Stiefel Whitney and Euler Classes by Universal Constructions

Every universal class gives a characteristic class. Let $u$ be a class of ${\gamma }_n$ on $BG$, where $BG$ is one of the three Grassmannian models. For a numerable bundle $E\to B$ over a base in both classes choose a classifying map $c_E$, that is, $c_E^{\ast }{\gamma }_n\cong E$; such a map exists by [F1] or [F2], since $B$ lies in the classification scope. Set $c_u(E)=c_E^{\ast }u$. If $c_E^{\mathrm{\prime }}$ is a second classifying map of the same bundle, both maps represent the same element of $[B,BG]$, because the bijection of [F1] or [F2] is defined on homotopy classes and has the same value on them; hence $c_E\simeq c_E^{\mathrm{\prime }}$ and [F3] gives $c_E^{\ast }u=c_E^{\mathrm{\prime }\ast }u$. Isomorphic bundles have the same classifying homotopy class by the same injectivity, so $c_u$ is well defined on isomorphism classes. For a map $f:B^{\mathrm{\prime }}\to B$ of such bases, $f^{\ast }E$ is classified by $c_{E}f$, since [F7] gives $(c_{E}f{)}^{\ast }{\gamma }_n\cong f^{\ast }{c}_E^{\ast }{\gamma }_n\cong f^{\ast }E$; therefore [F4] gives $c_u(f^{\ast }E)=(c_{E}f{)}^{\ast }u=f^{\ast }{c}_E^{\ast }u=f^{\ast }{c}_u(E)\mathrm{.}$ This clause uses AC exactly through [F1] and [F2].

Every characteristic class comes from a universal class. Let $c$ satisfy the two clauses of the Definition. For $BO(n)$ and $BU(n)$, [F5] supplies CW structures, so these models are paracompact Hausdorff CGWH and admissible. For $BSO(n)$ with $n\ge 1$, its double cover of $BO(n)$ has discrete two-point compact CW fiber. A subordinate partition on its trivializing cover makes it numerable by [F6]; the compact-fiber theorem in [F6] then makes its total space paracompact Hausdorff CGWH and of CW type. Thus the oriented model lies in both required classes without asserting that the Schubert theorem supplies its CW structure. For $n=0$ the model is a point. The tautological bundles have local linear charts from [F5] and are numerable by [F6] on these paracompact Hausdorff bases (the oriented one is also the pullback of the unoriented one). Hence $u:=c({\gamma }_n)$ is defined. For a numerable bundle $E\to B$ over a base in both classes with classifying map $c_E$ we have $c_E^{\ast }{\gamma }_n\cong E$, so isomorphism invariance and pullback naturality give $c(E)=c(c_E^{\ast }{\gamma }_n)=c_E^{\ast }c({\gamma }_n)=c_E^{\ast }u\mathrm{.}$ Thus $c$ is $c_u$ for the universal class $u=c({\gamma }_n)$.

The two passages are inverse and determine all values. Starting with $u$, the identity map classifies the universal bundle, so $c_u({\gamma }_n)={\mathrm{id}}^{\ast }u=u$ by [F4]. Starting with $c$, step 1.2 returns $c$ from $u=c({\gamma }_n)$. Hence the assignments $u\mapsto c_u$ and $c\mapsto c({\gamma }_n)$ are mutually inverse bijections between universal classes and characteristic classes, and a characteristic class is determined by its single value on the universal bundle. For $n=0$ all three Grassmannian models are points, so both sides are $H^k(\ast ;R)$: this group is zero for $k>0$ and is canonically $R$ for $k=0$. Thus the degree-zero rank-zero characteristic classes are the scalar classes $r\cdot 1$, one for each $r\in R$, rather than only the unit. The empty base carries the zero cohomology groups and the same formulas apply vacuously. Classifying-map ambiguity is absorbed in step 1.1. AC is inherited through classification and the admissibility/numeration arguments [F6]; it supplies DC where the partition theorem requires it.

