Intersection form

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1 Introduction

Let $N$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}N$ be an oriented $n$$n$-manifold. After Poincaré one studies the intersection number of transverse submanifolds (or immersions) in $N$$N$. This was generalized to a bilinear intersection product

$\displaystyle I_N=\cap_N=\cdot_N=\lambda_N \colon H_k(N;\Z)\times H_{n-k}(N;\Z) \to \Z$ $$\displaystyle I_N=\cap_N=\cdot_N=\lambda_N \colon H_k(N;\Z)\times H_{n-k}(N;\Z) \to \Z$$

defined on the homology of $N$$N$, when $N$$N$ is a PL manifold (recall that any smooth manifold $N$$N$ has a triangulation making $N$$N$ a PL manifold). For $n=2k$$n=2k$ this is the intersection form of $N$$N$ denoted by $q_N$$q_N$. For $n=4k$$n=4k$ and $N$$N$ closed the signature of this form is the signature $\sigma(N)$$\sigma(N)$ of $N$$N$. 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.

The exposition follows [Kirby1989, Chapter II], [Skopenkov2015b, $\S$$\S$6, $\S$$\S$10].

In this page $N$$N$ is a compact PL $n$$n$-manifold (possibly, with boundary).

2 A short direct definition of the intersection product

We use a simple definition of homology groups. In this section

• if the coefficients of a (co)homology group are omitted, then they are $\Zz_2$$\Zz_2$;
• $T$$T$ is a triangulation (or a cell subdivision) of $N$$N$, and $T^*$$T^*$ is the dual cell subdivision.

Define the modulo 2 intersection product

$\displaystyle \cap_{N,2}: H_k(N) \times H_{n-k}(N) \to \Z_2\quad\text{by}\quad [x]\cap_{N,2} [y] := |x\cap y|\mod2,$ $$\displaystyle \cap_{N,2}: H_k(N) \times H_{n-k}(N) \to \Z_2\quad\text{by}\quad [x]\cap_{N,2} [y] := |x\cap y|\mod2,$$

where $x$$x$ and $y$$y$ are modulo 2 $k$$k$-cycle in $T$$T$ and $(n-k)$$(n-k)$-cycle in $T^*$$T^*$.

Lemma 2.1. This product is well-defined.

Proof. The product $H_k(T)\times H_{n-k}(T^*)\to\Z_2$$H_k(T)\times H_{n-k}(T^*)\to\Z_2$ is well-defined because

(i) the intersection of a $k$$k$-cycle modulo 2 in $T$$T$ and the boundary of an $(n-k+1)$$(n-k+1)$-cell of $T^*$$T^*$ consists of an even number of points;

(ii) the intersection of the boundary of a $(k+1)$$(k+1)$-cell of $T$$T$ and an $(n-k)$$(n-k)$-cycle modulo 2 in $T^*$$T^*$ consists of an even number of points.

In this paragraph we prove assertion (i); assertion (ii) is proved analogously. Let $\sigma$$\sigma$ be a $k$$k$-face of $T$$T$, and $\tau^*$$\tau^*$ an $(n-k+1)$$(n-k+1)$-face of $T^*$$T^*$. Denote by $\tau$$\tau$ the $(k-1)$$(k-1)$-face of $T$$T$ dual to $\tau^*$$\tau^*$. We have $\sigma\cap\partial\tau^*\ne\emptyset$$\sigma\cap\partial\tau^*\ne\emptyset$ if and only if $\sigma\supset\tau$$\sigma\supset\tau$. So (i) is equivalent to the above definition of a modulo 2 $k$$k$-cycle in $T$$T$.

The proof is completed using the PL invariance of homology.

$\square$$\square$

Definition of the integer intersection product for oriented $N$$N$. Take oriented dual faces $\sigma$$\sigma$ of $T$$T$ and $\sigma^*$$\sigma^*$ of $T^*$$T^*$ intersecting at a point $S$$S$.

