Homology, Homotopy and Applications, vol.5(1), 2003, pp.423–436

THE SET OF RATIONAL HOMOTOPY TYPES WITH GIVEN COHOMOLOGY ALGEBRA

HIROO SHIGA and TOSHIHIRO YAMAGUCHI

(communicated by James Stasheff)

Abstract For a given commutative graded algebra A^{∗}, we study the set M_{A^{∗}} = {rational homotopy type of X | H^{∗}(X; Q) ≅ A^{∗}}. For example, we see that if A^{∗} is isomorphic to H^{∗}(S^{3} ∨ S^{5} ∨ S^{16}; Q), then M_{A^{∗}} corresponds bijectively to the orbit space P^{3}(Q)/Q^{∗} ∐{∗}, where P^{3}(Q) is the rational projective space of dimension 3 and the point {∗} indicates the formal space.

1. Introduction

For a given graded algebra over the rationals (abbreviated to G.A.) A^{∗}, there exists at least one rational homotopy type having A^{∗} as a cohomology algebra, namely the formal space. In general there are many rational homotopy types having isomorphic cohomology algebras. In [5] it was shown that there are two rational homotopy types with isomorphic cohomology algebras and isomorphic homotopy Lie algebras, and in [6] it was shown that there are infinitely many rationally elliptic homotopy types having isomorphic cohomology algebras. Set

M_{A^{∗}} = {rational homotopy type of X | H^{∗}(X; Q) ≅ A^{∗}}.

The set M_{A^{∗}} was studied by several authors([1],[2],[3],[7],[10]). For example, Lupton ([3]) showed that for any positive integer n there is a G.A. A^{∗} such that the cardinality of M_{A^{∗}} is n. Halperin and Stasheff studied M_{A^{∗}} by the set of perturbations of the differential of the formal differential graded algebra (abbreviated to D.G.A.). In particular they showed for A^{∗} = H^{∗}((S^{2} ∨ S^{2}) × S^{3}; Q), the set M_{A^{∗}} consists of two points. This example is also caluculated from our view point (see Section 3(4)). Schlessinger and Stasheff ([7]) extended the arguments in [2].

We study M_{A^{∗}} from a different point of view. Our strategy to study M_{A^{∗}} is as follows. We construct inductively 1-connected minimal algebras m_{n−1} such that there is a G.A.map

σ_{n} : (H^{∗}(m_{n−1})(n))^{∗} → A^{∗}

so that σ^{i} is isomorphic for i ⩽ n − 1 and monomorphic for i = n, where (H^{∗}(m_{n−1})(n))^{∗} is the sub G.A. of H^{∗}(m_{n−1}) generated by elements of degree

Received July 25, 2003, revised October 5, 2003; published on October 15, 2003. 2000 Mathematics Subject Classification: 55p62. Key words and phrases: rational homotopy type, minimal algebra, k-intrinsically formal (k-I.F.) \begin{matrix}c \\ ◯ \end{matrix} 2003, Hiroo Shiga and Toshihiro Yamaguchi. Permission to copy for private use granted.

⩽ n. Suppose we have constructed the pair (m_{n−1}, σ_{n−1}). Then there is a unique minimal algebras m_{D} containing m_{n−1} and a G.A.map

σ_{D} : (H^{∗}(m_{D})(n))^{∗} → A^{∗}

such that σ_{D}^{i} is isomorphic for i ⩽ n − 1, monomorphic for i = n and moreover σ_{D}^{n+1} induces an isomorphism on the decomposable part

σ_{D}^{n+1} : (H^{∗}(m_{D})(n))^{n+1} → (A(n))^{n+1},

where (A(n))^{n+1} is the degree n + 1 part of the subalgebra A(n) of A^{∗} generated by elements of degree ⩽ n. To construct m_{n} we choose a subspace W of H^{n+1}(m_{D}) satisfying certain conditions (see (2.3) and (2.4) in Section 2) so that H^{n+1}(m_{n}) ⊕ W = H^{n+1}(m_{D}).

Such a space W may be regarded as a rational point of a Grassmann manifold. The set of isomorphism classes of m_{n} containing m_{n−1} corresponds to the disjoint union of subsets of rational points of Grassmann manifolds modulo the action of D.G.A.automorphisms of m_{D} (see Theorem 2.1). We can show that any minimal algebra m with H^{∗}(m) ≅ A^{∗} is obtained in this way. For example if A^{∗} = H^{∗}(S^{3} ∨ S^{5} ∨ S^{16}; Q), then M_{A^{∗}} corresponds bijectively to P^{3}(Q)/Q^{∗} ∐{∗}, where P^{3}(Q) is the rational projective space of dimension 3 and the point {∗} corresponds to the formal space (see Section 3 (2)).

Throughout this paper we assume that G.A. A^{∗} satisfies that A^{0} = Q, A^{1} = 0 and dim_{Q} A^{i} < ∞ for any positive integer i.

2. Inductive construction of minimal models

In this section we construct inductively minimal algebras m_{n} and G.A. maps σ_{n} : H^{∗}(m_{n})(n + 1) → A^{∗} such that σ_{n}^{i} is isomorphic for i ⩽ n and monomorphic for i = n + 1.

Suppose that we constructed a minimal algebra m_{n−1} satisfying the following conditions.

(1)_{n−1} m_{n−1} is generated by elements of degree ⩽ n − 1.

(2)_{n−1} There is a G.A.-map

σ_{n−1} : (H^{∗}(m_{n−1})(n))^{∗} → A^{∗}

where σ_{n−1}^{i} is isomorphic for i ⩽ n − 1 and monomorphic for i = n.

