ON EQUIVARIANT ALGEBRAIC SUSPENSION

H. Blaine Lawson, Jr., P. C. Lima-filho, and M.-L. Michelsohn

Abstract.

Equivariant versions of the Suspension Theorem [L_{1}] for algebraic cycles on projective varieties are proved. Let G be a finite group, V a projective G-module, and X ⊂ _{}(V) an invariant subvariety. Consider the algebraic join Σ /^{V_{0}}X = X# _{}(V_{0}) of X with the regular representation V_{0} = ^{G} of G. The main result asserts that algebraic suspension induces a G-homotopy equivalence Z^{s}(X) ⟶ Z^{s}(Σ /^{V_{0}}X) of topological groups of algebraic cycles of codimension-s for all s ≤ dim X − e(X) where e(X) is the maximal dimension of g-fixed point sets in Σ /^{V_{0}}X for g ≠ 1. This leads to a Stability Theorem for equivariant algebraic suspension. The methods also yield a Quaternionic Suspension Theorem for cycles in _{}( ^{n}) under the antiholomorphic involution corresponding to scalar multiplication by the quaternion j. From this the homotopy type of spaces of quaternionic cycles is completely determined.

————————— 1991 Mathematics Subject Classification. Primary 14C05; Secondary 55P47, 55R45 Key words and phrases. Chow varieties, equivariant algebraic suspension,

quaternionic suspension.

All authors were supported by the NSF.

Table of Contents

1. Introduction

2. Suspension to the regular representation 3. Some immediate consequences

4. Important examples

5. Equivariant suspension for varieties

6. The quaternionic suspension theorem 7. Quaternionic suspension for varieties

§1. Introduction. In 1989 a Suspension Theorem for algebraic cycles on complex projective varieties was formulated and proved in [L_{1}]. This result proved to be the key to the development of an interesting theory for varieties. It also led to some new constructions and results in algebraic topology. (See [L_{2}] for a survey.) The point of this paper is to enunciate and prove an Algebraic Suspension Theorem in the presence of symmetry. The result is substantially more difficult than the non-equivariant analogue in [L_{1}]. It is also not universally valid; there are delicate necessary conditions involving the dimension of the cycles and the geometry of the action. However, the theorem yields a stable result which is foundational for building equivariant theories based on cycles and for applications to algebraic topology.

The methods also yield a quaternionic suspension theorem which is of independent geometric interest.

The first steps towards equivariant theories based on algebraic cycles were taken in [LM_{1}] where algebraic suspension was shown to be a homotopy equivalence after localizing away from the order of the group (i.e., after inverting all primes which divide the order of G). Such localization was necessary as shown by example. In this form the suspension has substantial content. However, in the world of finite groups, inverting the primes dividing |G| often kills the interesting invariants of G, and it would be quite useful if one could show that some form of suspension for algebraic cycles was a full G-homotopy equivalence.

Such a theorem is the main result of this paper. We show that the suspension of algebraic cycles by the regular representation of G is a full G-homotopy equivalence provided that the dimension of the cycles is sufficiently large with respect to the dimension of the g-fixed point set for each g ≠ 1 in G. In particular, if G acts freely, then suspension to the regular representation is a G- homotopy equivalence for cycles of any dimension ≥ 0. This result is sharp as shown by many examples (in §4). It also leads to a basic stability result.

We shall now be more explicit. Fix a finite group G and a finite dimensional complex G-module V, and consider a G-invariant algebraic subvariety X ⊂ (V). For each s ≤ dim(X), let Z^{s}(X) denote the free abelian group generated by the irreducible subvarieties of codimension-s in X. This group carries a natural topology (cf. [L_{1}], [Li_{5}]), and G acts naturally on Z^{s}(X) by continuous group automorphisms.

We now let V_{0} denote the regular representation of G, and we define the V_{0}-suspension of X to be the G-invariant subset Σ /^{V_{0}}X = X# (V_{0}) ⊂ (V ⊕ V_{0}) consisting of the union of all lines joining X to (V_{0}) in (V ⊕ V_{0}). (Here # denotes the algebraic join, [L_{1}], [H]). Suspension extends naturally to algebraic cycles on X and defines a continuous G-equivariant homomorphism

Σ /^{V_{0}} : Z^{s}(X) ⟶ Z^{s}(Σ /^{V_{0}}X)

Our first main results are the following. Consider the numerical invariants

e_{0}(X) = max_{g∈G−{1}} dim X^{g}  and  e(X) = max_{g∈G−{1}} dim (Σ /^{V_{0}}X)^{g}

of the action of G on X, where X^{g} =^{def} {x ∈ X : g(x) = x}.

Theorem A. (The Equivariant Suspension Theorem) If s ≤ dim X − e(X), then suspension to the regular representation

Σ /^{V_{0}} : Z^{s}(X) ⟶ Z^{s}(Σ /^{V_{0}}X)

is a G-homotopy equivalence.

Theorem B. (Stability) Let q > 1 be the smallest prime divisor of |G|. Then the suspension homomorphisms

Z^{s}(Σ /^{mV_{0}}X) ⟶^{Σ /^{V_{0}}} Z^{s}(Σ /^{(m+1)V_{0}}X)

are G-homotopy equivalences for all non-negative integers m such that

m ≥ \frac{q}{(q−1)|G|}(s + e_{0}(X) − dim X) + \frac{1}{(q−1)}

Our last set of results concerns “quaternionic subvarieties”. Let denote the quaternions with standard basis {1, i, j, k}, and write = ⊕ j where = span_{}{1, i}. Multiplication on the left by j in ^{n} induces a complex antilinear map j : ^{n} ⟶ ^{n} with j^{2} = −Id. This induces an antiholomorphic involution

\overline{j} : _{}( ^{n}) ⟶ _{}( ^{n})

which is fixed-point free (and which is covered by an antilinear map of O(1) whose square is -Id). For each s < 2n − 1 the map \overline{j} induces an involution

\overline{j}_{∗} : Z^{s}( _{}( ^{n})) ⟶ Z^{s}( _{}( ^{n}))

whose fixed-points Z_{}^{s}(n) can be considered the “quaternionic cycles” of codimension-s. The quotient group Z̃_{}^{s}(n) = Z_{}^{s}(n)/(1 + \overline{j}_{∗})Z^{s}( _{}( ^{n})) by the \overline{j}_{∗}-averaged cycles, can be considered the group of “irreducible quaternionic cycles” of codimension-s. It is a topological 2-torsion group. When the cycle-dimension is 1 for example, this group is the /2 -vector space generated by the irreducible \overline{j}_{∗}-invariant curves, which are, in a sense, the non-orientable projective algebraic curves.

Theorem C. (The Quaternionic Suspension Theorem) For each s < 2n − 1 the \overline{j}_{∗}-equivariant map

Σ /^{}: Z^{s}( _{}( ^{n})) ⟶ Z^{s}( _{}( ^{n+1}))

given by suspension to , is a ( /2 )-homotopy equivalence. It induces homotopy equivalences

Σ /^{}: Z_{}^{s}(n) ⟶ Z_{}^{s}(n + 1)  and  Σ /^{}: Z̃_{}^{s}(n) ⟶ Z̃_{}^{s}(n + 1).

