Intersection form

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1 Introduction

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

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

defined on the homology of $N$$N$. For $n=2k$$n=2k$ this is the intersection form of $N$$N$ denoted by $q_N$$q_N$. For $n=4k$$n=4k$ the signature of this form is the signature $\sigma(N)$$\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.

In this page $T$$T$ is a triangulation (or a cell subdivision) of $N$$N$, and $T^*$$T^*$ is the dual cell subdivision.

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

2 Definition of the intersection product

In this subsection we mostly omit $\Zz_2$$\Zz_2$-coefficients.

A short direct definition of a homology group with $\Zz_2$$\Zz_2$-coefficients. For an integer $s$$s$ denote by $C_s(T)=C_s(T;\Zz_2)$$C_s(T)=C_s(T;\Zz_2)$ the set (the $\Zz_2$$\Zz_2$-space) of arrangements of zeroes and units on the $s$$s$-dimensional cells of $T$$T$ (so $C_s(T)=\{0\}$$C_s(T)=\{0\}$ if there are no $s$$s$-dimensional cells in $T$$T$). Denote by $\partial=\partial_s\colon C_s(T) \to C_{s - 1}(T)$$\partial=\partial_s\colon C_s(T) \to C_{s - 1}(T)$ the extension over $C_s(T)$$C_s(T)$ of the map taking an $s$$s$-dimensional cell $\sigma$$\sigma$ of $T$$T$ to the boundary of $\sigma$$\sigma$. Denote

$\displaystyle Z_s(T)=Z_s(T;\Zz_2):=\ker\partial_s\quad\text{and}\quad B_s(T)=B_s(T;\Zz_2):=\mathrm{im}\partial_{s + 1}.$ $$\displaystyle Z_s(T)=Z_s(T;\Zz_2):=\ker\partial_s\quad\text{and}\quad B_s(T)=B_s(T;\Zz_2):=\mathrm{im}\partial_{s + 1}.$$

Then the $s$$s$-th homology group of $T$$T$ with $\Zz_2$$\Zz_2$-coefficients is defined as $H_s(T)=H_s(T;\Zz_2):=Z_s(T)/ B_s(T)$$H_s(T)=H_s(T;\Zz_2):=Z_s(T)/ B_s(T)$. This depends only on $N$$N$, not on $T$$T$.

A short direct definition of the modulo 2 intersection product. For modulo 2 chains $x\in C_k(T)$$x\in C_k(T)$ and $y\in C_{n-k}(T^*)$$y\in C_{n-k}(T^*)$ define the modulo 2 intersection number by the formula

$\displaystyle I_{T,2}(x,y)=x\cap_{T,2} y=\langle\, x \, , \, y\, \rangle := |x\cap y|\mod2\in\Zz_2.$ $$\displaystyle I_{T,2}(x,y)=x\cap_{T,2} y=\langle\, x \, , \, y\, \rangle := |x\cap y|\mod2\in\Zz_2.$$

Represent classes $[x]\in H_k(N;\Zz_2)$$[x]\in H_k(N;\Zz_2)$ and $[y]\in H_{n-k}(N;\Zz_2)$$[y]\in H_{n-k}(N;\Zz_2)$ by cycles $x$$x$ and $y$$y$ viewed as unions of $k$$k$-simplices of $T$$T$ and $(n-k)$$(n-k)$-simplices of $T^*$$T^*$, respectively. Define the modulo 2 intersection product

$\displaystyle I_{N,2}=\cap_{N,2}: H_k(N;\Zz_2) \times H_{n-k}(N;\Zz_2) \to \Zz_2\quad\text{by}\quad I_{N,2}([x],[y]):=I_{N,2}(x,y).$ $$\displaystyle I_{N,2}=\cap_{N,2}: H_k(N;\Zz_2) \times H_{n-k}(N;\Zz_2) \to \Zz_2\quad\text{by}\quad I_{N,2}([x],[y]):=I_{N,2}(x,y).$$

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).

Sketch of a short direct definition of the intersection product. Analogously, counting intersections with signs, one defines the intersection number

$\displaystyle I_N(x,y)=x\cdot y=\langle\, x \, , \, y\, \rangle \in \Z$ $$\displaystyle I_N(x,y)=x\cdot y=\langle\, x \, , \, y\, \rangle \in \Z$$

of integer chains $x\in C_k(T;\Zz)$$x\in C_k(T;\Zz)$ and $y\in C_{n-k}(T^*,\Zz)$$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$$I_N: H_k(N;\Zz)\times H_{n-k}(N;\Zz)\to\Zz$.

Remark 2.1. (a) Using the notion of transversality, one can give an equivalent (and `more general') definition as follows. Take a $k$$k$-chain $x\in C_k(N;\Zz)$$x\in C_k(N;\Zz)$ and an $(n-k)$$(n-k)$-chain $y\in C_{n-k}(N;\Zz)$$y\in C_{n-k}(N;\Zz)$ which are transverse to each other. The signed count of the intersections between $x$$x$ and $y$$y$ gives the intersection number $I_N(x,y)$$I_N(x,y)$. A particular case is intersection number of immersions. Then define the intersection product $I_N$$I_N$ by $I_N([x],[y]):=I_N(x,y)$$I_N([x],[y]):=I_N(x,y)$.

(b) 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 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

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

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 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 $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$$T$ and a chain $d\in Z_{n-s}(T^*)$$d\in Z_{n-s}(T^*)$, then $\partial d$$\partial d$ does not involve the cell $c^*$$c^*$ dual to $c$$c$.

5 Definition of signature

Let $q$$q$ be a symmetric bilinear form on a free $\mathbb{Z}$$\mathbb{Z}$-module. Denote by $b^+(q)$$b^+(q)$ ($b^-(q)$$b^-(q)$) the number of positive (negative) eigenvalues.

Note that since $q$$q$ is symmetric, it is diagonalisable over the real numbers, so $b^+(q)$$b^+(q)$ ($b^-(q)$$b^-(q)$) is the dimension of a maximal subspace on which the form is positive (negative) definite.

Then 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 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'$.

The rank of $q$$q$ is the rank of the underlying $\mathbb{Z}$$\mathbb{Z}$-module (or $\Zz_2$$\Zz_2$-vector space) $V$$V$.

A 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.

6.1 Classification over integers modulo 2

Theorem 6.1. (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.

6.2 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.

6.3 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 $(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$.

6.4 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 $\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

• [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

8 External links

• The Wikipedia page on Poincaré duality