If $T$$T$ is a triangulation of a smooth manifold $N$$N$, then $N$$N$ is contained in $\R^d$$\R^d$ for some $d$$d$. In the tangent space of $N$$N$ at $S$$S$ take a base of the tangent subspace corresponding to the orientation of $\sigma$$\sigma$. Take an analogous base for $\sigma^*$$\sigma^*$. If the ordered pair of these bases forms the orientation of $N$$N$, the orientations on $\sigma$$\sigma$ and on $\sigma^*$$\sigma^*$ are said to be agreeing.

Assume that $T$$T$ is a triangulation of a PL manifold $N$$N$. Denote $k:=\dim\sigma$$k:=\dim\sigma$. Let $T'$$T'$ be the barycentric subdivision of $T$$T$, one of whose vertices is $S$$S$. Take an ordering $(S,A_1,\ldots,A_k)$$(S,A_1,\ldots,A_k)$ of vertices of a $k$$k$-face of $T'$$T'$ contained in $\sigma$$\sigma$, corresponding to the orientation of $\sigma$$\sigma$. Analogously, take an ordering $(S,B_1,\ldots,B_{n-k})$$(S,B_1,\ldots,B_{n-k})$ of vertices of an $(n-k)$$(n-k)$-faces of $T'$$T'$ contained in $\sigma^*$$\sigma^*$, corresponding to the orientation of $\sigma^*$$\sigma^*$. The vertices of the $k$$k$-face and of the $(n-k)$$(n-k)$-face form together an $n$$n$-face of $T'$$T'$. Then $(S,A_1,\ldots,A_k,B_1,\ldots,B_{n-k})$$(S,A_1,\ldots,A_k,B_1,\ldots,B_{n-k})$ is an ordering of vertices of the $n$$n$-face. If this ordering forms the orientation of $N$$N$, the orientations on $\sigma$$\sigma$ and on $\sigma^*$$\sigma^*$ are said to be agreeing.

Analogously one defines agreeing orientations on faces of $T$$T$ and of $T^*$$T^*$ when $T$$T$ is a cellular decomposition.

Take agreeing orientations on faces of $T$$T$ and of $T^*$$T^*$. In this definition we make summations over oriented $k$$k$-faces $\sigma$$\sigma$ of $T$$T$. Take an integer $k$$k$-cycle $x = \sum_\sigma x_\sigma\sigma$$x = \sum_\sigma x_\sigma\sigma$ in $T$$T$. Analogously, take an integer $(n-k)$$(n-k)$-cycle $y = \sum_\sigma y_{\sigma^*}\sigma^*$$y = \sum_\sigma y_{\sigma^*}\sigma^*$ in $T^*$$T^*$. Define the integer intersection product

$\displaystyle \cap_{N;\Z}:H_k(N;\Z)\times H_{n-k}(N;\Z)\to\Z \quad\text{by}\quad [x]\cap_{N;\Z}[y]:= \sum_\sigma x_\sigma y_{\sigma^*}.$ $$\displaystyle \cap_{N;\Z}:H_k(N;\Z)\times H_{n-k}(N;\Z)\to\Z \quad\text{by}\quad [x]\cap_{N;\Z}[y]:= \sum_\sigma x_\sigma y_{\sigma^*}.$$

Analogously to the modulo 2 case, the product of an integer $k$$k$-cycle and a boundary of an $(n-k+1)$$(n-k+1)$-face is zero. This and the PL invariance of homology imply that the integer intersection product is well-defined.

Remark 2.2. Using the notion of cup product, one can give a dual (and so an equivalent) definition:

$\displaystyle I_N(x,y) = \langle x^*\smile y^*,[N]\rangle \in \Z,$ $$\displaystyle I_N(x,y) = \langle x^*\smile y^*,[N]\rangle \in \Z,$$

where $x^*\in H^{n-k}(N)$$x^*\in H^{n-k}(N)$, $y^*\in H^n(N)$$y^*\in H^n(N)$ are the Poincaré duals of $x$$x$, $y$$y$, and $[N]$$[N]$ is the fundamental class of the manifold $N$$N$. We can also define the cup (cohomology intersection) product

$\displaystyle I_N^*: H^k(N;\Zz) \times H^{n-k}(N;\Zz) \to \Zz \quad\text{by}\quad I_N^*(p,q) = \langle p \smile q , [N] \rangle .$ $$\displaystyle I_N^*: H^k(N;\Zz) \times H^{n-k}(N;\Zz) \to \Zz \quad\text{by}\quad I_N^*(p,q) = \langle p \smile q , [N] \rangle .$$

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. 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].

