Chern and Pontryagin Classes by Splitting and Complexification

The list $(v_1,i{v}_1,\dots ,v_n,i{v}_n)$ is a real basis of each fiber: since the $v_j$ form a complex basis, ${\sum }_j(a_j{v}_j+b_{j}i{v}_j)={\sum }_j(a_j+i{b}_j)v_j$ vanishes only when all $a_j+i{b}_j=0$, that is, all $a_j=b_j=0$. Hence the list orients the fibers of $E_{\mathbb{R}}$ over the chart, and by [F2] these local data are the candidate local orientations.

Compatibility on overlaps. Let $A\in {\mathrm{GL}}_n(\mathbb{C})$ be the complex change-of-frame matrix $w_j={\sum }_i{A}_{ij}{v}_i$. In the real bases of step 1.1 the change-of-frame matrix is the realification $R_A$ obtained by replacing every complex entry by its $2\times 2$ real block; realification is multiplicative in the sense $R_{AB}=R_A{R}_B$, because it is the matrix of the same complex-linear map read in real coordinates. By [F5] choose a complex basis in which $A$ is upper triangular; then $R_A$ is block upper triangular with diagonal blocks $\left(\begin{array}{cc}{a}_j& -b_j\\ b_j& a_j\end{array}\right)$ for the diagonal entries $a_j+i{b}_j$ of $A$. By [F6] its determinant is the product of the block determinants ${\prod }_j(a_j^2+b_j^2)=\mathrm{\mid }{\det }_{\mathbb{C}}A{\mathrm{\mid }}^2>0$. A positive determinant means the two ordered real frames induce the same orientation, and multiplicativity of realification reduces every frame pair to this comparison.

Complexification. Let $(e_1,\dots ,e_{2n})$ be an oriented real basis of a fiber of $V$. The vectors $e_1,\dots ,e_{2n}$ are a complex basis of the complexification, so by step 1.1 the complex orientation of $(V_{\mathbb{C}}{)}_{\mathbb{R}}$ is represented by the ordered real basis $e_1,i{e}_1,e_2,i{e}_2,\dots ,e_{2n},i{e}_{2n}$. Under the isomorphism $\phi$ of statement 3 this list becomes $(e_1,0),(0,e_1),(e_2,0),(0,e_2),\dots ,(e_{2n},0),(0,e_{2n})$, while the product orientation $o\oplus o$ is represented by the blocked list $(e_1,0),\dots ,(e_{2n},0),(0,e_1),\dots ,(0,e_{2n})$. Passing from the interleaved list to the blocked list is the shuffle of two length-$2n$ blocks; its inversion number is $0+1+\cdots +(2n-1)=2n(2n-1)\mathrm{/}2=n(2n-1)$, so the orientation sign is $(-1{)}^{n(2n-1)}=(-1{)}^n$.

Hence the local orientations of steps 1.1 and 2.1 agree on every overlap of a complex linear atlas, and [F3] glues them into a global integral orientation of $E_{\mathbb{R}}$, the complex orientation. The same determinant computation with $A$ the transition function of a pullback chart gives naturality under pullback, and with $A$ the matrix of a complex-linear isomorphism it gives invariance under complex-linear bundle isomorphisms.

For rank $n=0$ the frame list of step 1.1 is empty and the determinant computation of step 2.1 is vacuous, so the zero bundle carries its canonical orientation; [F7] supplies $e(0_B)=1$ for use below.

Taking Euler classes. The numeration of $V$ also numerates $V_{\mathbb{C}}$, $(V_{\mathbb{C}}{)}_{\mathbb{R}}$ and $V\oplus V$ by [F1], so every Euler class in this step lies in the scope of [F4]. If two orientations of a real bundle differ by a sign $\epsilon =\pm 1$ on positive frames, their Euler classes differ by the same $\epsilon$ by the orientation-sign clause of [F4]; for the ordered direct sum $V\oplus V$ the Whitney product clause of [F4] gives $e(V\oplus V)=e(V)e(V)$. Therefore $e((V_{\mathbb{C}}{)}_{\mathbb{R}})=(-1{)}^{n}e(V\oplus V)=(-1{)}^{n}e(V{)}^2$, which is statement 3.