There is also a theorem for quaternionic suspension of Z^{s}( _{}( ^{n} ⊕ )). These results allow a determination of the homotopy type of Z_{}^{s} and Z̃_{}^{s}. The spaces Z_{}^{s} fit in a surprizing and beautiful way with the classical theory of representations and characteristic classes.

Using results in this paper the authors have succeeded in calculating the coefficients of the new equivariant cohomology theories established in [LLM_{1}]. The results also play a role in defining cohomology operations on the morphic cohomology groups introduced in [FL_{1}].

Since writing this paper the authors have learned of independent results of Jacob Mostovoy on quaternion cycles. Among other things he has obtained Theorem 6.4 over the rationals.

§2. Suspension to the regular representation. Consider a finite group G of order γ, and let V be a finite dimensional G-module. Then G acts naturally on the projective space (V), and for each p ≥ 0, G also acts by automorphisms on the topological monoid C_{p}( (V)) of effective p-cycles on (V). Hence it also acts by automorphisms on the naïve topological group completion Z_{p}( (V)).

Let V_{0} = · G denote the regular representation of G. There is a continuous Gequivariant homomorphism

Σ /^{V_{0}} : Z_{p}( (V)) ⟶ Z_{p+γ}( (V ⊕ V_{0}))

given by associating to a cycle c in (V) its algebraic suspension Σ /^{V_{0}}(c) = c# (V_{0}). This map is always a homotopy equivalence (cf. [L_{1}]). However, it is far from being a Ghomotopy equivalence in general. This is discussed in detail in §4. The main result of this section asserts that if the dimension p is large enough that every (p + 1)-cycle in (V ⊕ V_{0}) is forced to meet the subset (V ⊕ V_{0})_{free} where G acts freely, then Σ /^{V_{0}} is a G-homotopy equivalence.

This produces the “stable” result that after sufficiently many iterations, the maps Σ /^{V_{0}} become G-homotopy equivalences.

Before stating the main result we examine the condition referred to above. Note that (V ⊕ V_{0})_{free} = (V ⊕ V_{0}) − Σ where

Σ = {x ∈ (V ⊕ V_{0}) : G_{x} ≠ {1}}

= {x ∈ (V ⊕ V_{0}) : ∃g ∈ G, g ≠ 1 and gx = x}

= ⋃_{g∈G−{1}} (V ⊕ V_{0})^{g}

where X^{g} denotes the fixed-point set of g on X.

Definition 2.1. Let Γ_{G} denote the set of conjucagy classes of subgroups of G. Given a finite dimensional G-module V, let d_{V} : Γ_{G} ⟶ denote the (equivariant) dimension function of (V) defined as d_{V} ([H]) = dim_{}(V)^{H}, where [H] denotes the conjugacy class of the subgroup H, and the dimension of an algebraic subset is defined to be the maximal dimension of its irreducible components. We then define

(2.1.1)  e(V) = e_{G}(V) =^{def} max_{[H]≠{1}∈Γ_{G}} d_{V ⊕V_{0}}([H]).

In the following technical result we use the same letter W to denote both a G-module W and its restriction Res_{H}^{G}(W) to a subgorup H ≤ G.

Lemma 2.2. The function e(V) has the following properties.

(1) If H < G, then e_{H}(V) ≤ e_{G}(V).

(2)

e_{G}(V) = max_{g≠1∈G} dim{ (V ⊕ V_{0})^{g}},

where (V ⊕ V_{0})^{g} denotes the set of points fixed by g ∈ G. Furthermore, one just needs to maximize over those points g ∈ G of prime order.

(3) If G is abelian, then

e(V) = max_{ρ∈_{G}} dim(V_{ρ})

where Ĝ denotes the character group of G and where V_{ρ} ⊂ V is the simultaneous eigenspace with character ρ. If, in addition, G is cyclic of prime order then

e(V) = dim { (V)^{g}} + 1 for any g ≠ 1.

(4) For any G let q be the smallest prime divisor of γ = |G|. Then

e(V) ≥ \frac{1}{q}\{dim V + γ\} − 1

and for all m > 0,

e(V ⊕ mV_{0}) ≤ dim V + \frac{γ}{q} (m + 1) − 1

Proof. Statement (1) is obvious. Given subgroup H < G and h ∈ H, let < h > denote the subgroup generated by h. Then (V ⊕ V_{0})^{H} ⊂ (V ⊕ V_{0})^{<h>} = (V ⊕ V_{0})^{h}, and the first assertion of (2) follows. Note that since dim (V ⊕ V_{0})^{g} ≤ (V ⊕ V_{0})^{g^{k}} for any k, it suffices to maximize over elements g of prime order in G.

For (3) recall that V_{0} ≅ ⨁_{ρ} ℓ_{ρ} where ℓ_{ρ} is the 1-dimensional eigenspace with character ρ. Then for any g ≠ 1 we have

(V ⊕ V_{0})^{g} = ∐_{ρ∈_{G}} ((V ⊕ V_{0})_{ρ}) = ∐_{ρ∈_{G}} (V_{ρ} ⊕ ℓ_{ρ}),

and (3) follows immediately from assertion (2).

To prove (4), consider g ∈ G of prime order q. Then we have

(V ⊕ V_{0})^{g} = ∐_{ρ∈_{/q}} ((V ⊕ V_{0})_{ρ}) = ∐_{ρ∈_{/q}} (V_{ρ} ⊕ \frac{γ}{q} ℓ_{ρ}) ,

from which the first assertion of (4) follows directly. For the second we note that

(V ⊕ (m + 1)V_{0})^{g} = ∐_{ρ∈_{/q}} (V_{ρ} ⊕ (m + 1)\frac{γ}{q} ℓ_{ρ}) ,

and

max_{ordg prime} max_{ρ} dim (V_{ρ} ⊕ (m + 1)\frac{γ}{ordg} ℓ_{ρ}) ≤ dim V + (m + 1)\frac{γ}{q}

We would like to point out that one can also give the following interpretation to the dimension functions that we considered, based on the previous lemma. Let χ_{V} denote the character of V, and for any subgroup H < G, let Ĥ denote the collection of equivalence classes of 1-dimensional representations of H, identifiable with the “character group” of H, consisting of all group homomorphisms H ⟶ ^{∗}. Then

(V)^{H} = (Res_{H}^{G}(V))^{H} = ∐_{ρ∈_{H}} (Res_{H}^{G}(V)_{ρ}).

On the other hand, dim Res_{H}^{G}(V)_{ρ} = (χ_{Res_{H}^{G}(V)} | χ_{ρ})_{H}, where (·|·)_{H} denotes the usual inner product for class functions on H. Hence, Fröbenius reciprocity

(χ_{Res_{H}^{G}(V)} | χ_{ρ})_{H} = (χ_{V} | χ_{Ind_{H}^{G}(ρ)})_{G}

implies that

e(V) = max {(χ_{V} | χ_{ξ})_{G} + [G : H] | ξ monomial induced by H } .

