2 The odd dimensional case

We begin by recalling the statement of Theorem 1.

Theorem 1   Let Q be a smooth quadric hypersurface in $\mbox{$\bf P$}$^{n + 1}, where n = 2k + 1. Then for any positive integer d $\equiv$ 0  (mod 2^k) there exist continuous maps f : $\mbox{$\bf P$}$ⁿ$\to$Q, where f^*($\mbox{${\cal O}_{Q}$}$(1)) = $\mbox{${\cal O}_{\P^n}$}$(d ).

Proof:     This is obvious for n = 1. We may assume, by induction, that the theorem holds for quadrics of dimension n - 2. $\mbox{$\bf 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' $\equiv$ 0  (mod 2^{k - 1}), the induction hypothesis gives us a map f' : $\mbox{$\bf P$}$^{n - 2}$\to$Q' satisfying f'^*($\mbox{${\cal O}_{Q'}$}$(1)) = $\mbox{${\cal O}_{\P^{n-2}}$}$(d'). Since $\mbox{$\bf P$}$^{n - 1} is the projective cone over $\mbox{$\bf P$}$^{n - 2}, Construction 1.1 yields a map C(f') : $\mbox{$\bf P$}$^{n - 1}$\to$C. The obstruction to extending this map to a map $\mbox{$\bf P$}$ⁿ$\to$Q is a class O(C(f')) $\in$ H²ⁿ($\mbox{$\bf P$}$ⁿ,$\pi_{2n-1}^{}$(Q)). By Lemma 4, the group $\pi_{2n-1}^{}$(Q) has exponent 4. Let $\widetilde{f}$ be the composite

$$\mbox{$\bf P$}$$^{n - 1}$$\;\stackrel{\alpha}{\rightarrow}\;$$$$\mbox{$\bf P$}$$^{n - 1}$$\;\stackrel{C(f')}{\rightarrow}\;$$C,

where $\alpha$ is the restriction of the map $\beta$ : $\mbox{$\bf P$}$ⁿ$\to$$\mbox{$\bf P$}$ⁿ preserving the cell structure, and given in suitable homogeneous coordinates by

(z₀ : ^... : z_n) $$\mapsto$$ (z₀² : ^... : z_n²).

The obstruction to extending $\widetilde{f}$ to a map $\mbox{$\bf P$}$ⁿ$\to$Q is

O($$\widetilde{f}$$) = $$\beta^{*}_{}$$(O(C(f'))) $$\in$$ H²ⁿ($$\mbox{$\bf P$}$$ⁿ,$$\pi_{2n-1}^{}$$(Q)).

But clearly $\beta^{*}_{}$ = 0 on this cohomology group. Let f : $\mbox{$\bf P$}$ⁿ$\to$Q be an extension of $\widetilde{f}$. We have a commutative diagram of integral cohomology groups

H2(Q)	$$\;\stackrel{f^*}{\longrightarrow}\;$$	H2($$\mbox{$\bf P$}$$n)
$$\downarrow$$ $$\wr$$		$$\downarrow$$ $$\wr$$
H2(Q')	$$\;\stackrel{f''^*}{\longrightarrow}\;$$	H2($$\mbox{$\bf P$}$$n - 2)

where f'' is the composite of f' with the restriction of $\beta$. Since f''^* is multiplication by 2d', so is f^*. $\Box$
We now prove the following refinement of Theorem 2, in the odd dimensional case. 1 1'

Theorem 2   Let Q $\subset$ $\mbox{$\bf P$}$^{n + 1} be a smooth quadric hypersurface with n = 2k + 1. Then there exists a continuous map f : Q$\to$Q of degree dⁿ whenever

(viii)

d $\equiv$ 0  (mod 2^k), or

(ix)

d = e^{2n - 1}, for some integer e.

22 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)}$\to$Q^{(n + 2)} extends to a map f : Q$\to$Q. By induction, we may assume that F_{d, n - 2} : Q'⁽ⁿ⁾$\to$Q'⁽ⁿ⁾ extends to a map f' : Q'$\to$Q', for Q' a smooth quadric hypersurface of dimension n - 2, and d = e^{2n - 3} for some e > 0. We have constructed (see (1.1)) a map C(f') : C$\to$C which satisfies C(f')$\mid_{Q^{(n+2)}}^{}$ = F_{d, n}. By computing C(f')_* on $\pi_{2n-1}^{}$(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 $\pi_{2n-1}^{}$(C) given as follows. Take F₀ = $\pi_{2n-1}^{}$(C); F₁ is the kernel of the composite

$$\pi_{2n-1}^{}$$(C)$$\;\stackrel{\simeq}{\rightarrow}\;$$$$\pi_{2n-1}^{}$$($$\widetilde{C}$$)$$\;\stackrel{\gamma}{\rightarrow}\;$$H_{2n - 1}($$\widetilde{C}$$,$$\mbox{$\bf Z$}$$),

where $\gamma$ is the Hurewicz map; and finally F₂ = im ($\pi_{2n-1}^{}$(Q⁽ⁿ⁾)$\to$$\pi_{2n-1}^{}$(C)). Clearly C(f')_* is compatible with this filtration and induces a map gr ^FC(f')_* on gr ^F$\pi_{2n-1}^{}$(C). This map may be computed as follows.

