PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 2, February 1998, Pages 595–606 S 0002-9939(98)04133-1

ELEMENTARY ABELIAN 2-GROUP ACTIONS ON FLAG MANIFOLDS AND APPLICATIONS

GOUTAM MUKHERJEE AND PARAMESWARAN SANKARAN

(Communicated by Thomas Goodwillie)

Abstract. Let N_{∗} denote the unoriented cobordism ring. Let G = (ℤ/2)^{n} and let Z_{∗}(G) denote the equivariant cobordism ring of smooth manifolds with smooth G-actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold M = O(m)/(O(m_{1}) × ⋯ × O(m_{s})) is in the subalgebra generated by ⨁_{i<2^{n}} N_{i}, where m = ∑ m_{j}, and 2^{n−1} < m ≤ 2^{n}. We obtain sufficient conditions for indecomposability of an element in Z_{∗}(G). We also obtain a sufficient condition for algebraic independence of any set of elements in Z_{∗}(G). Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map Z_{d}(G) → N_{d} in dimensions 2 ≤ d < n, for n > 2, and show that they generate a polynomial subalgebra of Z_{∗}(G).

1. Introduction

Let G = (ℤ_{2})^{n}, n ≥ 2. Denote by Z_{∗}(G) the equivariant cobordism ring of (smooth) closed manifolds with smooth G-actions having finite stationary point sets [2], [1]. The cobordism class of a manifold M, along with an action ϕ of G having finite stationary point set, will be denoted by [M, ϕ]. Let R_{q}(G) denote the vector space over ℤ_{2}, with basis the set of isomorphism classes of ℝG-modules of dimension q. If R_{∗}(G) = ∑_{q≥0} R_{q}(G), then R_{∗}(G) is a graded commutative ℤ_{2}-algebra with unit. The multiplication in R_{∗}(G) is given by [V] · [W] = [V ⊕ W]. One can identify R_{∗}(G) with the graded polynomial algebra over ℤ_{2} generated by Ĝ = Hom_{ℤ_{2}}(G, ℤ_{2}) (cf. [1]).

One has an algebra homomorphism η_{∗} : Z_{∗}(G) → R_{∗}(G) where η_{∗}([M, ϕ]) = ∑[T_{x}M], the sum being taken over the (finite) set of stationary points of M. By a theorem of Stong [11], we know that η_{∗} is a monomorphism. One has the ‘forgetful’ homomorphism ε_{∗} : Z_{∗}(G) → N_{∗}, the unoriented cobordism ring, [M, ϕ] ↦ [M]. Let T_{∗}^{n} denote the subalgebra of N_{∗} generated by ⨁_{i<2^{n}} N_{i}. Then tom Dieck [3] has shown that Im ε_{∗} = T_{∗}^{n} (cf. Kosniowski and Stong, section 4 of [5]). In this paper we prove

Theorem 1.1. Let G(m_{1}, …, m_{k}) denote the flag manifold

O(m)/(O(m_{1}) × ⋯ × O(m_{k})),

m = ∑ m_{i}. Let 2^{n−1} < m ≤ 2^{n}. Then [G(m_{1}, …, m_{k})] ∈ T_{∗}^{n}.

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors July 11, 1996.

1991 Mathematics Subject Classification. Primary 57R75, 57R85.

The problem of determining which flag manifolds are unoriented boundaries was addressed in [10]. The case of Grassmann manifolds was completely settled in [9], which also considers the case of complex and quaternionic Grassmannians. See also Stong [12]. The above theorem gives perhaps the best known result in general. It is easy to see that if two of the numbers m_{1}, …, m_{k} are equal, then G(m_{1}, …, m_{k}) admits a fixed point free involution and hence bounds. If m is even, then the above theorem does not yield the best possible result (cf. Theorem 2.1 in [10]). But by remark 2.3(iii) in [10], the general problem of determining which flag manifolds bound has been reduced to the consideration of only the case m = ∑ m_{i} is odd, m_{1}, …, m_{k} being distinct.

Theorem 1.1 is proved by exhibiting certain (ℤ_{2})^{n}-actions on G(m_{1}, …, m_{k}) with finitely many stationary points.

It is well-known that when n = 2, Z_{∗}(G) is isomorphic to the polynomial algebra with one generator [ℙ^{2}, ϕ], where the action is given as follows: t_{1}([x, y, z]) = [−x, y, z], t_{2}([x, y, z]) = [x, −y, z]. Here t_{1}, t_{2} denotes a set of generators of G. Hence ε_{∗} : Z_{∗}(G) → T_{∗}^{2} is an isomorphism (cf. [1]). However, for n ≥ 3, ε_{∗} is not a monomorphism [8] and the structure of Z_{∗}(G) is not known.

A G-manifold (M, ϕ) with finite stationary point set is equivariantly indecomposable if [M, ϕ] is an indecomposable element in Z_{∗}(G). Clearly, if [M, ϕ] ∈ Z_{∗}(G) and [M] ∈ N_{∗} is indecomposable, then [M, ϕ] ∈ Z_{∗}(G) is indecomposable. An important step towards understanding the structure of Z_{∗}(G) is to know the indecomposable elements, as they generate Z_{∗}(G) as a ℤ_{2}-algebra. In section 3 we obtain a sufficient criterion for an element in Z_{∗}(G) to be indecomposable. Since T_{∗}^{n} is a polynomial algebra, and since by tom Dieck’s theorem [3] Im ε_{∗} = T_{∗}^{n}, it follows that the exact sequence 0 → ker ε_{∗} → Z_{∗}(G) → T_{∗}^{n} → 0 splits. Therefore one can clearly ‘lift’ indecomposable elements from T_{∗}^{n} to obtain indecomposable elements in Z_{∗}(G). However, we apply our indecomposability criterion to construct indecomposable elements in Z_{∗}(G) which belong to K_{∗} = ker ε_{∗} in each dimension 2 ≤ m ≤ n except possibly in dimension n when n is even. For the precise statement see Theorem 3.6. This is in striking contrast to the situation in the unoriented cobordism ring N_{∗}, where there is no generator in dimensions 2^{j} − 1. Note that indecomposable elements in K_{∗} cannot arise by ‘lifting’ indecomposable elements from N_{∗}. We also prove a sufficient criterion for a set of elements in Z_{∗}(G) to be algebraically independent and use it to show that suitable indecomposable elements in K_{∗} generate a polynomial subalgebra of Z_{∗}(G). In the last section we generalize a result of Conner and Floyd regarding the number of stationary points x such that an irreducible representation occurs at the tangential representation T_{x}M with a given multiplicity.

