Stiefel-Whitney Characteristic Classes

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

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

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

We now generalize this further and consider the case where the fiber \(p^{-1}(x)\) carries additional structure (not merely 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 \(F\), a fiber bundle is a structure such that for each \(x\in B\) there exists an open set \(U\) and a homeomorphism \(\phi:U\times F\rightarrow p^{-1}(U)\) making the following diagram commute:

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

In particular, the two cases of interest 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 \(F\) is a topological group, it already carries a topology, so the topology on the product space \(U\times F\) in Definition 1 is clear; but when \(F\) is a vector space the situation is somewhat ambiguous. The most general setting would use a topological vector space \(V\) over a topological ring \(\mathbb{K}\), but for convenience we shall for now only consider the case where the base field of \(F\) is \(\mathbb{R}\) and \(F\) is equipped with the metric topology coming from the canonical inner product.

Definition 2 For a fiber bundle \(p:E \rightarrow B\), suppose the fiber space \(F\) is an \(\mathbb{R}\)-vector space with the topology described above, and moreover for any \(x\in B\) and the homeomorphism \(\phi:U\times F\rightarrow p^{-1}(U)\) from Definition 1, the map

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

is an isomorphism of vector spaces.

Through this, each fiber \(p^{-1}(x)\) is endowed with a vector space structure inherited 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 is a commutative diagram of continuous functions

where, for each \(x\in B_1\), the restriction of \(g\) to \(p^{-1}(x)\rightarrow p_2^{-1}(f(x))\) must be a linear map between vector spaces. It is obvious how to define an isomorphism between vector bundles.

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 on \(\mathbb{R}^n\) and the topology of \(\mathbb{R}\). But strictly speaking, the only information needed here is the topology of the vector space \(F\); when \(F\) is regarded 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 A non-trivial example is the orientation double cover of the Möbius strip. Also, in §Poincaré Duality, ⁋Example 5 we 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 \(\RP^n\). Since among the points on a line through the origin, the two points at distance \(1\) from the origin specify the same line, 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,

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

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 \(x\in \RP^n\) precisely the line that \(x\) originally belonged to in \(\mathbb{R}^{n+1}\).

This is not a trivial bundle. If it were trivial, there would exist a non-vanishing continuous section \(\RP^n\rightarrow E(\gamma_n^1)\). For instance, the map sending every point of \(B\) to the element \(1\) in the fiber would be such a section. But given any section \(s:\RP^n \rightarrow E(\gamma_n^1)\), 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 map sends \(\mathbf{x}\in S^n\subset\mathbb{R}^{n+1}\) to a scalar multiple of \(\mathbf{x}\). Let this scalar be \(t(\mathbf{x})\); 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, the intermediate value theorem implies that there exists \(\mathbf{x}_0\in S^n\) with \(t(\mathbf{x}_0)=0\).

More generally, the following holds.

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

Meanwhile, given any vector bundle \(p:E \rightarrow B\) and any continuous map \(f:B'\rightarrow B\), we define a new vector bundle \(f^\ast E \rightarrow B'\) by the formula

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

We call this the pullback bundle, and it is not difficult to see that if any vector bundle \(E' \rightarrow B'\) satisfies the above condition, then it factors through \(f^\ast E\).

Moreover, given any two vector bundles \(p_1:E_1\rightarrow B_1\) and \(p_2:E_2\rightarrow B_2\), 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 \(E_1\rightarrow B\) and \(E_2\rightarrow B\), 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 \(E_1\) and \(E_2\).

Although we have not given detailed proofs, in a similar way we can lift operations defined on each fiber (i.e., on vector spaces) to vector bundles. For example, given two vector bundles \(E_1\rightarrow B\) and \(E_2 \rightarrow B\), 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 sheaf cohomology (§Poincaré Duality, ⁋Definition 15), this is a cohomology for sheaves defined on a topological space, and it plays an important role in our discussion because via the étale space construction we can regard a sheaf whose stalks are vector spaces and a vector bundle as the same thing.

