Stiefel-Whitney Characteristic Classes

In the previous post, the map \(p:\Spe(\or_M^A)\rightarrow M\), which played an important role, was a covering space, and had the following properties.

1. For any \(x\in M\), \(p^{-1}(x)\cong \{x\}\times A^\times\).
2. Moreover, for any \(x\in M\), there exists a suitable open set \(U\) such that \(p^{-1}(U)\cong U\times A^\times\).

Now we generalize this further and examine the case where \(p^{-1}(x)\) carries additional structure (not a discrete set). The most general definition is as follows.

Definition 1 For a continuous surjection \(p:E \rightarrow B\) between topological spaces and a topological space \(F\), we say that \(p\) is a fiber bundle with fiber space \(F\) if for each \(x\in B\), there exists, containing \(x\), an open set \(U\) and a homeomorphism \(\phi:U\times F\rightarrow p^{-1}(U)\) making the following diagram

commute.

Here, \(B\) is called the base space, \(E\) the total space, and \(F\) the fiber of this bundle; if we can take \(U=B\), then this fiber bundle is called a trivial bundle. For instance, in the preceding example, \(M\) is the base space, \(\Spe(\or_M^A)\) is the total space, and \(A^\times\) equipped with the discrete topology is the fiber. More generally, any covering space can be regarded as a fiber bundle whose fiber carries the discrete topology.

The two cases of particular interest to us are when the fiber \(F\) is a vector space, and when it is a topological group. For convenience, we henceforth assume that \(B\) is connected.

Vector Bundles

First, we consider the case where \(F\) is a vector space. When the fiber \(F\) is a topological group, \(F\) is already equipped with a topology, so the topology on the product space \(U\times F\) in Definition 1 is clear, but the case of a vector space is somewhat ambiguous. In the most general setting, one could use the notion of a topological ring \(\mathbb{K}\) acting on a topological vector space \(V\), but for convenience we shall for now only consider the case where the base field of \(F\) is \(\mathbb{R}\) and \(F\) is a space equipped with the metric topology arising from the canonical inner product.

Definition 2 A fiber bundle \(p:E \rightarrow B\) is called a vector bundle if the fiber space \(F\) is an \(\mathbb{R}\)-vector space equipped with a topology as above, and moreover for each \(x\in B\) there exist, containing \(x\), an open set \(U\) and a homeomorphism \(\phi:U\times F\rightarrow p^{-1}(U)\) as in Definition 1 such that the following function

\[\phi(x,-):F \rightarrow p^{-1}(x);\qquad v\mapsto \phi(x,v)\]

is an isomorphism between vector spaces.

Through this, each fiber \(p^{-1}(x)\) inherits a vector space structure from \(F\). In general, given two vector bundles \(p_1:E_1 \rightarrow B_1\) and \(p_2:E_2\rightarrow B_2\), a morphism between them means a commutative diagram of continuous functions

where, restricting \(g\), for each \(x\in B_1\), to \(p_1^{-1}(x)\rightarrow p_2^{-1}(f(x))\), this function must be a linear map between vector spaces. How to define an isomorphism between vector bundles is then obvious.

Meanwhile, in Definition 2 above, we only considered the case where \(F\) is an \(\mathbb{R}\)-vector space, and we defined a topology on it using the inner product structure defined on \(\mathbb{R}^n\) and the topology on \(\mathbb{R}\). But strictly speaking, the only information needed here is the topology on the vector space \(F\), and when we view \(F\) as an inner product space, this is called a Euclidean bundle. In any case, since we will mostly consider \(\mathbb{R}\)-vector spaces, we shall gloss over this distinction.

Example 3 An example that is not a trivial bundle is the Möbius strip considered as a line bundle over \(S^1\). On the other hand, in §Poincaré Duality, ⁋Example 5 we also considered a non-trivial cover of \(S^1\), which can be generalized geometrically as follows.

For an \(n+1\)-dimensional vector space \(\mathbb{R}^{n+1}\), we call the space of lines through the origin the projective \(n\)-space and denote it by \(\RP^n\). Among the points on a line through the origin, the two points at distance \(1\) from the origin specify the same line, so we can think of this as the quotient space obtained from the unit \(n\)-sphere \(S^n\) by identifying antipodal points.

Now let us take this space \(\RP^n\) as the base space \(B\), and define a vector bundle \(E(\gamma_n^1)\) over it as follows. As a set, it is defined by

\[E(\gamma_n^1)=\{(x,v)\in \RP^n\times \mathbb{R}^{n+1}\mid v\in \span(x)\}\]

and the projection \(\gamma_n^1:E(\gamma_n^1)\rightarrow \RP^n\) is the projection onto the first coordinate. That is, \(\gamma_n^1\) attaches to each point \(x\in \RP^n\) exactly the line in \(\mathbb{R}^{n+1}\) that \(x\) originally belonged to.