2. Action of (ℤ_{2})^{n} on flag manifolds

Let 2 ≤ m ≤ 2^{n}, where n ≥ 2. In this section we shall exhibit certain G = (ℤ_{2})^{n} actions on the flag manifold G(m_{1}, …, m_{k}) ≅ O(m)/(O(m_{1}) × ⋯ × O(m_{k})), ∑ m_{i} = m. Indeed this is a routine generalization of the actions of G on projective spaces considered by tom Dieck in [3]. We recall the action of G on the Milnor manifold also considered by tom Dieck [3] as it will be needed in section 3.

Let \underline{n} = \{1, 2, …, n\}. We regard \{e_{α} | α ⊂ \underline{n}\} as the ‘standard basis’ of ℝ^{2^{n}}, with its usual innerproduct. For 1 ≤ i ≤ n, let t_{i} : ℝ^{2^{n}} → ℝ^{2^{n}} be the ℝ-linear map defined by

t_{i}(e_{α}) = \begin{cases}−e_{α} & if i ∈ α, \\ e_{α} & if i ∉ α. \end{cases}

Then it is readily checked that t_{i}^{2} = Id, and t_{i}t_{j} = t_{j}t_{i} for 1 ≤ i, j ≤ n. Therefore we obtain a linear action of G on ℝ^{2^{n}}.

Lemma 2.1. (i) The decomposition ℝ^{2^{n}} = ∑_{α⊂\underline{n}} ℝe_{α} expresses ℝ^{2^{n}} as a sum of mutually non-isomorphic irreducible G-submodules of ℝ^{2^{n}}.

(ii) If V is any G-submodule of ℝ^{2^{n}}, then V = ∑_{α∈S} ℝe_{α} for some S ⊂ P(\underline{n}) with dim V = #S.

Proof of (i). It is clear that each ℝe_{α} is a G-submodule. We only have to prove that if α ≠ β, then ℝe_{α} ≇ ℝe_{β}. Let α ≠ β, and let i ∈ αΔβ. Say, i ∈ α and i ∉ β. Then t_{i}e_{α} = −e_{α}, and t_{i}e_{β} = e_{β}. This shows that ℝe_{α} cannot be isomorphic to ℝe_{β} as G-modules. Part (ii) is an immediate consequence of part (i).

Let S ⊂ P(\underline{n}), #S = m. Let V = ∑_{α∈S} ℝe_{α} ⊂ ℝ^{2^{n}}. Since the G-action on ℝ^{2^{n}} preserves innerproduct and since V is a G-submodule, we obtain a G-action on any flag manifold G(m_{1}, …, m_{k}), ∑ m_{j} = m, modelled on V. Explicitly, if (A_{1}, …, A_{k}) ∈ G(m_{1}, …, m_{k}), and t ∈ G, then t(A_{1}, …, A_{k}) = (tA_{1}, …, tA_{k}). Clearly (A_{1}, …, A_{k}) is a stationary point if and only if each A_{j}, 1 ≤ j ≤ k, is a G-submodule of V. We conclude from Lemma 2.1 that there are only finitely many stationary points. We shall denote this action on G(m_{1}, …, m_{k}) by ϕ_{S} or simply by ϕ when there is no risk of confusion. Thus [G(m_{1}, …, m_{k}), ϕ_{S}] ∈ Z_{∗}(G) for every S ⊂ P(\underline{n}), #S = ∑ m_{j}. In the case of a Grassmann manifold G_{m,k} with G-action ϕ_{S}, the stationary points are E_{α} = ⟨e_{α_{1}}, …, e_{α_{k}}⟩, the span of e_{α_{1}}, …, e_{α_{k}} where α = {α_{1}, …, α_{k}} is any k-element subset of S.

Next, we recall the action of G on Milnor manifolds considered by tom Dieck [3]. Let S ⊂ P(\underline{n}), #S = k + 1, and let T ⊂ S, #T = l + 1. Then we can form the product (P^{l} × P^{k}, ϕ_{T} × ϕ_{S}). Let H_{l,k} be the Milnor manifold

{([∑_{α∈T} x_{α}e_{α}] , [∑_{β∈S} y_{β}e_{β}]) | ∑_{α∈T} x_{α}y_{α} = 0} .

Then H_{l,k} is a G-stable submanifold of (P^{l} × P^{k}, ϕ_{T} × ϕ_{S}). It is obvious that there are only finitely many stationary points for the G-action on H_{l,k}. We denote this action on H_{l,k} by ϕ_{T,S} or simply by ϕ.

Proof of Theorem 1.1. Since M = G(m_{1}, …, m_{k}), m = ∑ m_{i} ≤ 2^{n}, was shown to admit a G = (ℤ_{2})^{n} action with finite stationary point set, it follows from the result of tom Dieck [3] that [M] ∈ T_{∗}^{n}. This completes the proof.

Example 2.2. Take M = G(40, 42, 45). Then the dimension of M is 5370. The above theorem says that [M] ∈ T_{∗}^{7} = ℤ_{2}[x_{2}, x_{4}, …, x_{126}].

