Continuous Self Maps of Quadric Hypersurfaces

K. H. Paranjape and V. Srinivas

Introduction

Let G be a simply connected, semi-simple algebraic group over C. Let P ⊂ G be a parabolic subgroup, and let Y = G/P be the homogeneous space. In a recent paper [KS], we showed that

if Y = G/P is as above and f : Y → Y is a finite (algebraic) self map of degree > 1, then Y ≅ P^{n}.

The paper arose out of an attempt to understand the following problem of Lazarsfeld [L]:

Problem -99 Suppose G is a semi-simple algebraic group over C, P ⊂ G a maximal parabolic subgroup, Y = G/P. Let f : Y → X be a finite, surjective morphism of degree > 1 to a smooth variety X; then is X ≅ P^{n}? (n = dim X = dim Y)

Lazarsfeld (loc. cit.) answers this in the affirmative when Y = P^{n}, using the proof by S. Mori [M] of Hartshorne’s conjecture. We also showed that Lazarsfeld’s problem has an affirmative answer if Y is a smooth quadric hypersurface of dimension ≥ 3. This includes the case of the Grassmannian Y = G(2, 4). The general case seems to be open even for other Grassmann varieties.

Our goal in this paper is to study the analogous problems for continuous maps. Our homogeneous spaces are all complex submanifolds of complex projective spaces CP^{n}, with the usual topology; we drop the C to simplify notation. We show

Theorem 1 Let Q be a smooth quadric hypersurface in P^{n+1}, where n = 2k + 1. Then for any positive integer d ≡ 0 (mod 2^{k}) there exist continuous maps f : P^{n} → Q satisfying f^{∗}(O_{Q}(1)) = OP^{n}(d).

Note that such maps have degree greater than one whenever d > 1. We also show Theorem 2 Let Q be a smooth quadric hypersurface in P^{n+1}. Then there exists a positive integer m and continuous maps of degree (m · d)^{n} from Q to Q, for all d ∈ N.

Clearly Theorem 1 implies Theorem 2 in the case when n is odd, since there is an obvious map Q → P^{n} of degree 2. However, we also construct self maps of Q of odd degree, when n is odd. Observe that the degree of any self map of Q is an n-th power.

Our proofs are by obstruction theory, using the standard (Bruhat) cell decomposition, and by induction on the dimension. The cases when n is even and odd are dealt with separately, and the induction is in steps of 2. From the computation of the homotopy groups of quadrics, such a division seems natural.

In the paper [KS], we had also proved:

Proposition 0 Let k ≤ n, 2 ≤ l ≤ m be integers, such that there exists a finite surjective morphism between Grassmann varieties

f : G(k, k + n) → G(l, l + m).

Then k = l, m = n and f is an isomorphism.

We do not know whether there are continuous maps of degree bigger than one f : G(k, k + n) → G(l, l + m) with k ≠ l, m ≠ n and kn = lm.

We wish to thank P. Polo and A. R. Shastri for stimulating discussions.

1 Preliminaries

1.1 Cell structure

We begin by recalling the cell decomposition of a quadric hypersurface in a form convenient for us. We will proceed by induction on dimension. The two smallest dimensions are :

n = 1: Q is P^{1} ↪ P^{2} as a conic and has a natural cell decomposition Q = C ∪ {∞}. We have for d ∈ N maps F_{d} : Q → Q (z ↦ z^{d}) preserving the cell structure.

n = 2: Q is P^{1} × P^{1} ↪ P^{2} via the Segre embedding and has a natural cell decomposition

Q = (C × C) ∪ (C × {∞}) ∪ ({∞} × C) ∪ {(∞, ∞)}.

For all d ∈ N, we have maps F_{d} : Q → Q ((z, w) ↦ (z^{d}, w^{d})) which preserve the cell structure.

We henceforth assume that n ≥ 3.

Scholium 1 Q has a cell decomposition with cells in each even (real) dimension (the Bruhat cell decomposition) such that we have:

1. For n ≥ 3, Q^{(2n−2)} = C is the projective cone over Q′ ⊂ P^{n−1}, a smooth quadric hypersurface of complex dimension n − 2.

