Intersection form
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$$I_N=\cap_N=\cdot_N=\lambda_N\colon H_k(N;\Zz) \times H_{n-k}(N;\Zz) \to \Zz$$ | $$I_N=\cap_N=\cdot_N=\lambda_N\colon H_k(N;\Zz) \times H_{n-k}(N;\Zz) \to \Zz$$ | ||
defined on the homology of $N$. | defined on the homology of $N$. | ||
− | For $n=2k$ this is the '''intersection form''' of $N$. | + | For $n=2k$ this is the '''intersection form''' of $N$ denoted by $q_N$. |
− | For $n=4k$ the signature of this form is the '''signature''' $N$. | + | For $n=4k$ the signature of this form is the '''signature''' $\sigma(N)$ of $N$. |
− | The intersection product is related to [[Stiefel-Whitney_characteristic_classes|characteristic classes]]. | + | The intersection product is closely related to the notions of [[Stiefel-Whitney_characteristic_classes|characteristic classes]] and [[Linking_form|linking form]]. |
These are important invariants used in the classification of [[Manifold|manifolds]]. | These are important invariants used in the classification of [[Manifold|manifolds]]. | ||
+ | |||
+ | In this page $T$ is a triangulation (or a cell subdivision) of $N$, and $T^*$ is the [[Wikipedia:Poincare_duality#Dual_cell_structures|dual cell subdivision]]. | ||
+ | |||
+ | The exposition follows \cite[Chapter II]{Kirby1989}, \cite[$\S$6, $\S$10]{Skopenkov2015b}. | ||
</wikitex> | </wikitex> | ||
− | == Definition == | + | == Definition of the intersection product== |
<wikitex>; | <wikitex>; | ||
− | + | In this subsection we mostly omit $\Zz_2$-coefficients. | |
− | + | ||
− | Denote by $T | + | '''A short direct definition of a homology group with $\Zz_2$-coefficients.''' |
− | + | For an integer $s$ denote by $C_s(T)=C_s(T;\Zz_2)$ the set (the $\Zz_2$-space) of arrangements of zeroes and units on the $s$-dimensional cells of $T$ | |
− | + | (so $C_s(T)=\{0\}$ if there are no $s$-dimensional cells in $T$). | |
+ | Denote by $\partial=\partial_s\colon C_s(T) \to C_{s - 1}(T)$ the extension over $C_s(T)$ of the map taking an $s$-dimensional cell $\sigma$ of $T$ to the boundary of $\sigma$. | ||
+ | Denote | ||
+ | $$Z_s(T)=Z_s(T;\Zz_2):=\ker\partial_s\quad\text{and}\quad B_s(T)=B_s(T;\Zz_2):=\mathrm{im}\partial_{s + 1}.$$ | ||
+ | Then the $s$-th homology group of $T$ with $\Zz_2$-coefficients is defined as $H_s(T)=H_s(T;\Zz_2):=Z_s(T)/ B_s(T)$. | ||
+ | This depends only on $N$, not on $T$. | ||
+ | |||
+ | '''A short direct definition of the modulo 2 intersection product.''' | ||
+ | For modulo 2 chains $x\in C_k(T)$ and $y\in C_{n-k}(T^*)$ define the ''modulo 2 intersection number'' by the formula | ||
$$ | $$ | ||
− | I_{ | + | I_{T,2}(x,y)=x\cap_{T,2} y=\langle\, x \, , \, y\, \rangle := |x\cap y|\mod2\in\Zz_2. |
$$ | $$ | ||
− | + | Represent classes $[x]\in H_k(N;\Zz_2)$ and $[y]\in H_{n-k}(N;\Zz_2)$ by cycles $x$ and $y$ viewed as unions of $k$-simplices of $T$ and $(n-k)$-simplices of $T^*$, respectively. | |
+ | Define the ''modulo 2 intersection product'' | ||
$$ | $$ | ||
− | I_{N,2}=\cap_{N,2}: H_k(N;\Zz_2) \times H_{n-k}(N;\Zz_2) \to \Zz_2 | + | I_{N,2}=\cap_{N,2}: H_k(N;\Zz_2) \times H_{n-k}(N;\Zz_2) \to \Zz_2\quad\text{by}\quad |
− | $$ | + | I_{N,2}([x],[y]):=I_{N,2}(x,y).$$ |
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). | 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). | ||
+ | |||
+ | '''Sketch of a short direct definition of the intersection product.''' | ||
Analogously, counting intersections with signs, one defines the '''intersection number''' | Analogously, counting intersections with signs, one defines the '''intersection number''' | ||