When \(n\geq 1\), this is not a trivial bundle. If this were a trivial bundle, there would exist a non-vanishing continuous section \(\RP^n\rightarrow E(\gamma_n^1)\). For instance, the function sending every point of \(B\) to \(1\) in the fiber would be such a section. But suppose an arbitrary section \(s:\RP^n \rightarrow E(\gamma_n^1)\) is given; if we consider the following composition using the quotient map \(q:S^n \rightarrow \RP^n\):

\[S^n \overset{q}{\longrightarrow} \RP^n \overset{s}{\longrightarrow} E\overset{\pr_2}{\longrightarrow} \mathbb{R}^{n+1}\]

this function sends \(\mathbf{x}\in S^n\subseteq\mathbb{R}^{n+1}\) to a scalar multiple of \(\mathbf{x}\). If we denote this scalar by \(t(\mathbf{x})\), then \(t\) is a continuous function from \(S^n\) to \(\mathbb{R}\), and because of the quotient map \(q\) it satisfies

\[t(-\mathbf{x})=-t(\mathbf{x})\]

Now since \(S^n\) is connected, by the intermediate value theorem there exists \(\mathbf{x}_0\in S^n\) such that \(t(\mathbf{x}_0)=0\).

More generally, the following holds.

Proposition 4 For a topological space \(B\) and a vector bundle \(E\) of rank \(n\) defined on it, \(E\) is a trivial bundle if and only if there exist \(n\) everywhere linearly independent sections \(s_1,\ldots, s_n\).

Proof

If \(E\) is a trivial bundle, choose an isomorphism \(\psi:B\times\mathbb{R}^n\rightarrow E\) and set \(s_i(x)=\psi(x,e_i)\); then since for each \(x\), \(\psi(x,-)\) is an isomorphism, these become everywhere linearly independent sections.

Conversely, suppose such sections are given and set

\[\varphi:B\times\mathbb{R}^n\rightarrow E;\qquad (x,a)\mapsto \sum_i a_is_i(x)\]

Then \(\varphi\) is a continuous function covering \(\id_B\) and is linear on each fiber; since \(s_1(x),\ldots,s_n(x)\) are linearly independent, they form a basis of \(p^{-1}(x)\) and so \(\varphi(x,-)\) is an isomorphism. It remains to show the continuity of \(\varphi^{-1}\); picking a local trivialization \(\phi:U\times\mathbb{R}^n \rightarrow p^{-1}(U)\) from Definition 2 and looking at \(\phi^{-1}\circ\varphi\), this has the form \((x,a)\mapsto (x,A(x)a)\) where \(A:U\rightarrow \GL(n;\mathbb{R})\) is continuous. Since taking the inverse of a matrix is continuous, \(x\mapsto A(x)^{-1}\) is also continuous, and therefore \(\varphi^{-1}\) is continuous on each \(U\).

Meanwhile, given any vector bundle \(p:E \rightarrow B\) and any continuous map \(f:B'\rightarrow B\), by setting

\[f^\ast E=\{(x,v)\in B'\times E\mid f(x)=p(v)\}\subseteq B'\times E\]

we can define a new vector bundle \(f^\ast E \rightarrow B'\). We call this the pullback bundle, and it is not difficult to see that for any vector bundle \(E' \rightarrow B'\), if a bundle map covering \(f\), \(E'\rightarrow E\), is given, this factors uniquely through \(E'\rightarrow f^\ast E \rightarrow E\).

Meanwhile, given any two vector bundles \(p_1:E_1\rightarrow B_1\), \(p_2:E_2\rightarrow B_2\), we know that their product

\[p_1\times p_2: E_1\times E_2 \rightarrow B_1\times B_2\]

is also a vector bundle over \(B_1\times B_2\). Now if \(B_1=B_2=B\), then as usual, using the diagonal map

\[\Delta: B\rightarrow B\times B\]

the pullback bundle \(\Delta^\ast(p_1\times p_2)\) becomes a bundle over \(B\). We call this the Whitney sum of the two vector bundles \(E_1\rightarrow B\), \(E_2\rightarrow B\) and denote it by \(p_1\oplus p_2:E_1\oplus E_2\rightarrow B\). As the notation suggests, fiberwise this corresponds to the direct sum of the fibers of the two vector bundles \(E_1,E_2\).

