2 Proof of Theorem A

We first collect the required ingredients. Recall that R indicates a countable PID, and coefficients for cohomology are assumed to be taken in R, when unspecified.

We use the general tautness property of Čech cohomology: this says that for any closed subspace Y of a paracompact space Z and coefficients in any R-module M, the natural homomorphism

$$\begin{align*}\varinjlim H^k(U;{M}) \to H^k(Y;{M}) \end{align*}$$

is an isomorphism, where the direct limit is over all neighborhoods U of Y in Z (see [Reference SpanierSp66, Theorem 6.6.2]).

There is also the above-mentioned continuity property: for an inverse system of compact Hausdorff spaces $Z_n$ , with limit Z, the homomorphism

$$\begin{align*}\varinjlim H^k(Z_n;{M}) \to H^k(Z;{M}) \end{align*}$$

is an isomorphism, again for any R-module M [Reference SpanierSp66, Theorem 6.6.6].

We make use of the following general Künneth formula for Čech cohomology.

Proposition 2.1 Assume that P is locally contractible and that $P\times Y$ is paracompact Hausdorff. Then, for any $k\ge 0,$ there is a natural isomorphism

$$\begin{align*}H^k( P \times Y) \cong \bigoplus_{i=0}^k H^i(P; H^{k-i}(Y)), \end{align*}$$

functorial in P and Y.

When P is compact, a proof via the spectral sequence of a double complex may be found in Bredon’s book. (Use the short exact sequence of [Reference BredonBr97, Theorem IV.7.6] together with the universal coefficient theorem for sheaf cohomology [Reference BredonBr97, Theorem II.15.3].) The version stated above is due to Bartik, and uses a natural isomorphism between kth Čech cohomology and homotopy classes of maps into the Eilenberg–MacLane space $K({R},k)$ [Reference BartikBa68, Theorem C(b)].

In preparation for the proof of our main result, we state some lemmas. Writing $X_n^{(k)}$ for the $(k+2)$ -skeleton of $X_{n}$ , we have an isomorphism

(2.1) $$ \begin{align} H^i(X_n;M) \to H^i(X_n^{(k)};M) \end{align} $$

for $i\le k+1$ and $M=R$ , hence also for $i\le k$ and any M by the universal coefficient theorem for singular cohomology.

In addition to the spaces X and $X_{\le n}$ , we introduce

$$ \begin{align*} X_{>n} = \prod_{m>n} X_m, \end{align*} $$

and we define $X^{(k)}$ , $X_{\leq n}^{(k)}$ , and $X_{>n}^{(k)}$ as the analogous products involving the spaces $X_{m}^{(k)}$ . Note that the latter three products are all compact. Moreover, $X_{\leq n}^{(k)}$ contains the $(k+2)$ -skeleton of $X_{\leq n}$ , so in fact the $(k+2)$ -skeleta of $X_{\leq n}^{(k)}$ and of $X_{\leq n}$ are the same, which yields an isomorphism of the form (2.1) for $X_{\leq n}$ in place of $X_{n}$ .

CW complexes are not only paracompact, but in fact hereditarily paracompact in the sense that every subspace is again paracompact (see [Reference Lundell and WeingramLW69, Corollary II.4.4]). Because we can express the countable product X as the inverse limit

$$ \begin{align*} X = \varprojlim_{n} X_{\le n}, \end{align*} $$

this implies that X is again paracompact [Reference Dydak and GeogheganDG86, Lemma 7], and so are all $X_{>n}$ .

Lemma 2.2 The map

$$ \begin{align*} H^{i}(X) \to H^{i}(X_{\le n}^{(k)}\times X_{>n}) \end{align*} $$

is an isomorphism for any $n\ge 0$ and any $i\le k$ .

Proof By Proposition 2.1, this reduces to the fact that the map

$$ \begin{align*} H^{*}\bigl(X_{\le n};H^{*}(X_{>n})\bigr) \to H^{*}\bigl(X_{\le n}^{(k)};H^{*}(X_{>n})\bigr) \end{align*} $$

is an isomorphism in total degree $i\le k$ , which is a special case of what has been discussed above.

Lemma 2.3 For any $n\ge 0$ and any $i\le k,$ there is an isomorphism

$$ \begin{align*} \varinjlim_{V} H^{i}(V\times X_{>n}) = H^{i}(X^{(k)}_{\le n}\times X_{>n}), \end{align*} $$

where V ranges over the open neighborhoods of $X^{(k)}_{\le n}$ in $X_{\le n}$ .

Proof Note that $V\times X_{>n}$ is the inverse limit of the spaces $V\times X_{n+1}\times \dots \times X_{m}$ with $m\ge n+1$ . The latter are subspaces of CW complexes and therefore hereditarily paracompact, which ensures that $V\times X_{>n}$ itself is paracompact.