Lemma 6  

(x)

F₂, F₁/F₂ are vector spaces over $\mbox{$\bf Z$}$/2.

(xi)

The natural composite map

$$\pi_{2n}^{}$$(Q, C)$$\to$$$$\pi_{2n-1}^{}$$(C)$$\to$$F₀/F₁ $$\hookrightarrow$$ H_{2n - 1}($$\widetilde{C}$$,$$\mbox{$\bf Z$}$$)

is an isomorphism, giving a direct sum decomposition

$$\pi_{2n-1}^{}$$(C) $$\cong$$ F₁ $$\oplus$$ $$\pi_{2n}^{}$$(Q, C).

(xii)

gr ^FC(f')_* is multiplication by dⁿ.

Proof:     We have a commutative diagram with exact bottom row

	H2n($$\widetilde{Q}$$,$$\widetilde{C}$$)	$$\;\stackrel{\partial}{\rightarrow}\;$$	H2n - 1($$\widetilde{C}$$,$$\mbox{$\bf Z$}$$)
	$$\alpha$$ $$\uparrow$$		$$\uparrow$$ $$\gamma$$
0$$\to$$	$$\pi_{2n}^{}$$($$\widetilde{Q}$$,$$\widetilde{C}$$)	$$\to$$	$$\pi_{2n-1}^{}$$($$\widetilde{C}$$)	$$\to$$$$\pi_{2n-1}^{}$$($$\widetilde{Q}$$)$$\to$$ 0

The boundary homomorphism $\partial$ is an isomorphism by the long exact sequence of homology for the pair ($\widetilde{Q}$,$\widetilde{C}$), and $\alpha$ is an isomorphism by the relative Hurewicz theorem. This proves (ii) and gives an isomorphism F₁ $\cong$ $\pi_{2n-1}^{}$($\widetilde{Q}$). The self map $\widetilde{C(f')}$ of $\widetilde{C}$ is of degree dⁿ and this gives (iii) for F₀/F₁. Since Q⁽ⁿ⁾ = L is a linear projective subspace of $\mbox{$\bf P$}$^{n + 1}, $\widetilde{L}$ is an Sⁿ in $\widetilde{Q}$; it is easy to check that, in the fibration (*), this maps isomorphically to a great sphere S $\subset$ S^{n + 1}. In fact $\widetilde{L}$ is a section of the sphere bundle of the tangent bundle of S which is contained in $\widetilde{Q}$ by Scholium 3. Let D^- be a hemisphere capping S in S^{n + 1} and let U be its inverse image in $\widetilde{Q}$. We have the following

Sublemma 7   Let i : Sⁿ$\to$$\widetilde{Q}$ be a fibre of (*) lying over a point of D^-. A unit tangent vector field v on S $\cong$ Sⁿ gives a map v : Sⁿ$\to$$\widetilde{Q}$ 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 $\ell$ : S×[0,$\pi$]$\to$D^- given by

(x, t) $$\mapsto$$ sin(t) ^. p + cos(t) ^. x.

For all (x, t) $\in$ S×[0,$\pi$] let n(x, t) be the tangent vector at the point $\ell$(x, t) given by sin(t) ^. x - cos(t) ^. p. Then, d$\ell_{(x,t)}^{}$v(x) is orthogonal to n(x, t) in the tangent space of S^{n + 1} at $\ell$(x, t) so that we get a map H : S×[0,$\pi$]$\to$$\widetilde{Q}$ given by the formula

(x, t) $$\mapsto$$ d$$\ell_{(x,t)}^{}$$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. $\Box$
Thus we have isomorphisms

F₂ $$\cong$$ im ($$\pi_{2n-1}^{}$$($$\widetilde{L}$$)$$\to$$$$\pi_{2n-1}^{}$$($$\widetilde{Q}$$)) = im ($$\pi_{2n-1}^{}$$(Sⁿ)$$\to$$$$\pi_{2n-1}^{}$$($$\widetilde{Q}$$)),

where Sⁿ$\to$Q is the inclusion of the fibre of (*). Hence F₂ is a vector space over $\mbox{$\bf Z$}$/2, and by Scholium 5 the action of gr ^FC(f')_* on it is by d^{k + 1} $\equiv$ dⁿ  (mod 2). Further, we obtain an isomorphism