$S$ is Hausdorff. Let $s\mathrm{\ne }{s}^{\mathrm{\prime }}$ be points of $S$. If $p(s)\mathrm{\ne }p(s^{\mathrm{\prime }})$, choose disjoint open $U,V\subseteq B$ containing them, possible because $B$ is Hausdorff; then $p^{-1}(U)$ and $p^{-1}(V)$ are disjoint open subsets of $S$ containing $s$ and $s^{\mathrm{\prime }}$. If $p(s)=p(s^{\mathrm{\prime }})=b$, choose a chart domain $U_i$ containing $b$ and apply the homeomorphism ${\theta }_i$: the images lie in $U_i\times F$, which is Hausdorff by [F4] because $U_i\subseteq B$ and $F$ are Hausdorff; separating them there and applying ${\theta }_i^{-1}$ separates $s$ and $s^{\mathrm{\prime }}$ in $S$.

$S$ is paracompact. Let $\mathcal{W}$ be an open cover of $S$. For $b\in B$ the fiber $S_b\cong F$ is compact, so $\mathcal{W}$ has a finite subfamily covering $S_b$; fix such a finite list for every $b$ (this is the only choice made, and it is a choice of one set for every element of $B$). In a chart ${\theta }_i$ about $b$ the union of the listed open sets contains $\{b\}\times F$ in $U_i\times F$; the tube lemma [F2] applied with compact factor $F$ gives an open $V_b\ni b$ with $V_b\times F$ contained in that union, that is, $p^{-1}(V_b)$ is covered by the finitely many listed members of $\mathcal{W}$. The family $\{V_b\}$ is an open cover of the paracompact space $B$, so by [F3] it has a locally finite open refinement $\{V_{\beta }{\}}_{\beta \in J}$. Choose for each $\beta$ an index $b(\beta )$ with $V_{\beta }\subseteq V_{b(\beta )}$; then the family consisting of the open sets $W\cap p^{-1}(V_{\beta })$, for all $\beta$ and all $W$ in the finite list attached to $b(\beta )$, covers $S$ and refines $\mathcal{W}$, since $p^{-1}(V_{\beta })\subseteq p^{-1}(V_{b(\beta )})$ is covered by that finite list. It is locally finite: given $s\in S$ with $p(s)=b_0$, local finiteness of $\{V_{\beta }\}$ at $b_0$ supplies an open $N\ni b_0$ meeting only finitely many $V_{\beta }$, and then the neighborhood $p^{-1}(N)$ of $s$ meets $p^{-1}(V_{\beta })$ only for those finitely many $\beta$. Hence every open cover of $S$ has a locally finite open refinement, so $S$ is paracompact by [F3].

The compact-generation and CW-type clauses. Suppose first that $B$ is compactly generated. Since a paracompact Hausdorff space is regular, every point of a chart domain $U_i$ has an open neighborhood in $B$ whose closure lies in $U_i$; [F7] therefore makes $U_i$ compactly generated. The compact Hausdorff fiber $F$ is locally compact, so [F7] makes each ordinary product $U_i\times F$, and hence each open chart $p^{-1}(U_i)$, compactly generated. Compact generation is local on this open cover: if $A\subseteq S$ meets every compact subspace of $S$ in a closed set, then $A\cap p^{-1}(U_i)$ has the same property in the compactly generated chart and is closed there; the chart cover then makes $A$ closed in $S$. Thus $S$ is compactly generated. It is weak Hausdorff because it is Hausdorff by step 1.1, so it is CGWH. Independently, if $B$ and $F$ have the homotopy type of CW complexes, then [F6] makes $p:S\to B$ a Hurewicz fibration, and [F5] gives the homotopy type of a CW complex for $S$.