Let m_{D} be the minimal algebra obtained by adding generators to m_{n−1} whose differentials form a basis for the kernel of σ_{n−1}^{n+1}|(H(m_{n−1})(n))^{n+1} and σ_{D} : (H(m_{D})(n))^{∗} → A^{∗} be the induced map. We set

dim_{Q} A^{n+1} = u, dim_{Q} A^{n+1}/(A(n))^{n+1} = s
dim_{Q} H^{n+1}(m_{D}) = v

and

dim_{Q} \frac{H^{n+1}(m_{D})}{(H^{∗}(m_{D})(n))^{n+1}} = t.

Then we have

u − s = v − t.  (2.1)

Let l be an integer satisfying

max(0, t − s) ⩽ l ⩽ t  (2.2)

and W be a l-dimensional subspace of H^{n+1}(m_{D}) such that

W ∩ (H^{∗}(m_{D})(n))^{n+1} = {0}.  (2.3)

Let m^{W} be the minimal algebra obtained by adding l generators whose differentials span W. Note that H(m^{W})(n) = H(m_{D})(n), hence we have a G.A.map σ_{D} : (H(m^{W})(n))^{∗} → A^{∗} and

H^{n+1}(m^{W}) ⊕ W = H^{n+1}(m_{D})

so that

dim_{Q} \frac{H^{n+1}(m^{W})}{(H(m^{W})(n))^{n+1}} = t − l ⩽ s = dim_{Q} \frac{A^{n+1}}{(A(n))^{n+1}} .

Let m^{W}_{n} be a minimal algebra obtained by adding to m^{W} the cokernel of σ_{D}^{n} : (H(m^{W})(n))^{n} → A^{n}. Then we have a G.A. map

σ_{n} : (H(m^{W}_{n})(n))^{∗} → A^{∗}

such that σ_{n}^{i} is isomorphic for i ⩽ n. For a linear monomorphism

ψ : H^{n+1}(m^{W})/(H(m^{W})(n))^{n+1} → A^{n+1}/(A(n))^{n+1},

if the map σ_{n} ⊕ ψ can be extend to a G.A. map

σ^{W}_{n} : (H(m^{W}_{n})(n + 1))^{∗} → A^{∗},  (2.4)

then the pair (m^{W}_{n}, σ^{W}_{n}) satisfies the condition (1)_{n} and (2)_{n}. Remark that if we take W so that dim_{Q} W = t we can always construct a G.A. map (2.4).

Let m_{n} be a minimal algebra containg m_{n−1} (hence m_{D}) satisfying (1)_{n} and (2)_{n}. Then m_{n} is constructed from m_{D} by taking W as the kernel of i^{∗} : H^{∗}(m_{D}) → H^{∗}(m_{n}), where i is the inclusion.

By Plücker embedding Grassmann manifold is a projective variety defined over Q. Then the Q-subspace W corresponds to a rational point of the variety. Let Gr(v, l)(Q) be the set of rational points of the Grassmann manifold of l-dimensional Q-subspaces in a v-dimensional space H^{n+1}(m_{D}). Set

M_{l} = {W ∈ Gr(v, l)(Q)| W satisfies (2.3)}

satisfying (2.3). We take bases for H^{n+1}(m^{W})/(H^{∗}(m^{W})(n))^{n+1} and H^{∗}(m^{W})(n)^{n+1}. If we write a basis for W as a linear combinations of those bases, we see that M_{l} is a Zariski open set of Gr(v, l)(Q) (Compare with Example (3) in Section 3). Set

O_{l} = {W ∈ M_{l}| there is a G.A.map σ^{W}_{n} satisfying (2.4) for some linear map ψ}.

Let G be the group of D.G.A.automorphisms of m_{D}. Then G acts on H^{n+1}(m_{D}) and hence on Gr(v, l)(Q). Let W be an element of O_{l} and Φ be an element of G.

Then it is easy to see that Φ can be extended to a D.G.A.isomorphism

Φ : m^{W}_{n} → m^{Φ(W)}_{n}.

Hence G also acts on O_{l}.

Conversely let W_{1}, W_{2} be l-dimensional subspaces of H^{n+1}(m_{D}) such that there is a D.G.A.isomorphism

f : m^{W_{1}}_{n} → m^{W_{2}}_{n}.

Then f|m_{D} = Φ is an element of G and

Φ(W_{1}) = W_{2}.

Hence we have

Theorem 2.1. The set of isomorphism classes of minimal algebras m_{n} containing a minimal algebra m_{n−1} and satisfying (1)_{n}, (2)_{n} corresponds bijectively to the disjoint union of orbit spaces

X_{n} = ∐_{l=max(t−s,0)}^{t} O_{l}/G.

Note that X_{n} is not empty since O_{t} is not empty.

Definition 2.2. A G.A. A^{∗} is called k-intrinsically formal (abbreviated to k-I.F.) if for any minimal algebras m with H^{∗}(m) = A^{∗}, the sub D.G.A. m(k) is unique up to isomorphism.

Note that any G.A. A^{∗} is at least 2-I.F..

Let A^{∗} be (n−1)-I.F. and m be arbitrary minimal algebra with H^{∗}(m) ≅ A^{∗}. Set m_{n−1} = m(n − 1) and i_{n−1} : m_{n−1} → m be the inclusion. Then we can construct minimal algebras m_{D} and m^{W_{0}}_{n} as previous way where W_{0} is the kernel of the induced map

i_{D}^{∗} : H^{n+1}(m_{D}) → H^{n+1}(m).

The inclusion i_{D} can be extended to

i_{n} : m^{W_{0}}_{n} → m

so that m^{W_{0}}_{n} and i_{n}^{∗} satisfy (1)_{n}, (2)_{n}. Hence m can be constructed inductively as this way. Especially we have

Corollary 2.3. If A^{∗} is (n − 1)-I.F. and A^{j} = 0 for j > n + 1. Then O_{l} = M_{l} and M_{A^{∗}} = X_{n} = ∐_{max(t−s,0)⩽l⩽t} M_{l}/G.

