The signature of a closed oriented manifold of dimension divisible by four

Assume AC (The Axiom of Choice), inherited from the nondegeneracy of the middle form. Let $M$ be a closed oriented smooth manifold of dimension $4k$, $k\ge 0$, and let $Q_M$ be its middle-dimensional intersection form on $H^{2k}(M;\mathbb{R})$ (The middle-dimensional intersection form of a closed oriented 4k-manifold, The middle-dimensional intersection form is symmetric and nondegenerate). Let $(p,q,z)$ be the inertia data of the symmetric form $Q_M$ (Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form): $p$ the maximal dimension of a positive-definite subspace, $q$ the maximal dimension of a negative-definite subspace, and $z$ the dimension of the radical. By Sylvester's law of inertia (Sylvester's law of inertia: every real symmetric form is congruent to $\mathrm{diag}(I_p,-I_q,0_r)$, and $(p,q,r)$ is unique) these integers are intrinsic to $Q_M$ and $p+q+z={\dim }_{\mathbb{R}}{H}^{2k}(M;\mathbb{R})$. The signature of $M$ is $\sigma (M):=p-q\in \mathbb{Z}\mathrm{.}$

The middle form is nondegenerate by The middle-dimensional intersection form is symmetric and nondegenerate, so $z=0$ and $p+q={\dim }_{\mathbb{R}}{H}^{2k}(M;\mathbb{R})$; the signature can equivalently be read as the difference of the numbers of positive and negative diagonal entries in any basis diagonalizing $Q_M$. For $k=0$ the group $H^0(M;\mathbb{R})$ is free on the components of $M$ and the pairing is the signed count of components, so $\sigma (M^0)$ is the number of positively oriented components minus the number of negatively oriented components, and $\sigma (\mathrm{\varnothing })=0$.

The signature is defined by this item only for dimensions divisible by four; no value is assigned in other dimensions (see the closing bookkeeping remark on this page). The value is independent of the chosen diagonalizing basis and of reading the form over $\mathbb{Q}$ or over $\mathbb{R}$ by The signature is independent of the diagonalizing basis and unchanged by scalar extension from the rationals to the reals ↗, the lemma named in justified_by.