Let 1 ≤ k < m ≤ 2^{n}. For any S ⊂ P(\underline{n}), #S = m, we obtain an element [G_{m,k}, ϕ_{S}] in Z_{∗}(G), G = (ℤ_{2})^{n}. In general distinct choices of S do not necessarily lead to distinct elements [G_{m,k}, ϕ_{S}] in Z_{∗}(G). In fact we have the following rather amusing example.

Example 2.3. Let m = 2^{n} − 2, 1 ≤ k < m, k odd. Then for any S ⊂ P(\underline{n}), #S = m, one has [G_{m,k}, ϕ_{S}] = 0 in Z_{∗}(G).

Proof. First note that P(\underline{n}) has the structure of a Boolean algebra where addition is given by symmetric difference and multiplication, by intersection. For this reason, we write “λ + μ” to mean λΔμ in the proof. Let S ⊂ P(\underline{n}), #S = 2^{n} − 2 = m. Let α, β be elements of P(\underline{n}) not in S. Let γ = α + β. Then γ ≠ ∅. It is easy to see that f : δ ↦ δ + γ defines a bijection of S onto itself such that f^{2} = Id, and f is fixed point free. It follows that f induces a bijection from the set \binom{S}{k} of k-element subsets of S to itself, denoted by F, where F ({α_{1}, …, α_{k}}) = {α_{1} + γ, …, α_{k} + γ}. Since k is odd, and f : S → S is fixed point free, it follows that F is fixed point free.

Recall that the stationary points for the G-action ϕ_{S} on G_{m,k} are

E_{α} = ⟨e_{α_{1}}, …, e_{α_{k}}⟩,

where α = {α_{1}, …, α_{k}} ⊂ S, #α = k. From Lam’s [6] description of the tangent bundle of Grassmannians, it is easy to see that the tangential representation T_{α}G_{m,k} of G at E_{α} is isomorphic to

⨁_{\begin{matrix}1≤i≤k \\ 1≤j≤m−k \end{matrix}} ℝe_{α_{i}} ⊗ ℝe_{β_{j}} ≅ ⨁_{\begin{matrix}1≤i≤k \\ 1≤j≤m−k \end{matrix}} ℝe_{α_{i}+β_{j}},

where S\α = {β_{1}, …, β_{m−k}}. Since α_{i} + β_{j} = α_{i} + γ + β_{j} + γ = f(α_{i}) + f(β_{j}) for any α_{i}, β_{j} ∈ S, it is seen that T_{α}G_{m,k} ≅ T_{F (α)}G_{m,k}, as G-modules. Since F has no fixed point, η_{∗}[G_{m,k}, ϕ_{S}] = 0. By Stong’s theorem [11], it follows that [G_{m,k}, ϕ_{S}] = 0.

3. Indecomposability

In this section we obtain a sufficient condition for indecomposability of an element in Z_{∗}(G), G = (ℤ_{2})^{n}. We apply our criterion to show the existence of indecomposable elements in the kernel K_{∗} of the forgetful homomorphism ε_{∗} : Z_{∗}(G) → N_{∗}. We make use of the equivariant characteristic numbers of a G-manifold constructed by tom Dieck [4].

Let B = ℤ_{2}[b_{1}, …, b_{m}, …] be the graded ℤ_{2}-algebra with deg b_{m} = m, for m ≥ 1. Let L_{∗} denote the B-algebra B[[y_{1}, …, y_{n}]] of formal power series in y_{1}, …, y_{n} with deg y_{i} = 1, 1 ≤ i ≤ n. Let K_{∗} denote the fraction field of L_{∗}. Recall that R_{∗}(G) is the polynomial algebra over Ĝ = Hom_{ℤ_{2}}(G, ℤ_{2}).

As usual we denote the generators of G by t_{1}, …, t_{n}. For A ⊂ \underline{n}, denote by Y_{A} the irreducible representation of G given by the character χ_{A} ∈ Ĝ. Here

χ_{A}(t_{i}) = \begin{cases}1 & if i ∈ A, \\ 0 & if i ∉ A. \end{cases}

Thus R_{∗}(G) ≅ ℤ_{2}[Y_{A} | A ⊂ \underline{n}]. One has the subalgebra R̃_{∗}(G) of R_{∗}(G) generated by \{Y_{A} | A ⊂ \underline{n}, A ≠ ∅\}. Note that the algebra map η_{∗} : Z_{∗}(G) → R_{∗}(G) actually has image in R̃_{∗}(G), and that η_{∗} is well known to be a monomorphism [11].

Let γ : R̃_{∗}(G) → K_{∗} denote the ℤ_{2}-algebra homomorphism defined by γ(Y_{A}) = \frac{1}{y_{A}} ∑_{r≥0} b_{r}y_{A}^{r} ∈ K_{∗}, where b_{0} = 1, y_{A} = ∑_{i∈A} y_{i}. Then tom Dieck [4] shows that γ ∘ η is a monomorphism and that Im(γ ∘ η_{∗}) is contained in L_{∗}.

Let λ = λ_{1}, …, λ_{r}, r ≥ 0, be a non-increasing sequence of natural numbers. We denote by b_{λ} the product b_{λ_{1}}b_{λ_{2}} ⋯ b_{λ_{r}} ∈ B (b_{λ} = 1 if λ = ( ), the empty sequence). For any class [M, ϕ] ∈ Z_{d}(G), one can express γη([M, ϕ]) as ∑_{|λ|≥0} ψ_{λ}([M, ϕ]) · b_{λ}, with ψ_{λ}([M, ϕ]) ∈ ℤ_{2}(y_{1}, …, y_{n}) being a homogeneous element of degree |λ| − d, where |λ| = ∑_{j} λ_{j}. The ‘integrality theorem’ of tom Dieck [4] implies that ψ_{λ}([M, ϕ]) = 0 if |λ| < d, and that ψ_{λ}([M, ϕ]) ∈ ℤ_{2}[y_{1}, …, y_{n}] if |λ| ≥ d. One can regard ψ_{λ} as defining a ℤ_{2}-linear map ψ_{λ} : Z_{d}(G) → ℤ_{2}[y_{1}, …, y_{n}]. When |λ| = d, ψ_{λ}([M, ϕ]) coincides with the Stiefel-Whitney s-numbers s_{λ}[M] ∈ ℤ_{2} (cf. Kosniowski-Stong [5]). Just as the nonvanishing of s_{d}(M) implies the indecomposability of [M] in N_{∗}, one can expect that ψ_{k}([M, ϕ]), k any integer greater than d, may detect the indecomposability of [M, ϕ] ∈ Z_{∗}(G). In fact we have the following