Recall that a G-module ξ is said to be monomial if it has the form ξ = Ind_{H}^{G}(ρ), where ρ is a 1-dimensional H-module for some subgroup H < G.

2.3. Equivariant Algebraic Suspension Theorem I (Suspension to the regular representation). Let G be a finite group of order γ and denote by V_{0} = ^{G} the regular representation of G. Let V be any finite dimensional complex G-module. Then for each p ≥ e(V) the suspension homomorphism

Σ /^{V_{0}} : Z_{p}( (V)) ⟶ Z_{p+γ}( (V ⊕ V_{0}))

is a G-homotopy equivalence.

Proof. Consider the open G-invariant submonoid

T C_{p+γ} =^{def} {c ∈ C_{p+γ}( (V ⊕ V_{0})) : dim(|c| ∩ (V) = p}

of effective cycles of dimension p + γ which meet (V) in proper dimension, and let ZC_{p+γ} denote its naïve topological group completion. Algebraic suspension gives an equivariant embedding

Σ /^{V_{0}} : C_{p}( (V)) ↪ T C_{p+γ}

which extends to an injective homomorphism

Σ /^{V_{0}} : Z_{p}( (V)) ↪ T Z_{p+γ}

of group completions. The theorem is a consequence of the following two assertions.

Assertion 1. The image Σ /^{V_{0}} [Z_{p}( (V))] is a G-equivariant deformation retract of T Z_{p+γ}. (This assertion holds for any p ≥ 0.)

Assertion 2. The inclusion j : T Z_{p+γ} ⊂ Z_{p+γ}( (V ⊕ V_{0})) is a G-homotopy equivalence.

Proof of Assertion 1. This is a straightforward application of pulling to the normal cone. Consider the continuous family of G-equivariant automorphsims

Φ_{t} : (V ⊕ V_{0}) ⟶ (V ⊕ V_{0}) for 0 < t < ∞

defined by setting Φ_{t}([v : v_{0}]) = [tv : v_{0}] for v ∈ V and v_{0} ∈ V_{0}. This induces a family of G-equivariant continuous monoid automorphisms

(Φ_{t})_{∗} : C_{p+γ} ( (V ⊕ V_{0})) ⟶ C_{p+γ} ( (V ⊕ V_{0})) for 0 < t < ∞.

When restricted to T C_{p+γ}, this family extends continuously to t = 0. (See [L_{1}], [FL_{2}], or [Fu] for example.) When restricted to the image Σ /^{V_{0}}Z_{p} =^{def} Σ /^{V_{0}} [Z_{p}( (V))] , this family is the identity for all t ≥ 0. One checks directly (cf. [L_{1}], [FL_{2}]) that Image (Φ_{0})_{∗} = Σ /^{V_{0}}Z_{p} for all t. Thus the restriction of (Φ_{t})_{∗} to T C_{p+γ} gives an equivariant deformation retraction down to Σ /^{V_{0}}Z_{p}. Since each (Φ_{t})_{∗} for t ≥ 0 is a monoid homomorphism, this family extends to the group completions to give the desired result.

Proof of Assertion 2. This is based on the following Proposition. We abbreviate notation by setting C_{p+γ} = C_{p+γ}( (V ⊕ V_{0})).

Proposition 2.4. Let X ⊂ C_{p+γ} be any compact subset. Then for all integers d sufficiently large, there exists a continuous family Ψ_{t} : C_{p+γ} → C_{p+γ} of G-equivariant continuous monoid homomorphisms defined for 0 ≤ t ≤ 1 with the property that

(1) Ψ_{0} = d^{γ}·Id, and

(2) Ψ_{t}(X) ⊂ T C_{p+γ} for all t > 0.

Proof. We enlarge our G-space to V ⊕ V_{0} ⊕ V_{0} with V ⊕ V_{0} embedded as the first two factors. Consider the invariant subspaces

V′_{0} ⊂ V_{0} ⊕ V_{0}  and  V′′_{0} ⊂ V_{0} ⊕ V_{0}

where V′_{0} is the diagonal and V′′_{0} = {0} × V_{0}. Let

π′ : (V ⊕ V_{0} ⊕ V_{0}) − (V′_{0}) ⟶ (V ⊕ V_{0})

be the equivariant projection with vertex V′_{0}.

For each positive integer d > 0 let Div_{d} denote the space of effective divisors of degree d on (V ⊕ V_{0} ⊕ V_{0}), and consider the subset U′_{d} ⊂ Div_{d} of those divisors D for which

(2.4.1)  { ⋂_{g∈G} g|D|} ∩ (V′_{0}) = ∅.

Let U′′_{d} ⊂ Div_{d} be defined similarly with V′_{0} replaced by V′′_{0}, and set U_{d} = U′_{d} ∩ U′′_{d}.

Lemma 2.5. The set U_{d} is Zariski open and non-empty.

Proof. A straightforward argument (cf. [H]) shows that the set where (2.4.1) fails is Zariski closed. Thus U′_{d} is Zariski open, and by the same reasoning so is U′′_{d}. Thus U_{d} is Zariski open. To see that it is non-empty we consider the linear divisor

(2.5.1)  Δ =^{def} d · {V ⊕ V_{0} ⊕ ⨁_{g≠1} · g}

and observe that

⋂_{g∈G} g|Δ| = d^{γ}(V ⊕ V_{0}).

We now observe that for D ∈ U_{d} the intersection ⋂ g|D| has proper dimension by (2.4.1), for otherwise it would meet (V′_{0}). Consequently, the intersection product

=^{def} ∏_{g∈G} g_{∗}D

is well-defined and continuous on U_{d} (cf. [Fu]). Furthermore, for cycles c on (V ⊕ V_{0}), consider the suspension Σ /^{V′′_{0}}(c) = c# (V′′_{0}) in (V ⊕ V_{0} ⊕ V_{0}).

Lemma 2.6. The intersection product Σ /^{V′′_{0}}(c) • is well-defined and continuous on C_{p}(V ⊕ V_{0}) × U_{d}.

Proof. By [Fu] it suffices to show that for any subvariety Y ⊂ C_{p}(V ⊕V_{0}), the intersection Σ /^{V′′_{0}}(Y) ∩ | | has proper dimension. For this it suffices to show that if Y is a point y, then this intersection is finite. However, this last assertion is obvious since Σ /^{V′′_{0}}(y) is a linear subspace of dimension γ, and Σ /^{V′′_{0}}(y) ∩ | | is a subvariety which misses the hyperplane (V′′_{0}).

To each divisor D ∈ U_{d} we now construct a continuous, G-equivariant monoid endomorphism

Ψ_{D} : C_{p+γ}( (V ⊕ V_{0})) ⟶ C_{p+γ}( (V ⊕ V_{0}))

by setting

(2.6.1)  Ψ_{D}(c) = π′_{∗} {Σ /^{V′′_{0}}(c) • } .