$$I_N(x,y)=x\cdot y=\langle\, x \, , \, y\, \rangle \in \Z$$ | $$I_N(x,y)=x\cdot y=\langle\, x \, , \, y\, \rangle \in \Z$$ | ||
of integer chains $x\in C_k(T;\Zz)$ and $y\in C_{n-k}(T^*,\Zz)$. | of integer chains $x\in C_k(T;\Zz)$ and $y\in C_{n-k}(T^*,\Zz)$. | ||
− | + | Clearly, the product of a cycle and a boundary is zero. | |
− | + | Hence this defines the above ''intersection product'' | |
− | + | $I_N: H_k(N;\Zz)\times H_{n-k}(N;\Zz)\to\Zz$. | |
− | + | ||
− | + | ||
− | + | {{beginthm|Remark}} (a) Using the notion of ''transversality'', one can give an equivalent (and `more general') definition as follows. Take a $k$-chain $x\in C_k(N;\Zz)$ and an $(n-k)$-chain $y\in C_{n-k}(N;\Zz)$ which are ''transverse'' to each other. The signed count of the intersections between $x$ and $y$ gives the ''intersection number'' $I_N(x,y)$. A particular case is [[Intersection_number_of_immersions|intersection number of immersions]]. | |
+ | Then define the '''intersection product''' $I_N$ by $I_N([x],[y]):=I_N(x,y)$. | ||
− | Using the notion of ''cup product'', one can give a dual (and so an equivalent) definition: | + | (b) Using the notion of ''cup product'', one can give a dual (and so an equivalent) definition: |
$$I_N(x,y) = \langle x^*\smile y^*,[N]\rangle \in \Z,$$ | $$I_N(x,y) = \langle x^*\smile y^*,[N]\rangle \in \Z,$$ | ||
− | where $x^*\in H^{n-k}(N)$, $y^*\in H^n(N)$ are the [[ | + | where $x^*\in H^{n-k}(N)$, $y^*\in H^n(N)$ are the [[Poincare isomorphism|Poincaré duals]] of $x$, $y$, and $[N]$ is the fundamental class of the manifold $N$. We can also define the ''cup (cohomology intersection) product'' |
− | We can also define the ''cohomology intersection product'' | + | |
$$ | $$ | ||
I_N^*: H^k(N;\Zz) \times H^{n-k}(N;\Zz) \to \Zz | I_N^*: H^k(N;\Zz) \times H^{n-k}(N;\Zz) \to \Zz | ||
− | + | \quad\text{by}\quad | |
− | by | + | |
− | + | ||
I_N^*(p,q) = \langle p \smile q , [N] \rangle . | I_N^*(p,q) = \langle p \smile q , [N] \rangle . | ||
$$ | $$ | ||
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. | 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). | 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). | + | This is an advantage for mathematicians who are interested in complexes and topological manifolds (not only in PL and smooth manifolds). See \cite[Remark 2.3]{Skopenkov2005}. |
− | + | {{endthm}} | |
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</wikitex> | </wikitex> | ||
− | == | + | == Simple properties == |
<wikitex>; | <wikitex>; | ||
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The intersection product is bilinear. | The intersection product is bilinear. | ||
Hence it vanishes on torsion elements. | Hence it vanishes on torsion elements. | ||
− | Thus it descends to a bilinear pairing | + | Thus it descends to a bilinear ''(integer) intersection pairing'' |
$$H_k(N;\Zz) / \text{Torsion}\times H_{n-k}(N;\Zz) / \text{Torsion}\to\Zz.$$ | $$H_k(N;\Zz) / \text{Torsion}\times H_{n-k}(N;\Zz) / \text{Torsion}\to\Zz.$$ | ||
on the free modules. | on the free modules. | ||
− | |||
We have | We have | ||
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* If $k$ is even the form $q_N$ is symmetric: $q_N(x, y) = q_N(y, x)$. | * If $k$ is even the form $q_N$ is symmetric: $q_N(x, y) = q_N(y, x)$. | ||
* If $k$ is odd the form $q_N$ is skew-symmetric: $q_N(x, y) = - q_N(y, x)$. | * If $k$ is odd the form $q_N$ is skew-symmetric: $q_N(x, y) = - q_N(y, x)$. | ||
+ | </wikitex> | ||
+ | |||
+ | == Poincaré duality == | ||
+ | <wikitex>; | ||
+ | |||
+ | {{beginthm|Theorem}}[Poincaré duality] | ||
+ | (a) The modulo 2 intersection product is non-degenerate. | ||
+ | |||
+ | (b) The integer intersection pairing is unimodular (in particular non-degenerate). | ||