Consequences and boundary cases. The projective bundle of a numerable rank-$n$ bundle $E\to B$ with $n\ge 1$ is a numerable fiber bundle with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$, and the flag bundle is a composite of such bundles, each with the compact CW complex ${\mathbb{R}\mathbb{P}}^j$ as fiber; if the initial base is paracompact Hausdorff CGWH of CW type, the three preceding steps apply at each stage and give the same conclusion for every intermediate total space. A fiber product over $B$ of finitely many such total spaces is a numerable bundle over $B$ with compact fiber, so the same clauses apply. If $F=\mathrm{\varnothing }$ then $S=\mathrm{\varnothing }$; if $F$ is a point then $S\cong B$ over $B$; if $B=\mathrm{\varnothing }$ then $S=\mathrm{\varnothing }$. In each case paracompactness, Hausdorffness, compact generation and the CW-type conclusion hold under their stated hypotheses because the empty space and $B$ itself have the corresponding properties. For projective fibers, the one-point case is ${\mathbb{R}\mathbb{P}}^0=\{\ast \}$ and the empty convention is ${\mathbb{R}\mathbb{P}}^{-1}=\mathrm{\varnothing }$; both are compact and have CW type.

The projectivized transitions are well defined and obey the cocycle law. For $x\in U_i\cap U_j$ the linear isomorphism $g_{ji}(x)$ carries lines to lines and depends continuously on $x$, so the joint map $(x,\mathrm{\ell })\mapsto [g_{ji}(x)]\mathrm{\ell }$ is continuous. Indeed on a projective coordinate chart choose the representative with a specified coordinate equal to one, apply the continuous matrix, and take its nonzero-vector projective quotient. The identities of [F1] give $[g_{ii}(x)]=\mathrm{id}$ and $[g_{ki}(x)]=[g_{kj}(x)][g_{ji}(x)]$ for $x\in U_i\cap U_j\cap U_k$, because projectivization is functorial for composition of linear isomorphisms. Reading the displayed relation on the overlaps, the cocycle law makes $\sim$ reflexive, symmetric and transitive: it is the same calculation as in [F2] with $[g_{ji}]$ in place of $g_{ji}$. Therefore the quotient $P(E)$ exists, and the induced projection $p:P(E)\to B$ is continuous by [F4], since its composite with the quotient map is the first-coordinate projection on each summand.

The quotient is a numerable fiber bundle with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$. Let $Q:{\coprod }_i{U}_i\times {\mathbb{R}\mathbb{P}}^{n-1}\to P(E)$ be the quotient map. It is open: if $O$ is open in the disjoint union, then on the $j$-th summand the saturation $Q^{-1}Q(O)$ is the union, over $i$, of the images of $O\cap ((U_i\cap U_j)\times {\mathbb{R}\mathbb{P}}^{n-1})$ under the overlap homeomorphisms $(x,\mathrm{\ell })\mapsto (x,[g_{ji}(x)]\mathrm{\ell })$, and is therefore open. Thus $Q(O)$ is open by the definition of the quotient topology. It follows that the restrictions of $Q$ to the $i$-th summands are open onto $p^{-1}(U_i)$. The maps ${\mathrm{\Phi }}_i:p^{-1}(U_i)⟶U_i\times {\mathbb{R}\mathbb{P}}^{n-1},{\mathrm{\Phi }}_i[x,\mathrm{\ell },k]=(x,[g_{ik}(x)]\mathrm{\ell })$ are well defined, continuous by [F4], and inverse over $U_i$ to the maps $(x,\mathrm{\ell })\mapsto [x,\mathrm{\ell },i]$; the latter maps are open by the preceding calculation. Hence they are homeomorphisms over $U_i$, so $P(E)$ is a locally trivial fiber bundle with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$ in the sense of [F3]. The given numerating cover and partition of unity of $E$ serve unchanged, since the chart domains are the same $U_i$ and their supports are already subordinate; hence $P(E)$ is numerable.

The tautological quotient has the local bundle descriptions $U_i\times {\gamma }^{n-1}$: the same open-saturation argument as in step 2.1 applies to the overlap homeomorphisms $(x,\mathrm{\ell },v)\mapsto (x,[g_{ji}(x)]\mathrm{\ell },g_{ji}(x)v)$. Inside each such description refine the base by $D_{i,a}=\{(x,\mathrm{\ell }):v_a\mathrm{\ne }0\text{ for }0\mathrm{\ne }v\in \mathrm{\ell }\}$, for $1\le a\le n$. The unique vector $w_a(\mathrm{\ell })\in \mathrm{\ell }$ with coordinate $a$ equal to one is a continuous nonzero section. The map $(x,\mathrm{\ell },t)\mapsto (x,\mathrm{\ell },t{w}_a(\mathrm{\ell }))$ and its inverse, which reads the $a$-th coordinate of the vector, are continuous linear bundle charts. Thus the tautological quotient is a rank-one real bundle with the asserted fiber, not generally trivial over all of $p^{-1}(U_i)$.