Proposition 3.1. Let [M, ϕ] ∈ Z_{d}(G). Suppose that for some k > d, either ψ_{k}([M, ϕ]) ≠ 0 or ψ_{k−1,1}([M, ϕ]) ≠ 0; then [M, ϕ] ∈ Z_{∗}(G) is indecomposable.

We need the following lemma to prove the above proposition.

Lemma 3.2. Let [M, ϕ] = [M_{1}, ϕ_{1}] · [M_{2}, ϕ_{2}] ∈ Z_{d}(G), and let λ = λ_{1}, …, λ_{r}. Then

ψ_{λ}([M, ϕ]) = ∑_{λ=μ·ν} ψ_{μ}([M_{1}, ϕ_{1}]) · ψ_{ν}([M_{2}, ϕ_{2}])

where μ · ν denotes juxtaposition of μ and ν arranged in the non-increasing order.

Proof of Lemma 3.2. This is a straightforward consequence of the fact that γ is an algebra homomorphism. Indeed we have

∑_{λ} ψ_{λ}([M, ϕ])b_{λ} = γ([M, ϕ]) = γ([M_{1}, ϕ_{1}]) · γ([M_{2}, ϕ_{2}])

= (∑_{μ} ψ_{μ}([M_{1}, ϕ_{1}])b_{μ}) (∑_{ν} ψ_{ν}([M_{2}, ϕ_{2}])b_{ν})

= ∑_{μ} ∑_{ν} ψ_{μ}([M_{1}, ϕ_{1}]) · ψ_{ν}([M_{2}, ϕ_{2}])b_{μ}b_{ν}

= ∑_{λ} { ∑_{μ·ν=λ} ψ_{μ}([M_{1}, ϕ_{1}]) · ψ_{ν}([M_{2}, ϕ_{2}])} b_{λ}.

Hence, comparing the coefficients of b_{λ}, we obtain

ψ_{λ}([M, ϕ]) = ∑ ψ_{μ}([M_{1}, ϕ_{1}]) · ψ_{ν}([M_{2}, ϕ_{2}]) ∈ ℤ_{2}[y_{1}, …, y_{n}].

Proof of Proposition 3.1. Let λ = k − 1, 1. If λ = μ · ν, then either μ = 1 of ν = 1. Note that Z_{1}(G) = 0, so that for any [N, θ] ∈ Z_{∗}(G), ψ_{1}([N, θ]) = 0. By the above lemma it follows that ψ_{k−1,1} vanishes on any decomposable element of Z_{∗}(G). Therefore [M, ϕ] is indecomposable if ψ_{k−1,1}([M, ϕ]) ≠ 0. The proof for the case λ = k is similar.

The following three examples will be used in the proof of Theorem 3.6.

Example 3.3. Let 3 ≤ d = 2^{k} − 1 ≤ n. Let S = {∅, {1}, {2}, …, {d}}. Then σ = [P^{d}, ϕ_{S}] is indecomposable in Z_{∗}(G). (Note that [P^{d}] = 0 in N_{∗}.)

We prove this by showing that ψ_{d+2}([P^{d}, ϕ_{S}]) ≠ 0. Let us write Y_{i} for Y_{{i}}, and Y_{ij} for Y_{{i,j}}, 1 ≤ i, j ≤ d. A straightforward calculation shows that

η_{∗}(σ) = ∑_{i=1}^{d} ( ∏_{j=1,j≠i}^{d} Y_{ij}) Y_{i} + ∏_{i=1}^{d} Y_{i}.

Let D = ∏_{k=1}^{d} y_{k} ∏_{1≤p<q≤d}(y_{p} + y_{q}), and for 1 ≤ i ≤ d, let

N_{i} = ∑_{p=1}^{d} y_{p}^{d+2} + ∑_{p=1,p≠i}^{d}(y_{i}^{d+1}y_{p} + y_{p}^{d+1}y_{i}), C_{i} = ∏_{p=1,p≠i}^{d} y_{p} ∏_{\begin{matrix}1≤p<q≤d \\ p≠i,q≠i \end{matrix}}(y_{p} + y_{q}),

N_{0} = ∑_{p=1}^{d} y_{p}^{d+2}, C_{0} = ∏_{1≤p<q≤d}(y_{p} + y_{q}).

Then ψ_{d+2}(σ) = \frac{1}{D} (∑_{j=0}^{d} N_{j}C_{j}) and so

Dψ_{d+2}(σ) = ∑_{j=0}^{d} N_{j}C_{j}. (1)

One can show that the monomial y_{1}^{d+1}y_{2}^{d}y_{3}^{d−2} ⋯ y_{j}^{d−j+1} ⋯ y_{d} occurs exactly once on the right hand side of (1). In fact it occurs in N_{1}C_{1} exactly once and in no other N_{j}C_{j} (j ≠ 1). Hence ψ_{d+2}(σ) ≠ 0.

Example 3.4. Let d ≤ n, d even. As before, let S = {∅, {1}, …, {d}}. Let σ = [P^{d}, ϕ_{S}] ∈ Z_{∗}(G). Note that σ is indecomposable, since ε_{∗}(σ) = [P^{d}] ∈ N_{∗} is indecomposable. We claim that ψ_{N}([P^{d}, ϕ_{S}]) ≠ 0 for N = 2^{m}, m sufficiently large.

