Let $M^4$ be an orientable closed 4-manifold and $c\_1$ be the first Chern class of a complex line bundle on $M^4$. Let $b$ be the mod 2 reduction of $c\_1$, ie $b=c\_1$ mod 2.


We have a relation $w\_2 b = b^2$, where $w\_n$ is the $n^\text{th}$ Stiefel-Whitney class of the tangent bundle of $M^4$. This implies that if $M^4$ is spin, then the Chern number
on $M^4$ must be even, ie $\int\_{M^4} c\_1^2 =0$ mod 2.


My question is that for any $M^4$ that is not spin, can we always find a complex line bundle on $M^4$, such that the Chern number on $M^4$ is odd, ie $\int\_{M^4} c\_1^2 =1$ mod 2.



Chern number on non-spin manifold
The Enriques algebraic surface has even intersection form (i.e. for any class $\beta \in H^{2}(M,\mathbb{Z})$, $\int\_{M^{4}} \beta^2$ is even) but is not spin by Rokhlin's theorem since the signature of the intersection form is $8$.


A simply connected $4$-manifold is spin $\iff$ the intersection form is even (which doesn't apply to the Enriques surface which has $\pi\_{1} = \mathbb{Z}\_{2}$).