If $B$ is paracompact Hausdorff CGWH of CW type and AC is assumed, [F5] applies to the numerable projective bundle of step 2.1: the fiber is the compact Hausdorff finite CW space ${\mathbb{R}\mathbb{P}}^{n-1}$. It follows directly that $P(E)$ is paracompact Hausdorff, CGWH, and of CW type, which is the asserted total-space consequence.

An explicit refined numeration needs no choice. In chart $i$ put $s_{i,a}(\mathrm{\ell })=v_a^2\mathrm{/}{\sum }_b{v}_b^2$, independent of the nonzero representative, and $d_{i,a}=\max (s_{i,a}-1\mathrm{/}(2n),0)$. Since some $s_{i,a}\ge 1\mathrm{/}n$, the sum $D_i={\sum }_a{d}_{i,a}$ is positive. Put ${\theta }_{i,a}=d_{i,a}\mathrm{/}{D}_i$, and define ${\psi }_{i,a}=({\rho }_i\circ p){\theta }_{i,a}$ on $p^{-1}(U_i)$, extended by zero elsewhere. This extension is continuous since points outside $U_i$ have a neighborhood disjoint from the closed support of ${\rho }_i$. The family is locally finite, because the base family is locally finite and there are only $n$ coordinates for each $i$. Its sum is one. Its closed support lies in $p^{-1}(\mathrm{supp}{\rho }_i)$ and in the locus $s_{i,a}\ge 1\mathrm{/}(2n)$, hence inside $D_{i,a}$. It is therefore support-subordinate to the actual line charts of step 3.1, proving numerability of ${\gamma }_E$.

For $n=1$, the projective fiber is a point, so the displayed charts identify $P(E)$ with $B$ and ${\gamma }_E$ with $E$. Rank zero uses only the declared empty-space convention, not the formulas involving $1\mathrm{/}(2n)$. An empty base gives empty quotients. Steps 1.1, 2.1, 3.1 and 4.1 use the supplied charts and partition and finite coordinate operations only; AC enters solely in step 3.2 through [F5].

By [F1], ${\gamma }_E$ is a numerable rank-one real bundle on the paracompact Hausdorff CGWH space $P(E)$. Surjectivity of the classification bijection [F2] gives a map $c:P(E)\to {\mathrm{Gr}}_1({\mathbb{R}}^{\mathrm{\infty }})$ and a bundle isomorphism $c^{\ast }{\gamma }_1\cong {\gamma }_E$. This is all the classifying-map assertion needed here. AC is inherited from [F1] and [F2].

The class is well typed. The generator $a$ of [F3] is a class in $H^1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)$, and $c^{\ast }$ is defined on it, so $x_E=c^{\ast }a$ is a class in $H^1(P(E);{\mathbb{F}}_2)$. The definition has fixed one classifying map; it asserts nothing about other choices, and the next lemma shows that any other choice gives the same class. When $n=1$, $P(E)\cong B$ and ${\gamma }_E\cong E$ by the projective-bundle definition, so $x_E=c^{\ast }a$ is the degree-one cohomology class obtained from a classifying map of $E$; no injectivity of the assignment of cohomology classes to line bundles is asserted or used here. For the empty base, $P(E)=\mathrm{\varnothing }$ and $H^1(P(E);{\mathbb{F}}_2)=0$, so the unique value is $x_E=0$; the rank-zero convention of the projective-bundle definition is not used here because $n\ge 1$.

Let $c,c^{\mathrm{\prime }}:P(E)\to {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ be any two classifying maps as in the definition. They have isomorphic tautological pullbacks, both isomorphic to ${\gamma }_E$. By the projective definition $P(E)$ is paracompact Hausdorff CGWH and ${\gamma }_E$ is numerable, so injectivity of the bijection [F1] gives equality of the actual unbased homotopy classes $[c]=[c^{\mathrm{\prime }}]$. Thus [F2] gives $c^{\ast }a=c^{\mathrm{\prime }\ast }a$. This proves independence for all classifying maps, without limiting them to any particular embedding construction. AC is inherited from the stated projective and classification interfaces.

