"Signature" of manifold in singly even dimensional ($4k+2$) manifold? Signature of manifold in doubly even dimensional ($4k$) manifold
Given a connected and oriented manifold $M$ of dimension $4k$, the cup product gives rise to a quadratic form $Q$ on the 'middle' real cohomology group
$${ H^{2k}(M,\mathbf {R} )}.$$
The basic identity for the cup product
$${ \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}$$
shows that with $p = q = 2k$ the product is symmetric. It takes values in
$${  H^{4k}(M,\mathbf {R} )}.$$
If we assume also that M is compact, Poincaré duality identifies this with
$${\displaystyle H^{0}(M,\mathbf {R} )}$$
which can be identified with $\mathbf {R} $. Therefore cup product, under these hypotheses, does give rise to a symmetric bilinear form on $H^{2k}(M,\mathbf {R} )$; and therefore to a quadratic form $Q$. The form $Q$ is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself.  More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.
The signature of $M$ is by definition the signature of $Q$, an ordered triple according to its definition. If $M$ is not connected, its signature is defined to be the sum of the signatures of its connected components.
My question
is that how do we obtain an analogy of
"Signature" of manifold in singly even dimensional ($4k+2$) manifold?
It is said this is related to Arf invariant, but which Arf seems only defined in 2 dimensional, not higher 6, 10, 14 dimensional manifolds? What are some intuitions behind? Why are they related to
"Signature" of manifold?
 A: In dimension $4k+2$ there is no nontrivial invariant that can be defined only using the middle homology of $M$ that is invariant under surgery like the signature. This follows from the fact that the symmetric L group in dimension $4k+2$ vanishes. The symmetric L groups measure symmetric bilinear forms up to stable equivalence which corresponds to what happends to the middle homology under some trivial surgeries. Hence, all symmetric bilinear forms are trivial up to stable equivalence in dimension $4k+2$.
The Arf invariant is an invariant of quadratic forms. It is able to exist because the quadratic L group is nontrivial in dimension $4k+2$. However, it requires a quadratic refinement of the middle dimension bilinear form (roughly). This is usually derived from a stable framing of $M$. This Arf invariant works in any dimension $4k+2$.
Surely there are invariants that can be defined as long as we don't require invariance under surgery, but the primary use of signature is that it has this property, so anything that is analogous to signature should probably have this property.
