Intersection form
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1 Definition
After Poincaré, a closed oriented -manifold (PL or smooth) has a bilinear intersection product
defined on its homology.
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 product modulo 2
by the formula
This 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 (i.e. counting intersections with signs) one defines the intersection form
See yet another geometric interpretation, [Kirby1989, Chapter II], [Skopenkov2015b, 6].
Using the notion of transversality, one can give an equivalent (and `more general') definition as follows. Take a -chain and an -chain which is transverse to . The signed count of the intersections between and gives an intersection number
Using the notion of cup product, one can give a dual (and so an equivalent) definition as follows. Define the intersection product
by the formula
i.e. the cup product of and is evaluated on the fundamental cycle given by the manifold .
The definition of a cup product is `dual' (and so is analogous) to the above definition of the intersection form on homology of a manifold, 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 related important notion is linking form.
2 Properties
Clearly, 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 pairing is symmetric: .
- If is odd the pairing is skew-symmetric: .
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 .
3 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 some the sum of some number of hyperbolic forms:
In particular the rank of , in this case , is even.
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.
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 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.
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 .
5 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
- [Skopenkov2015b] A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2015, 2020. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part
6 External links
- The Wikipedia page on Poincaré duality
defined on its homology.
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 product modulo 2
by the formula
This 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 (i.e. counting intersections with signs) one defines the intersection form
See yet another geometric interpretation, [Kirby1989, Chapter II], [Skopenkov2015b, 6].
Using the notion of transversality, one can give an equivalent (and `more general') definition as follows. Take a -chain and an -chain which is transverse to . The signed count of the intersections between and gives an intersection number
Using the notion of cup product, one can give a dual (and so an equivalent) definition as follows. Define the intersection product
by the formula
i.e. the cup product of and is evaluated on the fundamental cycle given by the manifold .
The definition of a cup product is `dual' (and so is analogous) to the above definition of the intersection form on homology of a manifold, 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 related important notion is linking form.
2 Properties
Clearly, 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 pairing is symmetric: .
- If is odd the pairing is skew-symmetric: .
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 .
3 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 some the sum of some number of hyperbolic forms:
In particular the rank of , in this case , is even.
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.
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 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.
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 .
5 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
- [Skopenkov2015b] A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2015, 2020. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part
6 External links
- The Wikipedia page on Poincaré duality