By Lemma 2.6 these endomorphisms depend continuously on D ∈ U_{d} and for D = Δ (defined in (2.4.1)) we have

(2.6.2)  Ψ_{Δ}(c) = d_{γ} · c.

Given a subvariety Y ⊂ (V ⊕ V_{0}) we have Ψ_{D}(Y) ∩ (V) = π′{Σ /^{V′′_{0}}(Y) • } ∩ (V). From this we see that Ψ_{D}(Y) meets (V) properly if

Ỹ =^{def} (π′)^{−1}( (V)) ∩ Σ /^{V′′_{0}}(Y)

meets properly. Now projection from V′′_{0} induces an equivariant isomorphism

F : (V ⊕ V′_{0}) = (π′)^{−1}( (V)) ⟶ (V ⊕ V_{0})

and Ỹ = F^{−1}(Y). Thus given Y we have

(2.6.3)

B_{Y} =^{def} {D ∈ U_{d} : Ψ_{D}(Y) ∉ T C_{p+γ}}
= {D ∈ U_{d} : Ψ_{D}(Y) does not meet (V) properly }
= {D ∈ U_{d} : does not meet Ỹ properly}.

We note that by the upper-semicontinuity of dimension, the subset B_{Y} is Zariski closed.

Lemma 2.7. Let Y ⊂ (V ⊕ V_{0}) be an irreducible subvariety of dimension p + γ. Then the function

f(d) = codim(B_{Y}) in U_{d}

satisfies lim_{d→∞} f(d) = ∞.

Proof. We adopt the following notational conventions. Given any subvariety Z ⊂ ^{M} and any point z ∈ Z we denote by Z_{z} the germ of Z at z. We also denote by O_{Z,z} the local ring of germs at z of functions regular in a neighborhood of z on Z, and by 𝔪_{Z,z} the maximal ideal in O_{Z,z}. For convenience we shall drop the reference to Z when Z = (V ⊕ V_{0} ⊕ V_{0}).

We shall use the last characterization of B_{Y} in (2.6.3). Fix a regular point y ∈ Ỹ. We say that D ∈ U_{d} is bad at y if dim(| | ∩ Ỹ)_{y} > p, and we denote by B_{y} the subset of all such divisors. Note that B_{y} is a constructible subset of the linear subspace

Div_{d,y} =^{def} {D ∈ Div_{d} : gy ∈ D ∀ g ∈ G}.

Note that Div_{d,y} = (M_{d,y}) where

M_{d,y} =^{def} {σ ∈ O(d) : σ(gy) = 0 ∀g ∈ G}

Suppose now that y ∈ (V ⊕ V′_{0})_{free}. Write G = {g_{j}}_{j=1}^{γ} with g_{1} = 1. Then the elements y_{j} ≡ g_{j}(y), j = 1, …, γ are mutually distinct.

Sublemma 2.8. Fix an integer ℓ > 0 and let y_{1}, …, y_{γ} be distinct points in ^{M} = (V ⊕ V_{0} ⊕ V_{0}). Then the map

O(d) ⟶^{ρ} (O_{y_{1}}/𝔪_{y_{1}}^{ℓ+1}) × ⋯ × (O_{y_{γ}} /𝔪_{y_{γ}}^{ℓ+1})

defined by sending σ to the classes of its germs ([σ_{y_{1}}], …, [σ_{y_{γ}}]), is surjective for all d ≥ d_{ℓ} where d_{ℓ} depends only on ℓ, γ and M. Hence, the map

M_{d,y} ⟶^{ρ} (𝔪_{y_{1}}/𝔪_{y_{1}}^{ℓ+1}) × ⋯ × (𝔪_{y_{γ}} /𝔪_{y_{γ}}^{ℓ+1})

is also surjective.

Proof. By general position we may choose an affine chart ^{M} ⊂ ^{M} with coordinates (ζ_{1}, …, ζ_{M}) having the following properties:

(1) {y_{1}, ..., y_{γ}} ⊂ ^{M},

(2) y_{γ} = 0,

(3) {y_{1}, ..., y_{γ−1}} ∩ {ζ : ζ_{i} = 0} = ∅ for i = 1, ..., M. We want to show that there exists an integer d_{γ} such that

ρ_{γ,d} : [ζ]_{d} =^{def} {p(ζ) ∈ [ζ] : deg p ≤ d} ⟶ ∏_{i=1}^{γ} (O_{y_{i}}/𝔪_{y_{i}}^{ℓ+1})

is surjective for all d ≥ d_{γ}. We proceed by induction. The statement is clear for γ = 1 Assume it is proved for γ − 1, and let K_{d} = ker(ρ_{γ−1,d}). We want to find d_{γ} > d_{γ−1} so that ρ : K_{d} ⟶ O_{0}/𝔪_{0}^{ℓ+1} is surjective for all d ≥ d_{γ}.

For any multi-index α with |α| > ℓ we have P_{i,α}(ζ) =^{def} (ζ − y_{i})^{α} = 0 in O_{y_{i}}/𝔪_{y_{i}}^{ℓ+1} and so

P =^{def} ∏_{i=2}^{γ} P_{i,α(i)} ∈ K_{ℓ}

for any such choice of α(2), ..., α(γ) with |α(2)| + ⋯ + |α(γ)| ≤ d_{γ−1}. Choose one such P and note that P (0) ≠ 0 by our assumption (3).

Set K̂_{ℓ} ≡ span{ζ^{α}P (ζ) : |α| ≤ ℓ} = [ζ]_{ℓ} · P (ζ), and note that K̂_{ℓ} ⊂ K_{ℓ}. Furthermore, the map ρ : K̂_{ℓ} ⟶ O_{0}/𝔪_{0}^{ℓ} is evidently surjective. Thus, d_{γ} = d_{γ−1} + ℓ will work.

Sublemma 2.9. Let Z be a germ of a subvariety of ^{M} at a point z. Fix a function f ∈ 𝔪_{Z,z} and consider the (germ of the) subvariety Y = Z ∩ {f = 0} at z. Then for any ℓ > 0 the ring O_{Y,z}/𝔪_{Y,z}^{ℓ+1} depends only on the class of f in O_{Z,z}/𝔪_{Z,z}^{ℓ+1}

Proof. Note that O_{Y,z}/𝔪_{Y,z}^{ℓ+1} is just the quotient of O_{Z,z}/𝔪_{Z,z}^{ℓ+1} by the principal ideal (f).

We now begin our main estimate of the codimension of B_{Y}. To begin we fix ℓ and consider the composition

M_{d,y} ⟶ ⟶^{ρ} ∏_{i=1}^{γ} (𝔪_{y_{i}}/𝔪_{y_{i}}^{ℓ+1}) ⟶^{p×pg_{2}^{∗}×pg_{3}^{∗}×···×pg_{γ}^{∗}} ⟶ (𝔪_{_{Y},y}/𝔪_{_{Y},y}^{ℓ+1}) × ⋯ × (𝔪_{_{Y},y}/𝔪_{_{Y},y}^{ℓ+1})

which we denote by p_{1} × ⋯ × p_{γ}, and where d ≥ d_{ℓ}.

