Geometric Definition of the Kervaire Invariant On the wikipedia article for the Kervaire Invariant, it is stated that the relevant quadratic form can be defined geometrically via the self-intersections of immersions $S^{2m+1}\to M^{4m+2}$ determined by the framing.
How is this done?
References would be much appreciated!
 A: Maybe what they refere to on wikipedia is the approach Levine takes in "lectures on groups of homotopy spheres". The construction goes as follows: Suppose $M$ is $k-1$-connected of dimension 2k, $k$ odd. (It is only in dimension $2k$, $k$ odd the Kervaire invariant is defined, and it is "easy" to show that any framed manifold is framed cobordant to a $k-1$ connected one, see Kervaire and Milnor, Groups of Homotopy Spheres). Suppose you have an immersion $f:S^k\rightarrow M$. Then consider the pullback-bundle of the stable tangent bundle of $M$, 
$$
f^*(\epsilon^N\oplus TM)=\epsilon^N\oplus\nu(f)\oplus TS^k=\epsilon^{N-1}\oplus \nu(f)\oplus TD^{k+1}|_{S^k}
$$
where $\epsilon^N$ is the trivial $N$-plane bundle. The framing of $M$ gives you a framing of this bundle. The standard framing of $\epsilon^{N-1}\oplus TD^{k+1}$ gives you a framing of $TD^{k+1}|_{S^k}$. Together these two framings determine a map $S^k\rightarrow V_{N+2k,N+k}$, and the homotopy class is dependent only on the regular homtopy class of the immersion $f$. Since any two embeddings are regularly homotopic (homotopic through immersions) we get a well defined map $H_k(M)\rightarrow \pi_k(V_{N+2k,N+k})=\mathbb{Z}_2$ by defining $\phi_0(\alpha)=\phi_0(f)$ where $f$ is any embedding representing $\alpha$. This is a quadratic refinement of the mod 2 intersection pairing (it takes another page or two to show this). We define the Kervaire invariant as the Arf invariant:
$$
\Phi(M)=\sum_{i=1}^r\phi(x_i)\phi(y_i)
$$
where $\{x_i,y_i\}$ forms a symplectic basis for $H_k(M;\mathbb{Z}_2)$.