3 Bilinearity and supersymmetry

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 (for $\Z$$\Z$-coefficients). Thus it descends to a bilinear (integer) intersection pairing

$\displaystyle H_k(N;\Z)/\text{Torsion}\times H_{n-k}(N;\Z)/\text{Torsion} \to \Z.$ $$\displaystyle H_k(N;\Z)/\text{Torsion}\times H_{n-k}(N;\Z)/\text{Torsion} \to \Z.$$

on the free modules.

We have

$\displaystyle I_N(x,y) = (-1)^{k(n-k)}I_N(y,x).$ $$\displaystyle I_N(x,y) = (-1)^{k(n-k)}I_N(y,x).$$

Hence for $n=2k$$n=2k$

• If $k$$k$ is even the form $q_N$$q_N$ is symmetric: $q_N(x, y) = q_N(y, x)$$q_N(x, y) = q_N(y, x)$.
• If $k$$k$ is odd the form $q_N$$q_N$ is skew-symmetric: $q_N(x, y) = - q_N(y, x)$$q_N(x, y) = - q_N(y, x)$.

4 Superadditivity and transversal intersections

In this section the results and arguments work for both $\Z_2$$\Z_2$- and $\Z$$\Z$-coefficients, which we omit.

By the rank of a bilinear form on a $\Z$$\Z$-module we mean its rank over $\Q$$\Q$. We abbreviate

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${\rm rk}\cap_M$ to

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${\rm rk}M$. E.g. if $M$$M$ is the connected sum of $s$$s$ copies of $S^k\times S^k$$S^k\times S^k$, then

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${\rm rk}M=2s$; also

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${\rm rk}\C P^2 = {\rm rk}HP^2 = 1$. The rank (of the intersection form) of manifolds is not additive. Indeed, if $M_1=M_2=S^1 \times I$$M_1=M_2=S^1 \times I$ and $M_1\cup M_2=S^1 \times S^1$$M_1\cup M_2=S^1 \times S^1$, then

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${\rm rk}(M_1\cup M_2) =1 > 0= {\rm rk} M_1 + {\rm rk} M_2$.

Lemma 4.1. Let $M_1$$M_1$ and $M_2$$M_2$ be compact orientable $2k$$2k$-manifolds (possibly with boundary).

(a) (monotonicity) If $M_1\subset M_2$$M_1\subset M_2$, then

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${\rm rk}M_1 \le {\rm rk}M_2$.

(b) (superadditivity) Let $M_1\cup M_2$$M_1\cup M_2$ be the union along some boundary components. Then

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${\rm rk}(M_1\cup M_2) \ge {\rm rk}M_1 + {\rm rk}M_2$.

Proof. Part (a) is clear. Let us prove part (b). Let $M_1'$$M_1'$ be the complement in $M_1$$M_1$ to a collar of $\partial M_1$$\partial M_1$. Then by (a)

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$$\displaystyle {\rm rk}(M_1\cup M_2) \ge {\rm rk}(M_1'\sqcup M_2) = {\rm rk}M_1 + {\rm rk}M_2.$$

$\square$$\square$

Let $V$$V$ and $W$$W$ be $k$$k$- and $(n-k)$$(n-k)$-submanifolds of $N$$N$. They are (more precisely, the pair $V,W$$V,W$ is) called transversal if for any $x\in V\cap W$$x\in V\cap W$ there exists a closed neighborhood $Ox$$Ox$ of $x$$x$ in $N$$N$, and a PL homeomorphism $\varphi:Ox\to[-1,1]^n$$\varphi:Ox\to[-1,1]^n$ such that