Suppose A^{i} = 0 for i ⩽ n. Then X_{k} is one point for k < 3n + 1. Therefore m_{3n} is uniquely determined, i.e., A^{∗} is 3n-I.F.. This implies

Corollary 2.4. Any n-connected k-dimensional finite CW complex is formal if k ⩽ 3n + 1.

This result was noticed by Stasheff [8]. We see that Corollary 2.4 is best possible by the example A^{∗} = H^{∗}(S^{3} ∨ S^{3} ∨ S^{8}; Q).

The following examples are studied in the next section, where degree is denoted by suffix.

(1) A^{∗} = H^{∗}(S^{3}∨S^{7}∨S^{22}; Q),which is 20-I.F. and u = s = 1, v = t = 3 at n = 21.

(2) A^{∗} = H^{∗}(S^{3}∨S^{5}∨S^{16}; Q),which is 14-I.F. and u = s = 1, v = t = 4 at n = 15.

(3) A^{∗} = ∧(x_{3}, y_{5}) ⊗ Q[z_{8}]/(xy, xz^{2}, yz^{2}, z^{3}), which is 14-I.F. and u = 1, s = 0, v = 5, t = 4 at n = 15.

(4) A^{∗} = H^{∗}((S^{2} ∨ S^{2}) × S^{3}; Q), which is 3-I.F. and u = 2, s = 0, v = 4, t = 2 at n = 4.

(5) A^{∗} = H^{∗}((S^{3} ∨ S^{3}) × S^{5}; Q), which is 6-I.F. and u = 2, s = 0, v = 4, t = 2 at n = 7.

(6) A^{∗} = H^{∗}(S^{3} ∨ S^{5} ∨ S^{10} ∨ S^{16}; Q), which is 8-I.F. and u = s = v = t = 1 at n = 9.

(7) A^{∗} = H^{∗}(S^{5} ∨ (S^{3} × S^{10}); Q), which is 8-I.F. and u = s = v = t = 1 at n = 9.

(8) A^{∗} = H^{∗}((S^{3} × S^{8})♯(S^{3} × S^{8}); Q), which is 6-I.F. and u = s = v = t = 2 at n = 7. Here ♯ is connected sum.

3. Some examples

(1) A^{∗} = H^{∗}(S^{3} ∨ S^{7} ∨ S^{22}; Q) = ∧(x_{3}, y_{7}) ⊗ Q[z_{22}]/(xy, xz, yz, z^{2}) Then A^{∗} is 20-I.F. and by straightfoward calculation

m_{20} = (∧(x, y, θ_{9}, θ_{11}, θ_{13}, θ_{15}^{1}, θ_{15}^{2}, θ_{17}^{1}, θ_{17}^{2}, θ_{19}^{1}, θ_{19}^{2}), d)

with the differential is as follows : d(x) = d(y) = 0, dθ_{9} = xy, dθ_{11} = xθ_{9}, dθ_{13} = xθ_{11}, dθ_{15}^{1} = yθ_{9}, dθ_{15}^{2} = xθ_{13}, dθ_{17}^{1} = xθ_{15}^{1} + yθ_{11}, dθ_{17}^{2} = xθ_{15}^{2}, dθ_{19}^{1} = xθ_{17}^{1} + yθ_{13}, dθ_{19}^{2} = xθ_{17}^{2}.

Then at n = 21, u = s = 1 and v = t = 3. In fact m_{D} = m_{20} and H^{22}(m_{D}) = Q{e_{1}, e_{2}, e_{3}}, where e_{1} = [xθ_{19}^{2}], e_{2} = [xθ_{19}^{1} + yθ_{15}^{2}] and e_{3} = [yθ_{15}^{1}]. Let W be a 2 dimensional subspace of H^{22}(m_{D}) spanned by

a_{1,i}e_{1} + a_{2,i}e_{2} + a_{3,i}e_{3} (i = 1, 2),

with

rank [\begin{matrix}a_{1,1} & a_{2,1} & a_{3,1} \\ a_{1,2} & a_{2,2} & a_{3,2} \end{matrix}] = 2.

Let f ∈ Aut m_{D} = G be an element such that

f(x) = λx, f(y) = μy, λ, μ ∈ Q^{∗}.

Then we have

f(e_{1}) = λ^{7}μe_{1}, f(e_{2}) = λ^{4}μ^{2}e_{2}, f(e_{3}) = λμ^{3}e_{3}.

The set of W forms Gr(3, 2)(Q), the rational points of Grassmann manifold of 2-dimensional spaces in the 3-dimensional space H^{22}(m(20)). By the Plücker embedding i : Gr(3, 2)(Q) → P^{2}(Q),

i(W) = [|\begin{matrix}a_{1,1} & a_{2,1} \\ a_{1,2} & a_{2,2} \end{matrix}| , |\begin{matrix}a_{1,1} & a_{3,1} \\ a_{1,2} & a_{3,2} \end{matrix}| , |\begin{matrix}a_{2,1} & a_{3,1} \\ a_{2,2} & a_{3,2} \end{matrix}|],

G acts on P^{2}(Q) by f[x_{1}, x_{2}, x_{3}] = [λ^{11}μ^{3}x_{1}, λ^{8}μ^{4}x_{2}, λ^{5}μ^{5}x_{3}] = [ρx_{1}, x_{2}, ρ^{−1}x_{3}] with ρ = λ^{3}μ^{−1}. Hence by Corollary 2.3, we have

M_{A^{∗}} = M_{2}/G ∐ M_{3} ≃ P^{2}(Q)/Q^{∗} ∐{∗}.