Let K_{1} = ker(p_{1}). By the surjectivity of p_{1} from Sublemma 2.8 we have

codim K_{1} = \binom{p + γ + ℓ}{ℓ} − 1

We now fix f ∈ M_{d,y} with f̃ =^{def} p_{1}(f) ≠ 0. Set A_{2,f} =^{def} p_{1}^{−1}(f̃) = f + K_{1}. (This is an affine space which we consider to be a vector space with f as its origin.) Set Ỹ_{2} = Ỹ ∩ {f = 0} and consider the composition

A_{2,f} ⟶^{p_{2}} 𝔪_{_{Y},y}/𝔪_{_{Y},y}^{ℓ+1} ⟶^{π} 𝔪_{_{Y_{2}},y}/𝔪_{_{Y_{2}},y}^{ℓ+1}

where π is the natural projection. Denote this composition by p_{2,f}. Now by 2.9, this map depends only on the class f̃. By 2.8, the map p_{2} restricted to K_{1} is surjective, and hence the linear map p_{2,f} is also surjective.

Let K_{2,f} = ker(p_{2,f}). By the surjectivity of p_{2,f} we have

codim K_{2,f} in A_{2,f} ≥ \binom{p + γ − 1 + ℓ}{ℓ} − 1

If the germ Ỹ_{2} at y is reducible, we can apply the above construction to each of its irreducible components Ỹ_{2,α}. This will give us a finite number of “bad” linear subspaces K_{2,α,f}, each satisfying the codimension condition above. To keep the exposition simpler, we proceed as though the varieties were irreducible at each stage. Redoing the argument by looking at these distinct components at each stage does not change the proof qualitatively and the details are left to the reader.

We now fix f ∈ M_{d,y} with f̃_{1} =^{def} p_{1}(f) ≠ 0 and f̃_{2} =^{def} p_{2,f}(f) ≠ 0. Set A_{3,f} =^{def} (p_{1} × p_{2,f})^{−1}(f̃_{1}, f̃_{2}) = f + K_{2,f}. (This is an affine space which we consider to be a vector space with f as its origin.) Set Ỹ_{3} = Ỹ_{2} ∩ {g_{2}^{∗}f = 0} = Ỹ ∩ {f = 0} ∩ {g_{2}^{∗}f = 0}, and consider the composition

A_{3,f} ⟶^{p_{3}} 𝔪_{_{Y},y}/𝔪_{_{Y},y}^{ℓ+1} ⟶^{π} 𝔪_{_{Y_{3}},y}/𝔪_{_{Y_{3}},y}^{ℓ+1}

where π is the natural projection. Denote this composition by p_{3,f}. Now by 2.9, this map depends only on the classes f̃_{1} and f̃_{2}. By 2.8, the map p_{2} restricted to the dernel of (p_{1}, p_{2}) is surjective. Hence the linear map p_{3,f} is also surjective.

Let K_{3,f} = ker(p_{3,f}). By the surjectivity of p_{3,f} we have

codim K_{3,f} in A_{3,f} ≥ \binom{p + γ − 2 + ℓ}{ℓ} − 1

We now consider f ∈ M_{d,y} with f̃_{1} = p_{1}(f) ≠ 0, f̃_{2} = p_{2,f}(f) ≠ 0 and f̃_{3} = p_{3,f}(f) ≠ 0, and proceed as above. We repeat this process inductively up to the γ^{th} stage. where for f ∈ M_{d,y} having f̃_{1} ≠ 0, …, f̃_{γ−1} ≠ 0 we construct the spaces K_{γ,f} ⊂ A_{γ,f} with

codim K_{γ,f} in A_{γ,f} ≥ \binom{p + 1 + ℓ}{ℓ} − 1

The main observation now is the following. Set

B_{y} =^{def} {f ∈ M_{d,y} : meets Ỹ improperly at y, where D = Div(f)}.

Then we have

B_{y} ⊂ K_{1} ∪ ⋃_{f∉K_{1}} K_{2,f} ∪ ⋃_{f∉K_{1}∪K_{2,f}} K_{3,f} ∪ …

and the codimension estimates above show that

(2.9.1)  codim(B_{y}) in M_{d,y} ≥ \binom{p + 1 + ℓ}{ℓ} − 1

Suppose now that we have a divisor D ∈ B_{Y}. By definition (2.6.3) this means that dim Ỹ ∩ | | ≥ p + 1. Therefore, by our assumption that p ≥ e(V), there exists a point y ∈ Ỹ ∩ | | such that G_{y} = {1} (i.e., the orbit of y must have γ distinct points). Then we must have D ∈ B_{y}. It follows that

B_{Y} ⊂ ⋃_{(V ⊕V′_{0})_{free}∩_{Y}} B_{y}

and so

codim(B_{Y}) ≥ \binom{p + 1 + ℓ}{ℓ} − 1 − dim Ỹ = \binom{p + 1 + ℓ}{ℓ} − p − γ − 1,

for d ≥ d_{ℓ}. In particular, given any M > 0, one can find ℓ such that \binom{p+1+ℓ}{ℓ}−p−γ−1 > M and hence, for d ≥ d_{ℓ} one has f(d) = codim(B_{Y}) > M, proving Lemma 2.7.

We now conclude the proof of Proposition 2.4. Given a cycle c = ∑_{i} n_{i}Y_{i} ∈ C_{p+γ} we set

B_{c} =^{def} ⋃_{i} B_{Y_{i}}

and observe that Lemma 2.7 holds with B_{Y} replaced by B_{c}. Now in Proposition 2.4 it will suffice to let X be a finite union of components of C_{p+γ}. For such X

B_{X} =^{def} ⋃_{c∈X} B_{c}

is a constructible subset of Div_{d} with codim(B_{X}) ≥ f(d) − dim X. Hence,

(2.9.2)  codim(B_{X}) >> 0

for all d sufficiently large. For any such d consider the family of lines L_{Δ} in the projective space Div_{d} which pass through the point Δ (cf. (2.5.1)). For amost all lines ℓ ∈ L_{Δ}, the intersection ℓ ∩ B_{X} is finite (in fact, consists only of {Δ}) by (2.9.2). For any such ℓ, there exists D ∈ ℓ such that D_{t} ≡ (1 − t)Δ + tD ∉ B_{X} for all 0 < t ≤ 1. The family Ψ_{t} = Ψ_{D_{t}} has the desired property. This proves Proposition 2.4

To prove Assertion 2 we employ the fact that an equivariant map Ψ : A → B between G-CW-complexes is a G- homotopy equivalence if Ψ restricts to an ordinary homotopy equivalence of fixed-point sets

Ψ : A^{H} ⟶ B^{H}

for all subgroups H < G. (See [tD, p. 107]).