+ | {{endthm}} | ||
+ | |||
+ | ''Proof of (a).'' (This proof is well-known to specialists but is, in this short and explicit form, absent from textbooks.) | ||
+ | We use orthogonal complements with respect to the modulo 2 intersection product | ||
+ | $I_{T,2}:C_s(T)\times C_{n-s}(T^*)\to\Zz_2$. | ||
+ | It suffices to prove that | ||
+ | $$\phantom{}^\bot Z_{n-s}(T^*)= B_s(T)\quad\text{and}\quad Z_s(T)^\bot=B_{n-s}(T^*).$$ | ||
+ | Let us prove the left-hand equality; the right-hand equality is proved analogously. | ||
+ | Since $I_{T,2}$ is non-degenerate, we only need to check that $B_s(T)^\bot=Z_{n-s}(T^*)$. | ||
+ | The inclusion $B_s(T)^\bot \supset Z_{n-s}(T^*)$ is obvious. | ||
+ | The opposite inclusion follows because ''if $I_{T,2}(\partial c,d)=0$ for an $(s+1)$-cell $c$ of $T$ and a chain $d\in Z_{n-s}(T^*)$, then $\partial d$ does not involve the cell $c^*$ dual to $c$''. | ||
</wikitex> | </wikitex> | ||
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If $n$ is divisible by 4, the '''signature''' $\sigma(N)={\rm sign}(N)$ is defined to be the signature of the intersection form of $N$. | If $n$ is divisible by 4, the '''signature''' $\sigma(N)={\rm sign}(N)$ is defined to be the signature of the intersection form of $N$. | ||
</wikitex> | </wikitex> | ||
− | + | ||
− | == | + | == On classification of bilinear forms == |
<wikitex>; | <wikitex>; | ||
− | Let $q$ and $q'$ be | + | Let $q$ and $q'$ be bilinear forms on free $\mathbb{Z}$-modules (or $\Zz_2$-vector spaces) $V$ and $V'$ respectively. The forms $q$ and $q'$ are called ''equivalent'' or '''isomorphic''' if there is an isomorphism $f:V \to V'$ such that $q=f^*q'$. |
− | The ''rank'' of $q$ is the rank of the underlying $\mathbb{Z}$-module $V$. | + | The '''rank''' of $q$ is the rank of the underlying $\mathbb{Z}$-module (or $\Zz_2$-vector space) $V$. |
− | + | ||
− | + | A form $q$ is '''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 '''odd'''. | |
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− | $$ | + | |
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</wikitex> | </wikitex> | ||
− | == | + | === Classification over integers modulo 2 === |
− | <wikitex>; | + | <wikitex>; |
− | + | {{beginthm|Theorem}} | |
− | + | (a) Every odd symmetric non-degenerate bilinear form $q$ over $\Zz_2$ is isomorphic to the sum of some number of `unity' forms $(1)$. | |
− | + | (b) Every even symmetric non-degenerate bilinear form $q$ over $\Zz_2$ is isomorphic to the sum of some number of `hyperbolic' forms $H(\Zz_2)$ of rank $2$ defined by the following matrix | |
+ | $$ \left( \begin{array}{cc} ~0 & ~1 \\ 1 & ~0 \end{array} \right).$$ | ||
+ | In particular the rank of $q$ is even. | ||
+ | {{endthm}} | ||
− | + | Clearly, different sums from the theorem are not isomorphic. | |
</wikitex> | </wikitex> | ||
− | === Classification of | + | === Classification of skew-symmetric forms === |
− | <wikitex>; | + | <wikitex>; |
− | + | {{beginthm|Theorem}} | |
+ | Every skew-symmetric unimodular bilinear form $q$ over $\Zz$ is isomorphic to the sum of some number of hyperbolic forms $H_-(\Zz)$ of rank $2$ defined by the following matrix | ||
+ | $$\left( \begin{array}{cc} ~0 & ~1 \\ -1 & ~0 \end{array} \right).$$ | ||
+ | <!-- $$ q \cong \oplus_{i=1}^r H_-(\Zz).$$--> | ||
+ | In particular the rank of $q$ is even. | ||
+ | {{endthm}} | ||
− | {{ | + | {{beginproof}} The proof is by induction on the rank of the module $M$ on which $q$ is defined. |
− | + | Let $B$ be the matrix of $q$ in some basis. | |
− | + | Then $\det B =\pm1$. Thus expanding the determinant by the first column we obtain $\pm1$. | |
