next up previous
Next: 3 The even dimensional Up: Continuous Self Maps of Previous: 1.2 Computation of Homotopy

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 2k) 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 2k - 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

$\displaystyle \mbox{$\bf P$}$n - 1$\displaystyle \;\stackrel{\alpha}{\rightarrow}\;$$\displaystyle \mbox{$\bf P$}$n - 1$\displaystyle \;\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 : ... : zn) $\displaystyle \mapsto$ (z02 : ... : zn2).

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

O($\displaystyle \widetilde{f}$) = $\displaystyle \beta^{*}_{}$(O(C(f'))) $\displaystyle \in$ H2n($\displaystyle \mbox{$\bf P$}$n,$\displaystyle \pi_{2n-1}^{}$(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) $\displaystyle \;\stackrel{f^*}{\longrightarrow}\;$ H2($\displaystyle \mbox{$\bf P$}$n)
$\displaystyle \downarrow$ $\displaystyle \wr$   $\displaystyle \downarrow$ $\displaystyle \wr$
H2(Q') $\displaystyle \;\stackrel{f''^*}{\longrightarrow}\;$ H2($\displaystyle \mbox{$\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
d $ \equiv$ 0  (mod 2k), or
d = e2n - 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 Fd, n : Q(n + 2)$ \to$Q(n + 2) extends to a map f : Q$ \to$Q. By induction, we may assume that Fd, n - 2 : Q'(n)$ \to$Q'(n) extends to a map f' : Q'$ \to$Q', for Q' a smooth quadric hypersurface of dimension n - 2, and d = e2n - 3 for some e > 0. We have constructed (see (1.1)) a map C(f') : C$ \to$C which satisfies C(f')$ \mid_{Q^{(n+2)}}^{}$ = Fd, 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

$\displaystyle \pi_{2n-1}^{}$(C)$\displaystyle \;\stackrel{\simeq}{\rightarrow}\;$$\displaystyle \pi_{2n-1}^{}$($\displaystyle \widetilde{C}$)$\displaystyle \;\stackrel{\gamma}{\rightarrow}\;$H2n - 1($\displaystyle \widetilde{C}$,$\displaystyle \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  
F2, F1/F2 are vector spaces over $ \mbox{$\bf Z$}$/2.
The natural composite map

$\displaystyle \pi_{2n}^{}$(Q, C)$\displaystyle \to$$\displaystyle \pi_{2n-1}^{}$(C)$\displaystyle \to$F0/F1 $\displaystyle \hookrightarrow$ H2n - 1($\displaystyle \widetilde{C}$,$\displaystyle \mbox{$\bf Z$}$)

is an isomorphism, giving a direct sum decomposition

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

gr FC(f')* is multiplication by dn.

Proof:     We have a commutative diagram with exact bottom row

  H2n($\displaystyle \widetilde{Q}$,$\displaystyle \widetilde{C}$) $\displaystyle \;\stackrel{\partial}{\rightarrow}\;$ H2n - 1($\displaystyle \widetilde{C}$,$\displaystyle \mbox{$\bf Z$}$)  
  $\displaystyle \alpha$ $\displaystyle \uparrow$       $\displaystyle \uparrow$ $\displaystyle \gamma$  
0$\displaystyle \to$ $\displaystyle \pi_{2n}^{}$($\displaystyle \widetilde{Q}$,$\displaystyle \widetilde{C}$) $\displaystyle \to$ $\displaystyle \pi_{2n-1}^{}$($\displaystyle \widetilde{C}$) $\displaystyle \to$$\displaystyle \pi_{2n-1}^{}$($\displaystyle \widetilde{Q}$)$\displaystyle \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$ Sn + 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 Sn + 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) $\displaystyle \mapsto$ 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 Sn + 1 at $ \ell$(x, t) so that we get a map H : S×[0,$ \pi$]$ \to$$ \widetilde{Q}$ given by the formula

(x, t) $\displaystyle \mapsto$ d$\displaystyle \ell_{(x,t)}^{}$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 $\displaystyle \cong$ im ($\displaystyle \pi_{2n-1}^{}$($\displaystyle \widetilde{L}$)$\displaystyle \to$$\displaystyle \pi_{2n-1}^{}$($\displaystyle \widetilde{Q}$)) = im ($\displaystyle \pi_{2n-1}^{}$(Sn)$\displaystyle \to$$\displaystyle \pi_{2n-1}^{}$($\displaystyle \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 dk + 1 $ \equiv$ dn  (mod 2). Further, we obtain an isomorphism

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

so that F1/F2 is a $ \mbox{$\bf Z$}$/2-vector space. Let g : (Dn + 1, Sn)$ \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) $\displaystyle \to$ ($\displaystyle \widetilde{Q^{(n+1)}}$,$\displaystyle \widetilde{L}$)
$\displaystyle \varphi_{d}^{}$ $\displaystyle \downarrow$   $\displaystyle \downarrow$ $\displaystyle \widetilde{F_{d,n}}$
(Dn + 1, Sn) $\displaystyle \to$ ($\displaystyle \widetilde{Q^{(n+1)}}$,$\displaystyle \widetilde{L}$)

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

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

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

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

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

im ($\displaystyle \pi_{2n-1}^{}$(Dn + 1, Sn)$\displaystyle \to$$\displaystyle \pi_{2n-1}^{}$($\displaystyle \widetilde{C}$,$\displaystyle \widetilde{L}$))
= ker($\displaystyle \pi_{2n-1}^{}$($\displaystyle \widetilde{C}$,$\displaystyle \widetilde{L}$)$\displaystyle \to$H2n - 1($\displaystyle \widetilde{C}$,$\displaystyle \widetilde{L}$;$\displaystyle \mbox{$\bf Z$}$)).

In particular, F1/F2 is contained in this image; thus C(f')* acts by multiplication by dk + 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 + $\displaystyle \psi_{d}^{}$(b) + $\displaystyle \varphi_{d}^{}$(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$

next up previous
Next: 3 The even dimensional Up: Continuous Self Maps of Previous: 1.2 Computation of Homotopy
Kapil Hari Paranjape 2002-11-21