2. There is exactly one cell in each even (real) dimension except in dimension n for n = 2k.

3. Q^{(n)} may be explicitly described as follows:

n = 2k: Q^{(n)} = L′ ∪ L′′, where L′, L′′ ≅ P^{k} are linear subspaces of

P^{n+1} and L′ ∩ L′′ = L ≅ P^{k−1}.

n = 2k + 1: Q^{(n)} = L ≅ P^{k} is a linear subspace of P^{n+1}.

4. For n ≥ 3, Q^{(n+2)} ⊂ C is the Thom space over Q^{′(n)} of the complex line bundle O_{Q^{′(n)}}(1).

Proof: Let p be any point of Q and let H ⊂ P^{n+1} be a hyperplane tangent to Q at p. Then Q ∩ H = C is the projective cone over Q′ ⊂ P^{n−1}, a smooth quadric hypersurface of complex dimension n−2 (i.e. C is the Thom space of the complex line bundle O_{Q′}(1) = OP_{n−1}(1) |_{Q′}). By induction on dimension we get a cell decomposition of Q′. If Q^{′(m)} is the m-skeleton of Q′, and C(Q^{′(m)}) is the Thom space of O_{Q^{′(m)}}(1), then C(Q^{′(m)}) − C(Q^{′(m−1)}) is a union of cells of dimension m + 2. Thus, we obtain a cell decomposition of C. Since Q − C = C^{n}, we obtain the desired cell structure. □

We shall use the following construction. Construction 1 Let X, Y be compact topological spaces, L and M complex line bundles on X and Y respectively. Let f : X → Y be a continuous map, such that there is an isomorphism φ : L^{⊗d} → f^{∗}(M), for some positive integer d. Then there exists a map Φ : L → M giving a commutative diagram

\begin{matrix}L & ⟶^{Φ} & M \\ ↓ & & ↓ \\ X & ⟶^{f} & Y \end{matrix}

where Φ is the d-th power map on fibres of the vertical arrows. The restriction of Φ to the S^{1}-bundles X̃, Ỹ of L, M respectively induces a map f̃ : X̃ → Ỹ, which has degree d along the fibres. If C(X, L) and C(Y, M) denote the Thom spaces of L and M respectively, we have a map C(f) = C(f, φ) : C(X, L) → C(Y, M) induced by Φ.

In particular, if X ⊂ P^{N} is a projective variety, f : X → X an algebraic self-map, L = M = O_{X}(1), and φ is an isomorphism of algebraic line bundles, then C(X, L) is the projective cone (in P^{N+1}) of X and C(f) can be regarded as an algebraic self-map of C(X, L). Note that φ is unique upto a scalar multiple.

We had noted the existence of morphisms F_{d} : Q → Q for each d > 0, for a smooth quadric Q of dimension one or two. These maps satisfy F_{d_{1}} ∘F_{d_{2}} = F_{d_{1}d_{2}} for all d_{1}, d_{2} > 0. By repeatedly applying the above constructions, we obtain

Lemma 2 For each d > 0, there is an algebraic morphism

F_{d,n} : Q^{(n+2)} → Q^{(n+2)}

and an algebraic isomorphism

φ_{d,n} : O_{Q^{(n+2)}}(d) → F_{d,n}^{∗}(O_{Q^{(n+2)}}(1))

such that

(i) under the identification (by Scholium 1)

Q^{(n+2)} = C(Q^{′(n)}, O_{Q^{′(n)}}(1))

we have

F_{d,n} = C(F_{d,n−2}, φ_{d,n−2})

.

(ii) F_{d_{1},n} ∘ F_{d_{2},n} = F_{d_{1}d_{2},n}.

□

If F_{d,n−2} : Q^{′(n)} → Q^{′(n)} extends to a continuous map f′ : Q′ → Q′, then the isomorphism φ_{d,n−2} can be extended to an isomorphism (of topological complex line bundles)

φ′ : O_{Q′}(d) → f^{′∗}(O_{Q′}(1)).

Then the induced continuous map C(f′) = C(f′, φ′) : C → C restricts to F_{d,n} on Q^{(n+2)} ⊂ C. Since Q is obtained from C by attaching C^{n} via a map a : S^{2n−1} → C, the map C(f′) : C → C extends to a map f : Q → Q if and only if C(f′)_{∗}([a]) = m[a] ∈ π_{2n−1}(C), for some m ∈ Z. Thus, we need to compute π_{2n−1}(C), and the action of C(f′)_{∗} on it.