(2) A^{∗} = H^{∗}(S^{3} ∨ S^{5} ∨ S^{16}; Q) = ∧(x_{3}, y_{5}) ⊗ Q[z_{16}]/(xy, xz, yz, z^{2})

Then A^{∗} is 14-I.F. and by straightfoward calculation

m_{D} = m_{14} = (∧(x, y, θ_{7}, θ_{9}, θ_{11}^{1}, θ_{11}^{2}, θ_{13}^{1}, θ_{13}^{2}), d) (∗)

with the differential is as follows: d(x) = d(y) = 0, dθ_{7} = xy, dθ_{9} = xθ_{7}, dθ_{11}^{1} = yθ_{7}, dθ_{11}^{2} = xθ_{9}, dθ_{13}^{1} = xθ_{11}^{2}, dθ_{13}^{2} = xθ_{11}^{1} + yθ_{9}.

Then at n = 15, u = s = 1 and H^{16}(m_{D}) = Q{e_{1}, e_{2}, e_{3}, e_{4}}, where e_{1} = [xθ_{13}^{1}], e_{2} = [yθ_{11}^{1}], e_{3} = [xθ_{13}^{2} + θ_{7}θ_{9}] and e_{4} = [yθ_{11}^{2} + θ_{7}θ_{9}]. Hence at n = 15, v = t = 4. Let W be a 3-dimensional subspace of H^{16}(m_{D}) spanned by

a_{1,i}e_{1} + a_{2,i}e_{2} + a_{3,i}e_{3} + a_{4,i}e_{4} (i = 1, 2, 3),

where rank(a_{j,i})_{1⩽j⩽4,1⩽i⩽3} = 3.

Let f ∈ Aut m_{D} = G be an element such that

f(x) = λx, f(y) = μy, λ, μ ∈ Q^{∗}.

Then we have

f(e_{1}) = λ^{5}μe_{1}, f(e_{2}) = λμ^{3}e_{2}, f(e_{3}) = λ^{3}μ^{2}e_{3}, f(e_{4}) = λ^{3}μ^{2}e_{4}.

The set of W forms Gr(4, 3)(Q), which is isomorphic to P^{3}(Q) by the Plücker embedding i : Gr(4, 3)(Q) → P^{3}(Q),

i(W) = [|\begin{matrix}a_{1,1} & a_{2,1} & a_{3,1} \\ a_{1,2} & a_{2,2} & a_{3,2} \\ a_{1,3} & a_{2,3} & a_{3,3} \end{matrix}| , |\begin{matrix}a_{1,1} & a_{2,1} & a_{4,1} \\ a_{1,2} & a_{2,2} & a_{4,2} \\ a_{1,3} & a_{2,3} & a_{4,3} \end{matrix}| , |\begin{matrix}a_{1,1} & a_{3,1} & a_{4,1} \\ a_{1,2} & a_{3,2} & a_{4,2} \\ a_{1,3} & a_{3,3} & a_{4,3} \end{matrix}| , |\begin{matrix}a_{2,1} & a_{3,1} & a_{4,1} \\ a_{2,2} & a_{3,2} & a_{4,2} \\ a_{2,3} & a_{3,3} & a_{4,3} \end{matrix}|].

Then G acts on P^{2}(Q) by f[x_{1}, x_{2}, x_{3}, x_{4}] = [λ^{9}μ^{6}x_{1}, λ^{9}μ^{6}x_{2}, λ^{11}μ^{5}x_{3}, λ^{7}μ^{7}x_{4}] = [ρx_{1}, ρx_{2}, ρ^{2}x_{3}, x_{4}] by putting ρ = λ^{2}μ^{−1}. Hence by Corollary 2.3, we have

M_{A^{∗}} = M_{3}/G ∐ M_{4} ≃ P^{3}(Q)/Q^{∗} ∐{∗}.

(3) A^{∗} = ∧(x_{3}, y_{5}) ⊗ Q[z_{8}]/(xy, xz^{2}, yz^{2}, z^{3})

Then A^{∗} is 14-I.F. and at n = 15, u = 1, s = 0, and

m_{D} = m_{14} = m′_{14} ⊗ Q[z],

where m′_{14} is isomorphic to m_{14} in the example (2) and d(z) = 0. Then H^{16}(m_{D}) = Q{e_{1}, e_{2}, e_{3}, e_{4}, f_{1}}, where e_{1} = [xθ_{13}^{1}], e_{2} = [yθ_{11}^{1}], e_{3} = [xθ_{13}^{2} + θ_{7}θ_{9}], e_{4} = [yθ_{11}^{2} + θ_{7}θ_{9}] and f_{1} = [z^{2}]. Hence at n = 15, v = 5, t = 4. By Corollary 2.3,

M_{A^{∗}} = X_{15} = M_{4}/G.

Let W be an element of M_{4} spanned by

a_{1,i}e_{1} + a_{2,i}e_{2} + a_{3,i}e_{3} + a_{4,i}e_{4} + a_{5,i}f_{1} (i = 1, 2, 3, 4),

with

rank (a_{j,i})_{1⩽j⩽4,1⩽i⩽4} = 4 (∗).

By Plücker embedding, we see that the set of W satisfying (∗) corresponds bijectively to A^{4}(Q) = {[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}] ∈ P^{4}(Q)|x_{1} ≠ 0}.

Let f ∈ Aut m_{D} = G be an element such that

f(x) = λx, f(y) = μy, f(z) = κz, λ, μ, κ ∈ Q^{∗}.

Then we have

f^{∗}(e_{1}) = λ^{5}μe_{1}, f^{∗}(e_{2}) = λμ^{3}e_{2}, f^{∗}(e_{3}) = λ^{3}μ^{2}e_{3},
f^{∗}(e_{4}) = λ^{3}μ^{2}e_{4}, f^{∗}(f_{1}) = κ^{2}f_{1}.

Hence G acts on P^{4}(Q) by

f · [x_{1}, x_{2}, x_{3}, x_{4}, x_{5}] = [λ^{12}μ^{8}x_{1}, λ^{11}μ^{5}κ^{2}x_{2}, λ^{9}μ^{6}κ^{2}x_{3}, λ^{9}μ^{6}κ^{2}x_{4}, λ^{7}μ^{7}κ^{2}x_{5}].

