TITLE: Second Stiefel Whitney class of quotients of odd spheres
QUESTION [9 upvotes]: I don't know much of algebraic topology so the following question could be very silly. Let $G$ a finite subgroup of $U(n)$ that acts linearly (the action induced by the action of $U(n)$ on $\mathbb{C}^{n}$) and freely on the unit sphere $S^{2n-1}\subset \mathbb{C}^{n}$. Let $X$ be the quotient manifold $$X:=S^{2n-1}/G$$
I want to compute $w_{2}(X)\in H^{2}(X,\mathbb{Z}_{2})$ i.e. the second Stiefel-Whitney class of the tangent bundle of $X$. More precisely i'd like to know if it is $0$ or not and under which hypotheses on $G$ it vanishes.  Is this a standard problem? Is there some reference for this type of calculation or something similar? 
Thank you in advance!

REPLY [3 votes]: Because $G \subset U(n)$, the stabilised tangent bundle of $X = S^{2n-1}/G$ has an obvious complex structure. The determinant bundle $\lambda$ is the quotient of $S^{2n-1} \times \mathbb{C}$ by $G$ acting as $(p, z) \mapsto (g(p), (\det g) z)$. Now $\det : G \to \mathbb{R}/\mathbb{Z}$ descends to a homomorphism $H_1(X) \to \mathbb{R}/\mathbb{Z}$, so defines an element $\alpha \in H^1(X; \mathbb{R}/\mathbb{Z})$. Then $c_1(\lambda) \in H^2(X; \mathbb{Z})$ is the image of $\alpha$ under the Bockstein map associated to $0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z} \to 0$. Its mod 2 reduction equals $w_2(X) \in H^2(X; \mathbb{Z}/2\mathbb{Z})$.
Equivalently, $w_2(X)$ is the image of $\alpha$ under the Bockstein of $0 \to \mathbb{Z}/2\mathbb{Z} \to \mathbb{R}/2\mathbb{Z} \to \mathbb{R}/\mathbb{Z} \to 0$. Whether $\alpha$ is in the image of $H^1(X; \mathbb{R}/2\mathbb{Z})$ depends on whether one can make a consistent choice of square roots of $\det g$ for $g \in G$ (essentially the same lifting problem that appears in José's answer). I believe this is possible if and only if any $\bar g$ in the abelianisation of $G$ with $\det \bar g = -1$ has order divisible by 4 (or if you like: $g^2$ is not a commutator for any $g \in G$ with $\det g = -1$). That is then a criterion for $w_2(X)$ to vanish.