The middle-dimensional intersection form of a closed oriented 4k-manifold

Let $M$ be a closed oriented smooth manifold of dimension $4k$, $k\ge 0$, with orientation $o$ and fundamental class $[M]\in H_{4k}(M;\mathbb{Z})$ determined by $o$ (Fundamental class of a compact oriented manifold, Top homology of a connected manifold). The middle-dimensional intersection form of $M$ is $Q_M:H^{2k}(M;\mathbb{R})\times H^{2k}(M;\mathbb{R})⟶\mathbb{R},Q_M(x,y):=⟨x⌣y,[M]⟩,$ where $⌣$ is the singular cup product of Singular cohomology ring and $⟨-,-⟩$ is the Kronecker evaluation pairing of Kronecker evaluation pairing, read on the fundamental class. The pairing is well defined: the cup product is defined on cohomology classes and the evaluation on classes is independent of cocycle and cycle representatives by The kronecker pairing is independent of cocycle and cycle representatives. Since $\dim M=4k$ and both variables have degree $2k$, the product $x⌣y$ has degree $4k$ and pairs with the top-degree class $[M]$; the coefficient field is $\mathbb{R}$, with $\mathbb{Z}\subset \mathbb{R}$ as the coefficient image.

Integral form. On images of integral classes the formula restricts to the integral pairing $⟨x⌣y,[M]⟩$ on $H^{2k}(M;\mathbb{Z})$, valued in $\mathbb{Z}$. The torsion subgroup of $H^{2k}(M;\mathbb{Z})$ lies in the kernel of that integral pairing; this is proved in the symmetry and nondegeneracy lemma following on this page, not assumed here, and it is what lets the real form be controlled by the free quotient.

Geometric identification. For closed oriented embedded submanifolds $A,B\subset M$ of complementary dimension $2k$, with Poincaré duals $\mathrm{P}\mathrm{D}[A],\mathrm{P}\mathrm{D}[B]\in H^{2k}(M;\mathbb{Z})$ of their fundamental classes, one has $Q_M(\mathrm{P}\mathrm{D}[A],\mathrm{P}\mathrm{D}[B])=I(A,B)=⟨A,B{⟩}_M,$ the geometric intersection number, with the cohomology-first, front-evaluation cap and cup conventions and the geometric factor order fixed by The geometric intersection number is the Poincare-dual cup pairing; no third sign convention is introduced (The cap-product order is fixed by the AT convention, not minted here).

Disconnected and empty manifolds. If $M=M_1\bigsqcup \cdots \bigsqcup M_r$ is a disjoint union of closed oriented components, the orientation restricts to each component (orientability is componentwise, Every manifold is F2-orientable and orientability is componentwise) and $Q_M$ is defined componentwise, on the summands of $H^{2k}(M;\mathbb{R})={⨁}_j{H}^{2k}(M_j;\mathbb{R})$. For the empty manifold one sets $Q_{\mathrm{\varnothing }}=0$. The form is determined by this displayed formula alone; its symmetry, nondegeneracy and additivity properties are not part of the definition and are proved in the lemma following on this page. The pairing formula itself uses no choice principle. The geometric identification assumes AC (The Axiom of Choice), inherited from its Poincare-duality supplier.