Hence G acts on A^{4}(Q) by

f · (y_{1}, y_{2}, y_{3}, y_{4}) = (λ^{−1}μ^{−3}κ^{2}y_{1}, λ^{−3}μ^{−2}κ^{2}y_{2}, λ^{−3}μ^{−2}κ^{2}y_{3}, λ^{−5}μ^{−1}κ^{2}y_{4}),

where y_{i} = x_{i+1}/x_{1} for i = 1, .., 4. Then setting α = λ^{−7}κ^{2} and β = λ^{2}μ^{−1}, G acts on A^{4}(Q) by

f · (y_{1}, y_{2}, y_{3}, y_{4}) = (αβ^{3}y_{1}, αβ^{2}y_{2}, αβ^{2}y_{3}, αβy_{4}).

Since α and β take any non-zero rational numbers independently, we have

M_{A^{∗}} ≃ A^{4}(Q)/(Q^{∗} × Q^{∗}) ≃ P^{3}(Q)/Q^{∗} ∐{∗},

where Q^{∗} acts on P^{3}(Q) by

β · [z_{1}, z_{2}, z_{3}, z_{4}] = [β^{2}z_{1}, βz_{2}, βz_{3}, z_{4}]

and the point {∗} corresponds (0, 0, 0, 0) in A^{4}(Q), which corresponds a formal model. Thus M_{A^{∗}} is the same set as that of Example (2).

(4) A^{∗} = H^{∗}((S^{2} ∨ S^{2}) × S^{3}; Q) = Q[x_{2}, y_{2}] ⊗ Λ(z_{3})/(xy).

This example was studied by Halperin and Stasheff, see example 6.5 in [2]. It is 3-I.F. and at n = 4, s = 0 and t = 2. In fact

m_{D} = m_{3} = (∧(x, y, θ_{3}^{1}, θ_{3}^{2}, θ_{3}^{3}, z_{3}), d)

with d(x) = d(y) = d(z) = 0, dθ_{3}^{1} = x^{2}, dθ_{3}^{2} = xy, dθ_{3}^{3} = y^{2} and H^{5}(m_{3}) = Q{e_{1}, e_{2}, f_{1}, f_{2}}, where e_{1} = [yθ_{3}^{1} − xθ_{3}^{2}], e_{2} = [yθ_{3}^{2} − xθ_{3}^{3}], f_{1} = [xz] and f_{2} = [yz]. Then by Collorary 2.3,

M_{A^{∗}} = X_{4} = M_{2}/G.

Let W in M_{2} be spanned by

a_{1,i}e_{1} + a_{2,i}e_{2} + a_{3,i}f_{1} + a_{4,i}f_{2} (i = 1, 2),

where

rank (a_{j,i})_{1⩽j⩽2,1⩽i⩽2} = 2 .

By Plücker embedding, the set of W forms

{[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}] ∈ P^{5}(Q)|x_{1}x_{6} − x_{2}x_{5} + x_{3}x_{4} = 0, x_{1} ≠ 0}
≃ {(X_{1}, X_{2}, X_{3}, X_{4}, X_{5}) ∈ A^{5}(Q)|X_{5} − X_{2}X_{5} + X_{3}X_{4} = 0}
≃ {(X_{1}, X_{2}, X_{3}, X_{4}) ∈ A^{4}(Q)},

where X_{i} = x_{i+1}/x_{1} (i = 1, .., 5).

Let f ∈ Aut m_{D} = G be an element such that

f(x) = x, f(y) = y, f(z) = μz μ ∈ Q^{∗}
f(θ_{3}^{i}) = θ_{3}^{i} + λ_{i}z, λ_{i} ∈ Q, i = 1, 2, 3.

Then we have

f^{∗}(e_{1}) = e_{1} − λ_{2}f_{1} + λ_{1}f_{2}, f^{∗}(e_{2}) = e_{2} − λ_{3}f_{1} + λ_{2}f_{2},
f^{∗}(f_{1}) = μf_{1}, f^{∗}(f_{2}) = μf_{2},

and f^{∗} induces a map A_{f} defined by

A_{f}([x_{1}, .., x_{6}]) = [x_{1}, .., x_{6}] [\begin{matrix}1 & −λ_{3} & λ_{2} & λ_{2} & −λ_{1} & λ_{1}λ_{3} − λ_{2}^{2} \\ 0 & μ & 0 & 0 & 0 & −λ_{1}μ \\ 0 & 0 & μ & 0 & 0 & −λ_{2}μ \\ 0 & 0 & 0 & μ & 0 & −λ_{2}μ \\ 0 & 0 & 0 & 0 & μ & −λ_{3}μ \\ 0 & 0 & 0 & 0 & 0 & μ^{2} \end{matrix}] ,

hence f^{∗} induces a map Ã_{f} from A^{4}(Q) to itself defined by

Ã_{f} [\begin{matrix}X_{1} \\ X_{2} \\ X_{3} \\ X_{4} \end{matrix}] = [\begin{matrix}μ \\ & μ \\ & & μ \\ & & & μ \end{matrix}] [\begin{matrix}X_{1} \\ X_{2} \\ X_{3} \\ X_{4} \end{matrix}] + [\begin{matrix}−λ_{3} \\ λ_{2} \\ λ_{2} \\ −λ_{1} \end{matrix}] .

From this we see by varing λ_{i} ∈ Q (i = 1, 2, 3) and μ ∈ Q^{∗},

Ã_{f} [\begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix}] ∪ Ã_{f} [\begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix}] = A^{4}(Q).

Hence M_{A^{∗}} is at most two points.

