The orientation-reversed projective plane has signature minus one

Given: AC; the closed oriented smooth $4$-manifold ${\mathbb{C}\mathbb{P}}^2$ with the complex orientation and its orientation reversal $-{\mathbb{C}\mathbb{P}}^2$; the generator $y=c_1({\gamma }^{\ast })$.

For closed oriented smooth $4k$-manifolds, $\sigma (-M)=-\sigma (M)$, where $-M$ is $M$ with the reversed orientation (The signature is additive under disjoint union and negates under orientation reversal).

The fundamental class of the orientation-reversed manifold is $[-M]=-[M]$ in $H_{4k}(M;\mathbb{Z})$: the class determined by the reversed orientation is the negative of the class determined by the original orientation (Fundamental class of a compact oriented manifold).

Pontryagin classes are defined from the complexification of the bundle and require no orientation of the base; reversing the orientation of the manifold leaves the tangent bundle and its Pontryagin classes unchanged, and for ${\mathbb{C}\mathbb{P}}^2$ the supplier gives $p_1(T{\mathbb{C}\mathbb{P}}^2)=3y^2$ and $p_1[{\mathbb{C}\mathbb{P}}^2]=3$ (Pontryagin classes by complexification, The tangent bundle of complex projective space and its Pontryagin classes).

The first Pontryagin number is $p_1[M]=⟨p_1(TM),[M]⟩$ and the Kronecker pairing is additive in its homology variable (Pontryagin numbers of a closed oriented manifold, Kronecker evaluation pairing).

The middle form is $Q_M(x,y)=⟨x⌣y,[M]⟩$ and, for every closed oriented $4$-manifold, $\sigma (M)=p_1[M]\mathrm{/}3$ (The middle-dimensional intersection form of a closed oriented 4k-manifold, The four-dimensional signature formula).