Kervaire invariant

Definition

Let $k \ge 0$ and let $M$ be a closed smooth manifold of dimension $4k + 2$ with a framing of its stable normal bundle. Let $V = H^{2k+1}(M; \F_2)$ be its middle singular cohomology with coefficients in $\F_2$. The cup product followed by evaluation on the fundamental class gives a nondegenerate bilinear form $x \cdot y$ on $V$, and $x \cdot x = 0$ for all $x$. The framing determines a function $q \colon V \to \F_2$ such that $$q(x + y) = q(x) + q(y) + x \cdot y \quad \text{for all } x, y \in V.$$ Choose a basis $a_1, \dots, a_m, b_1, \dots, b_m$ of $V$ with $a_i \cdot b_j = \delta_{ij}$ and $a_i \cdot a_j = b_i \cdot b_j = 0$. The Kervaire invariant of $M$ is the Arf invariant of the quadratic form $q$: $$\Phi(M) = \sum_{i=1}^{m} q(a_i)\, q(b_i) \in \F_2.$$