F₁/F₂ $$\cong$$ $$\pi_{2n-1}^{}$$($$\widetilde{Q}$$)/im ($$\pi_{2n-1}^{}$$($$\widetilde{L}$$)) $$\cong$$ ₂($$\pi_{2n-1}^{}$$(S^{n + 1}, D^-)),

so that F₁/F₂ is a $\mbox{$\bf Z$}$/2-vector space. Let g : (D^{n + 1}, Sⁿ)$\to$($\widetilde{Q^{(n+1)}}$,$\widetilde{L}$) be the generator of $\pi_{n+1}^{}$($\widetilde{Q^{(n+1)}}$,$\widetilde{L}$) $\cong$ $\pi_{n+1}^{}$(Q^{(n + 1)}, L) $\cong$ $\mbox{$\bf Z$}$. We have a diagram, commutative upto homotopy,

(Dn + 1, Sn)	$$\to$$	($$\widetilde{Q^{(n+1)}}$$,$$\widetilde{L}$$)
$$\varphi_{d}^{}$$ $$\downarrow$$		$$\downarrow$$ $$\widetilde{F_{d,n}}$$
(Dn + 1, Sn)	$$\to$$	($$\widetilde{Q^{(n+1)}}$$,$$\widetilde{L}$$)

where $\varphi_{d}^{}$ is a map of degree d^{k + 1}. From the Scholium 5 we see that ($\varphi_{d}^{}$)_* induces multiplication by d^{k + 1} on $\pi_{2n-1}^{}$(D^{n + 1}, Sⁿ). By the sublemma we have isomorphisms

$$\pi_{n+1}^{}$$($$\widetilde{Q^{(n+1)}}$$,$$\widetilde{L}$$)$$\;\stackrel{\simeq}{\rightarrow}\;$$$$\pi_{n+1}^{}$$($$\widetilde{Q}$$,$$\widetilde{L}$$)$$\;\stackrel{\simeq}{\rightarrow}\;$$$$\pi_{n+1}^{}$$(S^{n + 1}, D^-),

so that the composite $\rho$ : (D^{n + 1}, Sⁿ)$\to$(S^{n + 1}, D^-) of g and the natural map ($\widetilde{Q^{(n+1)}}$,$\widetilde{L}$)$\to$(S^{n + 1}, D^-) is also a generator for $\pi_{n+1}^{}$(S^{n + 1}, D^-). By the Freudenthal suspension theorem, $\rho_{*}^{}$ is an isomorphism on $\pi_{2n-1}^{}$. From the diagram

	H2n($$\widetilde{Q}$$,$$\widetilde{C}$$;$$\mbox{$\bf Z$}$$)	$$\;\stackrel{\partial}{\rightarrow}\;$$	H2n - 1($$\widetilde{C}$$,$$\widetilde{L}$$;$$\mbox{$\bf Z$}$$)
	$$\alpha$$ $$\uparrow$$		$$\uparrow$$ $$\gamma$$
0$$\to$$	$$\pi_{2n}^{}$$($$\widetilde{Q}$$,$$\widetilde{C}$$)	$$\to$$	$$\pi_{2n-1}^{}$$($$\widetilde{C}$$,$$\widetilde{L}$$)	$$\to$$$$\pi_{2n-1}^{}$$($$\widetilde{Q}$$,$$\widetilde{L}$$)$$\to$$ 0

where $\alpha$, $\partial$ are isomorphisms,we see that

im ($$\pi_{2n-1}^{}$$(D^{n + 1}, Sⁿ)$$\to$$$$\pi_{2n-1}^{}$$($$\widetilde{C}$$,$$\widetilde{L}$$))
= ker($$\pi_{2n-1}^{}$$($$\widetilde{C}$$,$$\widetilde{L}$$)$$\to$$H_{2n - 1}($$\widetilde{C}$$,$$\widetilde{L}$$;$$\mbox{$\bf Z$}$$)).

In particular, F₁/F₂ is contained in this image; thus C(f')_* acts by multiplication by d^{k + 1} $\equiv$ dⁿ  (mod 2) on F₁/F₂,and this completes the proof of (iii). $\Box$
From this lemma we see that we have $\varphi_{d}^{}$ $\in$ Hom ($\pi_{2n}^{}$(Q, C), F₁) = F₁ and $\psi_{d}^{}$ $\in$ Hom (F₁/F₂, F₂) $\subset$ End (F₁) such that, for all pairs (a, b) in $\pi_{2n-1}^{}$(C) = $\pi_{2n}^{}$(Q, C) $\oplus$ F₁ we have the equation

C(f')_*(a, b) = (dⁿa, dⁿb + $$\psi_{d}^{}$$(b) + $$\varphi_{d}^{}$$(a)).

Since both F₁, F₁/F₂ are of exponent 2, it follows that the four-fold composite of C(f') satisfies

C(f')⁴_*(a, 0) = (d⁴ⁿa, 0),

and this proves the theorem. $\Box$