Direct sums. A local complex frame of $E\oplus F$ is the concatenation of a complex frame $(v_1,\dots ,v_m)$ of $E$ and a complex frame $(w_1,\dots ,w_k)$ of $F$, so the real frame of step 1.1 is $(v_1,i{v}_1,\dots ,v_m,i{v}_m,w_1,i{w}_1,\dots ,w_k,i{w}_k)$, exactly the ordered direct-sum frame of the complex-oriented summands $E_{\mathbb{R}}$ and $F_{\mathbb{R}}$. By [F2] the two orientations coincide, so statement 2 holds, and the rank-zero case is step 3.2.

Boundary cases. Rank zero is step 3.2. In statement 2, step 4.1 says that the complex orientation on $0\oplus F$ is the ordered sum of the canonical orientation on $0$ and the complex orientation on $F$; only after applying the Whitney product formula [F4] and $e(0)=1$ from [F7] does one obtain $e(0\oplus F)=1\cdot e(F)$. In statement 3 with $n=0$, both sides are the unit. For a complex line, $n=1$ in step 2.1 gives $\mathrm{\mid }{\det }_{\mathbb{C}}A{\mathrm{\mid }}^2>0$ directly, and for $V$ of rank $2$ step 2.2 has inversion number $2\cdot 1\mathrm{/}2=1$ and sign $(-1{)}^1=-1$. The argument uses no choice beyond the inherited numerability data recorded in [A1].

Identification: by [F1] and [F2] the projective space of ${\mathbb{C}}^{N+1}$ is the space of complex lines with the tautological bundle $\gamma$, and by [F3] the projective bundle of the trivial rank-$(N+1)$ bundle over a point is the same space with the same tautological line; hence $x=e({\gamma }_{\mathbb{R}})$ is the page's class for ${\mathbb{C}\mathbb{P}}^N$.

The sphere bundle: by [F4] the pair $(\mathrm{\ell },v)$ with $v$ a unit vector in the line $\mathrm{\ell }$ determines $v$ and is determined by it, so $S({\gamma }_{\mathbb{R}})\cong S^{2N+1}$; by [F6] and [F7] its integral cohomology is $\mathbb{Z}$ in degrees $0$ and $2N+1$ and vanishes in all other positive degrees.

Apply [F7] to the free homology groups in [F8]. It gives $H^{2k}({\mathbb{C}\mathbb{P}}^N;\mathbb{Z})\cong \mathbb{Z}$ for $0\le k\le N$ and zero cohomology in every other degree, in particular in degrees $2N+1$ and $2N+2$. The Gysin sequence [F5] of the rank-two oriented bundle ${\gamma }_{\mathbb{R}}$ reads $\cdots \to H^{i-2}({\mathbb{C}\mathbb{P}}^N)\stackrel{⌣x}{\to }{H}^i({\mathbb{C}\mathbb{P}}^N)\stackrel{p^{\ast }}{\to }{H}^i(S^{2N+1})\to \cdots$. For $2\le i\le 2N$, step 1.2 makes the middle sphere group zero, as well as the sphere group immediately preceding $H^{i-2}$; exactness therefore makes $⌣x$ an isomorphism. Starting from $H^0=\mathbb{Z}$, these isomorphisms show that $x^k$ generates $H^{2k}$ for every $0\le k\le N$. The independently established vanishing $H^{2N+2}=0$ gives $x^{N+1}=0$ without any circular appeal to the Gysin sequence.

Inclusion compatibility: the inclusion ${\mathbb{C}}^{m+1}\subseteq {\mathbb{C}}^{N+1}$ carries the tautological line of ${\mathbb{C}\mathbb{P}}^m$ to the restriction of the tautological line of ${\mathbb{C}\mathbb{P}}^N$ by [F2], so naturality of the Euler class [F3] gives $x_m={\iota }^{\ast }{x}_N$.

Ring structure: by step 2.1 the group $H^{2k}$ is the infinite cyclic group generated by $x^k$ for $0\le k\le N$, all other groups vanish, and $x^{N+1}=0$. The product satisfies $x^a\cdot x^b=x^{a+b}$, so every class is represented uniquely by a polynomial of degree at most $N$ and the graded ring is $\mathbb{Z}[x]\mathrm{/}(x^{N+1})$. With step 2.2 this is the assertion.