− | $$ | + | Hence $\gcd(B_{2,1},\ldots,B_{n,1})=1$ (observe that $B_{1,1}=0$). |
− | + | Denote by $C_{2,2},\ldots,C_{2,n}$ elements of $M$ such that | |
− | $$ | + | $$\gcd(B_{2,1},\ldots,B_{n,1})=C_{2,2}B_{2,1}+\ldots+C_{2,n}B_{n,1}.$$ |
− | + | Denote $C_{i,j} := \delta_i^j$ for $(i, j) \neq (2,2),\ldots,(n,2)$. Let $C := (C_{i,j})_{i, j = 1}^n$ be a $n \times n$ matrix. Then $(CB)_{2,1}=1$. For any matrix $A$ the first row of the matrix $CA$ is equal to the first row of the matrix $A$. Therefore $(CBC^t)_{2,1}=(CB)_{2,1}=1$ and $CBC^t$ represents the matrix of $q$ in a different basis $f_1,\ldots,f_n$. Since $q(f_2,f_1) = 1$, the restriction of $q$ onto the submodule $N:=\left<f_2, f_1\right>$ has matrix $H_-(\Zz)$. Since $H_-(\Zz)$ is non-degenerate and unimodular, $N$ is a direct summand in $M$, i.e., $M = N\oplus N^\bot$. Now we restrict $q$ to $N^\bot$ and apply the inductive hypothesis to the restriction.{{endproof}} | |
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− | + | Clearly, different sums from the theorem are not isomorphic. | |
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</wikitex> | </wikitex> | ||
− | === Examples | + | === Examples of symmetric indefinite forms === |
<wikitex>; | <wikitex>; | ||
+ | A form is called ''definite'' if it is positive or negative definite, otherwise it is called ''indefinite''. | ||
+ | Here we show that | ||
− | + | (O) for any pair $(r,s)$ there is an odd unimodular symmetric indefinite form of rank $r$ and signature $s$; | |
− | + | (E) for any pair $(r,8s)$ there is an even unimodular symmetric indefinite form of rank $r$ and signature | |
− | $$ | + | $8s$. |
− | b^+ (+1) \oplus b^- (-1) . | + | |
− | $$ | + | Cf. Theorem \ref{t:serre} and Proposition \ref{l:sign}. |
+ | |||
+ | All values (O) are realised by direct sums of the forms of rank 1, | ||
+ | $$b^+ (+1) \oplus b^- (-1).$$ | ||
An even positive definite form of rank 8 is given by the $E_8$ matrix | An even positive definite form of rank 8 is given by the $E_8$ matrix | ||
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k E_8 \oplus l H | k E_8 \oplus l H | ||
$$ | $$ | ||
− | with $k | + | with $k,l\in\Zz$, $l>0$ realise all forms from (E). |
+ | Here we use the convention that $k q$ is the $k$-fold direct sum of $q$ for $k>0$ and $kE_8$ is the $(-k)$-fold direct sum of $-E_8$ for $k<0$. | ||
+ | </wikitex> | ||
+ | === Classification of symmetric indefinite forms === | ||
+ | <wikitex>; | ||
+ | The classification of unimodular definite symmetric bilinear forms is a deep and difficult problem. | ||
+ | However the situation becomes much easier when the form is indefinite. | ||
+ | Fundamental invariants are rank, signature and being odd or even (aka '''type'''). | ||
+ | There is a simple classification result of indefinite forms \cite{Serre1970},\cite{Milnor&Husemoller1973}: | ||
+ | |||
+ | {{beginthm|Theorem}}\label{t:serre} (a) Every odd indefinite unimodular symmetric bilinear form over $\Zz$ is isomorphic to the sum of some number of `unity' forms $(1)$ and some number of `minis unity' forms $(-1)$. | ||
+ | |||
+ | (b) Two indefinite unimodular symmetric bilinear forms over $\mathbb{Z}$ are equivalent if and only if they have the same rank, signature and type. | ||
+ | {{endthm}} | ||
+ | |||
+ | Part (a) follows by part (b). | ||
+ | |||
+ | There is a restriction for values of the above invariants. | ||
+ | |||
+ | {{beginthm|Proposition|}}\label{l:sign} | ||
+ | The signature of an even (definite or indefinite) form is divisible by 8. | ||
+ | {{endthm}} | ||
+ | |||
+ | This follows from Proposition \ref{p:char} below. | ||
+ | |||