Conversely, any element g ∈ Aut m_{D} has the following form: g(x) = a_{1}x + a_{2}y, g(y) = b_{1}x + b_{2}y and g(z) = μz with

a_{1}, a_{2}, b_{1}, b_{2} ∈ Q, D = |\begin{matrix}a_{1} & a_{2} \\ b_{1} & b_{2} \end{matrix}| ≠ 0, μ ∈ Q^{∗}

and then g(θ_{1}) = a_{1}^{2}θ_{1} + 2a_{1}a_{2}θ_{2} + a_{2}^{2}θ_{3} + λ_{1}z, g(θ_{2}) = a_{1}b_{1}θ_{1} + (a_{1}b_{2} + a_{2}b_{1})θ_{2} + a_{2}b_{2}θ_{3} + λ_{2}z, g(θ_{3}) = b_{1}^{2}θ_{1} + 2b_{1}b_{2}θ_{2} + b_{2}^{2}θ_{3} + λ_{3}z for some λ_{i} ∈ Q. By straightfoward calculations we see that W_{1} = {e_{1}, e_{2}}, which corresponds to (0, 0, 0, 0) in A^{4}(Q), can not be mapped to W_{2} = {e_{1}, e_{2} + f_{2}} corresponding to (0, 1, 0, 0) in A^{4}(Q) by Aut m_{D}. In fact,

Ã_{g} [\begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix}] = \frac{1}{D^{2}} · [\begin{matrix}−b_{1}^{2}λ_{1} + 2a_{1}b_{1}λ_{2} − a_{1}^{2}λ_{3} \\ −b_{1}b_{2}λ_{1} + (a_{1}b_{2} + a_{2}b_{1})λ_{2} − a_{1}a_{2}λ_{3} \\ −b_{1}b_{2}λ_{1} + (a_{1}b_{2} + a_{2}b_{1})λ_{2} − a_{1}a_{2}λ_{3} \\ −b_{2}^{2}λ_{1} + 2a_{2}b_{2}λ_{2} − a_{2}^{2}λ_{3} \end{matrix}] = [\begin{matrix}∗ \\ α \\ α \\ ∗ \end{matrix}] ≠ [\begin{matrix}0 \\ 1 \\ 0 \\ 0 \end{matrix}] .

Thus we see that M_{A^{∗}} is just two points.

(5) A^{∗} = H^{∗}((S^{3} ∨ S^{3}) × S^{5}; Q) = Λ(x_{3}, y_{3}, z_{5})/(xy).

This example was considered by Schlessinger and Stasheff, see section 8 in [7]. It is 6-I.F. and

m_{D} = m_{6} = (∧(x_{3}, y_{3}, θ_{5}, z_{5}), d)

with d(x) = d(y) = d(z) = 0 and dθ_{5} = xy. Then H^{8}(m_{D}) = Q{e_{1}, e_{2}, f_{1}, f_{2}}, where e_{1} = [xθ_{5}], e_{2} = [yθ_{5}], f_{1} = [xz] and f_{2} = [yz]. Hence at n = 7, s = 0 and t = 2. By Corollary 2.3,

M_{A^{∗}} = X_{7} = M_{2}/G.

Let W in M_{2} be spanned by

a_{1,i}e_{1} + a_{2,i}e_{2} + a_{3,i}f_{1} + a_{4,i}f_{2} (i = 1, 2),

where rank(a_{j,i})_{1⩽j⩽2,1⩽i⩽2} = 2.

Let f ∈ Aut m_{D} = G be an element such that f(x) = a_{1}x+a_{2}y, f(y) = b_{1}x+b_{2}y, f(θ_{5}) = Dθ_{5} + λz and f(z) = μz, where

D = |\begin{matrix}a_{1} & a_{2} \\ b_{1} & b_{2} \end{matrix}| ≠ 0, λ ∈ Q, μ ∈ Q^{∗}.

Then

f^{∗}(e_{1}) = a_{1}De_{1} + a_{2}De_{2} + a_{1}λf_{1} + a_{2}λf_{2},
f^{∗}(e_{2}) = b_{1}De_{1} + b_{2}De_{2} + b_{1}λf_{1} + b_{2}λf_{2},
f^{∗}(f_{1}) = a_{1}μf_{1} + a_{2}μf_{2}, f^{∗}(f_{2}) = b_{1}μf_{1} + b_{2}μf_{2}.

By Plücker embedding the set of W forms

{[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}] ∈ P^{5}(Q)|x_{1}x_{6} − x_{2}x_{5} + x_{3}x_{4} = 0, x_{1} ≠ 0}
≃ {(X_{1}, X_{2}, X_{3}, X_{4}) ∈ A^{4}(Q)},

where X_{i} = x_{i+1}/x_{1} (i = 1, .., 4). Then G acts on A^{4}(Q) as follows:

Ã_{f} [\begin{matrix}X_{1} \\ X_{2} \\ X_{3} \\ X_{4} \end{matrix}] = \frac{μ}{D^{2}} [\begin{matrix}a_{1}^{2} & a_{1}b_{1} & a_{1}b_{1} & b_{1}^{2} \\ a_{1}a_{2} & a_{1}b_{2} & a_{2}b_{1} & b_{1}b_{2} \\ a_{1}a_{2} & a_{2}b_{1} & a_{1}b_{2} & b_{1}b_{2} \\ a_{2}^{2} & a_{2}b_{2} & a_{2}b_{2} & b_{2}^{2} \end{matrix}] [\begin{matrix}X_{1} \\ X_{2} \\ X_{3} \\ X_{4} \end{matrix}] + \frac{λ}{D} [\begin{matrix}0 \\ 1 \\ −1 \\ 0 \end{matrix}] .

First we show that any point (x_{1}, x_{2}, x_{3}, x_{4}) of A^{4}(Q) lies in the union of the orbit of (1, 0, 0, r) for some r ∈ Q and that of (0, 0, 0, 0) by decomposing A^{4}(Q) into the following pieces (a)∼(f).

