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## 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 1 2 as a conic and has a natural cell decomposition Q = {}. We have for d maps Fd : QQ ( z zd) preserving the cell structure.
n = 2:
Q is 1×1 2 via the Segre embedding and has a natural cell decomposition

Q = (×) (×{}) ({) {(,)}.

For all d , we have maps Fd : QQ ( (z, w) (zd, wd)) 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' 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'' k are linear subspaces of n + 1 and L' L'' = L k - 1.
n = 2k + 1:
Q(n) = L k is a linear subspace of n + 1.
4. For n 3, Q(n + 2) C is the Thom space over Q'(n) of the complex line bundle (1).

Proof:     Let p be any point of Q and let H n + 1 be a hyperplane tangent to Q at p. Then Q H = C is the projective cone over Q' n - 1, a smooth quadric hypersurface of complex dimension n - 2 (i.e. C is the Thom space of the complex line bundle (1) = (1)). 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 (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 = 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 : XY be a continuous map, such that there is an isomorphism : L df*(M), for some positive integer d. Then there exists a map : LM giving a commutative diagram

 L M X Y

where is the d-th power map on fibres of the vertical arrows. The restriction of to the S1-bundles , of L, M respectively induces a map : , 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,) : C(X, L)C(Y, M) induced by .

In particular, if X N is a projective variety, f : XX an algebraic self-map, L = M = (1), and is an isomorphism of algebraic line bundles, then C(X, L) is the projective cone (in N + 1) of X and C(f ) can be regarded as an algebraic self-map of C(X, L). Note that is unique upto a scalar multiple. We had noted the existence of morphisms Fd : QQ for each d > 0, for a smooth quadric Q of dimension one or two. These maps satisfy Fd1oFd2 = Fd1d2 for all d1, d2 > 0. By repeatedly applying the above constructions, we obtain

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

Fd, n : Q(n + 2)Q(n + 2)

and an algebraic isomorphism

: (d )Fd, n*((1))

such that
(i)
under the identification (by Scholium 1)

Q(n + 2) = C(Q'(n),(1))

we have

Fd, n = C(Fd, n - 2,)

.
(ii)
Fd1, noFd2, n = Fd1d2, n.

If Fd, n - 2 : Q'(n)Q'(n) extends to a continuous map f' : Q'Q', then the isomorphism can be extended to an isomorphism (of topological complex line bundles)

: (d )f'*((1)).

Then the induced continuous map C(f') = C(f',) : CC restricts to Fd, n on Q(n + 2) C. Since Q is obtained from C by attaching n via a map a : S2n - 1C, the map C(f') : CC extends to a map f : QQ if and only if C(f')*([a]) = m[a] (C), for some m . Thus, we need to compute (C), and the action of C(f')* on it.

Next: 1.2 Computation of Homotopy Up: 1 Preliminaries Previous: 1 Preliminaries
Kapil Hari Paranjape 2002-11-21