We calculated H∗ (EC ×C Fb , Z2 ) (see Proposition 5.2) using the Leray– Serre spectral sequence of the fibration Fb → EC ×C Fb → BC.

ALGEBRAIC REALIZATION WITH ONE ISOTROPY TYPE

17

The spectral sequence collapses at the E3 –level, so that H ∗ (EC ×C Fb , Z2 ) is a subquotient of the E2 –term of the spectral sequence, where E2p,q = H p (BC, Z2 ) ⊗ H q (Fb , Z2 ). p,q Let w be a class in E3p,q = E∞ , and suppose that it contains a typical p q element ζ ⊗ u ∈ H (BC, Z2 ) ⊗ H (Fb , Z2 ). To represent w we should find a map of fibrations:

(7.3)

X −−−−→   ιy

M   fy

−−−−→ Z  κ y

Fb −−−−→ EC ×C Fb −−−−→ BC

so that ι : X → Fb is a Steenrod representation of the dual of u, and κ : Z → BC is one of ζ. If we then lift f to the C–cover (cartesian square) and compose it with the projection: (7.4)

e

f f −→ EC × Fb −→ Fb , M

then we obtain a representative of a class z in N∗C [free](Fb ) that corresponds to ζ ⊗ u under the map in (7.1). We will distinguish three cases, p = 0, p = 1, and p > 1. Once we p.q constructed algebraic representatives for typical generators of E∞ in each of the cases, then our proof of Theorem 4.1 will be complete. 7.1. Algebraic representation for classes with p = 0. If p = 0, then we consider a class w = 1 ⊗ u ∈ H 0 (BC, Z2 ) ⊗ H q (Fb , Z2 ). Let ι : M → Fb represent the dual of u, and consider the diagram

(7.5)

j

π

e h

π

M −−−−→ C × M −−−1−→     e ιy fy

C   y

Fb −−−−→ EC × Fb −−−1−→ EC

The action is by left multiplication on the first factor of C × M , and it is on the left on both factors of EC × Fb . (We may also convert the left action on EC into a right action, applying h−1 on the right instead of h on the left, to follow more closely the conventions for the balanced product. These two approaches are the same.) The maps are defined by j(x) = (c0 , x), fe(c, x) = (ce, cι(x)), and e h(y) = (e, y) for the unit element c0 in C and a chosen point e ∈ EC.

18

KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN

We may divide out the action of C in the last two columns of the diagram to obtain

(7.6)

j

π

h

π

M −−−−→ C ×C M −−−1−→     ιy fy

∗  κ y

Fb −−−−→ EC ×C Fb −−−1−→ BC.

This diagram plays the role of the one in (7.3). It represents w = 1 ⊗ u, and the induced algebraic Steeenrod representation of the dual is the map fe : C × M → Fb , which represents a class in N∗C [free](Fb ). 7.2. Algebraic representation for classes with p = 1. In our next case, where p = 1, we consider a class w = ζ (1) ⊗ u ∈ H 1 (BC, Z2 ) ⊗ H q (Fb , Z2 ). We define a commutative diagram j

(7.7)

X −−−−→ S 1 × X −−−−→     e ιy fy h

S1  qe y

Fb −−−−→ EC × Fb −−−−→ EC

The map ι : X → Fb is chosen as an algebraic Steenrod representative of the dual of u. Without loss of generality, we suppose that EC = ES 1 , or that the action of C on EC is the restriction of an action of S 1 on EC. Pick a point e ∈ EC and define qe(t) = te. Recall from Proposition 2.9 that the action of G on Fb is the restriction of an action of S 1 . This allows us to define fe by setting fe(t, x) = (te, tι(x)). If 1 denotes the unit element in S 1 , then we set j(x) = (1, x). The map h (defined by setting h(y) = (e, y)) is chosen so that the first square commutes. As the right square in (7.7) is equivariant, we may divide out the C action in the last two columns, and we obtain the diagram (compare (7.3)): j

(7.8)

X −−−−→ S 1 ×C X −−−−→ S 1 /C    κ   ιy fy y Fb −−−−→ EC ×C Fb −−−−→ BC.

As we saw in (6.4), κ : S 1 /C → BC represents the dual of ζ (1) . It follows that (7.8) represents the dual of w = ζ (1) ⊗ u, and that fe : S 1 × M → Fb represents a class in N∗C [free](Fb ) that corresponds to the dual of w under the map in (7.1).