(a) If 4x_{1}x_{4} ≠ (x_{2} + x_{3})^{2} and x_{1} ≠ 0, set a_{1} = 0, a_{2} = −1, b_{1} = 1, b_{2} = −\frac{x_{2}+x_{3}}{2x_{1}} ,

μ = \frac{(x_{2}+x_{3})^{2}−4x_{1}x_{4}}{4x_{1}} , r = \frac{4x_{1}^{2}}{(x_{2}+x_{3})^{2}−4x_{1}x_{4}} and λ = \frac{1}{2}(x_{2} − x_{3}). Then we have

Ã_{f} [\begin{matrix}1 \\ 0 \\ 0 \\ r \end{matrix}] = [\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] .  (3.1)

(b) If 4x_{1}x_{4} ≠ (x_{2} + x_{3})^{2} and x_{4} ≠ 0, set a_{1} = 1, a_{2} = 0, b_{1} = −\frac{x_{2}+x_{3}}{2x_{4}} , b_{2} = 1,

μ = \frac{(x_{2}+x_{3})^{2}−4x_{1}x_{4}}{4x_{4}} , r = \frac{4x_{4}^{2}}{(x_{2}+x_{3})^{2}−4x_{1}x_{4}} and λ = \frac{1}{2}(x_{2} − x_{3}). Then we have (3.1).

(c) If 4x_{1}x_{4} ≠ (x_{2} + x_{3})^{2} and x_{1} = x_{4} = 0, set a_{1} = b_{1} = 1, a_{2} = −\frac{1}{2}, b_{2} = \frac{1}{2} μ = −\frac{x_{2}+x_{3}}{2} , r = −2 and λ = \frac{1}{2}(x_{2} − x_{3}). Then we have (3.1).

(d) If 4x_{1}x_{4} = (x_{2} +x_{3})^{2} and x_{1} ≠ 0, set a_{1} = x_{1}, a_{2} = −\frac{x_{2}+x_{3}}{2} , b_{1} = 0, b_{2} = \frac{1}{x_{1}} μ = −\frac{1}{x_{1}} , r = 0 and λ = \frac{1}{2}(x_{2} − x_{3}). Then we have (3.1).

(e) If 4x_{1}x_{4} = (x_{2} + x_{3})^{2} and x_{4} ≠ 0, set a_{1} = −\frac{x_{2}+x_{3}}{2} , a_{1} = x_{4}, b_{1} = −\frac{1}{x_{1}} , b_{2} = 0, μ = −\frac{1}{x_{4}} , r = 0 and λ = \frac{1}{2}(x_{2} − x_{3}). Then we have (3.1).

(f) If x_{1} = x_{4} = 0, x_{2} + x_{3} = 0, set a_{1} = 1, a_{2} = 0, b_{1} = 0, b_{2} = 1, μ = 1 and λ = x_{2}. Then we have

Ã_{f} [\begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix}] = [\begin{matrix}0 \\ x_{2} \\ x_{3} \\ 0 \end{matrix}] .

Thus we have a surjection

p : Q ∐{∗} → M_{A^{∗}} ≃ A^{4}(Q)/G

defined by p(∗) = the class of (0, 0, 0, 0) and p(r) = the class of (1, 0, 0, r).

If p(r_{1}) = p(r_{2}) then there is an element f ∈ G such that

Ã_{f} [\begin{matrix}1 \\ 0 \\ 0 \\ r_{1} \end{matrix}] = [\begin{matrix}1 \\ 0 \\ 0 \\ r_{2} \end{matrix}] .

By straightfoward calculations we have r_{1}r_{2} ∈ Q^{∗}^{2} if r_{1}r_{2} ≠ 0. Thus we have

M_{A^{∗}} ≃ Q^{∗}/Q^{∗}^{2} ∐{0} ∐{∗},

where {0} corresponds to (1, 0, 0, 0) and {∗} corresponds to the formal model. After tensoring with \overline{Q} the set of isomorphism classes consists of three points.

(6) A^{∗} = H^{∗}(S^{3} ∨ S^{5} ∨ S^{10} ∨ S^{16}; Q) = ∧(x_{3}, y_{5}) ⊗ Q[v_{10}, z_{16}]/(xy, xv, xz, yv, yz, v^{2}, vz, z^{2}).

Then m_{D} = m_{8} = (Λ(x, y, θ_{7}), d) with d(θ_{7}) = xy. Since H^{10}(m_{8}) = Q{xθ_{7}}, s = t = 1 at n = 9. Then since the condition (2)_{9} is satisfied

X_{9} = O_{0} ∐ O_{1} = M_{0} ∐ M_{1} ≃ {p_{0}, p_{1}},

where the corresponding model for p_{0} is

m^{(0)}_{9} = (Λ(x, y, θ_{7}), d)

with d(θ_{7}) = xy and the corresponding model for p_{1} is

m^{(1)}_{9} = (Λ(x, y, θ_{7}, θ_{9}), d)

with d(θ_{9}) = xθ_{7}.

Next consider X_{15} over each point. The model containing m^{(0)}_{9} is

m_{D} = m_{14} = (∧(x, y, θ_{7}, θ_{11}), d)

with d(θ_{11}) = yθ_{7}. Since H^{16}(m_{D}) = Q{yθ_{11}}, s = t = 1 at n = 15. Hence X_{15} consists of two points.

The model containing m^{(1)}_{9} is

m_{D} = m_{14} = (Λ(x, y, θ_{7}, θ_{9}, u_{10}, θ_{11}^{1}, θ_{11}^{2}, θ_{13}^{1}, θ_{13}^{2}), d) = (Q[u] ⊗ m, d)

where d(u_{10}) = 0 for a basis u_{10} of Coker(σ_{9}^{{xθ_{7}}})^{10} and m is the model (∗) in Example (2). Then H^{16}(m_{D}) = Q{e_{1}, e_{2}, e_{3}, e_{4}} is same as that of the above Example (2). Hence we have in this case

X_{15} ≃ M_{H^{∗}(S^{3}∨S^{5}∨S^{16})}.