To see this, note that

η_{∗}(σ) = ∑_{i=1}^{d} ( ∏_{j=1,j≠i}^{d} Y_{ij}) Y_{i} + ∏_{j=1}^{d} Y_{j}.

Let D = ∏_{p=1}^{d} y_{p} ∏_{1≤p<q≤d}(y_{p} + y_{q}) and for 1 ≤ i ≤ d, let

N_{i} = y_{i}^{N} + ∑_{p=1,p≠i}^{d}(y_{i} + y_{p})^{N} = ∑_{p=1,p≠i}^{d} y_{p}^{N},

as d is even and N is a power of 2, and

C_{i} = ∏_{p=1,p≠i}^{d} y_{p} ∏_{\begin{matrix}1≤p<q≤d \\ p≠i,q≠i \end{matrix}}(y_{p} + y_{q}).

Let N_{0} = ∑_{p=1}^{d} y_{p}^{N}, and C_{0} = ∏_{1≤p<q≤d}(y_{p} + y_{q}). Then D · ψ_{N}(σ) = ∑_{j=0}^{d} N_{j}C_{j}. One can show that for N sufficiently large—say N > \binom{d}{2}—the monomial y_{1}^{d−1}y_{2}^{d−2} ⋯ y_{d−1}y_{d}^{N} occurs in N_{0}C_{0} exactly once and does not occur in any other N_{j}C_{j}. Hence the aforementioned monomial survives in ψ_{N}(σ).

Example 3.5. Let S = {∅, {1}, …, {k}}, T = {∅, {1}, …, {l}}, l < k ≤ n, k, l both even. Recall that one has an action ϕ_{T,S} of G on the Milnor manifold H_{l,k}. Let σ = [H_{l,k}, ϕ_{T,S}] ∈ Z_{∗}(G). Further assume that \binom{k+l}{k} ≡ 1 (mod 2), so that [H_{l,k}] ∈ N_{∗} is indecomposable. We claim that for N = 2^{ν} sufficiently large, ψ_{N+1}(σ) ≠ 0.

To establish our claim, first note that the normal bundle to the imbedding H_{l,k} ↪^{i} P^{l} × P^{k} is i^{∗}(ξ_{l} ⊗ ξ_{k}), where ξ_{l} is the pull-back of the Hopf line bundle over P^{l} via the projection P^{l} × P^{k} → P^{l}. Using this it is easy to see that

η_{∗}(σ) = ∑_{\begin{matrix}0≤i≤l \\ 0≤j≤k \\ i≠j \end{matrix}} ( ∏_{p=0,p≠i}^{l} Y_{ip} ∏_{q=0,q≠i,j}^{k} Y_{jq}) ,

with the convention that Y_{0p} = Y_{{p}} = Y_{p0}. Let D = ∏_{p=1}^{k} y_{p} ∏_{1≤p<q≤k}(y_{p} + y_{q}). For 1 ≤ i ≤ l, 1 ≤ j ≤ k, and i ≠ j, write

N_{ij} = ∑_{p=l+1}^{k} y_{p}^{N+1} + ∑_{p=1,p≠i}^{l}(y_{i}^{N}y_{p} + y_{i}y_{p}^{N}) + ∑_{q=1,q≠i,j}^{k}(y_{j}^{N}y_{q} + y_{j}y_{q}^{N}),

C_{ij} = ∏_{p=1,p≠i,j}^{k} y_{p} · ∏_{\begin{matrix}1≤p<q≤k \\ q≠i,j, p≠i,j \end{matrix}}(y_{p} + y_{q}) · ∏_{p=l+1}^{k}(y_{i} + y_{p}).

For 1 ≤ j ≤ k, let

N_{0j} = ∑_{p=l+1}^{k} y_{p}^{N+1} + ∑_{p=1,p≠j}^{k}(y_{j}^{N}y_{p} + y_{j}y_{p}^{N}),

C_{0j} = ∏_{\begin{matrix}1≤p<q≤k \\ p≠j, q≠j \end{matrix}}(y_{p} + y_{q}) · ∏_{p=l+1}^{k} y_{p}.

For 1 ≤ i ≤ l, let

N_{i,0} = ∑_{p=l+1}^{k} y_{p}^{N+1} + ∑_{p=1,p≠i}^{l}(y_{i}^{N}y_{p} + y_{i}y_{p}^{N}),

C_{i0} = ∏_{\begin{matrix}1≤p<q≤k \\ p≠i,q≠i \end{matrix}}(y_{p} + y_{q}) · ∏_{q=l+1}^{k}(y_{i} + y_{q}).

Then a routine verification shows that

Dψ_{N+1}(σ) = ∑_{\begin{matrix}0≤i≤l \\ 0≤j≤k \\ i≠j \end{matrix}} N_{ij}C_{ij}. (2)

Consider the monomial y_{1}y_{2}^{k−2}y_{3}^{k−3} ⋯ y_{l−1}^{k−l+1}y_{l}^{k−l+1}y_{l+1}^{k−l} ⋯ y_{k−2}^{3}y_{k−1}^{2}y_{k}^{N}. We claim that this monomial survives in the right hand side of (2) for sufficiently large N—say N > \binom{k}{2}, N = 2^{ν}. First note that since N is large, the only monomials which involve y_{k}^{N} must come from the terms N_{ik}C_{ik}, 1 ≤ i ≤ l. In N_{ik} the only terms which contribute to the above monomial are y_{k}^{N}(∑k−1 ; \begin{matrix}p=1 \\ p≠i \end{matrix} y_{p}). It is not difficult

to show that the above monomial occurs (l−1)-times in y_{k}^{N}(∑_{p=2}^{k−1} y_{p})C_{1k}, and that it does not occur in y_{k}^{N}(∑k−1 ; \begin{matrix}p=1 \\ p≠i \end{matrix} y_{p})C_{ik} for i > 1. Since l is even, it follows that the

above monomial survives in Dψ_{N+1}(σ). Hence ψ_{N+1}(σ) ≠ 0.

