TITLE: Variants and Generalizations of Arf (-Brown-Kervaire) invariants
QUESTION [7 upvotes]: (1) I encounter the Arf invariants in Kirby-Taylor, Pin structures on low-dimensional manifolds. The form that I looked at was:
$$
S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[\pi \;i\; q(x)]
$$ 
The $M^2$ is an oriented 2 dimensional manifold with spin structures that has $\mathbb{Z}_2$ valued quadratic forms on $H_1(M^2,\mathbb{Z}_2)$, which obeys $$q(x + y) = q(x) +
x ∩ y + q(y) \mod 2,$$ here $x ∩ y$ denotes the $\mathbb{Z}_2$ intersection pairing. The bordism invariant is the Arf invariant.
(2) There is a generalization of the Arf invariant, the Arf-Brown-Kervaire
invariant:
$$S(q)=|H^1(M^2,\mathbb{Z}_2)|^{-1/2} \sum_{x\in H^1(M^2,\mathbb{Z}_2)} \exp[2\pi \;i\; q(x)/4].$$
It takes values in $\mathbb{Z}_2) \in  U(1)$. If $q(x)$ is even, $\forall x$, say $q$
is $\mathbb{Z}_2$-valued, then the manifold $M^2$ is orientable., it reduces to the Arf invariant.
I hope that I did not make wrong statements above, please correct me if I did it wrong.

Question: The context I know is only for 2 dimensional manifold. Do we have some analogous Arf and Arf-Brown-Kervaire invariants for (1) higher dimensional manifolds? and (2) analogous form of Arf and Arf-Brown-Kervaire invariants by considering higher homology group $H^d(M,\mathbb{Z}_2)$ or 
  $H^d(M,\mathbb{Z}_n)$? Do we have such variants and generalizations of the above?



Info from Kirby-Taylor, Pin structures on low-dimensional manifolds:

REPLY [4 votes]: The Arf invariant and its generalizations occur all over the place in high-dimensional topology, starting with Kervaire's use of his version to construct a topological manifold with no smooth structure. As in the versions you cite, one is dealing with a quadratic form whose underlying form is the intersection form. This was systematized by Kervaire and Milnor in their paper "Groups of homotopy spheres". The Arf invariant also plays a fundamental role in Sullivan's analysis of normal maps, which contains much of the information about a manifold beyond its homotopy type.  You can find this summarized nicely in Browder's book on simply-connected surgery, and further developments in Wall's book, Surgery Theory.
You can find a historical treatment of the subject in a paper by Ed Brown, and a discussion of its impact in homotopy theory in Snaith's article in the Notices of the AMS.