$\displaystyle \varphi(V\cap Ox)= [-1,1]^k\times0^{n-k} \quad\text{and}\quad \varphi(W\cap Ox)= 0^k\times[-1,1]^{n-k}.$ $$\displaystyle \varphi(V\cap Ox)= [-1,1]^k\times0^{n-k} \quad\text{and}\quad \varphi(W\cap Ox)= 0^k\times[-1,1]^{n-k}.$$

Lemma 4.2. Let $V$$V$ and $W$$W$ be closed transversal $k$$k$- and $(n-k)$$(n-k)$-submanifolds of $N$$N$. Then $[V]\cap_N [W] = V\cdot W$$[V]\cap_N [W] = V\cdot W$, where the right-hand part is the sum of signs of the intersection points of $V,W$$V,W$ for $\Z$$\Z$-coefficients, and is the parity of $|V\cap W|$$|V\cap W|$ for $\Z_2$$\Z_2$-coefficients.

Cf. Theorem 2.1 on intersection number of immersions. A simpler proof of Lemma 4.2 is given by

$\displaystyle [V]\bigcap\limits_N [W] = [V\cap OW]\bigcap\limits_{OW} [W] = [V\cap OW]\bigcap\limits_{OV\cap OW} [W\cap OV] = V\cdot W.$ $$\displaystyle [V]\bigcap\limits_N [W] = [V\cap OW]\bigcap\limits_{OW} [W] = [V\cap OW]\bigcap\limits_{OV\cap OW} [W\cap OV] = V\cdot W.$$

Here

• $OV,OW$$OV,OW$ are tubular (regular) neighborhoods of $V,W$$V,W$, and
• the intersection products

$\displaystyle \cap_{OW}:H_k(OW,\partial OW) \times H_{n-k}(OW) \to R$ $$\displaystyle \cap_{OW}:H_k(OW,\partial OW) \times H_{n-k}(OW) \to R$$

$\displaystyle \cap_{OV\cap OW}:H_k(OV\cap OW,\partial OW) \times H_{n-k}(OV\cap OW,\partial OV) \to R$ $$\displaystyle \cap_{OV\cap OW}:H_k(OV\cap OW,\partial OW) \times H_{n-k}(OV\cap OW,\partial OV) \to R$$

$\displaystyle \cap : H_k(B^k\times B^{n-k},S^{k-1}\times I^{n-k}) \times H_{n-k}(B^k\times B^{n-k},B^k\times S^{n-k-1}) \to R,$ $$\displaystyle \cap : H_k(B^k\times B^{n-k},S^{k-1}\times I^{n-k}) \times H_{n-k}(B^k\times B^{n-k},B^k\times S^{n-k-1}) \to R,$$

where $R$$R$ is either $\Z$$\Z$ or $\Z_2$$\Z_2$, are defined analogously to the above;

• the last equality holds by the transversality and because $[B^k\times0]\cap[0\times B^{n-k}]=1$$[B^k\times0]\cap[0\times B^{n-k}]=1$.

5 Poincaré duality

Theorem 5.1.[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 folklore, but in this short and explicit form is absent from textbooks.) Recall that $T$$T$ is a triangulation of a manifold $N$$N$, and $T^*$$T^*$ is the dual cell subdivision. We use orthogonal complements with respect to the modulo 2 intersection product $I_{T,2}:C_s(T)\times C_{n-s}(T^*)\to\Zz_2$$I_{T,2}:C_s(T)\times C_{n-s}(T^*)\to\Zz_2$. It suffices to prove that

$\displaystyle \phantom{}^\bot Z_{n-s}(T^*)= B_s(T)\quad\text{and}\quad Z_s(T)^\bot=B_{n-s}(T^*).$ $$\displaystyle \phantom{}^\bot Z_{n-s}(T^*)= B_s(T)\quad\text{and}\quad Z_s(T)^\bot=B_{n-s}(T^*).$$