Boundary cases. For $N=0$ the projective space is a point, $x=0$ and the ring is $\mathbb{Z}[x]\mathrm{/}(x)=\mathbb{Z}$, as required; the sphere $S^1$ has the stated cohomology by step 1.2. For $N=1$ the computation gives $H^{\ast }(S^2)=\mathbb{Z}[x]\mathrm{/}(x^2)$, the standard result. The coefficient ring $\mathbb{Z}$ is nonzero and the Gysin sequence is used only in degrees $\le 2N+2$, all of which lie in the range controlled by step 1.2. AC enters only through [A1].

Cell structure: by [F1] with $n=1$ the finite Grassmannians ${\mathbb{C}\mathbb{P}}^N={\mathrm{Gr}}_1({\mathbb{C}}^{N+1})$ have exactly one cell in each real dimension $0,2,\dots ,2N$, and by [F2] their union ${\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ is a CW complex with exactly one cell in each even dimension and none in odd dimensions.

The cellular complex: by [F3] the cellular cochain complex of ${\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ with $\mathbb{Z}$ coefficients has $C^{2k}=\mathbb{Z}$ and $C^{2k+1}=0$ for all $k\ge 0$; in particular every differential of the complex is zero, so $H^{2k}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}$ and $H^{2k+1}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=0$. By [F4] the cellular chain complex likewise gives $H_{2k}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}$, $H_{2k+1}=0$, hence freeness and finite generation in each degree.

Generators: by [F6], naturality of the Euler class identifies the restriction of $u=e({\gamma }_{\mathbb{R}})$ along ${\mathbb{C}\mathbb{P}}^k\hookrightarrow {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$ with the finite-stage class $x$. By [F5], $H^{\ast }({\mathbb{C}\mathbb{P}}^k;\mathbb{Z})=\mathbb{Z}[x]\mathrm{/}(x^{k+1})$ and $x^k$ generates $H^{2k}$. Hence the restriction of $u^k$ is $x^k\mathrm{\ne }0$, so $u^k\mathrm{\ne }0$ in the infinite cyclic group $H^{2k}({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})$ of step 2.1. Moreover the cellular restriction to the $2k$-skeleton sends the degree-$2k$ cellular coordinate isomorphically to the sole degree-$2k$ cell, so this nonzero restriction has coefficient $\pm 1$; thus $u^k$ is a generator.

Ring structure: the multiplication is generated by $u$ in degree two, and by step 3.1 each power $u^k$ is a generator of the infinite cyclic group $H^{2k}$; therefore $H^{\ast }({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}[u]$ as a graded ring, which with step 2.1 gives the full assertion.

Boundary cases. For $k=0$ the statement reads $H^0=\mathbb{Z}$, the class $u^0=1$ being a generator; the empty space does not occur, and the coefficient ring $\mathbb{Z}$ is nonzero. The degrees are unbounded above, but each degree is a single cyclic group, so no finiteness in dimension is asserted. The trivial line $u=0$ occurs only over the empty base, which is excluded. AC enters only through [A1].

Put $N=n-1$. The pullback $j_b^{\ast }{\gamma }_E$ has fiber precisely the line represented by each point of $P(E_b)$. The chosen complex-linear identification therefore identifies it with the usual tautological line over ${\mathbb{C}\mathbb{P}}^N$. The identification preserves the frames $(v,iv)$; changing a complex line frame by $a+ib\mathrm{\ne }0$ has positive real determinant $a^2+b^2$. Both bases are in Thom scope by [F1], also applied to the trivial rank-$n$ bundle over a point. Thus [F3] gives $j_b^{\ast }x=x_{\mathrm{t}\mathrm{a}\mathrm{u}\mathrm{t}}$ with exactly the stated orientation.

The integral cellular cochain complex of ${\mathbb{C}\mathbb{P}}^N$ has one copy of $\mathbb{Z}$ in each even degree $0,2,\dots ,2N$ and zero in every other degree. Every differential is zero. Hence its integral cohomology is $\mathbb{Z}$ in those degrees and zero elsewhere, including all degrees above $2N$. Its degree-zero unit is a generator.