Sheaf cohomology showed that cohomology encodes the obstruction to the existence of global sections of a sheaf. The Čech cohomology we now examine yields a similar result, but differs in that it answers this question by examining the process of gluing 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 originally knew.

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

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

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

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

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

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

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

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

Now any rank \(n\) vector bundle is determined by how its fibers are 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

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

If this condition were absent, then on the triple intersection \(U_i\cap U_j\cap U_k\), carrying the local trivialization from \(U_i\) to \(U_j\) via \(g_{ij}\), then to \(U_k\) via \(g_{jk}\), and back to \(U_i\) via \(g_{ki}\), would result in a different trivialization; but in reality this does not happen. Then the transition functions \(g_{ij}\) become Čech 1-cochains, and thus, fixing a local trivialization \(U_i\rightarrow \GL(n;\mathbb{R})\), we know there is a one-to-one correspondence between isomorphism classes of rank \(n\) vector bundles and 1-cochains. That is, there is a one-to-one correspondence between isomorphism classes of rank \(n\) vector bundles trivializable over an open cover \(U\) and \(\check{H}^1(\mathcal{U}, \GL(n;\mathbb{R}))\).

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

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

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 \(H^1(M;A)\). If this element is \(0\), then the monodromy action is the trivial action, which means that \(\Spe(\or_M^A)\) is a trivial covering space, and hence \(M\) becomes an \(A\)-orientable manifold. Moreover, for any commutative ring \(A\), since the initial object of \(\cRing\) is \(\mathbb{Z}\), for any manifold \(M\) a \(\mathbb{Z}\)-orientation \(H_1(M)\rightarrow \mathbb{Z}^\times\) determines an \(A\)-orientation \(H_1(M)\rightarrow A^\times\) by composing with \(\mathbb{Z}^\times\rightarrow A^\times\); thus the essential information about whether \(\Spe(\or_M^A)\) is a trivial cover lies 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 encodes information about covering spaces.

In this way, information about a vector bundle \(E\rightarrow B\) of rank \(k\) can be regarded as contained in \(\check{H}^1(B; \underline{\GL(k,\mathbb{R})})\). However, since the coefficients in the cohomology of \(B\) that we use are \(\mathbb{Z}\), 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, for any vector bundle \(p:E\rightarrow B\), this is an element \(w(p)\) of the cohomology ring \(H^\bullet(B;\mathbb{Z}/2)\), and as above, if \(w(p)=0\) then \(E\) becomes a trivial bundle. That is, if \(w(p)=0\), then by Proposition 4, there exist \(n=\rank(E)\) everywhere linearly independent continuous sections. Moreover, decomposing \(w(p)\) 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 \(w_n(p)\neq 0\), then not even one everywhere linearly independent section can exist, so any section must vanish somewhere.

For convenience, when the projection map \(p\) and base \(B\) are clear, we sometimes use notation such as \(w(E)\) instead of \(w(p)\). 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 \(F\rightarrow B\), 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 \(f:B'\rightarrow B\), we have \(w(f^\ast E)=f^\ast w(E)\).
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, we have \(w_1(\gamma_1^1)\neq 0\).

The following results are then obvious.

Proposition 6 For two vector bundles \(p_1:E_1\rightarrow B_1\) and \(p_2:E_2\rightarrow B_2\), if \(p_1\) and \(p_2\) are isomorphic then \(w(E_1)=w(E_2)\). In particular, if \(p:E\rightarrow B\) is a trivial bundle then \(w(E)=0\).

The first claim is obvious. For the second claim, it suffices to verify that a trivial bundle is given by the following pullback:

An interesting observation is that the isomorphism classes of line bundles over \(S^1\) are only two: the trivial line bundle and the line bundle of Example 3. Indeed, among line bundles 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 \(\mathbb{Z}/2\).

These are pullbacks of the tautological line bundle over \(\RP^1\). The trivial line bundle over \(S^1\) is the pullback via the continuous map sending every point of \(S^1\) to a fixed point of \(\RP^1\), and the nontrivial line bundle is the pullback of the line bundle via the quotient map \(S^1 \rightarrow \RP^1\).