Let us prove the left-hand equality; the right-hand equality is proved analogously. Since $I_{T,2}$$I_{T,2}$ is non-degenerate, we only need to check that $B_s(T)^\bot=Z_{n-s}(T^*)$$B_s(T)^\bot=Z_{n-s}(T^*)$. The inclusion $B_s(T)^\bot \supset Z_{n-s}(T^*)$$B_s(T)^\bot \supset Z_{n-s}(T^*)$ is obvious. The opposite inclusion follows because if $I_{T,2}(\partial c,d)=0$$I_{T,2}(\partial c,d)=0$ for an $(s+1)$$(s+1)$-cell $c$$c$ of $T$$T$ and a chain $d\in С_{n-s}(T^*)$$d\in С_{n-s}(T^*)$, then $\partial d$$\partial d$ does not involve the cell $c^*$$c^*$ dual to $c$$c$.

6 On classification of bilinear forms

Let $q$$q$ and $q'$$q'$ be bilinear forms on free $\mathbb{Z}$$\mathbb{Z}$-modules (or $\Zz_2$$\Zz_2$-vector spaces) $V$$V$ and $V'$$V'$ respectively. The forms $q$$q$ and $q'$$q'$ are called equivalent or isomorphic if there is an isomorphism $f:V \to V'$$f:V \to V'$ such that $q=f^*q'$$q=f^*q'$.

6.1 Invariants: rank, type, signature

The rank of a bilinear form $q$$q$ is the rank of the underlying $\Z$$\Z$-module (or $\Z_2$$\Z_2$-vector space) $V$$V$.

A bilinear form $q$$q$ is even if $q(x,x)$$q(x,x)$ is an even number for any element $x$$x$. Equivalently, if $q$$q$ is written as a square matrix in a basis, it is even if the elements on the diagonal are all even. Otherwise, $q$$q$ is odd.

Let $q$$q$ be a symmetric bilinear form on a free $\Z$$\Z$-module. Denote by $b^+(q)$$b^+(q)$ ($b^-(q)$$b^-(q)$) the number of positive (negative) eigenvalues. Note that $q$$q$ is symmetric, so is diagonalisable over the real numbers, so $b^+(q)$$b^+(q)$ ($b^-(q)$$b^-(q)$) is the dimension of a maximal subspace on which the form is positive (negative) definite.

The signature of $q$$q$ is defined to be

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$$\displaystyle \sigma(q)={\rm sign}(q) = b^+(q) - b^-(q).$$

If $n$$n$ is divisible by 4, the signature

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$\sigma(N)={\rm sign}(N)$ is defined to be the signature of the intersection form of $N$$N$.

6.2 Classification over integers modulo 2

Theorem 6.1. [AdrianAlbert1938], [MacWilliams1969] (a) Every odd symmetric non-degenerate bilinear form $q$$q$ over $\Zz_2$$\Zz_2$ is isomorphic to the sum of some number of `unity' forms $(1)$$(1)$.

(b) Every even symmetric non-degenerate bilinear form $q$$q$ over $\Zz_2$$\Zz_2$ is isomorphic to the sum of some number of `hyperbolic' forms $H(\Zz_2)$$H(\Zz_2)$ of rank $2$$2$ defined by the following matrix

$\displaystyle \left( \begin{array}{cc} ~0 & ~1 \\ 1 & ~0 \end{array} \right).$ $$\displaystyle \left( \begin{array}{cc} ~0 & ~1 \\ 1 & ~0 \end{array} \right).$$

In particular the rank of $q$$q$ is even.

Clearly, different sums from the theorem are not isomorphic.

6.3 Classification of skew-symmetric forms

Theorem 6.2. Every skew-symmetric unimodular bilinear form $q$$q$ over $\Zz$$\Zz$ is isomorphic to the sum of some number of hyperbolic forms $H_-(\Zz)$$H_-(\Zz)$ of rank $2$$2$ defined by the following matrix

$\displaystyle \left( \begin{array}{cc} ~0 & ~1 \\ -1 & ~0 \end{array} \right).$ $$\displaystyle \left( \begin{array}{cc} ~0 & ~1 \\ -1 & ~0 \end{array} \right).$$

In particular the rank of $q$$q$ is even.

