Chern and Pontryagin Classes by Splitting and Complexification — Examples

The base ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ is a path-connected paracompact Hausdorff CW complex and its tautological bundle is numerable. Under $P(\lambda )\cong {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$, the projective tautological line is $\lambda$, so the identity is a classifying map. Thus [F4] identifies $u=w_1(\lambda )$ with the nonzero polynomial generator. The real-linear map $({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}}\to \lambda \oplus \lambda$ given fiberwise by $v\otimes (a+ib)\mapsto (av,bv)$ has inverse $(x,y)\mapsto x\otimes 1+y\otimes i$ and commutes with the real transition functions, so it is a canonical real-bundle isomorphism by [F1]. Hence [F3] gives $w(({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}})=w(\lambda \oplus \lambda )=w(\lambda {)}^2=(1+u{)}^2=1+u^2$ over ${\mathbb{F}}_2$, where [F4] identifies $w(\lambda )=1+u$ and $2u=0$ in characteristic two.

Two-torsion: by [F5] with $j=0$ we have $2a=0$; multiplying by $a^{k-1}$ gives $2a^k=0$ for every $k\ge 1$.

The mod-two reduction of $a$: by [F2] applied to the complex bundle ${\lambda }_{\mathbb{C}}$, ${\rho }_2(a)=w_2(({\lambda }_{\mathbb{C}}{)}_{\mathbb{R}})=w_2(\lambda \oplus \lambda )=u^2$ by step 1.1.

The class $a^k$ is nonzero for every $k\ge 1$: the cochain formula [F6] commutes with coefficient reduction, since reduction preserves products of values on each pair of faces. It therefore induces a ring homomorphism, so ${\rho }_2(a^k)={\rho }_2(a{)}^k=u^{2k}$ by step 2.1, and $u^{2k}\mathrm{\ne }0$ because $H^{\ast }({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)={\mathbb{F}}_2[u]$ is a polynomial ring by [F4].

Exact order two: by step 3.1 the element $a^k$ is nonzero, and by step 1.2 it satisfies $2a^k=0$, so its additive order is exactly two.

Boundary cases. For $k=1$ the statements read $2a=0$, ${\rho }_2(a)=u^2\mathrm{\ne }0$ and $\mathrm{ord}(a)=2$. The nonvanishing assertion concerns this universal line on the fixed nonempty base; step 1.1 identifies its class as a polynomial generator. The zeroth power is outside the assertion: $a^0=1$ has infinite integral order, so the restriction $k\ge 1$ is necessary. The coefficient field ${\mathbb{F}}_2$ is nonzero, so the nonzero reduction genuinely certifies nonvanishing over $\mathbb{Z}$. No orientation of $\lambda$ is used, since $w_1$ and the complexification are orientation-free. AC is used only through [A1].

By [F3], $c_1(\gamma )=t$, and by [F1], $x=-t$. Since [F2] supplies a generator, rather than merely a nonzero class, $x$ is a generator as well. Restriction carries this identity to the standard ${\mathbb{C}\mathbb{P}}^1$. It remains to check the asserted sign on that complex-oriented sphere.

Write $M={\mathbb{C}\mathbb{P}}^1$ and $L={\gamma }^{\ast }{\mathrm{\mid }}_M$. The linear functional $(z_0,z_1)\mapsto z_1$ restricts on each line to a section $s$ of $L$. This section is continuous in bundle charts and vanishes precisely at $p=[1:0]$. On the affine chart $w=z_1\mathrm{/}{z}_0$, the tautological frame is $(1,w)$; its dual frame expresses $s$ as the scalar $w$. Both this base coordinate and the dual fiber coordinate have their complex orientations.

Let $E$ be the total space of $L$ and $E^{\times }$ the complement of its zero section. Since $L_{\mathbb{R}}$ is an integrally oriented numerable rank-two bundle over the CW complex $M$, [F4] supplies its unique normalized class $u_L\in H^2(D(L),S(L);\mathbb{Z})$. Inclusion $(D(L),S(L))\to (E,E^{\times })$ induces an isomorphism by the pair sequences, since $D(L)\to E$ and $S(L)\to E^{\times }$ are homotopy equivalences by radial contraction and radial normalization; let $U\in H^2(E,E^{\times };\mathbb{Z})$ be the unique class restricting to $u_L$. The section gives $v=s^{\ast }U\in H^2(M,M\setminus \{p\};\mathbb{Z})$. Its absolute image is $e(L_{\mathbb{R}})=c_1(L)$, because the section $s$ and the zero section are homotopic by fiberwise multiplication by a parameter in $[0,1]$, and relative-to-absolute maps commute with pullback.