Although we have not given a detailed proof, in a similar manner we can carry over operations defined on each fiber (that is, on vector spaces) to vector bundles. For instance, given two vector bundles \(E_1\rightarrow B\), \(E_2 \rightarrow B\), we can form their tensor product bundle \(E_1\otimes E_2 \rightarrow B\), and it is also possible to use operations such as \(\Hom\) or \(\bigwedge\).

Čech Cohomology

At this point we establish another cohomology theory. Like the sheaf cohomology in §Poincaré Duality, ⁋Definition 15, this is a cohomology for sheaves defined on a topological space, and it plays an important role in our story because via the étale space construction, a sheaf whose stalks are vector spaces can be identified with a vector bundle.

Sheaf cohomology showed that cohomology encodes the obstruction to the existence of global sections of a sheaf. The Čech cohomology we now examine gives a similar result, but differs in that it answers this question by examining the process of patching local sections together to form a global section. In any case, for nice cases including manifolds, Čech cohomology gives the same result as sheaf cohomology, and thus the Čech cohomology of a constant sheaf recovers the cohomology we already knew.

Consider a topological space \(X\), a sheaf \(\mathcal{F}\) defined on it, and an open cover of \(X\), \(\mathcal{U}=\{U_i\}_{i\in I}\). For each \(p\geq 0\), the group of Čech \(p\)-cochains is defined by the formula

\[\check{C}^p(\mathcal{U},\mathcal{F})=\prod_{i_0,\ldots,i_p}\mathcal{F}(U_{i_0}\cap \cdots\cap U_{i_p})\]

That is, this is the collection of sections defined over all \((p+1)\)-fold intersections. At this time, the differential

\[\check{C}^p(\mathcal{U},\mathcal{F})\rightarrow \check{C}^{p+1}(\mathcal{U}, \mathcal{F})\]

is given by the formula

\[(\delta c)_{i_0,\ldots, i_{p+1}}=\sum_{k=0}^{p+1} (-1)^k c_{i_0,\ldots,\hat{i}_k,\ldots,i_{p+1}}\vert_{U_{i_0}\cap\cdots\cap U_{i_{p+1}}}\]

Then Čech cohomology is given by the formula

\[\check{H}^p(\mathcal{U}, \mathcal{F})=\frac{\ker(\check{C}^p\rightarrow \check{C}^{p+1})}{\im(\check{C}^{p-1}\rightarrow \check{C}^{p})}\]

If \(\mathcal{U}\) is a sufficiently good cover (for instance, if every finite intersection is contractible, or is acyclic for \(\mathcal{F}\)), then we obtain the canonical isomorphism

\[H^p(X,\mathcal{F})\cong \check{H}^p(\mathcal{U},\mathcal{F})\]

Now any rank \(n\) vector bundle is determined by how its fiber is glued over an open cover. That is, it is determined by functions

\[g_{ij}:U_{ij}=U_i\cap U_j \rightarrow \GL(n;\mathbb{R})\]

These must satisfy the condition

\[g_{ij}\cdot g_{jk}\cdot g_{ki}=\id\]

and if this condition were absent, on a triple intersection \(U_i\cap U_j\cap U_k\), when carrying the local trivialization of \(U_i\) via \(g_{ij}\) to \(U_j\), then via \(g_{jk}\) to \(U_k\), and then via \(g_{ki}\) to \(U_i\), the trivialization would differ, but this condition means that in reality it does not. Then the transition functions \(g_{ij}\) form a Čech 1-cocycle by the above condition. At this time, if we change the local trivialization over each \(U_i\) by a function \(h_i:U_i\rightarrow \GL(n;\mathbb{R})\), then \(g_{ij}\) changes to \(h_ig_{ij}h_j^{-1}\), so cocycles giving the same vector bundle must be identified by this relation. That is, over an open cover \(\mathcal{U}\), there is a one-to-one correspondence between isomorphism classes of rank \(n\) vector bundles trivializable over it and \(\check{H}^1(\mathcal{U}, \GL(n;\mathbb{R}))\). However, since \(\GL(n;\mathbb{R})\) is a non-abelian group, the \(\check{H}^1\) here is not the cohomology group defined by the differential above, but rather a pointed set defined separately by the cocycle condition \(g_{ij}g_{jk}g_{ki}=\id\) and the equivalence relation just mentioned.

