The Hirzebruch Signature Theorem — Examples

Given: AC, the tautological complex line $\gamma \to {\mathbb{C}\mathbb{P}}^2$, the class $y=c_1({\gamma }^{\ast })$, and the complex orientation of ${\mathbb{C}\mathbb{P}}^2$.

The in-run supplier gives $H^{\ast }({\mathbb{C}\mathbb{P}}^n;\mathbb{Z})=\mathbb{Z}[x]\mathrm{/}(x^{n+1})$ with $x=c_1({\gamma }^{\ast })$, $⟨x^n,[{\mathbb{C}\mathbb{P}}^n]⟩=1$, and $p(T{\mathbb{C}\mathbb{P}}^n)=(1+x^2{)}^{n+1}$ in the truncated ring (The tangent bundle of complex projective space and its Pontryagin classes).

The middle-dimensional intersection form is $Q_M(x,y)=⟨x⌣y,[M]⟩$ on $H^{2k}(M;\mathbb{R})$, and the signature is the difference $p-q$ of the positive and negative inertia indices of the nondegenerate symmetric form $Q_M$ (The middle-dimensional intersection form of a closed oriented 4k-manifold, The signature of a closed oriented manifold of dimension divisible by four).

The first Pontryagin number is $p_1[M]=⟨p_1(TM),[M]⟩\in \mathbb{Z}$, the Kronecker evaluation of the Pontryagin class on the fundamental class (Pontryagin numbers of a closed oriented manifold, Kronecker evaluation pairing).

For every closed oriented $4$-manifold $M$, $\sigma (M)=p_1[M]\mathrm{/}3$ (The four-dimensional signature formula), and on projective spaces the signature and the L-genus agree: $\sigma ({\mathbb{C}\mathbb{P}}^{2k})=1=L[{\mathbb{C}\mathbb{P}}^{2k}]$ for every $k\ge 0$ (The signature and the L-genus agree on complex projective space).

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).

Given: AC; the standard orientation of $S^2$ with its orientation class $z$ and fundamental class $[S^2]$; the product orientation and product smooth structure on $S^2\times S^2$; the projections ${\mathrm{p}\mathrm{r}}_1,{\mathrm{p}\mathrm{r}}_2$.

For a commutative ring $R$ that is a PID with either every $H_q(Y;R)$ or every $H_p(X;R)$ finite free over $R$, the external product is a graded-ring isomorphism $H^{\ast }(X;R){\otimes }_R{H}^{\ast }(Y;R)\to H^{\ast }(X\times Y;R)$; this uses AC (Cohomological Kunneth cross product is a ring isomorphism).

For $n\ge 1$, ${\tilde{H}}_k(S^n;G)=G$ for $k=n$ and $0$ otherwise, so $H_2(S^2;\mathbb{Z})\cong \mathbb{Z}$ and $H_{\ast }(S^2;\mathbb{Z})$ is free (Homology of spheres). The fundamental class $[S^2]$ of Fundamental class of a compact oriented manifold restricts at every point to the local generator of the standard orientation, and for the compact connected boundaryless manifold $S^2$ restriction to every local stalk is injective with image $\mathbb{Z}$ (Top homology of a connected manifold); hence under the identification $H_2(S^2;\mathbb{Z})\cong \mathbb{Z}$ the class $[S^2]$ corresponds to $\pm 1$ and generates $H_2(S^2;\mathbb{Z})$.

For every space $X$ and $n\ge 0$ the universal coefficient sequence $0\to {\mathrm{Ext}}_{\mathbb{Z}}^1(H_{n-1}(X;\mathbb{Z}),G)\to H^n(X;G)\to {\mathrm{Hom}}_{\mathbb{Z}}(H_n(X;\mathbb{Z}),G)\to 0$ is natural with evaluation as the second map; this uses AC (Topological universal coefficient short exact sequence for cohomology). Over a field the evaluation map alone is an isomorphism (Cohomology over a field is dual to homology over that field).

With the product orientation and product smooth structure, $S^2\times S^2$ is a closed oriented smooth $4$-manifold and $[S^2\times S^2]=[S^2]\times [S^2]$ (Product orientations, The fundamental class of a product is the cross product of the fundamental classes, Products of smooth manifolds have a canonical product smooth structure).

The Kronecker pairing is multiplicative under cross products: $⟨\alpha \times \beta ,c\times d{⟩}_{X\times Y}=⟨\alpha ,c{⟩}_X⟨\beta ,d{⟩}_Y$ (The Kronecker pairing is multiplicative under cross products).