Fix H < G. Let us abbreviate notation by letting C_{p+γ,d} denote the cycles of degree d in C_{p+γ}, and setting Z_{p+γ} = Z_{p+γ}( (V ⊕ V_{0} ⊕ V_{0}) and T Z_{p+γ} = T Z_{p+γ}( (V ⊕ V_{0} ⊕ V_{0}), etc..

We want to prove that the homomorphism

j_{∗} : (T Z_{p+γ})^{H} ⟶ (Z_{p+γ})^{H}

induced by the inclusion j is an isomorphism for all k ≥ 0.

Fix k and suppose that f : S^{k} ⟶ (Z_{p+γ})^{H} is a continuous map. Then, there is a D > 0 such that f(S^{k}) ⊂ Z_{p+γ,≤D}^{H}, where Z_{p+γ,≤D} is the image of the compact set ∐_{r+s≤D} C_{p+γ,r} × C_{p+γ,s} under the canonical projection π : C_{p+γ} × C_{p+γ} ⟶ Z_{p+γ}. Denote X = ∐_{r≤D} C_{p+γ,r} and let Ψ̃_{t} : Z_{p+γ} ⟶ Z_{p+γ} be the extension of the monoid morphism Ψ_{t} : C_{p+γ} ⟶ C_{p+γ} from Proposition 2.4 to a G-equivariant group homomorphism. Then the composition Ψ̃_{t} ∘ f gives a homotopy F_{t} : S^{k} ⟶ (Z_{p+γ})^{H} satisfying

F_{0} = d^{γ} · f  and  F_{t}(S^{k}) ⊂ (T Z_{p+γ})^{H} ∀ t > 0,

where d is as in Proposition 2.4.

This proves that for any element α ∈ π_{k} (Z_{p+γ})^{H}, we have d^{γ}α ∈ Image(j_{∗}) for all d sufficiently large. If we choose d_{1}, d_{2} relatively prime, then d_{1}^{γ}, d_{2}^{γ} are relatively prime, and this proves that α ∈ Image(j_{∗}). Thus j_{∗} is surjective.

Consider now a continuous map f : D^{k+1} ⟶ (Z_{p+γ})^{H} such that f(∂D^{k+1}) ⊂ (T Z_{p+γ})^{H}. Using Proposition 2.4 and the same argument as above one finds d_{0} such that for each d ≥ d_{0} there is a homotopy F_{t} : D^{k+1} ⟶ (Z_{p+γ})^{H} with

F_{0} = d^{γ} · f  and  F_{t}(D^{k+1}) ⊂ (T Z_{p+γ})^{H} ∀t > 0.

This proves that for any element β ∈ π_{k} (T Z_{p+γ})^{H} with j_{∗}β = 0, there exists d_{0} such that d^{γ}β = 0 for all d ≥ d_{0}. Choosing d_{1}, d_{2} relatively prime as above shows that β = 0. Thus j_{∗} is injective. This completes the proof of Assertion 2. Assertions 1 and 2 clearly imply the theorem.

§3. Some immediate consequences. Let G be a finite group of order γ and let V be a finite dimensional G-module. Denote by V_{0} the regular representation of G. Then beginning with any p < dim V − 1 we have a sequence of G-equivariant homomorphisms

(3.0.1)  Z_{p}( (V)) ⟶^{Σ /^{V_{0}}} Z_{p+γ}( (V ⊕ V_{0})) ⟶^{Σ /^{V_{0}}} Z_{p+2γ}( (V ⊕ 2V_{0})) ⟶^{Σ /^{V_{0}}} …

given by algebraic suspension to V_{0}. We shall see in the next section that these homomorphisms are definitely not G-equivalences in general. However, Lemma 2.2 (3) and the Theorem 2.3 immediately imply the following.

Theorem 3.1. (The Stability Theorem) Let q be the smallest prime divisor of γ = |G|. Then the homomorphism

Σ /^{V_{0}} : Z_{p+mγ}( (V ⊕ mV_{0})) ⟶ Z_{p+(m+1)γ}( (V ⊕ (m + 1)V_{0}))

is a G-homotopy equivalence for all

m ≥ \frac{q}{γ(q−1)} \{dim V − (p + 1) + \frac{γ}{q} \} .

For groups of prime order Lemma 2.2(2) and Theorem 2.3 give a more refined result.

Theorem 3.2. Let G = /q where q is prime. Then

Σ /^{V_{0}} : Z_{p}( (V)) ⟶ Z_{p+q}( (V ⊕ V_{0}))

is a G-homotopy equivalence for all p ≥ max_{ρ∈_{G}} dim V_{ρ}. In particular, Σ /^{V_{0}} gives the series of G-homotopy equivalences

Z_{p}( (kV_{0})) ⟶^{≅} Z_{p+q}( ((k + 1)V_{0})) ⟶^{≅} Z_{p+2q}( ((k + 2)V_{0})) ⟶^{≅} …

whenever p ≥ k.

§4. Important examples. At first glance the hypothesis that p > e(V) in Theorem 2.3 seems awkward and technical since the function e(V) is not easy to compute in general. However there is very good evidence to indicate that this hypothesis is sharp.

Certainly without the hypothesis the proof fails. When p ≤ e(V), cycles can be “trapped” in fixed-point sets of subgroups of G so that no equivariant move will displace them to be “transversal”. Furthermore, calculations show that often when this hypothesis fails, such cycles cannot in any way be equivariantly displaced. In such cases Σ /^{V_{0}} is not a G-equivalence, and these cycles are contributing to new elements in π_{0}Z^{G}. We summarize some of these calculations here.

Let G be a abelian of order γ and for m ≥ 0, k > 0, set Z(m, k) =^{def} Z_{mγ}(( (k + m)V_{0})). Consider the sequence of G-equivariant suspension maps

Z(0, k) ⟶^{Σ /_{V_{0}}} Z(1, k) ⟶^{Σ /_{V_{0}}} Z(2, k) ⟶^{Σ /_{V_{0}}} Z(3, k) ⟶^{Σ /_{V_{0}}} …

By Theorem 2.3 these maps become G-equivalences starting with the first Z(m, k) where m(γ − 1) > k. However, these maps are certainly not G-equivalences at the outset. This is seen already at the level of π_{0}

In [LLM_{1}] direct calculations of π_{0}(Z(m, k)^{G}) were carried out, and in particular, the following was established.

Fact 1. The inclusions (kV_{0}) ⊂ ((k + 1)V_{0}) induce isomorphisms

P_{G} =^{def} π_{0}(Z(0, 1)^{G}) ⟶^{≅} π_{0}(Z(0, 2)^{G}) ⟶^{≅} π_{0}(Z(0, 3)^{G}) ⟶^{≅} …

and P_{G} ≅ ⊕P̃_{G} where P̃_{G} is a torsion group of rank < γ. In particular, for prime integers q,

P̃_{( /q )} ≅ ( /q )^{q−1}.

