Intersection form
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$q$ 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. | $q$ 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. | ||
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== Algebraic invariants == | == Algebraic invariants == | ||
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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 $q = f^* q'$. | 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 $q = f^* q'$. | ||
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{{endthm}} | {{endthm}} | ||
Proof: | Proof: | ||
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== Classification of indefinite forms == | == Classification of indefinite forms == | ||
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There is a simple classification result 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}} | {{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}} | ||
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== Examples == | == Examples == | ||
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</wikitex> | </wikitex> |
Revision as of 15:07, 7 June 2010
Contents |
1 Introduction
For a closed (topological or smooth) manifold of dimension the intersection form
is obtained by the formula
i.e. the cup product of and is evaluated on the fundamental cycle given by the manifold .
descends to a bilinear pairing on the free module . It is a symmetric and unimodular (in particular non-degenerate) pairing, the latter follows from Poincaré duality.
2 Algebraic invariants
Let and be unimodular symmetric bilinear forms on underlying free -modules and respectively. The two forms and are said equivalent if there is an isomorphism such that .
A form 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 -module V.
As is symmetric it is diagonalisable over the real numbers. If denotes the dimension of a maximal subspace on which the form is positive definite, and if is the dimension of a maximal subspace on which the form is negative definite, then the signature of is defined to be
The form may have two different types. It is of type even if is an even number for any element . Equivalently, if is written as a square matrix in a basis, it is even if the elements on the diagonal are all even. Otherwise, is said of type odd.
Proposition 2.1. If the unimodular symmetric bilinear form is even then its signature is divisible by 8.
Proof:
3 Classification of indefinite forms
There is a simple classification result of indefinite forms:
4 Examples
5 References
This page has not been refereed. The information given here might be incomplete or provisional. |