The middle-dimensional intersection form is $Q_M(x,y)=⟨x⌣y,[M]⟩$ on $H^{2k}(M;\mathbb{R})$, it is symmetric and nondegenerate, and $\sigma (M)$ is the positive minus the negative inertia index of $Q_M$ (The middle-dimensional intersection form of a closed oriented 4k-manifold, The middle-dimensional intersection form is symmetric and nondegenerate, The signature of a closed oriented manifold of dimension divisible by four).

For every closed oriented smooth $4$-manifold $M$, $p_1[M]=3\sigma (M)$ (The four-dimensional signature formula).

Given: AC; the closed oriented smooth $4$-manifold ${\mathbb{C}\mathbb{P}}^2$ with the complex orientation; the product ${\mathbb{C}\mathbb{P}}^2\times {\mathbb{C}\mathbb{P}}^2$ with the product orientation and product smooth structure.

For closed oriented smooth manifolds $M^{4a}$ and $N^{4b}$ with the product orientation, $\sigma (M\times N)=\sigma (M)\sigma (N)$ (The signature is multiplicative under Cartesian products).

The complex projective plane satisfies $\sigma ({\mathbb{C}\mathbb{P}}^2)=1$ (The signature and the L-genus agree on complex projective space).

For $P={\mathbb{C}\mathbb{P}}^{2k_1}\times \cdots \times {\mathbb{C}\mathbb{P}}^{2k_r}$ with the product orientation, $k_i\ge 1$, one has $\sigma (P)=1=L[P]$ (The signature and the L-genus agree on products of complex projective spaces).

The products ${\mathbb{C}\mathbb{P}}^{2j_1}\times \cdots \times {\mathbb{C}\mathbb{P}}^{2j_r}$ over partitions form a $\mathbb{Q}$-basis of ${\mathrm{\Omega }}_{4k}^{SO}\otimes \mathbb{Q}$; equivalently ${\mathrm{\Omega }}_{\ast }^{SO}\otimes \mathbb{Q}$ is the polynomial algebra on the classes $[{\mathbb{C}\mathbb{P}}^2],[{\mathbb{C}\mathbb{P}}^4],[{\mathbb{C}\mathbb{P}}^6],\dots$ (Products of complex projective spaces span rational oriented bordism).

Products of closed oriented smooth manifolds carry the product orientation and the canonical product smooth structure, hence are again closed oriented smooth (Product orientations, Products of smooth manifolds have a canonical product smooth structure).

Given: AC; the complex projective plane ${\mathbb{C}\mathbb{P}}^2$ with the complex orientation and its orientation reversal $-{\mathbb{C}\mathbb{P}}^2$; the Grassmannian ${\mathrm{Gr}}_1({\mathbb{C}}^3)$ with its Schubert stratification.

$\mathrm{Gr}(1,V)$ is the set of $1$-dimensional linear subspaces of $V$, and ${\mathbf{P}}_{\mathbb{C}}^2$ is the quotient of ${\mathbb{C}}^3\setminus \{0\}$ by nonzero scalar multiplication; a point of either is exactly a complex line in ${\mathbb{C}}^3$, so ${\mathbb{C}\mathbb{P}}^2={\mathrm{Gr}}_1({\mathbb{C}}^3)$ (The Grassmannian of r-dimensional subspaces of a finite-dimensional vector space, projective space points).

A Schubert symbol for ${\mathrm{Gr}}_n({\mathbb{F}}^N)$ is a strictly increasing sequence $1\le a_1<\cdots <a_n\le N$, its cell satisfies $e(a)\cong {\mathbb{F}}^{d(a)}$ with $d(a)={\sum }_i(a_i-i)$, of real dimension $d(a)$ for $\mathbb{F}=\mathbb{R}$ and $2d(a)$ for $\mathbb{F}=\mathbb{C}$, and the Schubert strata form a finite CW structure on ${\mathrm{Gr}}_n({\mathbb{F}}^N)$ (Schubert cells in real and complex Grassmannians, Schubert cells give the stable Grassmannian CW structure).

For a finite CW complex with $c_n$ cells in dimension $n$, the Euler characteristic is $\chi (X)={\sum }_n(-1{)}^n{c}_n(X)$ (Euler characteristic of a finite CW complex).

$\sigma ({\mathbb{C}\mathbb{P}}^2)=1$ and $\sigma (-{\mathbb{C}\mathbb{P}}^2)=-1$ (The signature and the L-genus agree on complex projective space, the orientation computation in step 1.1).

$-{\mathbb{C}\mathbb{P}}^2$ is ${\mathbb{C}\mathbb{P}}^2$ with the reversed orientation, so the two have the same underlying space and the same finite CW structures (Fundamental class of a compact oriented manifold).