ALGEBRAIC REALIZATION WITH ONE ISOTROPY TYPE

19

7.3. Algebraic representation for classes with p ≥ 2. Let us make a preliminary remark about Schubert cycles. Consider the last situation described in Proposition 2.8, where α is bad and a = 2b is even. Then i

GR (Ωα , aα)C = GC (C∞ , b) = BU (b) ֒→ GR (R∞ , a) = BO(a). Proposition 7.1. Let w ∈ H ∗ (BO(a), Z2 ) be a homogeneous polynomial in even Stiefel–Whitney classes. Then there exists an equivariant embedding ι : X ֒→ BU (b) ⊂ BO(a) of a nonsingular real algebraic C–variety (with trivial action) so that ι∗ [X] is dual to w. Proof. For the given class w ∈ H ∗ (BO(a)), Z2 ), we have c ∈ H ∗ (BU (b)), Z2 ), so that i∗ (w) = c, see (2.7). We have a union of nonsingular closed complex Schubert varieties that represent its dual: c∗ = (ι1 )∗ [eC (σC1 )] + · · · + (ιr )∗ [eC (σCr )] for Schubert symbols σC1 , . . . , σCr and inclusion ιj : eC (σCj ) ֒→ GC (C∞ , b). (We are using the notation and facts from [23, Chapter 6 and 14]. For a given symbol, we denote the real Schubert cell by e, the complex one by eC , and their closures by e and eC . As in other instances, [·] denotes the fundamental class of the inclosed expression.) There are polynomial equations that describe each eC (σC ) as a subvariety of GC (C∞ , b). Compare [7, p. 236] for the equations in the real setting. The complex situation is similar. For a symbol σC = (σ1 , . . . , σb ) in the complex context, we define the related real symbol be setting (7.9)

σR = (2σ1 − 1, 2σ1 , . . . , 2σb − 1, 2σb ).

Suppose that X is a b–dimensional complex subspace X of CN and XR is its underlying real vector space. If X ∈ eC (σC ), then XR ∈ e(σR ). For the Schubert cells we see (7.10)

eC (σC ) = e(σR ) ∩ GC (CN , b) &

eC (σC ) = e(σR ) ∩ GC (CN , b).

It follows that eC (σC ) is a subvariety of GR (R2N , 2b). It will be nonsingular if N is large compared to b. It follows from (7.10) that
(7.11) X = e(σR1 ) ⊔ · · · ⊔ e(σRr ) ∩ GC (C∞ , b) ⊂ GR (R∞ , a) is a nonsingular C–fixed subvariety that represents the dual of w.



Remark 7.2. One may use the Schubert calculus [17, p. 197ff] to express w as a linear combination of real Schubert cycles, but we do not know that this would provide us with a C–invariant representation of w. Stated differently, we do not know whether X in (7.11) is a union of real Schubert varieties. In this third and last case, p ≥ 2, there will be surviving terms in the spectral sequence only if all entries of A = (a1 , . . . , ak ) are even (use Propositions 5.1 and 5.2) and a typical generator p,q w ∈ E3p,q = E∞

20

KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN

2 . As in Proposiis of the form w = ζ (p) ⊗u, where u is a monomial in the wj,2s tion 5.1, wj,k is the k–th Stiefel–Whitney class of the factor H ∗ (BO(aj ), Z2 ) of H ∗ (Fb , Z2 ) = H ∗ (BO(a1 ), Z2 ) ⊗ · · · ⊗ H ∗ (BO(ak ), Z2 ).

Accordingly, express u as a product, u = u1 ⊗ · · · ⊗ uk and apply Proposition 7.1 to each factor. We obtain C–equivariant algebraic Steenrod representatives ιj : Xj → BO(aj ) for 1 ≤ j ≤ k. Their product ι=

k Y

j=1

ιj : X =

k Y

b

Xj −→ F =

k Y

BO(aj )

j=1

j=1

is a C–equivariant algebraic Steenrod representative of u. Let κ : Z → BC be an algebraic representation of ζ(p) , the dual of ζ (p) , as described in Proposition 6.1. We may now write down a commutative diagram

(7.12)

e×X X −−−−→ Z     ιy κ e×ιy

−−−−→

Ze   κ ey

Fb −−−−→ EC × Fb −−−−→ EC. The last two vertical maps are C–equivariant, and dividing out the action we obtain e ×C X −−−−→ Z X −−−−→ Z       (7.13) ιy κy κ e × C ιy Fb −−−−→ EC ×C Fb −−−−→ BC.

The diagram in (7.13) is a representation of the class w, and the map Ze × X → EC × Fb → Fb

represents a class in N∗C [free](Fb ) that corresponds to the dual of w under the map in (7.1). By construction, it is algebraic. References [1] S. Akbulut and H. King, The Topology of Real Algebraic Sets with Isolated Singularities, Ann. of Math. 113 (1981), 425–446. [2] S. Akbulut and H. King, On Approximating Submanifolds by Algebraic Sets and a Solution to the Nash Conjecture, Invent. Math. 107 (1992), 87–89. [3] S. Akbulut and H. King, Submanifolds and homology of nonsingular algebraic varieties, Amer. J. Math.107 (1985) 45–83. [4] S. Akbulut and H. King, All compact manifolds are homeomorphic to totally algebraic real algebraic sets, Comment. Math. Helv. 66 (1991), 139–149. [5] R. Benedetti and A. Tognoli, On Real Algebraic Vector Bundles, Bull. Sci. Math. 104 (1980), 89–112.

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KARL HEINZ DOVERMANN AND ARTHUR G. WASSERMAN

Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822 E-mail address: [email protected] Department of Mathematics, University of Michigan, Ann Arbor, MI E-mail address: [email protected]