Intersection form

1 Introduction

For a closed (topological or smooth) manifold $X$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; For a closed (topological or smooth) manifold $X$ of dimension n$ the intersection form $$ q_X: H^{2n}(X;\mathbb{Z}) \times H^{2n}(X;\mathbb{Z}) \to \mathbb{Z} $$ is obtained by the formula $$ q_X(x,y) = \langle x \smile y , [X] \rangle , $$ i.e. the cup product of $x$ and $y$ is evaluated on the fundamental cycle given by the manifold $X$. $q$ descends to a bilinear pairing on the free module $H^{2n}(X;\mathbb{Z}) / \text{Torsion}$. It is a symmetric and unimodular (in particular non-degenerate) pairing, the latter follows from Poincaré duality. == Algebraic invariants == Let $q$ and $q'$ be unimodular symmetric bilinear forms on underlying free $\mathbb{Z}$-modules $V$ and $V'$ respectively. The two forms $q$ and $q'$ are said equivalent if there is an isomorphism $f:V \to V'$ such that $q = f^* q'$. A form $q$ is called ''definite'' if it is positive or negative definite, otherwise it is called ''indefinite''. The ''rank'' of q is the rank of the underlying $\mathbb{Z}$-module V. As $q$ is symmetric it is diagonalisable over the real numbers. If $b^+$ denotes the dimension of a maximal subspace on which the form is positive definite, and if $b^-$ is the dimension of a maximal subspace on which the form is negative definite, then the ''signature'' of $q$ is defined to be $$ \text{sign}(q) = b^+ - b^-. $$ The form $q$ may have two different ''types''. It is of type ''even'' if $q(x,x)$ is an even number for any element $x$. Equivalently, if $q$ is written as a square matrix in a basis, it is even if the elements on the diagonal are all even. Otherwise, $q$ is said of type ''odd''. {{beginthm|Proposition|}} If the unimodular symmetric bilinear form $q$ is even then its signature is divisible by 8. {{endthm}} Proof: == Classification of indefinite forms == There is a simple classification result of indefinite forms: {{beginthm|Theorem|(Serre?)}} Two indefinite unimodular symmetric bilinear forms $q, q'$ over $\mathbb{Z}$ are equivalent if and only if $q$ and $q'$ have the same rank, signature and type. {{endthm}} == Examples == </wikitex> == References == {{#RefList:}} [[Category:Theory]] {{Stub}}X$ of dimension $4n$$4n$ the intersection form

$\displaystyle q_X: H^{2n}(X;\mathbb{Z}) \times H^{2n}(X;\mathbb{Z}) \to \mathbb{Z}$ $$\displaystyle q_X: H^{2n}(X;\mathbb{Z}) \times H^{2n}(X;\mathbb{Z}) \to \mathbb{Z}$$

is obtained by the formula

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

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

$q$$q$ descends to a bilinear pairing on the free module $H^{2n}(X;\mathbb{Z}) / \text{Torsion}$$H^{2n}(X;\mathbb{Z}) / \text{Torsion}$. It is a symmetric and unimodular (in particular non-degenerate) pairing, the latter follows from Poincaré duality.

1 Algebraic invariants

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

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

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

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

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

Proposition 3.1. If the unimodular symmetric bilinear form $q$$q$ is even then its signature is divisible by 8.

Proof:

2 Classification of indefinite forms

There is a simple classification result of indefinite forms:

Theorem 5.1 (Serre?). Two indefinite unimodular symmetric bilinear forms $q, q'$$q, q'$ over $\mathbb{Z}$$\mathbb{Z}$ are equivalent if and only if $q$$q$ and $q'$$q'$ have the same rank, signature and type.

2 References

This page has not been refereed. The information given here might be incomplete or provisional.