Let D denote the ideal of decomposable elements in Z_{∗}(G). Let \overline{K}_{m} denote the ℤ/2-vector space K_{m}/(K_{m}∩D). The dimension of \overline{K}_{m} is the number of ℤ/2-linearly independent indecomposable elements in K_{m}.

Theorem 3.6. Let 2^{s−1} < n ≤ 2^{s}, n ≥ 3. Let d = dim_{ℤ/2} \overline{K}_{m}. Then (i) d ≥ n − 2^{r} + 2, when m = 2^{r} − 1, m ≤ n, (ii) d ≥ n − 2^{r}, when 2^{r} < m + 1 < 2^{r+1} ≤ 2^{s}, and m is odd, (iii) d ≥ n − m, when m is even, 2 ≤ m < n.

Proof. We shall only prove (ii), proofs for other parts being exactly analogous, and make use of Examples 3.3 and 3.4. Let m be odd and 2^{r} < m + 1 < 2^{r+1}, r ≤ s − 1. Then [H_{l,k}] ∈ N_{∗} is indecomposable where k = 2^{r}, l = m + 1 − k. Note that l ≥ 2, and that k < n. Now let σ = [H_{l,k}, ϕ_{S,T}] be as in Example 3.5. By Example 3.5, ψ_{N+1}(σ) ∈ ℤ_{2}[y_{1}, …, y_{k}] is not zero. Suppose that ψ_{N+1}(σ) = ∑_{r≥0} P_{r}y_{t}^{r}, with 1 ≤ t ≤ k, P_{r} ∈ ℤ_{2}[y_{1}, …, y_{t−1}, y_{t+1}, …, y_{k}], is the expression for ψ_{N+1}(σ) as a polynomial in y_{t} with positive degree.

Now, for 1 ≤ j ≤ n − 2^{r}, let A_{j} = {∅, {1}, …, {t − 1}, {t + 1}, …, {k}, {k + j}}, and let

B_{j} = \begin{cases}T & if l < t, \\ T ∪ \{k + j\}\\{t\} & if t ≤ l. \end{cases}

Then, writing σ_{j} = [H_{l,k}, ϕ_{A_{j},B_{j}}], we see that ψ_{N+1}(σ_{j}) = ∑_{r≥0} P_{r}y_{k+j}^{r}. Therefore, ψ_{N+1}(σ + σ_{j}) = ∑ P_{r}(y_{t}^{r} + y_{k+j}^{r}) ≠ 0. Hence by Proposition 3.1, it follows that σ + σ_{j} is indecomposable. Clearly ε_{∗}(σ + σ_{j}) = 2[H_{l,k}] = 0. Therefore σ + σ_{j} ∈ K_{m}. Writing u_{j} = σ + σ_{j}, for 1 ≤ j ≤ n − 2^{r}, we see that for any sequence of 1 ≤ j_{1} < ⋯ < j_{p} ≤ n−2^{r}, one has ψ_{N+1}(u_{j_{1}} +⋯+u_{j_{p}}) = ∑_{1≤q≤p} ψ_{N+1}(u_{j_{q}}) = ∑ P_{r}(py_{t}^{r} + ∑_{1≤q≤p} y_{k+j_{q}}^{r}) ≠ 0. This proves that u_{1}, …, u_{n−2^{r}} are linearly independent in \overline{K}_{m}, completing the proof.

Remark 3.7. Note that in the course of the above proof one could as well work with the smallest integer M such that ψ_{M}(σ) ≠ 0 in the place of the integer N + 1.

The above theorem suggests the following conjecture. Note that the group Aut(ℤ_{2})^{n} ≅ SL_{n}(ℤ_{2}) acts on Z_{∗}(G) as ℤ_{2}-algebra automorphisms. Indeed if w ∈ SL_{n}(ℤ_{2}), then w([M, ϕ]) = [M, ϕ^{w}], where ϕ^{w}(t, x) = ϕ(w(t), x) for all x ∈ M. In particular, let σ ∈ Z_{d}(G) be indecomposable. Then so is w(σ), and σ + w(σ) ∈ K_{d}.

Conjecture. If w(σ) ≠ σ, then σ + w(σ) is an indecomposable element in K_{d}.

Let u_{1}, u_{2}, …, u_{k} be homogeneous elements in Z_{∗}(G). Assume that there exists an integer N_{i} such that f_{i} := ψ_{N_{i}}(u_{i}) ≠ 0. We may suppose that N_{i} is the smallest such integer. By relabelling the u_{i}’s if necessary, we assume that N_{1} ≤ N_{2} ≤ ⋯ ≤ N_{k}. Let i_{0} = 0 < i_{1} < ⋯ < i_{p} = k be such that N_{j} = N_{i_{r}} for i_{r−1} + 1 ≤ j ≤ i_{r}, 1 ≤ r ≤ p.

Proposition 3.8. With the above notation, assume that f_{1}, …, f_{k} ∈ ℤ_{2}[y_{1}, …, y_{n}] are algebraically independent. Then u_{1}, …, u_{k} are algebraically independent.