Earlier, in §Poincaré Duality, ⁋Proposition 7, we saw that for a manifold \(M\), \(A\)-orientability is defined by the group homomorphism

\[\pi_1(M,x)\rightarrow A^\times\]

Meanwhile, since \(A\) is a commutative ring, the above group homomorphism factors through the abelian group homomorphism

\[H_1(M)\rightarrow A^\times\]

and by §Cohomology, ⁋Proposition 3 (Universal coefficient theorem for cohomology) this is an element of \(H^1(M;A^\times)\). If this element is \(0\), this is equivalent to the monodromy action being trivial, which in turn means that \(\Spe(\or_M^A)\) is a trivial covering space and so \(M\) becomes an \(A\)-orientable manifold. Meanwhile, for any commutative ring \(A\), since the initial object of \(\cRing\) is \(\mathbb{Z}\), for any manifold \(M\) once a \(\mathbb{Z}\)-orientation \(H_1(M)\rightarrow \mathbb{Z}^\times\) is determined, we can compose it with \(\mathbb{Z}^\times\rightarrow A^\times\) to determine an \(A\)-orientation \(H_1(M)\rightarrow A^\times\); thus we know that the essential information about whether \(\Spe(\or_M^A)\) is a trivial cover is contained in \(H^1(M;\mathbb{Z}/2)\), and thinking of \(\mathbb{Z}/2\) as \(\GL(1;\mathbb{Z})\), this is an example of how first cohomology contains information about covering spaces.

In this manner, information about a vector bundle \(E\rightarrow B\) of rank \(k\) can be regarded as being contained in \(\check{H}^1(B; \underline{\GL(k;\mathbb{R})})\). However, since the coefficient group of the cohomology of \(B\) that we use is \(\mathbb{Z}\), we do not have all the data contained there. Instead, our goal is to find weaker substitutes for this, namely invariants, in the cohomology ring \(H^\bullet(B)\).

Stiefel-Whitney Classes

The first characteristic class we examine is the Stiefel-Whitney class. First of all, whenever any vector bundle \(p:E\rightarrow B\) is given, this is an element of the cohomology ring \(H^\bullet(B;\mathbb{Z}/2)\), \(w(p)\), and just as above, if \(E\) is a trivial bundle then \(w(p)=1\). In fact, by Proposition 4, a trivial bundle has \(n=\rank(E)\) everywhere linearly independent continuous sections, and the extent to which \(w(p)\) deviates from \(1\) measures the obstruction to choosing such sections. To see this, if we decompose \(w(p)\) according to degree in the cohomology ring as

\[w(p)=w_0(p)+w_1(p)+\cdots\]

each \(w_i(p)\) becomes an obstruction class to choosing \(n-i+1\) everywhere linearly independent sections. That is, if \(w_i(p)\neq 0\), then \(n-i+1\) everywhere linearly independent sections cannot exist. In particular, if \(w_n(p)\neq 0\), not even \(1\) everywhere linearly independent section can exist, so any section must become \(0\) somewhere.

For convenience, when the projection map \(p\) and base \(B\) are clear, instead of \(w(p)\) we also use notation such as \(w(E)\). We now present the axioms that \(w(E)\) satisfies.

Definition 5 For a vector bundle \(E \rightarrow B\) of rank \(n\) and a vector bundle \(F\rightarrow B\), the cohomology classes \(w_i(E)\in H^i(B;\mathbb{Z}/2)\) satisfying the following axioms are called the Stiefel-Whitney classes.

1. (Rank) \(w_0(E)=1\), and if \(i>n\), then \(w_i(E)=0\).
2. (Naturality) For any \(f:B'\rightarrow B\), \(w(f^\ast E)=f^\ast w(E)\) holds.
3. (Whitney product formula) \(w(E\oplus F)=w(E)w(F)\) holds.
4. (Normalization) For the tautological line bundle \(\gamma_1^1:E(\gamma_1^1)\rightarrow \RP^1\) of Example 3, \(w_1(\gamma_1^1)\neq 0\).

Then we obtain the following results.

Proposition 6 Over a topological space \(B\), for two vector bundles \(p_1:E_1\rightarrow B\), \(p_2:E_2\rightarrow B\), if \(p_1,p_2\) are isomorphic, then \(w(E_1)=w(E_2)\). In particular, if \(p:E\rightarrow B\) is a trivial bundle, then \(w(E)=1\).