Grassmannians

More generally, any rank \(k\) vector bundle over any space is obtained by pulling back the universal bundle \(\gamma^k_\infty:E(\gamma_\infty^k)\rightarrow \Gr_k(\mathbb{R}^\infty)\) over the infinite Grassmannian. That is, given any vector bundle \(p:E \rightarrow B\), there exists a unique bundle map from \(p\) to \(\gamma_k^\infty\) 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 the Stiefel-Whitney class \(w(\gamma^k_\infty)\) of the universal bundle \(\gamma^k_\infty\).

In the sense that this single bundle realizes every rank \(k\) bundle as its pullback without omission, we call \(\gamma^k_\infty\) the universal family of rank \(k\) vector bundles. Soon this one 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, and 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 complicated task, in this section we shall content ourselves with an introduction to these properties and, where possible, brief 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 \(k\)-dimensional linear subspaces of \(\mathbb{R}^{n}\). 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 (i.e., two \(k\)-dimensional subspaces of \(\mathbb{R}^{n}\)) are to each other. This is the same phenomenon as, for instance, in \(\RP^n\), points corresponding to two lines in \(\mathbb{R}^{n+1}\) with similar “slopes” being close; this can be defined rigorously using \(n\times k\) matrices, and with this topology \(\Gr_k(\mathbb{R}^{n})\) becomes a \(k(n-k)\)-dimensional compact topological manifold.

Now let us examine the cohomology rings of these spaces. Since we are using \(\mathbb{Z}/2\)-coefficients anyway, by §Poincaré Duality, ⁋Theorem 11, we may think in terms of homology cycles of \(\Gr_k(\mathbb{R}^n)\).

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

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

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

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

These \(\sigma_i\) indicate when the space \(X\cap F_i\) grows. That is, they measure 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 encode 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, when the flag

\[F_0\subset F_1\subset\cdots\subset F_n\]

is fixed, when the dimensions of \(X\cap F_i\) jumped. For example, when \(X=F_k\), the corresponding partition is \((0,0,\ldots,0)\), meaning that the dimension jumps are completed without delay in the first \(k\) terms as \(i\) increases. For instance, \((0,1,0,\ldots,0)\) means that \(X\cap F_1\) is 1-dimensional, \(X\cap F_2\) is also 1-dimensional, \(X\cap F_3\) is 2-dimensional, and thereafter the dimensions jump one by one without delay.

Now based on this, consider the subset

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

These are open submanifolds 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 \(\Omega_\lambda(F_\bullet)\) define homology classes of \(\Gr_k(\mathbb{R}^n;\mathbb{Z}/2)\) via the inclusion

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

We call these Schubert cycles, and their Poincaré duals \(\sigma_\lambda\) Schubert classes. These are cohomology classes of degree \(\lvert \lambda\rvert=\sum \lambda_i\). At this point, each subspace \(\Omega_\lambda(F_\bullet)\) depends on the choice of flag \(F_\bullet\), but their images in homology, the Schubert cycles, do not depend on the choice of \(F_\bullet\), and therefore neither do the Schubert classes. Moreover, \(H^\bullet(\Gr_k(\mathbb{R}^n);\mathbb{Z}/2)\) is a polynomial algebra generated by partitions \(\lambda\) satisfying the above conditions, and thus it suffices for us to examine only the cup product structure among them.

Example 7 For example, consider \(\Gr_2(\mathbb{R}^4;\mathbb{Z}/2)\). 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 use our geometric intuition, as in §Poincaré Duality, ⁋Example 16, we think of this as an intersection of Schubert cycles. For this we must consider two subspaces in general position corresponding to \(\sigma_{(1,0)}\), 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 effective condition is essentially \(\dim(X\cap F_2)\geq 1\). 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 \(F_2\).