Proof. Suppose that P (u_{1}, …, u_{k}) = 0 where P is a homogeneous polynomial of degree d. Then we can write P as P = ∑ ε\underline{r}u_{1}^{r_{1}} ⋯ u_{k}^{r_{k}} where the sum is over all sequences \underline{r} = (r_{1}, …, r_{k}) ∈ ℕ^{k} with ∑_{i=1}^{k} r_{i}|u_{i}| = d, ε\underline{r} ∈ \{0, 1\}, ℕ being the set of non-negative integers. Let |\underline{r}| = r_{1} + ⋯ + r_{k}. Let S = \{\underline{r} | ε\underline{r} = 1\}, and let s = min\{|\underline{r}| | \underline{r} ∈ S\}. Note that P = ∑_{\underline{r}∈S} u_{1}^{r_{1}} ⋯ u_{k}^{r_{k}}. For \underline{r} ∈ S, let \underline{r}^{j} = r_{i_{j−1}+1} + ⋯ + r_{i_{j}}, 1 ≤ j ≤ p. Let S_{p} = \{\underline{r} ∈ S | |\underline{r}| = s\}, and let s_{p} = min\{\underline{r}^{p} | \underline{r} ∈ S_{p}\}. Similarly, let S_{p−1} = \{\underline{r} ∈ S_{p} | \underline{r}^{p} = s_{p}\}, and set s_{p−1} = min\{\underline{r}^{p−1} | \underline{r} ∈ S_{p−1}\}. Having defined S_{p}, S_{p−1}, …, S_{p−j+1} and s_{p}, s_{p−1}, …, s_{p−j+1}, we define S_{p−j} and s_{p−j} as S_{p−j} = \{\underline{r} ∈ S_{p−j+1} | \underline{r}^{p−j+1} = s_{p−j+1}\} and s_{p−j} = min\{\underline{r}^{p−j} | \underline{r} ∈ S_{p−j}\}. Note that S_{1} consists precisely of those \underline{r} ∈ S_{p} such that the sequence (\underline{r}^{p}, …, \underline{r}^{1}) = (s_{p}, …, s_{1}) is the smallest in the lexicographic ordering of ℕ^{p}, as \underline{r} varies in S_{p}. Now let

λ = N_{i_{1}}^{s_{1}} … N_{i_{p}}^{s_{p}} = \underbrace{N_{i_{1}}, …, N_{i_{1}}}_{s_{1}}, ⋯, \underbrace{N_{i_{p}}, …, N_{i_{p}}}_{s_{p}} .

Claim. For \underline{r} ∈ S,

ψ_{λ}(u_{1}^{r_{1}} ⋯ u_{k}^{r_{k}}) = \begin{cases}f_{1}^{r_{1}} ⋯ f_{k}^{r_{k}} & if \underline{r} ∈ S_{1}, \\ 0 & otherwise. \end{cases}

It is easily seen that ψ_{λ}(u_{1}^{r_{1}} ⋯ u_{k}^{r_{k}}) = 0 if |\underline{r}| > s = |λ|. Also it is trivial to see that ψ_{N_{i_{j}}^{s_{j}}} (u_{i_{j−1}+1}^{r_{j−1}+1} ⋯ u_{i_{j}}^{r_{j}}) = f_{i_{j−1}+1}^{r_{j−1}+1} ⋯ f_{i_{j}}^{r_{j}}, 1 ≤ j ≤ p. Furthermore

ψ_{M_{1}···M_{q}}(u_{i_{j−1}+1}^{r_{1}} ⋯ u_{i_{j}}^{r_{q}}) = 0, q = i_{j} − i_{j−1}, if some M_{t} < N_{i_{j}} by minimality of N_{j}. By Lemma 3.2 the claim follows. To complete the proof of the lemma, we apply ψ_{λ} to P (u_{1}, …, u_{k}) = 0. By the claim above

0 = ψ_{λ}(P (u_{1}, …, u_{k})) = ∑_{\underline{r}∈S_{1}} ψ_{λ}(u_{1}^{r_{1}} ⋯ u_{k}^{r_{k}}) = ∑_{\underline{r}∈S_{1}} f_{1}^{r_{1}} ⋯ f_{k}^{r_{k}}

which contradicts the hypothesis that the f_{i}’s are algebraically independent.

Let u_{1}, …, u_{r} be homogeneous elements in K_{∗} constructed as in the proof of Theorem 3.6. By Remark 3.7 there exist M_{i} smallest such that ψ_{M_{i}}(u_{i}) ≠ 0. Write ψ_{M_{i}}(u_{i}) = P_{i}(y_{1}, …, y_{k_{i}}) where P_{i} involves y_{k_{i}}.

Theorem 3.9. With the above notation, suppose u_{1}, …, u_{r} ∈ K_{∗} are such that 1 ≤ k_{1}, …, k_{r} ≤ n are all distinct. Then u_{1}, …, u_{r} are algebraically independent.

Proof. The theorem follows immediately from the above proposition as ψ_{M_{i}}(u_{i}) = P_{i}(y_{1}, …, y_{k_{i}}), i = 1, …, r, are clearly algebraically independent.

In particular, it follows from Theorem 3.6 that K_{m} has at least r elements which generate a polynomial algebra on r variables where

(i) r = n − 2^{q} + 2, when m = 2^{q} − 1, m ≤ n,

(ii) r = n − 2^{q}, when 2^{q} < m + 1 < 2^{q+1} ≤ 2^{s}, and m is odd,

(iii) r = n − m, when m is even, 2 ≤ m < n.

We conclude this section with the following questions. Let as usual G = (ℤ_{2})^{n}.

Question 1. Is Z_{∗}(G) finitely generated as a ℤ_{2}-algebra?

Question 2. Are there indecomposable elements in Z_{∗}(G) beyond dimension 2^{n} − 2?

4. Tangential representations

An important step in determining the structure of Z_{∗}(ℤ_{2}^{2}) is the following observation of Conner and Floyd (Lemma 32.3, [1]). Let (M, ϕ) be a smooth closed ℤ_{2} × ℤ_{2}-manifold, and let α ⊂ {1, 2}. If Y_{α} occurs in the representation T_{x}M with multiplicity p, at some stationary point x. Then it occurs with the same multiplicity at an even number of stationary points. We shall show in this section that this phenomenon happens for any (ℤ_{2})^{n}-manifold with finite stationary point set.