For $N\ge 1$, the sphere bundle of its tautological complex line is $S^{2N+1}$: the homeomorphism sends $(\mathrm{\ell },v)$ with $v\in \mathrm{\ell }$ of unit length to $v$, and its inverse sends $v$ to $(\mathbb{C}v,v)$. Both maps are continuous in the quotient and bundle charts of [F1]. For $2\le k\le 2N$, both flanking groups in [F2] vanish by [F5], since $1\le k-1<k<2N+1$. Thus multiplication by $x_{\mathrm{t}\mathrm{a}\mathrm{u}\mathrm{t}}$ is an isomorphism $H^{k-2}\to H^k$ in this range. Starting at the unit, its powers generate all the even groups in step 1.2. This proves the integral basis assertion and degree-two generation; vanishing above the top degree comes from step 1.2, not from an induction through the exceptional top sphere group.

With ${\mathbb{F}}_p$ coefficients the same cellular cochain complex has one copy of ${\mathbb{F}}_p$ in each even degree and zero differentials. The natural coefficient map reduces each integral cell coordinate modulo $p$. Each power in step 2.1 is an integral generator, hence has coordinate $+1$ or $-1$ and reduces to a basis vector over ${\mathbb{F}}_p$. Together with step 1.1 this proves the reduction assertion.

When $n=1$, step 1.2 with $N=0$ gives $H^2=0$ and the single basis element $1$, integrally and after reduction. No use of step 2.1 is needed. An empty base contributes no fibers. Rank zero is excluded; the separate convention in [F1] gives empty projectivization with no $x$. The determinant argument in step 1.1 proves independence of each complex-linear identification, so no varying sign is introduced across components. AC is inherited from the stated suppliers; choosing a frame for one fixed fiber introduces no further choice assumption.

Since $E$ is numerable, choose a subordinate partition of unity on a vector-bundle trivializing cover. By [F1] that same cover and the same partition trivialize and numerate $P(E)\to B$, whose fiber ${\mathbb{C}\mathbb{P}}^{n-1}$ is compact Hausdorff. Thus [F4] makes $p$ a Hurewicz, hence Serre, fibration over the path-connected CW complex $B$.

By [F2] the classes $1,x_R,\dots ,x_R^{n-1}$ restrict on every fiber to an $R$-basis of $H^{\ast }({\mathbb{C}\mathbb{P}}^{n-1};R)$: for $R=\mathbb{Z}$ directly, and for $R={\mathbb{F}}_p$ through the coefficient reductions.

Construction on bases of CW type. Let $B^{\mathrm{\prime }}$ be paracompact Hausdorff CGWH of CW type. Projectivizing the given linear charts and their transitions produces a fiber bundle with the same numeration, by exactly the quotient-chart maps in [F1]. By [F5] its total space $T=P(E)$ is paracompact Hausdorff, CGWH, and of CW type. On each projective coordinate domain $v_j\mathrm{\ne }0$, the representative with $v_j=1$ trivializes the tautological line; [F7] numerates this chart cover. The real frames $(v,iv)$ agree in orientation because multiplication by $a+ib\mathrm{\ne }0$ has determinant $a^2+b^2>0$. Thus the real rank-two tautological bundle is oriented, numerable and in Thom scope, so [F7] defines $x$ on $T$. Its restriction on each fiber is the tautological Euler class by oriented naturality, and applying [F2] to the trivial rank-$n$ bundle over a point gives the required integral and prime-field fiber bases.

Choose a homotopy equivalence $w:W\to B^{\mathrm{\prime }}$ with $W$ a CW complex and form $T_W=w^{\ast }T$ with projection $q:T_W\to T$. Both bundle projections are Serre fibrations by [F4] and [F8]. In their natural homotopy sequences, the fiber map is the identity on ${\mathbb{C}\mathbb{P}}^{n-1}$ and the base maps induce isomorphisms. Hence $q$ induces isomorphisms on every positive homotopy group. Explicitly, for surjectivity of the middle map, lift a base class through the base isomorphism; its boundary vanishes by injectivity on the fiber group, so exactness lifts it to the source total group. Correct the difference from the target class using surjectivity on the fiber group. For injectivity, an element killed in the target has zero base image, hence comes from a fiber element. That fiber element maps to a base boundary in the target; lift that boundary class through the base isomorphism and use injectivity on the fiber group to conclude that the original element is zero. This group argument also works in degree one with multiplication in place of addition: the fiber is path connected, so both boundary maps to its component set vanish. Path lifting and path-connected fibers identify the components of each total space with those of its base, giving a bijection on components as well. Thus $q$ is a weak homotopy equivalence. By [F9], $q^{\ast }$ is a cohomology isomorphism with coefficients $\mathbb{Z}$ or ${\mathbb{F}}_p$; its ring structure is preserved by pullback.

