Intersection form
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[edit] 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 .
By linearity vanishes on torsion elements, hence the map descends to a bilinear pairing on the free module which we also denote . This pairing is uni-modular (in particular non-degenerate) by Poincaré duality.
- If is even the pairing is symmetric: .
- If is odd the pairing is skew-symmetric: .
[edit] 2 Uni-modular bilinear forms
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 is the rank of the underlying -module .
[edit] 3 Skew-symmetric bilinear forms
The skew-symmetric hyperbolic form of rank , , is defined by the following intersection matrix
Proposition 3.1. Every skey-symmetri uni-modular bilinear form over , , isomorphic to some the sum of some number of hyperbolic forms:
In particular the rank of , in this case , 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 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.
[edit] 4.1 Classification of indefinite forms
There is a simple classification result of indefinite forms [Serre1970],[Milnor&Husemoller1973]:
There is a further invariant of a unimodular symmetric bilinear form on : An element is called a characteristic vector of the form if one has
for all elements . Characteristic vectors always exist. In fact, when reduced modulo 2, the map is linear. By unimodularity there therefore exists an element such that the map equals this linear map.
The form is even if and only if is a characteristic vector. If and are characteristic vectors for , then there is an element with . This follows from unimodularity. As a consequence, the number is independent of the chosen characteristic vector modulo 8. One can be more specific:
Proposition 4.2. For a characteristic vector of the unimodular symmetric bilinear form one has
Proof: Suppose is a characteristic vector of . Then is a characteristic vector of the form
where form basis elements of the additional summand with square . We notice that
However, the form is indefinite, so the above classification theorem applies. In particular, is odd and has the same signature as , so it is equivalent to the diagonal form with summands of (+1) and summands of . This diagonal form has a characteristic vector that is simply a sum of basis elements in which the form is diagonal. Of course . 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,
An even positive definite form of rank 8 is given by the matrix
Likewise, the matrix represents a negative definite even form of rank 8.
On the other hand, the matrix given by
determines an indefinite even form of rank 2 and signature 0. It is easy to see that the direct sums
with realise all unimodular symmetric indefinite even forms that are allowed by the above classification result. Here we use the convention that is the -fold direct sum of for positive and is the -fold direct sum of the negative definite form .
[edit] 5 References
- [Milnor&Husemoller1973] J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973. MR0506372 (58 #22129) Zbl 0292.10016
- [Serre1970] J. Serre, Cours d'arithmétique, Presses Universitaires de France, Paris, 1970. MR0255476 (41 #138) Zbl 0432.10001
[edit] 6 External links
- The Wikipedia page on Poincaré duality