1.2 Computation of Homotopy groups

We have the S^{1}-fibration (Hopf fibration) S^{2n+3} → P^{n+1}. Let Q̃ → Q be the restriction of this to Q.

Scholium 3 Q̃ is the total space of the unit sphere bundle of the tangent bundle of S^{n+1}. Hence we have a fibration

S^{n} → Q̃ → S^{n+1}.  (∗)

Proof: If (z_{0} : ⋯: z_{n}) are homogeneous coordinates on P^{n+1}, u_{i} = ℜ(z_{i}) and v_{i} = ℑ(z_{i}), then we may take

S^{2n+3} = {(u_{i}, v_{i}) | ∑ u_{i}^{2} + ∑ v_{i}^{2} = 2}.

We may assume that Q is defined in these coordinates by the equation ∑ z_{i}^{2} = 0, so that

Q̃ = {(u_{i}, v_{i}) | ∑ u_{i}^{2} = ∑ v_{i}^{2} = 1 and ∑ u_{i}v_{i} = 0}.

Hence, the projection to the u_{i}’s is a fibration of the required sort. □

For any subset A ⊂ Q, let Ã denote its inverse image in Q̃, so that there is an induced S^{1}-fibration Ã → A. For any abelian group G let _{2}G denote its 2-torsion subgroup and let G/2 = G ⊗ Z/2Z.

Lemma 4 (i) If n is odd, then for i ≤ 2n − 1, we have a short exact

sequence

0 → π_{i}(S^{n})/2 → π_{i}(Q̃) → _{2}π_{i}(S^{n+1}) → 0.

(ii) If n is even, then we have a split exact sequence for each i

0 → π_{i}(S^{n}) → π_{i}(Q̃) → π_{i}(S^{n+1}) → 0.

Proof: In the long exact sequence

⋯ → π_{i}(S^{n}) → π_{i}(Q̃) → π_{i}(S^{n+1}) →^{∂_{i}} π_{i−1}(S^{n}) → ⋯

the boundary maps ∂_{i} are the maps on π_{i} induced by a map (well defined upto homotopy) Δ : ΩS^{n+1} → S^{n} coming from the fibration (∗). Suppose s : S^{n} → ΩS^{n+1} = ΩΣS^{n} is the map inducing the suspension homomorphisms Σ_{i} : π_{i}(S^{n}) → π_{i+1}(S^{n+1}). Since (∗) is the spherical fibration associated to the tangent bundle of S^{n+1}, it is well known (see [W] IV, (10.4)) that if θ = Δ ∘ s,

[θ] = 1 + (−1)^{n+1} ∈ π_{n}(S^{n})

where 1 ∈ π_{n}(S^{n}) is the standard generator. From the Freudenthal Suspension theorem, Σ_{i} is an isomorphism for i ≤ 2n−2, and Σ_{2n−1} is a surjection.

We shall often make use of the following well known result (see [W] XI, (1.11), (1.12), (1.16)).

Scholium 5 Let f : S^{n} → S^{n} be any continuous map of degree d, where n > 1. Then the induced map

f_{∗} : π_{i}(S^{n}) → π_{i}(S^{n})

is

(i) multiplication by d if i < 2n − 1

(ii) multiplication by d on the torsion subgroup, for i = 2n − 1 (and in particular on π_{2n−1}, if n is odd)

(iii) multiplication by d^{2} on π_{2n−1}(S^{n}) ⊗ Q.

□

If n is even, then [θ] = 0 so that θ_{∗} = 0. Hence, ∂_{n+1} vanishes. Thus (∗) has a homotopy section and the long exact sequence for this fibration splits into short exact sequences as asserted in (ii).

If n is odd, Scholium 5 implies that θ_{∗} acts as multiplication by 2 on π_{i}(S^{n}) in the range i ≤ 2n − 1. This proves (i). □

2 The odd dimensional case

We begin by recalling the statement of Theorem 1.

Theorem 1 Let Q be a smooth quadric hypersurface in P^{n+1}, where n = 2k + 1. Then for any positive integer d ≡ 0 (mod 2^{k}) there exist continuous maps f : P^{n} → Q, where f^{∗}(O_{Q}(1)) = OP^{n}(d).