Choose a coordinate disk $V$ about $p$, centered at $w=0$. Excision identifies $H^2(M,M\setminus \{p\};\mathbb{Z})$ with $H^2(V,V\setminus \{0\};\mathbb{Z})$: remove $M\setminus V$, a closed subset of the open relative subspace. In the trivialization $E{\mathrm{\mid }}_V=V\times \mathbb{C}$, the Thom class is the pullback of the positive generator of $H^2(\mathbb{C},\mathbb{C}\setminus \{0\};\mathbb{Z})$ by the fiber projection. Indeed contraction of $V$ to its center gives an equivalence of pairs with the central fiber, where [F4] fixes that generator. The section is $w\mapsto (w,w)$, so its pullback is the positive local orientation cohomology class: the composite with the fiber projection is the identity coordinate $w\mapsto w$. Thus the local class $v$ evaluates to $+1$ on the complex-oriented local homology generator.

The fundamental class $[M]$ maps to that positive local generator by [F6]. Evaluation therefore gives $⟨c_1(L),[M]⟩=1$: the absolute image of $v$ is $c_1(L)$ by step 2.1, and evaluation commutes with the relative quotient on chains. Explicitly, a relative cocycle is a cochain vanishing on chains in $M\setminus \{p\}$, so evaluating its absolute image on a cycle equals evaluating the relative cocycle on that cycle's relative image. This also shows that excision and restriction preserve this pairing. Hence $x$ is positive in the stated convention, and $c_1(\gamma )=-x$ is negative.

Since ${\gamma }^{\ast }$ is a line, [F3] gives $c({\gamma }^{\ast })=1+c_1({\gamma }^{\ast })=1+x$. For $n=1$ the restriction used above is the identity. The empty base does not occur; $n=0$ is excluded from the generator claim because $H^2({\mathbb{C}\mathbb{P}}^0;\mathbb{Z})=0$. All bundles used are over finite CW complexes and their complex orientations supply the integral Thom normalization. AC is inherited as stated in [A1].

The ordinary finite product base is a path-connected CW complex. To verify the topology qualification, exhaust two countable CW factors by increasing finite subcomplexes $X_j,Y_j$. Their product cells have finite closures. If $W$ is open in the product cell topology and $(a,b)\in W$, choose compact product neighborhoods $K_1\times M_1\subset W$ in the first finite stages containing the point. Given $K_j\times M_j\subset W$, compactness of $M_j$ gives for each $x\in K_j$ compact neighborhoods $K_x$ of $x$ and $M_x$ of $M_j$ in the next finite stages with $K_x\times M_x\subset W$. A finite collection of the interiors of $K_x$ covers $K_j$; take their union for $K_{j+1}$ and the intersection of the corresponding $M_x$ for $M_{j+1}$. The unions of the relative interiors of $K_j$ and $M_j$ are open in the weak CW topologies: on each finite stage their tails are an increasing union of open sets. Their product lies in $W$. Thus the ordinary product and cell topologies agree. Product characteristic maps give the CW structure (a product of two disks is a disk with its product boundary), and the resulting product still has countably many cells. Induction proves the assertion using [F5]. Path connectivity follows coordinatewise.

For $n\ge 1$, step 1.1 supplies the CW base. The coordinate universal circle bundles of [F4] are numerable; their associated complex lines and their pullbacks are numerable by pulling back the same local partitions. Thus the hypotheses of [F1] hold; a finite sum remains numerable by multiplying the finitely many local partition functions. Each $L_i$ is a complex line bundle and $E={⨁}_i{L}_i$ by [F3]; by multiplicativity and line normalization [F1], $c(E)={\prod }_{i}c(L_i)={\prod }_i(1+x_i)$.