Proof. The proof is by induction on the rank of the module $M$$M$ on which $q$$q$ is defined. Let $B$$B$ be the matrix of $q$$q$ in some basis. Then $\det B =\pm1$$\det B =\pm1$. Thus expanding the determinant by the first column we obtain $\pm1$$\pm1$. Hence $\gcd(B_{2,1},\ldots,B_{n,1})=1$$\gcd(B_{2,1},\ldots,B_{n,1})=1$ (observe that $B_{1,1}=0$$B_{1,1}=0$). Denote by $C_{2,2},\ldots,C_{2,n}$$C_{2,2},\ldots,C_{2,n}$ elements of $M$$M$ such that

$\displaystyle \gcd(B_{2,1},\ldots,B_{n,1})=C_{2,2}B_{2,1}+\ldots+C_{2,n}B_{n,1}.$ $$\displaystyle \gcd(B_{2,1},\ldots,B_{n,1})=C_{2,2}B_{2,1}+\ldots+C_{2,n}B_{n,1}.$$

Denote $C_{i,j} := \delta_i^j$$C_{i,j} := \delta_i^j$ for $(i, j) \neq (2,2),\ldots,(n,2)$$(i, j) \neq (2,2),\ldots,(n,2)$. Let $C := (C_{i,j})_{i, j = 1}^n$$C := (C_{i,j})_{i, j = 1}^n$ be a $n \times n$$n \times n$ matrix. Then $(CB)_{2,1}=1$$(CB)_{2,1}=1$. For any matrix $A$$A$ the first row of the matrix $CA$$CA$ is equal to the first row of the matrix $A$$A$. Therefore $(CBC^t)_{2,1}=(CB)_{2,1}=1$$(CBC^t)_{2,1}=(CB)_{2,1}=1$ and $CBC^t$$CBC^t$ represents the matrix of $q$$q$ in a different basis $f_1,\ldots,f_n$$f_1,\ldots,f_n$. Since $q(f_2,f_1) = 1$$q(f_2,f_1) = 1$, the restriction of $q$$q$ onto the submodule $N:=\left<f_2, f_1\right>$$N:=\left<f_2, f_1\right>$ has matrix $H_-(\Zz)$$H_-(\Zz)$. Since $H_-(\Zz)$$H_-(\Zz)$ is non-degenerate and unimodular, $N$$N$ is a direct summand in $M$$M$, i.e., $M = N\oplus N^\bot$$M = N\oplus N^\bot$. Now we restrict $q$$q$ to $N^\bot$$N^\bot$ and apply the inductive hypothesis to the restriction.

$\square$$\square$

Clearly, different sums from the theorem are not isomorphic.

6.4 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 $(r,s)$$(r,s)$ there is an odd unimodular symmetric indefinite form of rank $r$$r$ and signature $s$$s$;

(E) for any pair $(r,8s)$$(r,8s)$ there is an even unimodular symmetric indefinite form of rank $r$$r$ and signature $8s$$8s$.

Cf. Theorem 6.3 and Proposition 6.4.

All values (O) are realised by direct sums of the forms of rank 1,

$\displaystyle b^+ (+1) \oplus b^- (-1).$ $$\displaystyle b^+ (+1) \oplus b^- (-1).$$

An even positive definite form of rank 8 is given by the $E_8$$E_8$ matrix

$\displaystyle E_8 = \left( \begin{array}{c c c c c c c c} \ 2 \ & \ 1\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 \\ \end{array} \right) .$ $$\displaystyle E_8 = \left( \begin{array}{c c c c c c c c} \ 2 \ & \ 1\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ & \ 0\ \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 \\ \end{array} \right) .$$

Likewise, the matrix $-E_8$$-E_8$ represents a negative definite even form of rank 8.