+ | An element $c \in V$ is called a ''characteristic vector'' of the form $q$ if | ||
+ | $$ | ||
+ | q(c,x) \equiv q(x,x) \ (\text{mod} \ 2) | ||
+ | $$ | ||
+ | for all elements $x \in V$. Characteristic vectors always exist. In fact, when reduced modulo 2, the map $x \mapsto q(x,x) \in\Z/2$ is linear. | ||
+ | Hence by unimodularity there exists an element $c$ such that the map $q(c,-)$ equals this linear map. | ||
+ | |||
+ | The form $q$ is even if and only if $0$ is a characteristic vector. | ||
+ | If $c$ and $c'$ are characteristic vectors for $q$, then by unimodularity there is an element $h$ with | ||
+ | $c' = c + 2h$. | ||
+ | Hence the number $q(c,c)$ is independent of the chosen characteristic vector $c$ modulo 8. | ||
+ | One can be more specific: | ||
+ | |||
+ | {{beginthm|Proposition|}}\label{p:char} | ||
+ | For a characteristic vector $c$ of the unimodular symmetric bilinear form $q$ one has | ||
+ | $$ | ||
+ | q(c,c) \equiv \text{sign}(q) \ (\text{mod} \ 8) | ||
+ | $$ | ||
+ | {{endthm}} | ||
+ | {{beginproof}} | ||
+ | Suppose $c$ is a characteristic vector of $q$. Then $c + e_+ + e_-$ is a characteristic vector of the form | ||
+ | $q':= q \oplus H_-(\Zz)$, where $e_+, e_-$ form basis elements of the additional $\mathbb{Z}^2$ summand with? square $\pm 1$. We notice that | ||
+ | $$ | ||
+ | q(c,c) = q'(c+e_+ + e_-, c+e_+ + e_-) . | ||
+ | $$ | ||
+ | However, the form $q'$ is indefinite, so the above classification theorem applies. In particular, $q'$ is odd and has the same signature as $q$, so it? is equivalent to the diagonal form with $b^+ + 1$ summands of (+1) and $b^- + 1$ summands of $(-1)$. This diagonal form has a characteristic vector $c'$ that is simply a sum of basis elements in which the form is diagonal. Of course $q'(c',c') = b^+ - b^-$. The claim now follows from the fact that the square of a characteristic vector is independent of the chosen characteristic vector modulo 8. | ||
+ | {{endproof}} | ||
</wikitex> | </wikitex> | ||
Latest revision as of 12:32, 16 December 2023
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[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 denoted by . For the signature of this form is the signature of . The intersection product is closely related to the notions of characteristic classes and linking form. These are important invariants used in the classification of manifolds.
In this page is a triangulation (or a cell subdivision) of , and is the dual cell subdivision.
The exposition follows [Kirby1989, Chapter II], [Skopenkov2015b, 6, 10].
[edit] 2 Definition of the intersection product
In this subsection we mostly omit -coefficients.
A short direct definition of a homology group with -coefficients. For an integer denote by the set (the -space) of arrangements of zeroes and units on the -dimensional cells of (so if there are no -dimensional cells in ). Denote by the extension over of the map taking an -dimensional cell of to the boundary of . Denote
Then the -th homology group of with -coefficients is defined as . This depends only on , not on .
A short direct definition of the modulo 2 intersection product. For modulo 2 chains and define the modulo 2 intersection number by the formula
Represent classes and by cycles and viewed as unions of -simplices of and -simplices of , respectively. 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).
Sketch of a short direct definition of the intersection product. Analogously, counting intersections with signs, one defines the intersection number
of integer chains and . Clearly, the product of a cycle and a boundary is zero. Hence this defines the above intersection product .