Each $x_i$ has cohomological degree two, so the degree-$2k$ component of the product is $c_k(E)={\sum }_{1\le i_1<\cdots <i_k\le n}{x}_{i_1}\cdots x_{i_k}=e_k(x_1,\dots ,x_n)$. The empty product for $k=0$ is $1$ and the empty sum for $k>n$ is $0$. Equivalently these are the coefficients of $z^k$ in the formal polynomial ${\prod }_i(1+x_{i}z)$; polynomial degree in $z$ is distinct from cohomological degree.

By naturality and line normalization in [F1], $x_i=p_i^{\ast }u$, for the specific generator $u=e({\gamma }_{\mathbb{R}})$ of [F2]. Apply the Kunneth ring isomorphism repeatedly, taking one new ${\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ factor each time: that factor has finite free integral homology in every degree, which suffices for the supplier even though the full cohomology is not finitely generated. The cross product sends its coordinate generators to the $p_i^{\ast }u=x_i$. Every generator has even degree, so all graded tensor signs are $+1$. This identifies the ring with $\mathbb{Z}[x_1,\dots ,x_n]$, with no relations among the $x_i$.

Boundary cases. For $n=0$ the base is a point, $E$ is the rank-zero bundle, the empty product is $1$, and the ring is $\mathbb{Z}$ with no variables; [F1] gives exactly these Chern conventions. For $n=1$ the product is $1+x_1$ and $E=L_1$; for $k>n$ the elementary symmetric polynomial $e_k$ vanishes, matching the rank cutoff. If a line is replaced by a trivial summand, its first class is zero and its factor is $1$ by [F1]; this is a specialization, not a claim that one of the given universal coordinate lines is trivial. The coefficient ring $\mathbb{Z}$ is nonzero and the product is finite, so no convergence question arises. AC is used only through [A1].

Multiplication of clutching functions corresponds to tensor product: $E_g\otimes E_h\cong E_{gh}$, because the transition functions multiply in the clutching convention; in particular $E_d\otimes E_{d^{\mathrm{\prime }}}\cong E_{d+d^{\mathrm{\prime }}}$ and $E_{-d}\cong E_d^{\ast }$, since $z^{-d}=(z^d{)}^{-1}$ and dualizing inverts transition functions.

The class $u=c_1(E_1)$ is a generator of $H^2(S^2;\mathbb{Z})\cong \mathbb{Z}$. By [F1] and [F4], clutching identifies the isomorphism classes of complex lines with $\mathbb{Z}$, with $E_1$ corresponding to $1$. Step 1.1 shows that this bijection carries addition of degrees to tensor product, so $E_1$ generates the Picard group. The group isomorphism $c_1$ of [F2] therefore carries $E_1$ to a generator $u$ of cohomology. By [F3] this is the tautological Hopf line in the fixed convention.

For $d\ge 0$, iterating step 1.1 and additivity [F2] gives $c_1(E_d)=d{c}_1(E_1)=du$; for $d<0$, $E_d\cong E_{\mathrm{\mid }d\mathrm{\mid }}^{\ast }$ by step 1.1 and the dual formula of [F5] gives $c_1(E_d)=-\mathrm{\mid }d\mathrm{\mid }u=du$.

The pairing with the fundamental class: by step 2.1 the class $u$ generates $H^2(S^2;\mathbb{Z})$, so there is a unique orientation $[S^2]$ with $⟨u,[S^2]⟩=+1$; this is the Hopf normalization of the statement, and then $⟨c_1(E_d),[S^2]⟩=d$ by step 3.1.

Boundary cases. For $d=0$ the clutching map is constant, $E_0$ is trivial and $c_1=0=0\cdot u$. For $d=1$ we have $c_1(E_1)=u$ by definition; for $d=-1$ the bundle is the dual of the Hopf line and $c_1=-u$. Negative exponents are covered by the dual computation in step 3.1 and by [F4], which includes $d<0$. The coefficient ring $\mathbb{Z}$ is nonzero, so a nonzero multiple $d\cdot u$ of a generator is nonzero exactly when $d\mathrm{\ne }0$, giving the claimed classification. AC is used only through [A1].

The Euler comparison: [F4] supplies the complex orientation of $L_{\mathbb{R}}$, so [F1] with $n=1$ gives $e(L_{\mathbb{R}})=c_1(L)$.

Orientation: $L_{\mathbb{R}}$ is oriented by [F4], hence orientable, so $w_1(L_{\mathbb{R}})=0$ by [F3].