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

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

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

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

Then there are two cases. First, one case is when the plane spanned by the two lines of \(F_2\) and \(F_2'\) is not contained in \(G_3\). For example, if \(V\) meets \(F_2\) in \(\span(e_1+e_2)\) and \(V\) meets \(F_2'\) in \(\span(e_3+e_4)\), this is the case. Then \(V\) can be written exactly as \(\span(e_1+e_2,e_3+e_4)\), which meets \(G_0,G_1\) in dimension 0, \(G_2,G_3\) in dimension 1, and \(G_4\) in dimension 2. That is, this corresponds to the case \((1,1)\).

The other case is when the plane spanned by the two lines of \(F_2\) and \(F_2'\) is contained in \(G_3\). For example, if we consider the case where \(V\) meets \(F_2\) in \(\span(e_2)\) and \(V\) meets \(F_3\) in \(\span(e_3)\), then \(V=\span(e_2,e_3)\) and this is contained in \(G_3\). This case corresponds to \((2,0)\).

More generally, we represent these partitions by Young diagrams, and using this we can compute the coefficient in front of \(\sigma_\nu\) when computing the cup product \(\sigma_\lambda\smile\sigma_\mu\) of two Schubert classes, for \(\nu\) satisfying \(\lvert\nu\rvert=\lvert\lambda\rvert+\lvert\mu\rvert\).

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 way as Example 3, the following bundle attaching to each point of \(\Gr_k(\mathbb{R}^{n+k})\) the vector space corresponding to that point

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

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

Now for each \(n\), the map

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

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 its direct limit

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

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

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

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 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 each \(\Gr_k(\mathbb{R}^{k+n})\). Moreover, the tautological bundles \(E(\gamma^k_{n+k})\) also glue together compatibly with this structure.

Carrying the Schubert classes of finite Grassmannians over to the infinite Grassmannian is not the right direction. However, as explained above, the infinite Grassmannian is a space having finite Grassmannians as subcomplexes, and the Schubert cycles we constructed above behave well under these inclusions. That is, pushing the Schubert cycle of \(\Gr_k(\mathbb{R}^{k+i})\) corresponding to partition \(\lambda\) into \(\Gr_k(\mathbb{R}^{k+i+1})\) via \(\Gr_k(\mathbb{R}^{k+i})\rightarrow \Gr_k(\mathbb{R}^{k+i+1})\) gives the same result as intersecting the Schubert cycle of \(\Gr_k(\mathbb{R}^{k+i+1})\) directly corresponding to partition \(\lambda\) with \(\Gr_k(\mathbb{R}^n)\subset \Gr_k(\mathbb{R}^{k+i+1})\).

Now consider the \(k\) partitions

\[\lambda_1=(1,0,\cdots, 0),\quad \lambda_2=(2,0,\cdots,0),\qquad \lambda_k=(k,0,\cdots,0)\]

Then by the above argument these become homology classes of \(\Gr_k(\mathbb{R}^\infty)\), and the functions

\[w_i: H_\bullet(\Gr_k(\mathbb{R}^\infty);\mathbb{Z}/2)\rightarrow \mathbb{Z}/2; \qquad \text{$w_i(\Omega_{\lambda_i}(F_\bullet))=1$ and is $0$ otherwise}\]

lie in the \(i\)-th cohomology class \(H^i(\Gr_k(\mathbb{R}^\infty);\mathbb{Z}/2)\), and thus we know that

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

Then \(H^\bullet(\Gr_k(\mathbb{R}^\infty);\mathbb{Z}/2)\) is generated as a polynomial algebra by these \(w_i\). For example, any partition \((a_1,\cdots,a_n)\) corresponds to the monomial

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

and this becomes one of the (infinitely many) generators of \(H^\bullet(\Gr_k(\mathbb{R}^\infty);\mathbb{Z}/2)\) as a \(\mathbb{Z}/2\)-module, computed by the Littlewood-Richardson rule mentioned above. Now these \(w_i\) satisfy all the axioms that Stiefel-Whitney classes must satisfy, and their existence is proved from the fact that this is preserved under pullback.