Restriction to a fiber. Fix $b\in B$ and identify the fiber $P(E_b)$ with ${\mathbb{R}\mathbb{P}}^{n-1}$ through a linear isomorphism $E_b\cong {\mathbb{R}}^n$; write $i:P(E_b)\hookrightarrow P(E)$ for the inclusion. By [F7] the pullback $i^{\ast }{\gamma }_E$ is the tautological line ${\gamma }^1$ over $P(E_b)\cong {\mathbb{R}\mathbb{P}}^{n-1}$, and the standard inclusion $j:{\mathbb{R}\mathbb{P}}^{n-1}\hookrightarrow {\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ satisfies $j^{\ast }{\gamma }_1\cong {\gamma }^1$ by [F5]. On the other hand $(ci{)}^{\ast }{\gamma }_1\cong i^{\ast }{c}^{\ast }{\gamma }_1\cong i^{\ast }{\gamma }_E\cong {\gamma }^1$, so $ci$ and $j$ are two maps to ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ with isomorphic pullbacks of the tautological line. The fiber is a compact Hausdorff finite CW complex, hence paracompact by [F8] and CGWH. Its tautological line is numerable by the same finite coordinate partition used in the projective definition. Injectivity of the classification bijection [F1] therefore gives an actual homotopy $ci\simeq j$ on this fiber. Therefore [F2] and the definition of $x_E$ give $i^{\ast }{x}_E=(ci{)}^{\ast }a=j^{\ast }a,$ which is the standard generator of $H^1({\mathbb{R}\mathbb{P}}^{n-1};{\mathbb{F}}_2)$ when $n-1\ge 1$ and zero when $n-1=0$, by [F3]. This proves clause 2.

Fiber basis. Restriction $i^{\ast }:H^{\ast }(P(E);{\mathbb{F}}_2)\to H^{\ast }(P(E_b);{\mathbb{F}}_2)$ is a unital ring homomorphism, so [F6] gives $i^{\ast }(x_E^k)=(i^{\ast }{x}_E{)}^k$. By step 2.1 this is $a^k$ for every $k\ge 0$, with the convention that $a^0=1$ and that $a=0$ when $n=1$; in particular $i^{\ast }(1)=1$. By [F3], restriction is an isomorphism in every degree from zero to $n-1$, so its images $1,a,\dots ,a^{n-1}$ are nonzero and span their respective one-dimensional groups. By [F4] there are no groups in higher degrees. They therefore form an ${\mathbb{F}}_2$-basis; they are the restrictions of $1,x_E,\dots ,x_E^{n-1}$. For $n=1$ the fiber is ${\mathbb{R}\mathbb{P}}^0$, a point, and the list reduces to $1$. For $n\ge 2$ the class $i^{\ast }{x}_E=a$ is nonzero and generates $H^1$ of the fiber. This proves clause 3.

The projection $P(E)\to B$ is a Serre fibration with fiber ${\mathbb{R}\mathbb{P}}^{n-1}$ whose classes $1,x,\dots ,x^{n-1}$ restrict to a basis of the fiber cohomology. This is [F1] combined with [F2]: the fiber over $b$ is $P(E_b)$ and the restrictions of the displayed classes are a basis.