Proof: This is obvious for n = 1. We may assume, by induction, that the theorem holds for quadrics of dimension n − 2.

P^{n} has a cell decomposition where the skeleta are linear projective subspaces of smaller dimension. By Scholium 1 we have a cell decomposition of Q whose n − 1 skeleton is the projective cone C over a smooth quadric Q′ of dimension n − 2. For any d′ ≡ 0 (mod 2^{k−1}), the induction hypothesis gives us a map f′ : P^{n−2} → Q′ satisfying f^{′∗}(O_{Q′}(1)) = OP_{n−2}(d′). Since P^{n−1} is the projective cone over P^{n−2}, Construction 1.1 yields a map C(f′) : P^{n−1} → C.

The obstruction to extending this map to a map P^{n} → Q is a class O(C(f′)) ∈ H^{2n}(P^{n}, π_{2n−1}(Q)). By Lemma 4, the group π_{2n−1}(Q) has exponent 4. Let f̃ be the composite

P^{n−1} →^{α} P^{n−1} →^{C(f′)} C,

where α is the restriction of the map β : P^{n} → P^{n} preserving the cell structure, and given in suitable homogeneous coordinates by

(z_{0} : ⋯: z_{n}) ↦ (z_{0}^{2} : ⋯: z_{n}^{2}).

The obstruction to extending f̃ to a map P^{n} → Q is

O(f̃) = β^{∗}(O(C(f′))) ∈ H^{2n}(P^{n}, π_{2n−1}(Q)).

But clearly β^{∗} = 0 on this cohomology group.

Let f : P^{n} → Q be an extension of f̃. We have a commutative diagram of integral cohomology groups

H^{2}(Q) ⟶^{f^{∗}} H^{2}(P^{n})
↓ ≀  ↓ ≀
H^{2}(Q′) ⟶^{f^{′′∗}} H^{2}(P^{n−2})

where f′′ is the composite of f′ with the restriction of β. Since f^{′′∗} is multiplication by 2d′, so is f^{∗}. □

We now prove the following refinement of Theorem 2, in the odd dimensional case.

Theorem 2′ Let Q ⊂ P^{n+1} be a smooth quadric hypersurface with n = 2k+1. Then there exists a continuous map f : Q → Q of degree d^{n} whenever

(i) d ≡ 0 (mod 2^{k}), or
(ii) d = e^{2^{n−1}}, for some integer e.

Proof: Clearly (i) follows from Theorem 1. We prove (ii) by showing that for the chosen integers d the map F_{d,n} : Q^{(n+2)} → Q^{(n+2)} extends to a map f : Q → Q. By induction, we may assume that F_{d,n−2} : Q^{′(n)} → Q^{′(n)} extends to a map f′ : Q′ → Q′, for Q′ a smooth quadric hypersurface of dimension n − 2, and d = e^{2^{n−3}} for some e > 0. We have constructed (see (1.1)) a map C(f′) : C → C which satisfies C(f′) |_{Q^{(n+2)}}= F_{d,n}. By computing C(f′)_{∗} on π_{2n−1}(C) we will show that the obstruction to extending the four-fold composite C(f′)^{4} to a self map of Q vanishes.

There is a filtration F on π_{2n−1}(C) given as follows. Take F_{0} = π_{2n−1}(C); F_{1} is the kernel of the composite

π_{2n−1}(C) →^{≃} π_{2n−1}(C̃) →^{γ} H_{2n−1}(C̃, Z),

where γ is the Hurewicz map; and finally F_{2} = im (π_{2n−1}(Q^{(n)}) → π_{2n−1}(C)).

Clearly C(f′)_{∗} is compatible with this filtration and induces a map gr^{F} C(f′)_{∗} on gr^{F} π_{2n−1}(C). This map may be computed as follows.

Lemma 6 (i) F_{2}, F_{1}/F_{2} are vector spaces over Z/2.

(ii) The natural composite map

π_{2n}(Q, C) → π_{2n−1}(C) → F_{0}/F_{1} ↪ H_{2n−1}(C̃, Z)

is an isomorphism, giving a direct sum decomposition

π_{2n−1}(C) ≅ F_{1} ⊕ π_{2n}(Q, C).