Fact 2. The inclusion-suspension maps

… ↪ π_{0}(Z(k, 2k)^{G}) ↪ π_{0}(Z(k + 1, 2(k + 1))^{G}) ↪ π_{0}(Z(k + 2, 2(k + 2))^{G}) ↪ … are injective and not surjective for all k. Set Q_{G} =^{def} lim_{k→∞} π_{0}(Z(k, 2k)^{G}). Then Q_{G} can be written as the localization of a graded ring

Q_{G} = ^{∗}(G)_{M}

where ^{∗} is a functor on abelian groups with the following properites:

(1) ^{0} = and ^{1} ≅ G.

(2) ^{∗}(G_{1} ⊕ G_{2}) ≅ ^{∗}(G_{1}) ⊗ ^{∗}(G_{2})

(3) If G is cyclic of prime order q, then there is a natural isomorphism

∗(G) ≅ H^{2∗}(G; )

with cohomology with coefficients in the trivial G-module . and where M is the multiplicative system generated by 1 + x for x ∈ ^{1}(G) ≅ G.

Note that Q_{G} embeds in the completion ^{ ˜ ∗} (G) of ^{∗}(G) with repsect to the aumentation ideal.

Clearly P_{G} and Q_{G} are radically different. P_{G} is a finitely generated abelian group and Q_{G} is a ring which is infinitely generated additively. When G = /q for q prime, there is a short exact sequence

0 ⟶ ⟶ Q_{G} ⟶ Q̃_{G} ⟶ 0

where

Q̃_{G} ≅ ( /q )[x]_{(1+x)}.

At intermediate stages there is a short exact sequence

0 ⟶ ⟶ π_{0}(Z(k, 2k)^{G}) ⟶ Q̃_{G}^{(k)} ⟶ 0

and an additive isomorphism

Q̃_{G}^{(k)} ≅ ( /q )[x, x̄]/ ⟨(1 + x)^{k+1}, (1 + x̄)^{k+1}, (1 + x)(1 + x̄) − 1⟩

Another set of examples comes from of divisors. Suppose G is abelian and let V be a complex G-module of dimension n+1. Then it can be shown that Z_{n−1}( (V))_{o} = \underrightarrow{lim}_{d} Div_{d} =

\underrightarrow{lim}_{d} (S^{d}V) where S^{d}V is the d^{th} symmetric tensor power of V and Z_{n−1}( (V))_{o} denote

the divisors of degree zero in (V). As in the proof of 2.2 we see that #(π_{0} (S^{d}V)^{G}) is exactly the number of distinct characters of G which appear in the representation S^{d}V.

If G = /q and V is non-trivial, then all characters occur in S^{d}V for d > 2q. To see this note that if S^{d}V has characters a and b, then S^{d}V has all characters na + mb for non-negative integers n, m with n + m = d. One can solve the system

na + mb ≡ c(mod q)
n + m ≡ d(mod q)

for all c, and for d large we can solve the second equation in positive integers. Therefore if V has at least two distinct characters, then π_{0}Z_{n−1}( (V))^{G} ≅ /q . Thus if V is the trivial 2-dimensional module, we have

{0} = π_{0}Z_{0}( (V))^{G} ⟶^{Σ /^{V_{0}}} π_{0}Z_{q}( (V ⊕ V_{0}))^{G} = /q .

More generally one has π_{0}Z_{n+γ−1}( (V ⊕V_{0}))^{G} ≅ G, but π_{0}Z_{n+γ−1}( (V))^{G} ≠ G if say V is a representation of a non-trivial quotient. These examples show again that our hypothesis p ≥ e(V) in Theorem 2.3 is sharp.

§5. Equivariant suspension for varieties. The proof of Theorem 2.3 goes through without change if one replaces (V) with a G-invariant subvariety of (V).

Theorem 5.1. Let G be a finite group and V a finite-dimensional complex G-module. Suppose that X ⊂ (V) is a G-invariant algebraic subvariety, and set

e(X) =^{def} max_{g∈G}_{g≠{1}} dim (Σ /^{V_{0}}X)^{g}

Let V_{0} denote the regular representation of V_{0} and γ = dim V_{0} = |G|. Then for any p ≥ e(X), the suspension homomorphism

Σ /^{V_{0}} : Z_{p}(X) ⟶ Z_{p+γ}(Σ /^{V_{0}}X)

is a G-homotopy equivalence.

Note 5.2. In Theorem 5.1 we can take dim ∅ = −∞. That is, if G acts freely on X, then the equivariant suspension theorem holds for all p ≥ 0. One needs only to check the case p = 0. The proof goes through here because the action of G on (V ⊕ V′_{0}) is free outside (V′_{0}) and the subvarietes Ỹ ∩ never meet (V′_{0}) because D is chosen so that ∩ (V′_{0}) = ∅.

Proof of Theorem B. The Stability Theorem B stated in the introduction is proved as follows. There are two G-equivariant projections

π : Σ /^{V_{0}}X − (V_{0}) → X  and  π′ : Σ /^{V_{0}}X − X → (V_{0}).

If v ∈ Σ /^{V_{0}}X is fixed by g ∈ G and if v ∉ (V_{0}) ∪ X, then both π(v) and π′(v) are also gfixed and belong to the same g-character spaces of V ⊕V_{0}. We estimate e(X) by observing that

dim (Σ /^{V_{0}}X)^{g} ≤ max_{ρ} dim Σ /^{V_{ρ}}(X^{g}) = dim X^{g} + \frac{γ}{q}

where V_{ρ} denotes the subspace of V_{0} with g-character ρ. Therefore,

dim (Σ /^{(m+1)V_{0}}X)^{g} ≤ dim X^{g} + \frac{(m+1)γ}{q} ,

from which it follows that

e(Σ /^{mV_{0}}X) ≤ e_{0}(X) + \frac{(m+1)γ}{q} .

Theorem B now follows from Theorem A, which is equivalent to Theorem 5.1 above.

§6. The quaternionic suspension theorem. The proof of Theorem 2.3 given above carries over with no essential change to prove a “quaternionic suspension theorem”. The algebraic subvarieties if _{}( ^{n}) which are invariant under the involution induced by multiplication by j, have interesting geometric structure and arise naturally for example in certain twistor constructions. Families of such j-invariant cycles also play an interesting role in topology. For example, using them one can make quaternionic analogues of the constructions in [BLLMM] concerning the total Chern and Stiefel- Whitney classes. A key to computing the topological structure of these spaces are Theorems 6.3 and 6.5 below. These applications are carried out in [LLM_{2}].

Let denote the quaternions with standard basis 1, i, j, k, and let ^{2} ⟶^{≅} be the canonical isomorphism which associates to a pair (u, v) the quaternion u + vj. For any n this gives a canonical complex isomorphism

2n ⟶^{≅} ^{n}.

Under this identification left scalar multiplication by j in ^{n} becomes the complex-antilinear map

j : ^{2n} ⟶ ^{2n}

with j^{2} = −Id, given in coordinates (u, v) = (u_{1}, ..., u_{n}, v_{1}, ..., v_{n}) by

j(u, v) = (−\overline{v}, \overline{u}).