Remark 2.1. (a) 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. Then define the intersection product by .
(b) 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 cup (cohomology intersection) product
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). See [Skopenkov2005, Remark 2.3].
[edit] 3 Simple 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 (integer) intersection pairing
on the free modules.
We have
Hence for
- If is even the form is symmetric: .
- If is odd the form is skew-symmetric: .
[edit] 4 Poincaré duality
Theorem 4.1.[Poincaré duality] (a) The modulo 2 intersection product is non-degenerate.
(b) The integer intersection pairing is unimodular (in particular non-degenerate).
Proof of (a). (This proof is well-known to specialists but is, in this short and explicit form, absent from textbooks.) We use orthogonal complements with respect to the modulo 2 intersection product . It suffices to prove that
Let us prove the left-hand equality; the right-hand equality is proved analogously. Since is non-degenerate, we only need to check that . The inclusion is obvious. The opposite inclusion follows because if for an -cell of and a chain , then does not involve the cell dual to .
[edit] 5 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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Tex syntax erroris defined to be the signature of the intersection form of .
[edit] 6 On classification of bilinear forms
Let and be bilinear forms on free -modules (or -vector spaces) 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 (or -vector space) .
A form is 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 odd.
[edit] 6.1 Classification over integers modulo 2
Theorem 6.1. (a) Every odd symmetric non-degenerate bilinear form over is isomorphic to the sum of some number of `unity' forms .
(b) Every even symmetric non-degenerate bilinear form over is isomorphic to the sum of some number of `hyperbolic' forms of rank defined by the following matrix
In particular the rank of is even.
Clearly, different sums from the theorem are not isomorphic.
[edit] 6.2 Classification of skew-symmetric forms
Theorem 6.2. Every skew-symmetric unimodular bilinear form over is isomorphic to the sum of some number of hyperbolic forms of rank defined by the following matrix
In particular the rank of is even.
Proof. The proof is by induction on the rank of the module on which is defined. Let be the matrix of in some basis. Then . Thus expanding the determinant by the first column we obtain . Hence (observe that ). Denote by elements of such that
Clearly, different sums from the theorem are not isomorphic.
[edit] 6.3 Examples of symmetric indefinite forms
A form is called definite if it is positive or negative definite, otherwise it is called indefinite. Here we show that
(O) for any pair there is an odd unimodular symmetric indefinite form of rank and signature ;
(E) for any pair there is an even unimodular symmetric indefinite form of rank and signature .
Cf. Theorem 6.3 and Proposition 6.4.
All values (O) 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 forms from (E). Here we use the convention that is the -fold direct sum of for and is the -fold direct sum of for .
[edit] 6.4 Classification of symmetric indefinite forms
The classification of unimodular 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 being odd or even (aka type). There is a simple classification result of indefinite forms [Serre1970],[Milnor&Husemoller1973]:
Theorem 6.3. (a) Every odd indefinite unimodular symmetric bilinear form over is isomorphic to the sum of some number of `unity' forms and some number of `minis unity' forms .
(b) Two indefinite unimodular symmetric bilinear forms over are equivalent if and only if they have the same rank, signature and type.
Part (a) follows by part (b).
There is a restriction for values of the above invariants.
Proposition 6.4. The signature of an even (definite or indefinite) form is divisible by 8.
This follows from Proposition 6.5 below.
An element is called a characteristic vector of the form if
for all elements . Characteristic vectors always exist. In fact, when reduced modulo 2, the map is linear. Hence by unimodularity there 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 by unimodularity there is an element with . Hence the number is independent of the chosen characteristic vector modulo 8. One can be more specific:
Proposition 6.5. 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.
[edit] 7 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, 2020. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part
[edit] 8 External links
- The Wikipedia page on Poincaré duality
defined on the homology of . For this is the intersection form of denoted by . For the signature of this form is the signature of . The intersection product is closely related to the notions of characteristic classes and linking form. These are important invariants used in the classification of manifolds.
In this page is a triangulation (or a cell subdivision) of , and is the dual cell subdivision.
The exposition follows [Kirby1989, Chapter II], [Skopenkov2015b, 6, 10].
[edit] 2 Definition of the intersection product
In this subsection we mostly omit -coefficients.