(iii) gr^{F} C(f′)_{∗} is multiplication by d^{n}.

Proof: We have a commutative diagram with exact bottom row

H_{2n}(Q̃, C̃) →^{∂} H_{2n−1}(C̃, Z)
α ↑  ↑ γ
0 → π_{2n}(Q̃, C̃) → π_{2n−1}(C̃) → π_{2n−1}(Q̃) → 0

The boundary homomorphism ∂ is an isomorphism by the long exact sequence of homology for the pair (Q̃, C̃), and α is an isomorphism by the relative Hurewicz theorem. This proves (ii) and gives an isomorphism F_{1} ≅ π_{2n−1}(Q̃). The self map { \atop C(f′)} of C̃ is of degree d^{n} and this gives (iii) for F_{0}/F_{1}.

Since Q^{(n)} = L is a linear projective subspace of P^{n+1}, L̃ is an S^{n} in Q̃; it is easy to check that, in the fibration (∗), this maps isomorphically to a great sphere S ⊂ S^{n+1}. In fact L̃ is a section of the sphere bundle of the tangent bundle of S which is contained in Q̃ by Scholium 3. Let D^{−} be a hemisphere capping S in S^{n+1} and let U be its inverse image in Q̃. We have the following

Sublemma 7 Let i : S^{n} → Q̃ be a fibre of (∗) lying over a point of D^{−}. A unit tangent vector field v on S ≅ S^{n} gives a map v : S^{n} → Q̃ which is homotopic within U to the inclusion i.

Proof: Let p be the point of D^{−} orthogonal to S (i.e. the “pole”). We have a map ℓ : S × [0, π] → D^{−} given by

(x, t) ↦ sin(t) · p + cos(t) · x.

For all (x, t) ∈ S × [0, π] let n(x, t) be the tangent vector at the point ℓ(x, t) given by sin(t) · x − cos(t) · p. Then, dℓ_{(x,t)}v(x) is orthogonal to n(x, t) in the tangent space of S^{n+1} at ℓ(x, t) so that we get a map H : S × [0, π] → Q̃ given by the formula

(x, t) ↦ dℓ_{(x,t)}v(x) + sin(t) · n(x, t).

Clearly H(x, 0) = v(x) and H(x, t) = x considered as a tangent vector at p. □

Thus we have isomorphisms

F_{2} ≅ im (π_{2n−1}(L̃) → π_{2n−1}(Q̃)) = im (π_{2n−1}(S^{n}) → π_{2n−1}(Q̃)),

where S^{n} → Q is the inclusion of the fibre of (∗). Hence F_{2} is a vector space over Z/2, and by Scholium 5 the action of gr^{F} C(f′)_{∗} on it is by d^{k+1} ≡ d^{n} (mod 2). Further, we obtain an isomorphism

F_{1}/F_{2} ≅ π_{2n−1}(Q̃)/im (π_{2n−1}(L̃)) ≅ _{2}(π_{2n−1}(S^{n+1}, D^{−})),

so that F_{1}/F_{2} is a Z/2-vector space.

Let g : (D^{n+1}, S^{n}) → ({ \atop Q^{(n+1)}}, L̃) be the generator of π_{n+1}({ \atop Q^{(n+1)}}, L̃) ≅

π_{n+1}(Q^{(n+1)}, L) ≅ Z. We have a diagram, commutative upto homotopy,

(D^{n+1}, S^{n}) → ({ \atop Q^{(n+1)}}, L̃)
φ_{d} ↓  ↓ F̃_{d,n}
(D^{n+1}, S^{n}) → ({ \atop Q^{(n+1)}}, L̃)

where φ_{d} is a map of degree d^{k+1}. From the Scholium 5 we see that (φ_{d})_{∗} induces multiplication by d^{k+1} on π_{2n−1}(D^{n+1}, S^{n}).

