TITLE: Criterion for alternation of the linking form
QUESTION [5 upvotes]: I was recently informed by a source of the following fact: 
Theorem 1: The linking form on an orientable smooth 5-manifold $M$ is alternating if and only if $M$ is spin$^{\mathbb{C}}$.
Question 1: Does anybody know a reference/attribution for this fact? I've poked around online and in the library, and asked my usual go-to experts, but found nothing. (This page gives a reference to a criterion of Wall in the simply-connected case, but not the general one.)
I'm not in a position to be able to ask my source for a reference, but I'm pretty confident that Theorem 1 is correct because I think I have a proof. In fact the proof gives a criterion for the linking form on any orientable, odd-dimensional topological manifold to be alternating, which specializes to the above fact for smooth 5-manifolds. 
Question 2: Is such a criterion already known? Written down? Is it remotely interesting?

REPLY [4 votes]: The final sentence of Theorem 1 of Browder's Remark on the Poincaré Duality Theorem states that a five-dimensional space $X$ with Poincaré duality has $H_2(X) = F + T + T + \mathbb{Z}_2$, where $F$ is free and $T$ is torsion, if and only if $w_3 \neq 0$. If $b$ denotes the linking form and $x$ is the generator of $\mathbb{Z}_2$, then $b(x, x) = \frac{1}{2}$ so $b$ is not alternating. Therefore, if $b$ is alternating, then $w_3 = 0$ which is equivalent to $w_2$ having an integral lift (which is equivalent to spin${}^c$ in the smooth case).
I don't know of a reference for the converse. However, if $H_2(X)$ has no two-torsion, then it is spin${}^c$ and the fact that the linking form is alternating follows from the classification of non-singular anti-symmetric linking forms; see here.