Intersection form

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[edit] 1 Introduction

For a closed oriented manifold (topological or smooth) X of dimension 2n the intersection form

\displaystyle  q_X: H^{n}(X;\mathbb{Z}) \times H^{n}(X;\mathbb{Z}) \to \mathbb{Z}

is obtained by the formula

\displaystyle  q_X(x,y) = \langle x \smile y , [X] \rangle ,

i.e. the cup product of x and y is evaluated on the fundamental cycle given by the manifold X.

By bilinearity q_X vanishes on torsion elements, hence the map q_X descends to a bilinear pairing on the free module H^{n}(X;\mathbb{Z}) / \text{Torsion} which we also denote q_X. This pairing is uni-modular (in particular non-degenerate) by Poincaré duality.

  • If n is even the pairing q_X is symmetric: q_X(x, y) = q_X(y, x).
  • If n is odd the pairing q_X is skew-symmetric: q_X(x, y) = - q_X(y, x).

[edit] 2 Uni-modular bilinear forms

Let q and q' be unimodular symmetric bilinear forms on underlying free \mathbb{Z}-modules V and V' respectively. The two forms q and q' are said equivalent if there is an isomorphism f:V \to V' such that q = f^* q'.

A form q is called definite if it is positive or negative definite, otherwise it is called indefinite. The rank of q is the rank of the underlying \mathbb{Z}-module V.

[edit] 3 Skew-symmetric bilinear forms

The skew-symmetric hyperbolic form of rank 2, H_-(\Zz), is defined by the following intersection matrix

\displaystyle  \left( \begin{array}{cc} ~0 & ~1 \\ -1 & ~0 \end{array} \right) .

Proposition 3.1. Every skey-symmetri uni-modular bilinear form over \Zz, q, isomorphic to some the sum of some number of hyperbolic forms:

\displaystyle  q \cong \oplus_{i=1}^r H_-(\Zz).

In particular the rank of q, in this case 2r, is even.

[edit] 4 Symmetric bilinear forms

The classification of uni-modular definite symmetric bilinear forms is a deep and difficult problem. However the situation becomes much easier when the form is indefinite. We begin by stating some fundamental invariants.

Since q is symmetric it is diagonalisable over the real numbers. If b^+ denotes the dimension of a maximal subspace on which the form is positive definite, and if b^- is the dimension of a maximal subspace on which the form is negative definite, then the signature of q is defined to be

\displaystyle  \text{sign}(q) = b^+ - b^-.

The form q may have two different types. It is of type even if q(x,x) is an even number for any element x. Equivalently, if q is written as a square matrix in a basis, it is even if the elements on the diagonal are all even. Otherwise, q is said of type odd.

[edit] 4.1 Classification of indefinite forms

There is a simple classification result of indefinite forms [Serre1970],[Milnor&Husemoller1973]:

Theorem 4.1 (Serre (?)). Two indefinite unimodular symmetric bilinear forms q, q' over \mathbb{Z} are equivalent if and only if q and q' have the same rank, signature and type.

There is a further invariant of a unimodular symmetric bilinear form q on V: An element c \in V is called a characteristic vector of the form if one has

\displaystyle  q(c,x) \equiv q(x,x) \ (\text{mod} \ 2)

for all elements x \in V. Characteristic vectors always exist. In fact, when reduced modulo 2, the map x \mapsto q(x,x) \in \mathbb{Z}/2 is linear. By unimodularity there therefore exists an element c such that the map q(c,-) equals this linear map.

The form q is even if and only if 0 is a characteristic vector. If c and c' are characteristic vectors for q, then there is an element h with c' = c + 2h. This follows from unimodularity. As a consequence, the number q(c,c) is independent of the chosen characteristic vector c modulo 8. One can be more specific:

Proposition 4.2. For a characteristic vector c of the unimodular symmetric bilinear form q one has

\displaystyle  q(c,c) \equiv \text{sign}(q) \ (\text{mod} \ 8)

Proof: Suppose c is a characteristic vector of q. Then c + e_+ + e_- is a characteristic vector of the form

\displaystyle  q' = q \oplus \begin{pmatrix} 1  & \ 0 \\ 0 & -1 \end{pmatrix},

where e_+, e_- form basis elements of the additional \mathbb{Z}^2 summand with square \pm 1. We notice that

\displaystyle  q(c,c) = q'(c+e_+ + e_-, c+e_+ + e_-) .

However, the form q' is indefinite, so the above classification theorem applies. In particular, q' is odd and has the same signature as q, so it is equivalent to the diagonal form with b^+ + 1 summands of (+1) and b^- + 1 summands of (-1). This diagonal form has a characteristic vector c' that is simply a sum of basis elements in which the form is diagonal. Of course q'(c',c') = b^+ - b^-. The claim now follows from the fact that the square of a characteristic vector is independent of the chosen characteristic vector modulo 8.

Corollary 4.3. The signature of an even (definite or indefinite) form is divisible by 8.

[edit] 4.2 Examples, Realisations of indefinite forms

We shall show that any indefinite form permitted by the above theorem and corollary can be realised.

All possible values of rank and signature of odd forms are realised by direct sums of the forms of rank 1,

\displaystyle  b^+ (+1) \oplus b^- (-1) .

An even positive definite form of rank 8 is given by the E_8 matrix

\displaystyle  E_8 = \left( \begin{array}{c c c c c c c c} \ 2 \ & \ 1\  & \ 0\  & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 \\ \end{array}  \right) .

Likewise, the matrix -E_8 represents a negative definite even form of rank 8.

On the other hand, the matrix H given by

\displaystyle  H = \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0  \end{pmatrix}

determines an indefinite even form of rank 2 and signature 0. It is easy to see that the direct sums

\displaystyle  k E_8 \oplus l H

with k \in \mathbb{Z}, l \in \mathbb{N}=\{1,2,\dots\} realise all unimodular symmetric indefinite even forms that are allowed by the above classification result. Here we use the convention that k E_8 is the k-fold direct sum of E_8 for positive k and (-k) (-E_8) is the |k|-fold direct sum of the negative definite form -E_8.

[edit] 5 References

[edit] 6 External links

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