By the sublemma we have isomorphisms

π_{n+1}({ \atop Q^{(n+1)}}, L̃) →^{≃} π_{n+1}(Q̃, L̃) →^{≃} π_{n+1}(S^{n+1}, D^{−}),

so that the composite ρ : (D^{n+1}, S^{n}) → (S^{n+1}, D^{−}) of g and the natural

map ({ \atop Q^{(n+1)}}, L̃) → (S^{n+1}, D^{−}) is also a generator for π_{n+1}(S^{n+1}, D^{−}). By the Freudenthal suspension theorem, ρ_{∗} is an isomorphism on π_{2n−1}. From the diagram

H_{2n}(Q̃, C̃; Z) →^{∂} H_{2n−1}(C̃, L̃; Z)
α ↑  ↑ γ
0 → π_{2n}(Q̃, C̃) → π_{2n−1}(C̃, L̃) → π_{2n−1}(Q̃, L̃) → 0

where α, ∂ are isomorphisms,we see that

im (π_{2n−1}(D^{n+1}, S^{n}) → π_{2n−1}(C̃, L̃)) = ker(π_{2n−1}(C̃, L̃) → H_{2n−1}(C̃, L̃; Z)).

In particular, F_{1}/F_{2} is contained in this image; thus C(f′)_{∗} acts by multiplication by d^{k+1} ≡ d^{n} (mod 2) on F_{1}/F_{2},and this completes the proof of (iii). □

From this lemma we see that we have φ_{d} ∈ Hom (π_{2n}(Q, C), F_{1}) = F_{1} and ψ_{d} ∈ Hom (F_{1}/F_{2}, F_{2}) ⊂ End (F_{1}) such that, for all pairs (a, b) in π_{2n−1}(C) = π_{2n}(Q, C) ⊕ F_{1} we have the equation

C(f′)_{∗}(a, b) = (d^{n}a, d^{n}b + ψ_{d}(b) + φ_{d}(a)).

Since both F_{1}, F_{1}/F_{2} are of exponent 2, it follows that the four-fold composite of C(f′) satisfies

C(f′)_{∗}^{4}(a, 0) = (d^{4n}a, 0),

and this proves the theorem. □

3 The even dimensional case

In this section we prove the following theorem Theorem 2 Let Q be a smooth quadric hypersurface of dimension n = 2k. There is an integer m such that for any positive d ≡ 0 (mod m), the map F_{d,n} : Q^{(n+2)} → Q^{(n+2)} extends to a continuous map f : Q → Q (of degree d^{n}).

Proof: Suppose that for some d > 0, the map F_{d,n−2} : Q^{′(n)} → Q^{′(n)} extends to a map f′ : Q′ → Q′. Then the map C(f′) : C → C, obtained by the construction of (1.1), restricts to F_{d,n} : Q^{(n+2)} → Q^{(n+2)}. We compute below the obstruction to extending C(f′) to a map f : Q → Q.

As in the odd dimensional case, we begin by observing that we have the diagram with exact bottom row

H_{2n}(Q̃, C̃) →^{∂} H_{2n−1}(C̃, Z)
α ↑  ↑ γ
0 → π_{2n}(Q̃, C̃) → π_{2n−1}(C̃) → π_{2n−1}(Q̃) → 0

where the vertical maps are Hurewicz maps. Since ∂, α are isomorphisms, the composite

π_{2n}(Q, C) → π_{2n−1}(C) ≅ π_{2n−1}(C̃) →^{γ} H_{2n−1}(C̃)

is an isomorphism, and

π_{2n−1}(C) ≅ π_{2n}(Q, C) ⊕ π_{2n−1}(Q) ≅ π_{2n}(Q̃, C̃) ⊕ π_{2n−1}(Q̃).

From Lemma 4, we have a split exact sequence for each i

0 → π_{i}(S^{n}) → π_{i}(Q̃) → π_{i}(S^{n+1}) → 0,

where the splitting is obtained from a homotopy section of (∗). Since Q^{(n)} = Q^{(n+1)}, the map π_{i}(Q^{(n)}) → π_{i}(Q) is an isomorphism for i = n, and a surjection for i = n + 1. In particular, the homotopy section of (∗), and the inclusion of the fibre S^{n} ⊂ Q̃ of (∗), factor through Q^{(n)}. Hence π_{2n−1}(Q^{(n)}) → π_{2n−1}(Q) is surjective. Thus

π_{2n−1}(C) ≅ π_{2n}(Q, C) ⊕ im (π_{2n−1}(Q^{(n)})).