Theorem 4.1. Let n ≥ 2 and (M, ϕ) be a closed (ℤ_{2})^{n}-manifold with finite stationary point set. Suppose that for some stationary point x, and α ⊂ \underline{n}, Y_{α} occurs with multiplicity p in T_{x}M. Then it occurs with the same multiplicity at an even number of stationary points.

Proof. Let d = dim M. We use the notation of section 3. Let V = Y_{α_{1}}^{p_{1}} ⋯ Y_{α_{k}}^{p_{k}}, ∑ p_{i} = d, α_{1}, …, α_{k} distinct. Then

γ(V) = \frac{1}{y_{α_{1}}^{p_{1}} ⋯ y_{α_{k}}^{p_{k}}} (1 + b_{1}y_{α_{1}} + b_{2}y_{α_{1}}^{2} + ⋯)^{p_{1}} ⋯ (1 + b_{1}y_{α_{k}} + b_{2}y_{α_{k}}^{2} + ⋯)^{p_{k}}.

Let r < d. The coefficient C_{d−r}(V) of b_{1}^{d−r} in the expression for γ(V) can be calculated as follows: If p_{1} =: p ≥ r, then

C_{d−r}(V) = \frac{\binom{p}{r}}{y_{α_{1}}^{r}} + P_{α_{1}}(V)

where P_{α_{1}}(V) is a polynomial in (1/y_{α_{1}}) of degree less than r with coefficients in ℤ_{2}[1/y_{α_{j}} | 2 ≤ j ≤ k]. If p < r, then C_{d−r}(V) = Q_{α_{1}}(V), a polynomial in (1/y_{α_{1}}) of degree less than r with coefficients in ℤ_{2}[1/y_{α_{j}} | j ≠ 1]. Let S_{α,r} = {x ∈ S | Y_{α} occurs with multiplicity r}. Fix α = α_{1}. We need to show that |S_{α,r}| is even for each r, 1 ≤ r ≤ d. Now suppose that r = s is the largest integer such that Y_{α} occurs with multiplicity s, among T_{x}M, as x varies over the (finite) set S of stationary points. One knows that s < d. Then

0 = \underbrace{ψ_{1,...,1}}_{d−s}(M, ϕ) = ∑_{x∈S} C_{d−s}(T_{x}M). (3)

Multiplying both sides of (3) by y_{α}^{s} we obtain

0 = ∑_{x∈S_{α,s}}(1 + y_{α}^{s}P_{α}(T_{x}M)) + ∑_{x∈S\S_{α,s}} y_{α}^{s}Q_{α}(T_{x}M). (4)

Note that y_{α}^{s}P_{α}(T_{x}M), y_{α}^{s}Q_{α}(T_{x}M) are polynomials in y_{α} without constant terms having coefficients in ℤ_{2}[1/y_{β} | β ≠ α]. Clearing the denominators in (4), we obtain an expression

0 = ( ∑_{x∈S_{α,s}} 1) y_{β_{1}}^{k_{1}} ⋯ y_{β_{t}}^{k_{t}} + ( ∑_{x∈S_{α,s}} y_{α}^{s}P_{α}(T_{x}M)) y_{β_{1}}^{k_{1}} ⋯ y_{β_{t}}^{k_{t}}

+ ( ∑_{x∈S\S_{α,s}} y_{α}^{s}Q_{α}(T_{x}M)) y_{β_{1}}^{k_{1}} ⋯ y_{β_{t}}^{k_{t}}

in ℤ_{2}[y_{1}, …, y_{n}], where β_{1}, …, β_{t} ∈ P(\underline{n}), α ≠ β_{i}, 1 ≤ i ≤ t, k_{1}, …, k_{t} > 0. Put y_{α} = 0 in the above expression. Then we obtain 0 = |S_{α,s}| · y_{β_{1}}^{k_{1}} ⋯ y_{β_{t}}^{k_{t}} in ℤ_{2}[y_{1}, …, y_{n}]/⟨y_{α}⟩. As ℤ_{2}[y_{1}, …, y_{n}]/⟨y_{α}⟩ is again a polynomial algebra in (n−1)- variables, we see that |S_{α,s}| must be even.

Inductively assume that Y_{α}^{p} occurs in T_{x}M, if at all, at an even number of stationary points for any d > p > r. We must show that Y_{α}^{r} occurs in T_{x}M, if at all, for an even number of stationary points. Proceeding exactly as before we see that

0 = ∑_{p≥r} ( ∑_{x∈S_{α,p}} \binom{p}{r} + y_{α}^{r}P_{α}(T_{x}M)) + ∑ ; x∈S\ ⋃_{p≥r} S_{α,p} y_{α}^{r}Q_{α}(T_{x}M).

Again note that y_{α}^{r}P_{α}(T_{x}M) and y_{α}^{r}Q_{α}(T_{x}M) are polynomials in y_{α} without constant terms and having coefficients in ℤ_{2}[1/y_{β} | β ≠ α]. As before we see that ∑_{p≥r} |S_{α,p}|\binom{p}{r} = 0 (mod 2). Hence |S_{α,r}| ≡ ∑_{p>r} |S_{α,p}|\binom{p}{r} ≡ 0 (mod 2), since by the induction hypothesis each |S_{α,p}|, p > r, is even.

Remark 4.2. It may be noted that our method can be used to interpret the vanishing of ψ_{λ}(M, ϕ) for |λ| < d = dim M in terms of the irreducible representations Y_{α}, α ⊂ \underline{n}, which occur in T_{x}M as x varies in the (finite) set of stationary points of (M, ϕ).

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Stat-Math Division, Indian Statistical Institute, 203 B. T. Road, Calcutta-700 035, India

E-mail address: goutam@isical.ernet.in

SPIC Mathematical Institute, 92 G. N. Chetty Road, Madras-600 017, India

E-mail address: sankaran@smi.ernet.in