For the first claim, an isomorphism between \(E_1\) and \(E_2\) gives \(E_1\cong \id_B^\ast E_2\), so by naturality in Definition 5, we have \(w(E_1)=\id_B^\ast w(E_2)=w(E_2)\). For the second claim, it suffices to verify that a trivial bundle is given by the following pullback

An interesting observation is that there are only two isomorphism classes of line bundles over \(S^1\), namely the trivial line bundle and the line bundle of Example 3; indeed, among the line bundles defined over \(S^1\), one can verify that the line bundle obtained by “twisting twice” is isomorphic to the trivial line bundle. This is to some extent predictable from Proposition 6, because the Stiefel-Whitney class of a line bundle over \(S^1\) must lie in \(H^1(S^1;\mathbb{Z}/2)\), which is isomorphic to \(\mathbb{Z}/2\).

Another observation is that these are pullbacks of the tautological line bundle over \(\RP^1\). In the case of the trivial line bundle over \(S^1\), it is the pullback via a continuous map sending every point of \(S^1\) to a fixed point of \(\RP^1\), while the nontrivial line bundle is the pullback of the line bundle via a homeomorphism \(S^1 \rightarrow \RP^1\).

Grassmannians

More generally, any rank \(k\) vector bundle over a paracompact space is obtained by pulling back, from the infinite Grassmannian \(\Gr(k,\mathbb{R}^\infty)\), the universal bundle \(\gamma^k_\infty:E(\gamma_\infty^k)\rightarrow \Gr(k,\mathbb{R}^\infty)\). That is, given a paracompact space \(B\) and any vector bundle \(p:E \rightarrow B\) over it, there exists a bundle map from \(p\) to \(\gamma^k_\infty\), unique up to homotopy, making the following diagram

commute, and this is isomorphic to the following pullback diagram

Moreover, the Stiefel-Whitney class of the vector bundle \(E\) is also obtained by pulling back, from the universal bundle \(\gamma^k_\infty\), the Stiefel-Whitney class \(w(\gamma^k_\infty)\).

In the sense that it realizes all rank \(k\) bundles without exception as its pullbacks, \(\gamma^k_\infty\) is called the universal family of rank \(k\) vector bundles. That is, this single bundle parametrizes all rank \(k\) bundles, and the isomorphism class of a bundle corresponds one-to-one with the homotopy class of the classifying map \(B\rightarrow\Gr(k,\mathbb{R}^\infty)\).

Therefore, we must examine the (infinite) Grassmannian and the universal bundle over it, as well as the cohomology ring \(H^\bullet(\Gr(k,\mathbb{R}^\infty), \mathbb{Z}/2)\) of the infinite Grassmannian in which the Stiefel-Whitney class of this bundle lives. Since rigorously proving all properties of Grassmannians is a complex task, in this section we shall content ourselves with an introduction to these properties and, where possible, giving simple explanations.

First, we examine the basic properties and cohomology ring of \(\Gr(k,\mathbb{R}^n)\). By definition, \(\Gr(k,\mathbb{R}^{n})\) is the space of all linear subspaces of \(\mathbb{R}^{n}\) of dimension \(k\). For example, \(\Gr(1,\mathbb{R}^{n+1})\) is by definition the projective space \(\RP^n\). Since each point of \(\Gr(k,\mathbb{R}^{n})\) is a subspace of \(\mathbb{R}^{n}\), we intuitively know how close two points (that is, two subspaces of \(\mathbb{R}^{n}\) of dimension \(k\)) are to each other. This is the same as, for example, points corresponding to two lines with similar “slopes” in \(\mathbb{R}^{n+1}\) being close to each other in \(\RP^n\); this can be defined rigorously using \(n\times k\) matrices, and under this topology \(\Gr(k,\mathbb{R}^{n})\) becomes a \(k(n-k)\)-dimensional compact topological manifold.

Now let us examine their cohomology ring. Since we are using \(\mathbb{Z}/2\)-coefficients anyway, by §Poincaré Duality, ⁋Theorem 11, we may consider homology cycles of \(\Gr(k,\mathbb{R}^n)\).

To this end, let us fix a full flag of \(\mathbb{R}^n\)

\[F_\bullet:\qquad 0=F_0\subseteq F_1\subseteq F_2\subseteq\cdots\subseteq F_n=\mathbb{R}^n\]

Then, in \(\mathbb{R}^n\), any \(k\)-plane \(X\) defines

\[0=\dim (X\cap F_0)\leq\dim(X\cap F_1)\leq\cdots\leq \dim(X\cap F_n)=k\]

and this sequence shows how \(X\) sits inside \(\mathbb{R}^n\). To track this, we define a Schubert symbol \(\sigma=(\sigma_1,\ldots, \sigma_k)\) as a sequence satisfying the condition

\[1\leq \sigma_1<\sigma_2<\cdots<\sigma_k\leq n\]