We can refine this a little. Since π_{n}(Q^{(n)}) ≅ π_{n}(Q) ≅ Z, there is a map g̃ : S^{n} → Q̃^{(n)}, inducing a map g : S^{n} → Q^{(n)}, such that g̃, g represent generators of π_{n}, and g̃ is homotopic to the inclusion of the fibre of (∗); further,

π_{n}(Q̃^{(n)}) ≅ H_{n}(Q̃^{(n)}, Z) ↪ H_{n}(Q^{(n)}, Z),

and (F_{d,n})_{∗} acts by multiplication by d^{k} on H_{n}(Q^{(n)}, Z). Then there is an inclusion h : π_{2n−1}(S^{n}) ↪ π_{2n−1}(C) induced by g and a homotopy commutative diagram

S^{n} →^{μ} S^{n}
g ↓  ↓ g
Q^{(n)} →^{F_{d,n}} Q^{(n)}

where μ has degree d^{k}.

Next, an easy computation shows that L̃′ ≅ S^{n+1} ⊂ Q̃^{(n)} ⊂ Q̃ maps isomorphically onto S^{n+1} in the fibration (∗). Further F_{d,n} restricts to a self map of L′ of degree d^{k}.

Thus we have a decomposition

im (π_{2n−1}(Q^{(n)}) → π_{2n−1}(C)) = h_{∗}(π_{2n−1}(S^{n})) ⊕ π_{2n−1}(L′).

The action of C(f′)_{∗} on the left is compatible with this decomposition, and induces μ_{∗} on π_{2n−1}(S^{n}), and (F_{d,n})_{∗} on π_{2n−1}(L′). Note that since n is even, π_{2n−1}(S^{n}) ≅ Z ⊕ π_{2n−1}(S^{n})_{tors} (where the subscript “tors” denotes the torsion subgroup).

Lemma 8 (i) C(f′)_{∗} acts by multiplication by d^{n} on π_{2n−1}(C) ⊗ Q.

(ii) With respect to the direct sum decomposition

π_{2n−1}(C) = π_{2n}(Q, C) ⊕ (Z ⊕ π_{2n−1}(S^{n})_{tors}) ⊕ π_{2n−1}(L′),

C(f′)_{∗} has a matrix of the form

( \begin{matrix}d^{n} & 0 & 0 & 0 \\ 0 & d^{n} & 0 & 0 \\ φ_{1} & φ_{3} & d^{k} & 0 \\ φ_{2} & 0 & 0 & d^{k+1} \end{matrix} )

where

φ_{1} ∈ Hom (π_{2n}(Q, C), π_{2n−1}(S^{n})_{tors})
φ_{2} ∈ Hom (π_{2n}(Q, C), π_{2n−1}(S^{n+1}))
φ_{3} ∈ Hom (Z, π_{2n−1}(S^{n})_{tors})

Proof: From the Scholium 5, the action of μ_{∗} on π_{2n−1}(S^{n})_{tors} is by d^{k}, while it is by d^{2k} = d^{n} on π_{2n−1}(S^{n}) ⊗ Q. The action of (F_{d,n})_{∗} on

π_{n+1}(L̃′) ≅ H_{n+1}(L̃′, Z) ≅ H_{n}(L′, H_{1}(S^{1}))

is by d^{k+1}. From the Scholium, this implies that (F_{d,n})_{∗} acts by d^{k+1} on π_{2n−1}(L′). Hence (ii) follows, once we prove (i).

Now Q^{(n)} = L′ ∪ L′′ where L′ ∩ L′′ = L ≅ P^{k−1}. Since π_{n}(L) is finite, we see that π_{n}(Q^{(n)}) → π_{n}(Q^{(n)}, L) is injective. We have a diagram, whose vertical arrows are Hurewicz maps,

H_{n}(Q^{(n)}, Z) →^{≅} H_{n}(Q^{(n)}, L; Z)
↑  ↑ ≀
π_{n}(Q^{(n)}) → π_{n}(Q^{(n)}, L)

so that the Hurewicz map on π_{n}(Q^{(n)}) is injective.

Consider the quotient map

Q^{(n)} = L′ ∪ L′′ → (L′ ∪ L′′)/L ≅ S^{n} ∨ S^{n}.