Since A^{>16} = 0, M_{A^{∗}} is the disjoint union of two points and P^{3}(Q)/Q^{∗} ∐{∗}. See Fig 1.

(7) A^{∗} = H^{∗}(S^{5} ∨ (S^{3} × S^{10}); Q) = Λ(x_{3}, y_{5}) ⊗ Q[z_{10}]/(xy, xz, z^{2}).

Then m_{D} = m_{8} = (Λ(x, y, θ_{7}), d) with d(θ_{7}) = xy. Since H^{10}(m_{8}) = Q{xθ_{7}}, W = 0 or W = Q{xθ_{7}} at n = 9. If W = {0}, (σ^{W}_{9})^{13} : H^{3}(m^{W}_{9}) · H^{10}(m^{W}_{9}) = 0 → A^{3} · A^{10} ≠ 0 can not be a G.A.map. Hence the condition (2)_{9} is not satisfied. Hence W must be Q{xθ_{7}}.

Next consider X_{12}. Then

m_{D} = m_{12} = (Λ(x, y, θ_{7}, θ_{9}, u_{10}, θ^{1}_{11}, θ^{2}_{11}), d)

with d(θ_{7}) = xy, d(θ_{9}) = xθ_{7}, d(θ^{1}_{11}) = yθ_{7}, d(θ^{2}_{11}) = xθ_{9}. Since H^{13}(m_{D}) = (H^{+}(m_{D})(12))^{13} and A^{>13} = 0, M_{A^{∗}} is an one point.

(8) A^{∗} = H^{∗}((S^{3} × S^{8})♯(S^{3} × S^{8}); Q) = Λ(x_{3}, y_{3}) ⊗ Q[u_{8}, w_{8}]/(xy, xu, xw + yu, yw, u^{2}, uw, w^{2}).

It is 6-intrinsically formal Poincaré algebra of formal dimension 11 such that m_{6} = (Λ(x, y, θ_{5}), d) with d(x) = d(y) = 0 and d(θ_{5}) = xy. There is a map σ_{6} : (H^{∗}(m_{6})(7))^{∗} → A^{∗} given by σ_{6}(x) = x, σ_{6}(y) = y and sending other elements to zero. Since u = s = v = t = 2 at n = 7, we have 0 ⩽ l ⩽ 2. Consider the each cases of l = 0, 1, 2 at n = 7 in the followings.

Case of l = 0.

Since W = 0, H^{8}(m^{W}) = H^{8}(m_{6}) = Q{[xθ_{5}], [yθ_{5}]}. Put σ^{W} (x) = x, σ^{W} (y) = y, σ^{W} ([xθ_{5}]) = u and σ^{W} ([yθ_{5}]) = w. Then the condition (1)_{7} and (2)_{7} are satisfied. Since σ^{W} : H^{∗}(m^{W}) → A^{∗} is isomorphic, this one point set M_{0} = O_{0}, corresponding the elliptic model (Λ(x, y, θ_{5}), d), is a component of M_{A^{∗}}.

Case of l = 1.

For H^{8}(m_{6}) = Q{e_{1} = [xθ_{5}], e_{2} = [yθ_{5}]}, W is spanned by ae_{1} + be_{2} for [a, b] ∈ P^{1}(Q) = M_{1}. Then m^{W}_{8} = (Λ(x, y, θ_{5}, θ_{7}, u_{8}), d) where d(θ_{7}) = ae_{1} + be_{2} and d(u_{8}) = 0. But (σ^{W}_{8})^{11} : H^{3}(m^{W}_{8}) · H^{8}(m^{W}_{8}) → A^{3} · A^{8} can not be a G.A.map since x · (bxθ_{5} + ayθ_{5}) = d(yθ_{7}) and y · (bxθ_{5} + ayθ_{5}) = d(xθ_{7}). Hence the condition (2)_{7} is not satisfied.

Case of l = 2.

Since W = Q{xθ_{5}, yθ_{5}},

m^{W} = (Λ(x, y, θ_{5}, θ^{1}_{7}, θ^{2}_{7}), d)

where d(θ^{1}_{7}) = xθ_{5} and d(θ^{2}_{7}) = yθ_{5} and

m^{W}_{8} = (Λ(x, y, θ_{5}, θ^{1}_{7}, θ^{2}_{7}, u^{1}_{8}, u^{2}_{8}), d)

where du^{i}_{8} = 0 (i = 1, 2). Since t = 0 at 8 ⩽ n ⩽ 11 and A^{>11} = 0, it is one point corresponding to the formal model.

Thus M_{A^{∗}} is two points. See Fig 2.

Homology, Homotopy and Applications, vol. 5(1), 2003 435 In the following figures, numbers mean degrees. Fig 1

[figure: (6); 0; 8; 9; 9; 14; 15; 15; 16; 16; 17; 17; ⋯; ⋯; 14; 15; 16; 17; ⋯] The set P^{3}(Q)/Q^{∗} ∐{∗} is indicated by one circle. Fig 2

[figure: (8); 0; 2; 3; 4; 5; 6; 7; ⋯] Here ⨀ implies that there exists an elliptic minimal model generated by elements

of degree ⩽ 5 satisfying H^{∗}(m) ≅ A^{∗}.

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2003/n1a18/v5n1a18.(dvi,ps,pdf)

Hiroo Shiga shiga@sci.u-ryukyu.ac.jp

Department of Mathematical Sciences, Colledge of Science, Ryukyu University, Okinawa 903-0213, Japan

Toshihiro Yamaguchi tyamag@cc.kochi-u.ac.jp

Department of Mathematics Education, Faculty of Education, Kochi University, Kochi 780-8520, Japan