Intersection form
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[edit] 1 Introduction
Let be a closed oriented -manifold (PL or smooth). After Poincaré one studies the intersection number of transverse submanifolds or chains in . The intersection number gives a bilinear intersection product
defined on the homology of . For this is the intersection form of . For the signature of this form is the signature of . The intersection product is related to characteristic classes. These are important invariants used in the classification of manifolds.
[edit] 2 Definition
Take a triangulation of . Denote by the dual cell subdivision. Represent classes and by cycles and viewed as unions of -simplices of and -simplices of , respectively. Define the modulo 2 intersection number by the formula
Define the modulo 2 intersection product
This product is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary).
Analogously, counting intersections with signs, one defines the intersection number
of integer chains and .
Using the notion of transversality, one can give an equivalent (and `more general') definition as follows. Take a -chain and an -chain which are transverse to each other. The signed count of the intersections between and gives the intersection number . A particular case is intersection number of immersions.
Define the intersection product by .
See [Kirby1989, Chapter II], [Skopenkov2005, Remark 2.3], [Skopenkov2015b, 6].
Using the notion of cup product, one can give a dual (and so an equivalent) definition:
where , are the Poincaré duals of , , and is the fundamental class of the manifold . We can also define the cohomology intersection product
by the formula
The definition of a cup product is `dual' (and so is analogous) to the above definition of the intersection product on homology, but is more abstract. However, the definition of a cup product generalizes to complexes (and so to topological manifolds). This is an advantage for mathematicians who are interested in complexes and topological manifolds (not only in PL and smooth manifolds).
If , the intersection product is the intersection form denoted by .
A closely related important notion is linking form.
[edit] 3 Properties
The following properties are easy to check using the simple direct definition; they also follow from simple properties of the cup product.
The intersection product is bilinear. Hence it vanishes on torsion elements. Thus it descends to a bilinear pairing
on the free modules. This pairing is unimodular (in particular non-degenerate) by Poincaré duality.
We have
Hence for
- If is even the form is symmetric: .
- If is odd the form is skew-symmetric: .
[edit] 4 Definition of signature
Let be a symmetric bilinear form on a free -module. Denote by () the number of positive (negative) eigenvalues.
Note that since is symmetric, it is diagonalisable over the real numbers, so () is the dimension of a maximal subspace on which the form is positive (negative) definite.
Then the signature of is defined to be
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[edit] 5 Equivalence of bilinear forms
Let and be unimodular bilinear forms on underlying free -modules and respectively. The forms and are called equivalent or isomorphic if there is an isomorphism such that .
The rank of is the rank of the underlying -module .
[edit] 6 Skew-symmetric bilinear forms
The skew-symmetric hyperbolic form of rank , , is defined by the following intersection matrix
Proposition 6.1. Every skey-symmetri uni-modular bilinear form over , , isomorphic to the sum of some number of hyperbolic forms:
In particular the rank of , in this case , is even.
[edit] 7 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. Fundamental invariants are rank, signature and the following two.
A form is called definite if it is positive or negative definite, otherwise it is called indefinite.
A 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] 7.1 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 7.2. 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 7.3. The signature of an even (definite or indefinite) form is divisible by 8.
[edit] 7.2 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] 8 References
- [Kirby1989] R.C. Kirby, The topology of 4-manifolds, Lecture Notes in Math. 1374, Springer-Verlag, 1989. MR1001966 (90j:57012)
- [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
- [Skopenkov2005] A. Skopenkov, A classification of smooth embeddings of 4-manifolds in 7-space, Topol. Appl., 157 (2010) 2094-2110. Available at the arXiv:0512594.
- [Skopenkov2015b] A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2015.
http://www.mccme.ru/circles/oim/home/combtop13.htm#photo
[edit] 9 External links
- The Wikipedia page on Poincaré duality