This induces an isomorphism on H_{n}, and hence an injection on π_{n}. We have a diagram

Q^{(n)}  ⟶^{F_{d,n}} Q^{(n)}
↓  ↓
S^{n} ∨ S^{n} ⟶^{ρ} S^{n} ∨ S^{n}

where ρ = ρ′ ∨ ρ′′, and ρ′, ρ′′ are self maps of S^{n} of degree d^{k}.

Let (S^{n} ∨S^{n})⊗Q be the space obtained from S^{n} ∨S^{n} by localising at Q. The map Q^{(n)} → (S^{n} ∨ S^{n}) ⊗ Q extends to a map ψ : C → (S^{n} ∨ S^{n}) ⊗ Q, since π_{i}(S^{n} ∨ S^{n}) ⊗ Q = 0 for n < i < 2n − 1. Further, the diagram

C  →^{C(f′)}  C
ψ ↓  ↓ ψ
S^{n} ∨ S^{n} ⊗ Q →^{ρ} S^{n} ∨ S^{n} ⊗ Q

commutes upto homotopy, since there are no obstructions to extending the constant homotopy on Q^{(n)}.

The map ψ_{∗} : π_{2n−1}(C) ⊗ Q → π_{2n−1}((S^{n} ∨ S^{n}) ⊗ Q) is injective on the summand

(π_{2n−1}(S^{n}) ⊕ π_{2n−1}(L′)) ⊗ Q = π_{2n−1}(S^{n}) ⊗ Q

by construction. Hence the map

ν : π_{2n−1}(C) ⊗ Q → H_{2n−1}(C̃, Z) ⊕ π_{2n−1}((S^{n} ∨ S^{n}) ⊗ Q)

is injective. The action of C(f′)_{∗} ⊗ Q is obtained by restricting the action of { \atop C(f′)}_{∗} ⊕ (ρ ⊗ Q)_{∗} to the image of ν.

We have an isomorphism (see [W] XI (1.6), (1.7))

π_{2n−1}(S^{n} ∨ S^{n}) ≅ π_{2n}(S^{n} × S^{n}) ⊕ π_{2n−1}(S^{n}) ⊕ π_{2n−1}(S^{n}),

and ρ_{∗} acts by (ρ′ × ρ′′)_{∗} on the first summand, and by ρ′_{∗} = ρ′′_{∗} on the other two summands. Since ρ′, ρ′′ have degree d^{k}, one easily computes from Scholium 5 that (ρ ⊗ Q)_{∗} acts by d^{2k} = d^{n} on π_{2n−1}((S^{n} ∨ S^{n}) ⊗ Q).

Finally, we have an isomorphism H_{2n−1}(C̃, Z) ≅ H_{2n−2}(C, H_{1}(S^{1})), so that { \atop C(f′)}_{∗} acts on H_{2n−1}(C̃, Z) by d^{n}. This completes the proof of (i). □

We now easily complete the proof of the Theorem. Assume by induction that, for all d ≡ 0 (mod m′), the map F_{d,n−2} : Q^{′(n)} → Q^{′(n)} extends to a map f′ : Q′ → Q′. Let m = (m′N)^{2}, where N annihilates π_{2n−1}(C)_{tors}. If d ≡ 0 (mod m), then d = d_{1}d_{2} where d_{1} = em′N and d_{2} = m′N for some integer e. We then have self maps f′_{1}, f′_{2} extending F_{d_{1},n−2}, F_{d_{2},n−2} respectively. Then C(f′_{1} ∘ f′_{2}) = C(f′_{1}) ∘ C(f′_{2}) is an extension of F_{d,n}, and one readily computes from the above lemma that it acts by multiplication by d^{n} on π_{2n−1}(C). Hence C(f′_{1} ∘ f′_{2}) extends to a self map of Q. □

References

[L] R. Lazarsfeld Some applications of the theory of Positive vector bundles in Complete Intersections, C.I.M.E., Acireale 1983. Lect. Notes. Math. 1092, Springer-Verlag, Berlin, Heidelberg, New York (1984).

[M] S. Mori Projective manifolds with ample tangent bundles, Ann. Math. 110 (1979) 593-606.

[KS] K. H. Paranjape and V. Srinivas Self maps of homogeneous spaces. preprint.

[W] G. W. Whitehead Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer-Verlag, Berlin, Heidelberg, New York (1978).