Parity of self-linking For a class $x$ in $H_{2k}$ of a 4k-manifold $M$, the self-intersection $x.x$ agrees mod 2 with the cap product of $x$ with the Wu class $v_{2k}$. If instead $x$ is a torsion element of $H_{2k}$ of a (4k+1)-manifold, the self-linking of $x$ defines an element of order 2. Is this equal to the inner product of $x$ with $v_{2k}$?
 A: Yes. For any space $M$ there is defined a torsion linking pairing on the $Q/Z$-coefficient cohomology groups
$$L:H^m(M;Q/Z) \times H^n(M;Q/Z) \to H^{m+n+1}(M;Q/Z);(a,b) \mapsto \delta a \cup b = a \cup \delta b$$
with $\delta:H^p(M;Q/Z)\to H^{p+1}(M)$ the Bockstein coboundaries for $p=m,n$, such that $$\text{image}(\delta)=\text{kernel}(H^{p+1}(M)\to H^{p+1}(M;Q))=\text{torsion}(H^{p+1}(M)).$$ 
This pairing is $(-1)^{(m+1)(n+1)}$-symmetric
$$L(a,b)=(-1)^{(m+1)(n+1)}L(b,a) \in H^{m+n+1}(M;Q/Z).$$
If $M$ is a closed connected oriented $(m+n+1)$-dimensional manifold this gives a  $(-1)^{(m+1)(n+1)}$-symmetric torsion linking pairing
$$L:H^m(M;Q/Z) \times H^n(M;Q/Z) \to H^{m+n+1}(M;Q/Z)=Q/Z$$
which induces the usual nonsingular linking pairing on the torsion groups
$$L:{\rm torsion}(H^{m+1}(M)) \times {\rm torsion}(H^{n+1}(M)) \to Q/Z.$$
For a $(4k+1)$-dimensional manifold $M$ and $m=n=2k$ this linking pairing is skew-symmetric. The self-linking defines a linear map
$$H^{2k}(M;Q/Z) \to Q/Z~ ;~ a \mapsto L(a,a)=0~\text{or}~1/2$$
which is represented by the image of the Wu class $v_{2k}(M) \in H^{2k}(M;Z_2)$ under the injection $Z_2 \to Q/Z~;~1 \mapsto 1/2$. The Poincare dual of a torsion homology class $x \in H_{2k}(M)$ is of the form $\delta a\in H^{2k+1}(M)$ for some $a \in H^{2k}(M;Q/Z)$ and
$$\text{self-linking}(x)=L(a,a)= \delta a \cup v_{2k}(M) = \langle x,v_{2k}(M) \rangle \in Z_2\subset Q/Z.$$
[Subsequent editing 17.1.2015]
On page 49 of the 1976 AMS Memoir "A product formula for surgery obstructions" by John Morgan it is stated without proof that for a $(4k+1)$-dimensional manifold $M$
$$L(x,x) ~=~\langle x,v_{2k}(M)\rangle \in Z_2 \subset Q/Z~{\rm for}~ x \in {\rm torsion}(H_{2k}(M))~.$$
It must be confessed that the above "proof" does not go into sufficient detail. So here is a direct proof of Morgan's formula. Let $C=C(M)$ be the chain complex of $M$, $C^*={\rm Hom}_Z(C(M),Z)$ the cochain complex, and consider the cup$_0$ chain map and the cup$_1$ chain homotopy
$$\phi_0 ~=~ [M]\cap -~ :~ C^{4k+1-*} \to C~,~\phi_1
 ~:~ \phi_0~ \simeq~ T\phi_0~ :~ C^{4k+1-*} \to  C$$
given by the evaluation of an extended diagonal chain map on a fundamental cycle $[M] \in C_{4k+1}$, with $T$ the signed transposition involution on ${\rm Hom}_Z(C^*,C) = C\otimes_ZC$.  In particular there are defined $Z$-module morphisms
$$\phi_0~:~C^{2k+1} \to C_{2k}~,~ \phi'_0:C^{2k} \to  C_{2k+1}~,~ \phi_1~:~C^{2k+1} \to C_{2k+1}$$
such that
$$d \phi'_0~=~ \phi_0 d^* ~:~ C^{2k} \to C_{2k}~,$$
$$\phi_0 + \phi'{_0}^*~ =~ d\phi_1+\phi_1 d^*~ :~ C^{2k+1} \to C_{2k}$$
$$d\phi_0 + (d\phi_0)^*~ =~ d\phi_1d^*~ :~ C^{2k} \to C_{2k}~.$$
The intersection number of cohomology classes $x \in H^{2k+1}(M)$, $y \in H^{2k}(M)$
is given by
$$\langle x \cup y,[M]\rangle ~=~\phi_0(x)(y) \in Z~ .$$
Also
$$\langle Sq^{2k}(x),[M]\rangle ~ =~ \phi_1(x)(x)~=~\langle v_{2k}(M) \cup x,[M]\rangle  \in Z_2~.$$
The linking number of torsion cohomology classes  $x,z \in {\rm torsion}(H^{2k+1}(M))$ is
$$L(x,z)~ =~ \phi_0(d^*)^{-1}(x)(z) \in  Q/Z~.$$
In view of the formal identity
$$\phi_0(d^*)^{-1} + (\phi_0(d^*)^{-1})^*~ =~ \phi_1$$
the self-linking number of $x=z$
$$L(x,x)~ =~ \phi_0(d^*)^{-1}(x)(x) \in Q/Z$$
is represented by a rational number $r \in Q$ such that
$$2r~ =~ \phi_1(x)(x) \in Z \subset Q$$
so that $r=0$ or $1/2 \in Q/Z$, according to the value of $\phi_1(x)(x) \in Z_2$, and
$$L(x,x)~ =~ \langle v_{2k}(M) \cup x,[M]\rangle  \in Z_2 \subset Q/Z~.$$
For a fuller account of the generalized Wu-Thom relationship between the symmetric structure on a manifold (resp. geometric Poincare complex) and the Wu classes of the normal bundle (resp. Spivak normal fibration) see section 9 of my 1980 Proc. LMS paper The algebraic theory of surgery II. http://www.maths.ed.ac.uk/~aar/papers/ats2.pdf 
For a fuller account of the relationship between intersection and linking numbers and the Wu classes check out section 3.3  of my 1981 Princeton book Exact sequences in the algebraic theory of surgery http://www.maths.ed.ac.uk/~aar/books/exacsrch.pdf 