This induces a fixed-point free, antiholomorphic map

(6.0.1)  \overline{j} : _{}^{2n−1} ⟶ _{}^{2n−1}

with \overline{j}^{2} =Id. There is a natural fibration _{}^{2n−1} ⟶ _{}^{n−1} whose fibres are \overline{j}-invariant projective lines, and on these lines \overline{j} is linearly equivalent to the antipodal map.

The involution \overline{j} carries algebraic subvarieties of _{}^{2n−1} to themselves and induces a continuous, antiholomorphic involution

\overline{j}_{∗} : C_{p,d}( _{}^{2n−1}) ⟶ C_{p,d}( _{}^{2n−1})

for all p and d. This involution is additive on the monoid C_{p}( _{}^{2n−1}) = ∐_{d} C_{p,d}( _{}^{2n−1}) and extends to a continuous involutive automorphism

\overline{j}_{∗} : Z_{p}( _{}^{2n−1}) ⟶ Z_{p}( _{}^{2n−1}).

There is a \overline{j}-equivariant algebraic suspension homomorphism

(6.0.2)  Σ /^{}: Z_{p}( _{}^{2n−1}) ⟶ Z_{p+2}( _{}^{2n+1})

defined as follows. Write ^{n+1} = ^{n} ⊕ and set _{}^{2n−1} = _{}( ^{n} ⊕ {0}), _{}( ) = ({0} ⊕ ). Then given a p-cycle c on _{}^{2n−1}, we set

Σ /^{}(c) = c# _{}( ).

If V ⊂ _{}^{2n−1} is an irreducible subvariety with homogeneous cone Ṽ =^{def} ⋃{ℓ : ℓ ∈ V } ⊂ ^{n}, then the homogeneous cone of Σ /^{}(V) in ^{n+1} is just

Σ̃ /^{}(V) = Ṽ ⊕ ⊂ ^{n} ⊕ .

From this one sees that Σ /^{}commutes with \overline{j}_{∗}. Our first main result is the following

Theorem 6.1. For all p and n with 0 ≤ p < 2n − 1, the algebraic suspension map

Σ /^{}: Z_{p}( _{}^{2n−1}) ⟶ Z_{p+2}( _{}^{2n+1}),

as a map of /2 -spaces under the action of \overline{j}_{∗} is a /2 -homotopy equivalence.

Corollary 6.2. Let Z_{p}^{}(n) ⊂ Z_{p}( _{}^{2n−1}) denote the \overline{j}_{∗}-fixed point set, i.e., the group of \overline{j}_{∗}-invariant algebraic p-cycles. Then the algebraic suspension homomorphism

Σ /^{}: Z_{p}^{}(n) ⟶ Z_{p+2}^{}(n + 1)

is a homotopy equivalence.

Proof. The proof of 6.1 follows exactly the arguments given for the proof of Theorem 2.3. Certain homomorphisms of local rings which were complex linear are complex antilinear in this case. However, all dimension estimates go through without change. Since the action of G = /2 on _{}^{2n−1} is free, there are no restrictions on the dimension p. Details are left to the reader.

In analogy with [Lam] and [LM_{2}] we now consider the “Galois quotient”

Z̃_{p}^{}(n) =^{def} Z_{p}^{}(n)/A_{p}(n)

of the \overline{j}_{∗}-fixed cycles by the subgroup A_{p}(n) =^{def} (1 + \overline{j}_{∗})Z_{p}( _{}^{2n+1}) of \overline{j}_{∗}- averaged ones. Note that this is the (topological) 2-torsion group generated by the irreducible quaternionic subvarieties of dimension p. For p = 1 these subvarieties can be thought of as the “nonorientable algebraic curves”.

Corollary 6.3. For all p, n with 0 ≤ p < 2n − 1 the algebraic suspension homomorphism

Σ /^{}: Z̃_{p}^{}(n) ⟶ Z̃_{p+2}^{}(n + 1)

is a homotopy equivalence.

Proof. This is a direct consequence of the following two statements (1) Σ /^{}: A_{p}(n) ⟶ A_{p+2}(n + 1) is a homotopy equivalence.

(2) A_{p}(n) ⟶ Z_{p}^{}(n) ⟶ Z̃_{p}^{}(n) is a quasi-fibration sequence. To establish (1), one checks that the arguments for Theorem 2.3 can be carried through for the subgroup A_{p}(n). Statement (2) follows from [LLM_{1},§3.4] and [M], once one identifies Z̃_{p}^{}(n) with the homotopy quotient B(Z_{p}^{}(n), A_{p}(n), ∗).

It is useful, in light of Theorem 6.1, to introduce the following notation. Let Z_{}^{q} (n) denote the \overline{j}_{∗}-fixed cycles of complex codimension q in _{}^{2n−1}, and let Z̃_{}^{q} (n) denote its analogue for the quotient. (Thus Z_{}^{q} (n) = Z_{2n−1−q}^{}(n).) The corollaries above assert that the homotopy type of these spaces depends on q but not on n, so we abreviate the notation to Z_{}^{q} and Z̃_{}^{q}.

Theorem 6.4. For all integers q ≥ 0 there are a homotopy equivalences:

Z_{}^{2q+1} ≅ ∏_{i=0}^{q} K( , 4i) × ∏_{i=0}^{q} K( /2 , 4i + 1) and

Z̃_{}^{2q+1} ≅ pt

Proof. By Corollary 6.2 there are homotopy equivalences

Z_{}^{2q+1} ≅ Z_{0}^{}( _{}^{2q+1})  and  Z̃_{}^{2q+1} ≅ Z_{0}^{}( _{}^{2q+1})/(1 + \overline{j}_{∗})Z_{0}( _{}^{2q+1}).

Now \overline{j} acts freely on _{}^{2q+1}. Hence the group of \overline{j}_{∗}-fixed 0-cycles coincides with the group of averaged 0-cycles and is homeomorphic to Z_{0}( _{}^{2q+1}/ _{2}). By [DT] we have that

π_{∗}Z_{0}( _{}^{2q+1}/ _{2}) ≅ H_{∗}( _{}^{2q+1}/ _{2}, )

and these latter groups are easily computed from the Serre spectral sequence.

§7. Quaternionic suspension for varieties. As noted in §5, the arguments given for cycles in projective space carry over to cycles in any invariant subvariety. This gives us a the following general version of Theorem 6.1.

Theorem 7.1. Let X ⊂ _{}^{2n−1} be an algebraic subvariety which is invariant under the involution \overline{j}. (Such subvarieties can be thought of as “quaternionic”.) Then for all p with 0 ≤ p ≤ dim X, the quaternionic algebraic suspension homomorphism

Σ /^{}: Z_{p}(X) ⟶ Z_{p+2}(Σ /^{}X),

as a map of /2 -spaces under the action of \overline{j}_{∗} is a /2 -homotopy equivalence.

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