Applying [F3] to the fibration of step 1.1 with the classes of step 1.2 gives the $H^{\ast }(B;R)$-module isomorphism ${⨁}_{i=0}^{n-1}{H}^{\ast -2i}(B;R)\to H^{\ast }(P(E);R)$ sending $(a_i)$ to ${\sum }_i{p}^{\ast }{a}_i⌣x_R^i$. In particular $1,x_R,\dots ,x_R^{n-1}$ are a basis of the free module $H^{\ast }(P(E);R)$ over $H^{\ast }(B;R)$.

The monic relation. Apply step 2.1 with $R=\mathbb{Z}$ to the element $x^n\in H^{2n}(P(E);\mathbb{Z})$: there are unique classes $b_i\in H^{2n-2i}(B;\mathbb{Z})$, $0\le i\le n-1$, with $x^n={\sum }_{i=0}^{n-1}{b}_i{x}^i$; setting $a_j:=(-1{)}^{j+1}{b}_{n-j}$ for $1\le j\le n$ turns this into $x^n-a_1{x}^{n-1}+a_2{x}^{n-2}-\cdots +(-1{)}^n{a}_n=0$, with $a_j\in H^{2j}(B;\mathbb{Z})$ by the grading. Uniqueness of the $a_j$ is uniqueness of the coefficients $b_i$ in the basis of step 2.1.

All relations. Let $f(t)=t^n-a_1{t}^{n-1}+\cdots +(-1{)}^n{a}_n\in H^{\ast }(B;\mathbb{Z})[t]$ and let $P\in H^{\ast }(B;\mathbb{Z})[t]$ satisfy $P(x)=0$. All coefficients of $f$ and the degree-two variable are central by [F10]. Successively subtracting the leading coefficient times the appropriate power of $t$ times $f$ reduces the degree, over this possibly noncommutative coefficient ring. Thus monic division gives $P=Qf+R$ with $\mathrm{deg}R<n$, and evaluating at $x$ gives $R(x)=0$, say $R(t)={\sum }_{i<n}{r}_i{t}^i$ with $r_i\in H^{\ast }(B;\mathbb{Z})$; then ${\sum }_i{p}^{\ast }{r}_i⌣x^i=0$ in $H^{\ast }(P(E);\mathbb{Z})$. By the basis property of step 2.1 all $r_i=0$, so $R=0$ and $P=Qf$ lies in the ideal generated by $f$.

Apply [F3] on each CW component of $W$ with the classes $q^{\ast }(x_R^i)$. Their restrictions form the bases proved in step 1.3; no Euler construction on $W$ is needed here. The natural square of cup-product maps has vertical isomorphisms $w^{\ast }$ (by [F6]) and $q^{\ast }$ (by step 1.4), so the module map for $B^{\mathrm{\prime }}$ is an isomorphism. For disconnected $W$, singular cohomology is the product of component cohomologies in each degree: singular simplices lie in a single component. The finite direct sum indexed by $0\le i<n$ commutes with this product. Therefore the same module map is an isomorphism without connectedness. Steps 3.1 and 4.1 apply to this module map and give the monic relation and its full relation ideal on $B^{\mathrm{\prime }}$.

Boundary cases. For $n\ge 1$ the basis $1,x,\dots ,x^{n-1}$ is nonempty and begins with the unit $1$; in the rank-one case $n=1$ the module is $H^{\ast }(B;R)$ itself and the relation reads $x-a_1=0$, so $a_1=x$ and no higher $a_i$ occurs. The zero bundle is excluded by $n\ge 1$, an empty base gives the unique zero cohomology groups; a disconnected base is handled by step 5.1, and the coefficient rings $\mathbb{Z}$ and ${\mathbb{F}}_p$ are nonzero by hypothesis. The relation has leading term $x^n$ and constant term $(-1{)}^n{a}_n$, with all intermediate coefficients verified in step 3.1; the ring $H^{\ast }(B;\mathbb{Z})[t]$ admits monic division by the leading-term subtraction of step 4.1 with central even coefficients, so no domain hypothesis is used. AC is inherited through [A1] in the numerable-fibration, Thom, partition, Leray–Hirsch and coefficient suppliers.