Choose a homotopy equivalence $g:K\to B$ from a CW complex, put $E^{\mathrm{\prime }}=g^{\ast }E$, and form $g^{\mathrm{\prime }}:P(E^{\mathrm{\prime }})\to P(E)$. Projectivization commutes with this pullback: in a pulled-back linear chart the identification sends $(k,\mathrm{\ell })$ to $(k,(g(k),\mathrm{\ell }))$, and the formulas agree on overlaps since both have the same pulled-back linear transitions. Hence $P(E^{\mathrm{\prime }})\cong g^{\ast }P(E)$, with $g^{\mathrm{\prime }}$ restricting to the identity on each projective fiber. Both projective totals are paracompact Hausdorff of CW type by [F9]. Both projections are Serre fibrations by [F1]. In their natural homotopy sequences [F5], the fiber and base maps are isomorphisms on all homotopy groups. A direct exactness chase gives the same for the total map, including degree one: to lift a target total class, lift its base image by the base isomorphism; its fiber boundary vanishes by fiber injectivity, so it lifts to the source total group. Correct the difference using surjectivity on the fiber group. For injectivity, a source class killed in the target has zero base image, so comes from a fiber class. Its image is a target base boundary; lift that boundary through the base isomorphism and use fiber injectivity to see the original total class vanishes. The degree-one chase uses products rather than sums and the fact that the fiber is path connected, so the boundary to its component set is trivial. Path lifting with path-connected fibers identifies total components with base components. Thus $g^{\mathrm{\prime }}$ is a weak homotopy equivalence.

Suppose first that $B$ is a CW complex. Then [F3] applies to the Serre fibration of step 1.1 with the classes $e_i=x^i$, $i=0,\dots ,n-1$, which are homogeneous of degree $i$ and restrict to a basis by step 1.1, provided the base is path-connected. If $B$ is path-connected, the conclusion is that $\mathrm{\Phi }:{⨁}_{i=0}^{n-1}{H}^{\ast -i}(B;{\mathbb{F}}_2)⟶H^{\ast }(P(E);{\mathbb{F}}_2),(a_i)⟼{\sum }_{i=0}^{n-1}{p}^{\ast }{a}_i⌣x^i,$ is an $H^{\ast }(B;{\mathbb{F}}_2)$-module isomorphism, so $H^{\ast }(P(E);{\mathbb{F}}_2)$ is free over $H^{\ast }(B;{\mathbb{F}}_2)$ on $1,x,\dots ,x^{n-1}$. If $B$ is a general CW complex, its path components $B_{\alpha }$ are open and closed subcomplexes, because a CW complex is locally path-connected and cells are connected; restricting $E$ gives a numerable bundle over the path-connected CW complex $B_{\alpha }$, and $P(E)={\coprod }_{\alpha }P(E{\mathrm{\mid }}_{B_{\alpha }})$. For a disjoint union $X={\coprod }_{\alpha }{X}_{\alpha }$ of open and closed pieces the singular chain complex is the direct sum of the piece complexes, since the connected simplex ${\mathrm{\Delta }}^k$ maps into a single piece; dualising gives a product of cochain complexes, whose cycles are exactly the families of cycles and whose coboundaries are exactly the families of coboundaries, the latter using AC to select one primitive at each index. Hence $H^k(X;R)\cong {\prod }_{\alpha }{H}^k(X_{\alpha };R)$, and applying this to $B$ and to $P(E)$ turns the componentwise isomorphisms into the displayed $H^{\ast }(B;{\mathbb{F}}_2)$-module isomorphism.

By [F7], $g^{\mathrm{\prime }}$ is an integral homology isomorphism, and by [F6] and the module five lemma [F10], $g^{\mathrm{\prime }\ast }$ is an isomorphism on ${\mathbb{F}}_2$-cohomology. Also $g^{\ast }$ is a cohomology isomorphism by [F10]. Apply Leray–Hirsch over each CW component of $K$ using the specified classes $e_i=g^{\mathrm{\prime }\ast }(x^i)$, not an unproved identification with a separately defined tautological class on $P(E^{\mathrm{\prime }})$. They restrict to a fiber basis because $g^{\mathrm{\prime }}$ is the identity on each fiber and [F2] supplies that basis. The component argument of step 2.1 gives an isomorphism ${\mathrm{\Phi }}_K$ for these classes. Naturality [F4] gives $g^{\mathrm{\prime }\ast }{\mathrm{\Phi }}_B={\mathrm{\Phi }}_K({⨁}_i{g}^{\ast })$. The other three maps are isomorphisms, so ${\mathrm{\Phi }}_B$ is an isomorphism. Finite sums indexed by $0\le i<n$ commute with the degreewise component products. This proves the module claim on the stated general base.