The mod-two comparison: [F2] with $i=1$ gives $w_2(L_{\mathbb{R}})={\rho }_2{c}_1(L)$ and $w_3(L_{\mathbb{R}})=0$; by [F5] all higher Chern classes of $L$ vanish, so $w_k(L_{\mathbb{R}})=0$ for $k\ge 3$ as well.

Boundary cases. The trivial line $L={\epsilon }^1$ has $c_1=0$, $e=0$ and $w_1=w_2=0$; the restriction to rank one is the exact range in which [F5] applies, and the general-rank versions are the cited theorems. The coefficient field ${\mathbb{F}}_2$ is nonzero, so ${\rho }_2$ is the standard reduction. AC is used only through [A1].

Chern stability: by [F1] and [F3], $c({\epsilon }^r)=1$ and multiplicativity gives $c(E\oplus {\epsilon }^r)=c(E)c({\epsilon }^r)=c(E)$; the cutoff $c_i=0$ for $i>m$ is part of the definition.

Pontryagin stability: by [F2] the classes $p_i$ are unchanged when a trivial summand is adjoined, and $p_i(F)=0$ whenever $2i>n$ by the defining cutoff.

Consistency of the two parities: in the complex case the total class is unchanged, so all components are unchanged; in the real case the total Pontryagin class as a finite sum ${\sum }_i{p}_i$ is unchanged because each $p_i$ is.

Boundary cases. For $r=0$ both identities are trivial; for $E$ or $F$ trivial of rank zero, $c(0)=1$ and $p(0)=1$ by the conventions, so the identities read $c({\epsilon }^r)=1$ and $p({\epsilon }^r)=1$. The rank cutoffs are strict: they force $c_i(E)=0$ only for $i>m$ and $p_i(F)=0$ only for $2i>n$. They do not include $i=m$ or $2i=n$, where the top Chern or Pontryagin class may be nonzero. The coefficient ring $\mathbb{Z}$ is nonzero and the sums are finite. AC is used only through [A1].

The Pontryagin class of the line: $\lambda$ has real rank one, so by the cutoff in [F1] we have $p_i(\lambda )=0$ for all $i\ge 1$, while $p_0(\lambda )=1$; hence $p(\lambda )=1$.

The complexification of $E=\lambda \oplus \lambda$: by [F2], the two bundles $E_{\mathbb{C}}$ and ${\lambda }_{\mathbb{C}}\oplus {\lambda }_{\mathbb{C}}$ have identical complex block diagonal transition matrices and hence a canonical isomorphism, and by multiplicativity [F3] applied to the two lines, $c(E_{\mathbb{C}})=(1+a{)}^2=1+2a+a^2=1+a^2$ in $H^{\ast }({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})$, because $2a=0$ by the two-torsion relation of [F2].

The top component of $E_{\mathbb{C}}$ in degree four is $c_2(E_{\mathbb{C}})=a^2$, so by [F1] with $i=1$, $p_1(E)=(-1{)}^1{c}_2(E_{\mathbb{C}})=-a^2$, which is nonzero by the exact-order-two statement of [F2].

The right-hand side of the refuted formula is $p(\lambda )p(\lambda )=1\cdot 1=1$ by step 1.1, whose degree-four component is $0$; the left-hand side $p(E)$ has degree-four component $-a^2\mathrm{\ne }0$ by step 2.1.

Hence $p(\lambda \oplus \lambda )\mathrm{\ne }p(\lambda )p(\lambda )$ integrally: the degree-four components differ by the nonzero two-torsion class $-a^2$. The witness pair is $(\lambda ,\lambda )$; its Whitney sum is the bundle $E=\lambda \oplus \lambda$ used above. The argument uses no additive universal-coefficient computation beyond the nonvanishing $a^2\mathrm{\ne }0$ proved on the companion $A$-page lemma.

Boundary cases. After tensoring the integral cohomology ring with $\mathbb{Z}[1\mathrm{/}2]$, $a\otimes 1=(2a)\otimes (1\mathrm{/}2)=0$, so the two sides for this witness both become $1$; the trivial bundle case $a=0$ gives no failure, and the rank cutoffs are used at rank one ($p(\lambda )=1$) and rank two ($p_1$ is the first nonzero Pontryagin class). The base ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ is path connected and the coefficient group $\mathbb{Z}$ is nonzero. AC is used only through [A1].