These \(\sigma_i\) show when the space \(X\cap F_i\) grows. That is, they can encode the information measuring where the dimension jumps via

\[\dim(X\cap F_{\sigma(i)})=i, \qquad \dim(X\cap F_{\sigma(i)-1})=i-1\]

Reversing this, we can capture this information by assigning to a suitable partition

\[\lambda:\qquad \lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k,\qquad \lambda_1\leq n-k\]

the condition

\[\dim(X\cap F_{n-k+i-\lambda_i})\geq i\]

These partitions show, once the flag

\[F_0\subseteq F_1\subseteq\cdots\subseteq F_n\]

is fixed, how early the dimension of \(X\cap F_i\) jumped. That is, \(\lambda_i=n-k+i-\sigma_i\) measures how far the \(i\)-th jump was moved forward from the latest possible position \(\sigma_i=n-k+i\), and because of this \(\lambda_i-\lambda_{i+1}=\sigma_{i+1}-\sigma_i-1\geq 0\), so \(\lambda\) automatically becomes a decreasing sequence. For example, \(\lambda=(0,0,\ldots,0)\) is the generic case where all jumps occur as late as possible, and since the condition \(\dim(X\cap F_{n-k+i})\geq i\) is satisfied by any \(k\)-plane, \(\Omega_{(0,\ldots,0)}\) becomes the entire Grassmannian. Conversely, when \(X=F_k\), since \(\dim(F_k\cap F_j)=\min(k,j)\), we have \(\sigma_i=i\), that is \(\lambda_i=n-k\), and the corresponding partition is the full rectangle \((n-k,\ldots,n-k)\), which is the smallest case corresponding to a single point of the Grassmannian.

Now based on this, consider the following subsets

\[\Omega_\lambda^\circ(F_\bullet)=\left\{V\in\Gr(k,F_n)\mid\text{$\dim(V\cap F_{n-k+i-\lambda_i})= i$ and $\dim(V\cap F_{n-k+i-\lambda_i-1})= i-1$ for all $1\leq i\leq k$}\right\}\]

These are each dense open subsets inside their closures

\[\Omega_\lambda(F_\bullet)=\left\{V\in\Gr(k,F_n)\mid\text{$\dim(V\cap F_{n-k+i-\lambda_i})\geq i$ for all $1\leq i\leq k$}\right\}\]

and these \(\Omega_\lambda(F_\bullet)\) push their mod \(2\) fundamental class forward along the inclusion

\[\Omega_\lambda(F_\bullet)\hookrightarrow \Gr(k,\mathbb{R}^n)\]

to define homology classes in \(H_\bullet(\Gr(k,\mathbb{R}^n);\mathbb{Z}/2)\). We call the subspace \(\Omega_\lambda(F_\bullet)\) a Schubert cycle, and the Poincaré dual \(\sigma_\lambda\) of the homology class thus obtained a Schubert class. These are cohomology classes of degree \(\lvert \lambda\rvert=\sum \lambda_i\). Here, the Schubert cycle itself depends on the choice of flag \(F_\bullet\), but the Schubert class it defines does not depend on the choice of \(F_\bullet\). Also, \(H^\bullet(\Gr(k,\mathbb{R}^n);\mathbb{Z}/2)\) has, for partitions \(\lambda\) satisfying the above conditions, their Schubert classes \(\sigma_\lambda\) as a basis as a \(\mathbb{Z}/2\)-module, and therefore it suffices for us to examine only the cup product structure among these.

Example 7 For example, let us look at \(H^\bullet(\Gr(2,\mathbb{R}^4);\mathbb{Z}/2)\). We shall examine the square of the Schubert class \(\sigma_{(1,0)}\) corresponding to the partition \((1,0)\):

\[\sigma_{(1,0)}\smile\sigma_{(1,0)}=\sigma_{(1,1)}+\sigma_{(2,0)}\]

To utilize our geometric intuition, let us think of this as an intersection of Schubert cycles, just as in §Poincaré Duality, ⁋Example 16. For this, we need to consider two subspaces in general position corresponding to the homology class of \(\sigma_{(1,0)}\), which is possible by changing the choice of flag.

For a fixed flag \(F_\bullet\), let us explicitly write out what condition the partition \(\lambda=(1,0)\) represents:

\[\dim(X\cap F_{4-2+1-1})=\dim(X\cap F_2)\geq 1,\qquad \dim(X\cap F_{4-2+2-0})=\dim (X\cap F_4)\geq 2\]

That is, the only effectively valid condition is \(\dim(X\cap F_2)\geq 1\). This means that \(X\) meets \(F_2\) in dimension at least 1, which can be rephrased as the condition that \(X\) contains a suitable line \(L\) contained in \(F_2\).