A short direct definition of a homology group with -coefficients. For an integer denote by the set (the -space) of arrangements of zeroes and units on the -dimensional cells of (so if there are no -dimensional cells in ). Denote by the extension over of the map taking an -dimensional cell of to the boundary of . Denote
Then the -th homology group of with -coefficients is defined as . This depends only on , not on .
A short direct definition of the modulo 2 intersection product. For modulo 2 chains and define the modulo 2 intersection number by the formula
Represent classes and by cycles and viewed as unions of -simplices of and -simplices of , respectively. 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).
Sketch of a short direct definition of the intersection product. Analogously, counting intersections with signs, one defines the intersection number
of integer chains and . Clearly, the product of a cycle and a boundary is zero. Hence this defines the above intersection product .
Remark 2.1. (a) 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. Then define the intersection product by .
(b) 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 cup (cohomology intersection) product
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). See [Skopenkov2005, Remark 2.3].
[edit] 3 Simple 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 (integer) intersection pairing
on the free modules.
We have
Hence for
- If is even the form is symmetric: .
- If is odd the form is skew-symmetric: .
[edit] 4 Poincaré duality
Theorem 4.1.[Poincaré duality] (a) The modulo 2 intersection product is non-degenerate.
(b) The integer intersection pairing is unimodular (in particular non-degenerate).
Proof of (a). (This proof is well-known to specialists but is, in this short and explicit form, absent from textbooks.) We use orthogonal complements with respect to the modulo 2 intersection product . It suffices to prove that
Let us prove the left-hand equality; the right-hand equality is proved analogously. Since is non-degenerate, we only need to check that . The inclusion is obvious. The opposite inclusion follows because if for an -cell of and a chain , then does not involve the cell dual to .
[edit] 5 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] 6 On classification of bilinear forms
Let and be bilinear forms on free -modules (or -vector spaces) 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 (or -vector space) .
A form is 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 odd.
[edit] 6.1 Classification over integers modulo 2
Theorem 6.1. (a) Every odd symmetric non-degenerate bilinear form over is isomorphic to the sum of some number of `unity' forms .
(b) Every even symmetric non-degenerate bilinear form over is isomorphic to the sum of some number of `hyperbolic' forms of rank defined by the following matrix
In particular the rank of is even.
Clearly, different sums from the theorem are not isomorphic.
[edit] 6.2 Classification of skew-symmetric forms
Theorem 6.2. Every skew-symmetric unimodular bilinear form over is isomorphic to the sum of some number of hyperbolic forms of rank defined by the following matrix
In particular the rank of is even.
Proof. The proof is by induction on the rank of the module on which is defined. Let be the matrix of in some basis. Then . Thus expanding the determinant by the first column we obtain . Hence (observe that ). Denote by elements of such that
Clearly, different sums from the theorem are not isomorphic.
[edit] 6.3 Examples of symmetric indefinite forms
A form is called definite if it is positive or negative definite, otherwise it is called indefinite. Here we show that
(O) for any pair there is an odd unimodular symmetric indefinite form of rank and signature ;
(E) for any pair there is an even unimodular symmetric indefinite form of rank and signature .
Cf. Theorem 6.3 and Proposition 6.4.
All values (O) 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 forms from (E). Here we use the convention that is the -fold direct sum of for and is the -fold direct sum of for .
[edit] 6.4 Classification of symmetric indefinite forms
The classification of unimodular 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 being odd or even (aka type). There is a simple classification result of indefinite forms [Serre1970],[Milnor&Husemoller1973]:
Theorem 6.3. (a) Every odd indefinite unimodular symmetric bilinear form over is isomorphic to the sum of some number of `unity' forms and some number of `minis unity' forms .
(b) Two indefinite unimodular symmetric bilinear forms over are equivalent if and only if they have the same rank, signature and type.
Part (a) follows by part (b).
There is a restriction for values of the above invariants.
Proposition 6.4. The signature of an even (definite or indefinite) form is divisible by 8.
This follows from Proposition 6.5 below.
An element is called a characteristic vector of the form if
for all elements . Characteristic vectors always exist. In fact, when reduced modulo 2, the map is linear. Hence by unimodularity there 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 by unimodularity there is an element with . Hence the number is independent of the chosen characteristic vector modulo 8. One can be more specific:
Proposition 6.5. 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.
[edit] 7 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, 2020. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part
[edit] 8 External links
- The Wikipedia page on Poincaré duality