On the other hand, the matrix $H$$H$ given by

$\displaystyle H = \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0 \end{pmatrix}$ $$\displaystyle H = \begin{pmatrix} \ 0 \ & \ 1 \ \\ 1 & 0 \end{pmatrix}$$

determines an indefinite even form of rank 2 and signature 0. It is easy to see that the direct sums

$\displaystyle k E_8 \oplus l H$ $$\displaystyle k E_8 \oplus l H$$

with $k,l\in\Zz$$k,l\in\Zz$, $l>0$$l>0$ realise all forms from (E). Here we use the convention that $k q$$k q$ is the $k$$k$-fold direct sum of $q$$q$ for $k>0$$k>0$ and $kE_8$$kE_8$ is the $(-k)$$(-k)$-fold direct sum of $-E_8$$-E_8$ for $k<0$$k<0$.

6.5 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 $\Zz$$\Zz$ is isomorphic to the sum of some number of `unity' forms $(1)$$(1)$ and some number of `minis unity' forms $(-1)$$(-1)$.

(b) Two indefinite unimodular symmetric bilinear forms over $\mathbb{Z}$$\mathbb{Z}$ 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 $c \in V$$c \in V$ is called a characteristic vector of the form $q$$q$ if

$\displaystyle q(c,x) \equiv q(x,x) \ (\text{mod} \ 2)$ $$\displaystyle q(c,x) \equiv q(x,x) \ (\text{mod} \ 2)$$

for all elements $x \in V$$x \in V$. Characteristic vectors always exist. In fact, when reduced modulo 2, the map $x \mapsto q(x,x) \in\Z/2$$x \mapsto q(x,x) \in\Z/2$ is linear. Hence by unimodularity there exists an element $c$$c$ such that the map $q(c,-)$$q(c,-)$ equals this linear map.

The form $q$$q$ is even if and only if $0$$0$ is a characteristic vector. If $c$$c$ and $c'$$c'$ are characteristic vectors for $q$$q$, then by unimodularity there is an element $h$$h$ with $c' = c + 2h$$c' = c + 2h$. Hence the number $q(c,c)$$q(c,c)$ is independent of the chosen characteristic vector $c$$c$ modulo 8. One can be more specific:

Proposition 6.5. For a characteristic vector $c$$c$ of the unimodular symmetric bilinear form $q$$q$ one has

$\displaystyle q(c,c) \equiv \text{sign}(q) \ (\text{mod} \ 8)$ $$\displaystyle q(c,c) \equiv \text{sign}(q) \ (\text{mod} \ 8)$$

Proof. Suppose $c$$c$ is a characteristic vector of $q$$q$. Then $c + e_+ + e_-$$c + e_+ + e_-$ is a characteristic vector of the form $q':= q \oplus H_-(\Zz)$$q':= q \oplus H_-(\Zz)$, where $e_+, e_-$$e_+, e_-$ form basis elements of the additional $\mathbb{Z}^2$$\mathbb{Z}^2$ summand with? square $\pm 1$$\pm 1$. We notice that

$\displaystyle q(c,c) = q'(c+e_+ + e_-, c+e_+ + e_-) .$ $$\displaystyle q(c,c) = q'(c+e_+ + e_-, c+e_+ + e_-) .$$

However, the form $q'$$q'$ is indefinite, so the above classification theorem applies. In particular, $q'$$q'$ is odd and has the same signature as $q$$q$, so it? is equivalent to the diagonal form with $b^+ + 1$$b^+ + 1$ summands of (+1) and $b^- + 1$$b^- + 1$ summands of $(-1)$$(-1)$. This diagonal form has a characteristic vector $c'$$c'$ that is simply a sum of basis elements in which the form is diagonal. Of course $q'(c',c') = b^+ - b^-$$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.

$\square$$\square$

7 References

• [AdrianAlbert1938] A. Adrian Albert, Symmetric and alternate matrices in an arbitrary field, I, Trans. Amer. Math. Soc., 43(3) (1938) 386-436.
• [Kirby1989] R.C. Kirby, The topology of 4-manifolds, Lecture Notes in Math. 1374, Springer-Verlag, 1989. MR1001966 (90j:57012)
• [MacWilliams1969] J. MacWilliams, Orthogonal matrices over finite fields, Amer. Math. Monthly, 76 (1969) 152--164.
• [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. Preprint of a part in English

8 External links

• The Wikipedia page on Poincaré duality