Now to compute the cup product \(\sigma_{(1,0)}\smile\sigma_{(1,0)}\), we need to consider two flags \(F_\bullet\) and \(F_\bullet'\) in general position. For instance,

\[F_\bullet:\quad \langle e_1\rangle\subseteq \langle e_1,e_2\rangle\subseteq \langle e_1,e_2,e_3\rangle,\qquad F_\bullet':\quad \langle e_4\rangle\subseteq \langle e_3,e_4\rangle\subseteq \langle e_2,e_3,e_4\rangle\]

are such flags. Now the \(V\) we consider must meet both \(\langle e_1,e_2\rangle\) and \(\langle e_3,e_4\rangle\) in dimension \(1\). For this, consider another flag

\[G_\bullet:\quad \langle e_1+e_4\rangle\subseteq\langle e_1+e_4,e_2+e_3\rangle\subseteq \langle e_1+e_4,e_2+e_3,e_2-e_3\rangle\]

First, since \(F_2\cap F_2'=0\), a \(2\)-dimensional subspace satisfying both conditions \(\dim(V\cap F_2)\geq 1\) and \(\dim(V\cap F_2')\geq 1\) is exactly represented as the sum of a line \(L\) in \(F_2\) and a line \(L'\) in \(F_2'\). That is, the intersection of the two Schubert cycles is

\[S=\left\{L\oplus L'\mid L\subseteq F_2,\ L'\subseteq F_2'\right\}\cong \mathbb{P}^1\times\mathbb{P}^1\]

and the coefficient \(1\) attached to each of the two terms on the right-hand side is because \(S\) meets each of the two Schubert cycles determined by \(G_\bullet\) in exactly one point.

Indeed, the condition for \(\Omega_{(1,1)}(G)\) is \(V\subseteq G_3=\left\{x\mid x_1=x_4\right\}\), so for \(L\oplus L'\) to belong here, \(x_1=x_4=0\) is forced from each of \(L\) and \(L'\), leaving only \(L=\span(e_2)\) and \(L'=\span(e_3)\). On the other hand, the condition for \(\Omega_{(2,0)}(G)\) is \(G_1=\langle e_1+e_4\rangle\subseteq V\); decomposing \(e_1+e_4\) along \(F_2\oplus F_2'\) gives \(e_1\) and \(e_4\), so this time \(L=\span(e_1)\) and \(L'=\span(e_4)\) are forced. Thus the two intersection points are \(\span(e_2,e_3)\) and \(\span(e_1,e_4)\) respectively, one each, and this is why \(\sigma_{(1,1)}\) and \(\sigma_{(2,0)}\) appear with coefficient \(1\).

More generally, we represent these partitions by Young diagrams, and using this we can compute, when calculating the cup product \(\sigma_\lambda\smile\sigma_\mu\) of two Schubert classes, the coefficient appearing in front of \(\sigma_\nu\) for \(\nu\) satisfying \(\lvert\nu\rvert=\lvert\lambda\rvert+\lvert\mu\rvert\). The rule for reading off this coefficient from the Young diagram is called the Littlewood-Richardson rule.

Now we must define \(\Gr(k,\mathbb{R}^\infty)\) and the universal bundle over it. For this, we first define the tautological bundle over \(\Gr(k,\mathbb{R}^n)\). In the same manner as Example 3, the following bundle attaching to each point of \(\Gr(k,\mathbb{R}^{n+k})\) the vector space corresponding to that point

\[E(\gamma^k_n)=\left\{([V], x)\in \Gr(k,\mathbb{R}^{n+k})\times \mathbb{R}^{n+k}\mid \text{$V$ a $k$-dimensional subspace of $\mathbb{R}^{n+k}$ and $x\in V$}\right\}\]

exists, and we call this the tautological bundle over \(\Gr(k,\mathbb{R}^{n+k})\).

Now for each \(n\), the formula

\[\mathbb{R}^{k+n} \rightarrow \mathbb{R}^{k+n+1};\qquad (x_1,\ldots,x_{k+n}) \mapsto (x_1,\ldots,x_{k+n},0)\]

defines an inclusion of \(\mathbb{R}^{k+n}\) into \(\mathbb{R}^{k+n+1}\), and through this we can view a \(k\)-dimensional subspace of \(\mathbb{R}^{k+n}\) as a \(k\)-dimensional subspace of \(\mathbb{R}^{k+n+1}\). That is, the above inclusion induces an inclusion \(\Gr(k,\mathbb{R}^{k+n})\rightarrow \Gr(k,\mathbb{R}^{k+n+1})\) between Grassmannians. Now considering the directed system

\[\Gr(k,\mathbb{R}^k)\hookrightarrow \Gr(k,\mathbb{R}^{k+1})\hookrightarrow\cdots\]

we call their direct limit

\[\Gr(k,\mathbb{R}^\infty)=\varinjlim_{n\geq 0}\Gr(k,\mathbb{R}^{k+n})\]

the infinite Grassmannian. In the same manner, the direct limit of total spaces

\[E(\gamma_\infty^k)=\varinjlim_{n\geq 0} E(\gamma^k_{n})\]

is defined, and this defines a rank \(k\) vector bundle over \(\Gr(k,\mathbb{R}^\infty)\). These of course do not depend on the choice of the inclusion \(\mathbb{R}^{k+n}\hookrightarrow \mathbb{R}^{k+n+1}\).

Intuitively, \(\Gr(k,\mathbb{R}^\infty)\) can be thought of as giving a complex structure by gluing together the various \(\Gr(k,\mathbb{R}^{k+n})\). Moreover, the tautological bundles \(E(\gamma^k_{n})\) also become attached compatibly with this structure.

Carrying over the Schubert cycles of finite Grassmannians to the infinite Grassmannian is not the right direction. In our convention, \(\Omega_\lambda(F_\bullet)\) has codimension \(\lvert\lambda\rvert\), so its dimension grows with \(n\). However, as explained above, the infinite Grassmannian is a space having finite Grassmannians as subcomplexes, and the Schubert classes constructed above behave well under these inclusions. That is, if we attach a new direction at the bottom of the flag, letting \(F'_1\) be that direction and \(F'_{j+1}=F'_1\oplus F_j\), then the condition defining \(\Omega_\lambda(F'_\bullet)\) read for an element of \(\Gr(k,\mathbb{R}^{k+i})\) becomes the same as the condition defining \(\Omega_\lambda(F_\bullet)\), so for the inclusion \(\iota:\Gr(k,\mathbb{R}^{k+i})\hookrightarrow \Gr(k,\mathbb{R}^{k+i+1})\) we have \(\iota^\ast\sigma_\lambda=\sigma_\lambda\). In this way, for each \(\lambda\) a cohomology class \(\sigma_\lambda\) of \(\Gr(k,\mathbb{R}^\infty)\) is determined.

Now consider the \(k\) partitions

\[\lambda_1=(1,0,\cdots, 0),\quad \lambda_2=(1,1,0,\cdots,0),\qquad \lambda_k=(1,\cdots,1)\]

Then we obtain the corresponding Schubert classes

\[w_1\in H^1(\Gr(k,\mathbb{R}^\infty);\mathbb{Z}/2),\cdots, w_k\in H^k(\Gr(k,\mathbb{R}^\infty);\mathbb{Z}/2)\]

The condition imposed by \(\lambda_i\) in \(\Gr(k,\mathbb{R}^n)\) collapses to the single condition \(\dim(V\cap F_{n-k+i-1})\geq i\), which is where \(k-i+1\) sections of the tautological bundle lose independence; thus we see that what we earlier read as \(w_i\) being the obstruction class to choosing such sections is exactly this. On the other hand, the Schubert class of the single-row partition \((i,0,\cdots,0)\) is, in \(w(\gamma^k_\infty)\)’s formal inverse \(\bar{w}\), the degree \(i\) component, so as in \(\bar{w}_2=w_1^2+w_2\), for \(i\geq 2\) it differs from \(w_i\).

Then \(H^\bullet(\Gr(k,\mathbb{R}^\infty);\mathbb{Z}/2)\) is generated by these \(w_i\) as a polynomial algebra. For instance, the monomials

\[w_1^{a_1}w_2^{a_2}\cdots w_k^{a_k}\]

form an (infinite) basis of this ring as a \(\mathbb{Z}/2\)-module, and the Schubert classes \(\sigma_\lambda\) constructed above also form such a basis. However, although the number of elements in each degree is the same for the two bases, they are different bases, so one cannot simply read the parts of a partition as exponents to correspond \(\sigma_\lambda\) to a monomial; the cup product between them is computed by the Littlewood-Richardson rule mentioned above. Now these \(w_i\) satisfy all the axioms that Stiefel-Whitney classes satisfy, and existence is proved from the fact that this is preserved under pullback.

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References

[Hat] A. Hatcher, Vector Bundles and K-Theory, online notes, 2017.
[MS] J. W. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, 1974.