We may therefore appeal to Proposition 2.1, which together with tautness gives isomorphisms

$$ \begin{align*} \varinjlim_{V} H^{i}(V\times X_{>n}) &= \varinjlim_{V} \bigoplus_{j} H^{j}\bigl(V;H^{i-j}(X_{>n})\bigr) \\ &= \bigoplus_{j} H^{j}\bigl(X^{(k)}_{\le n};H^{i-j}(X_{>n})\bigr) = H^{i}(X^{(k)}_{\le n}\times X_{>n}).\\[-45pt] \end{align*} $$

Lemma 2.4 For any $i\le k,$ the inclusion $\iota \colon X^{(k)}\hookrightarrow X$ induces an isomorphism

$$ \begin{align*} \iota^{*}\colon H^{i}(X) = H^{i}(X^{(k)}). \end{align*} $$

Proof Consider the open neighborhoods of $X^{(k)} \subseteq X$ that are of the form

$$ \begin{align*} U = V \times X_{>n}, \end{align*} $$

where V is an open neighborhood of $X_{\leq n}^{(k)}$ in $X_{\leq n}$ , for some n. The collection $\mathcal {U}$ of all such U is cofinal in the directed set of all open neighborhoods of $X^{(k)}$ , by compactness of $X^{(k)}$ and the definition of the product topology. Let $\mathcal {U}_n\subseteq \mathcal {U}$ be the subset of all U that can be written in the above form for a given n, so $\mathcal {U}_n \subset \mathcal {U}_{n'}$ for $n\leq n'$ . In this notation, Lemma 2.3 reads

$$ \begin{align*} \varinjlim_{U\in \mathcal{U}_n} H^i(U) = H^{i}(X^{(k)}_{\le n}\times X_{>n}) \end{align*} $$

for all $i\leq k$ .

Using this identity together with Lemma 2.2 and the tautness property for $X^{(k)} \subseteq X$ gives

$$ \begin{align*} H^i(X^{(k)}) &\cong \varinjlim_{U\in \mathcal{U}} H^i(U) = \varinjlim_n \varinjlim_{U\in \mathcal{U}_n} H^i(U) \\ &\cong \varinjlim_n H^i(X_{\leq n}^{(k)} \times X_{>n}) = \varinjlim_n H^i(X) = H^{i}(X) \end{align*} $$

for all $i\leq k$ , which completes the proof.

Now, we will show that the natural homomorphism $\alpha \colon \varinjlim _n H^kX_{\leq n} \to H^kX$ is an isomorphism for any fixed degree k, proving Theorem A.

We have a commuting square

(2.2)

Since $X^{(k)} = \varprojlim _n X_{\leq n}^{(k)}$ is a limit of compact spaces, the homomorphism $\beta $ is an isomorphism by the continuity property. The homomorphism j is a colimit of isomorphisms $H^k(X_{\leq n}) \to H^k(X_{\leq n}^{(k)})$ , so it is an isomorphism itself. Finally, the map $\iota ^{*}$ is an isomorphism by Lemma 2.4, and we conclude that so is $\alpha $ .

3 Flag manifolds

Infinite-dimensional flag manifolds are closely related to products of projective spaces. Let $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ be the space parametrizing chains of subspaces $V_1 \subset \cdots \subset V_n \subset \mathbb {C}^\infty $ with $\dim V_{i}=i$ , topologized as a union

$$\begin{align*}{Fl} (1,\ldots,n;\mathbb{C}^\infty) = \bigcup_m {Fl} (1,\ldots,n;\mathbb{C}^m). \end{align*}$$

A standard calculation shows that

$$\begin{align*}H^* {Fl} (1,\ldots,n;\mathbb{C}^\infty) = \mathbb{Z}[t_1,\ldots,t_n] \end{align*}$$

is again a polynomial ring in n variables of degree $2$ (see, e.g., [Reference Anderson and FultonAF24, Section 2.5]). Since the space $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ is locally contractible, paracompact, and Hausdorff, any usual cohomology theory will do: in particular, singular and Čech cohomology agree.

The infinite flag manifold $ {Fl} (\mathbb {C}^\infty )$ parametrizes chains of subspaces $V_1 \subset V_2 \subset \cdots \subset \mathbb {C}^\infty $ , with $\dim V_i = i$ , and is topologized as an inverse limit of partial flag manifolds:

$$\begin{align*}{Fl} (\mathbb{C}^\infty) = \varprojlim_n {Fl} (1,\ldots,n;\mathbb{C}^\infty), \end{align*}$$

where the maps in the inverse system are projections onto the first part of a chain. As with $(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ , this space is no longer locally contractible, so the choice of cohomology theory matters.

Theorem B Using Čech cohomology with coefficients in a countable principal ideal domain R, we have

$$\begin{align*}H^* {Fl} (\mathbb{C}^\infty) = H^*(\mathbb{C}\mathbb{P}^\infty)^{\times\mathbb{N}} = R[t_1,t_2,\ldots], \end{align*}$$

a polynomial ring in countably many variables of degree $2$ .

With $\mathbb {Q}$ coefficients, it seems feasible to prove this directly by following the pattern of the previous argument, using a version of the Leray–Hirsch theorem in place of the Künneth formula, but we have not pursued this line of reasoning. Instead, we will deduce Theorem B from the calculation of $H^*(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ by constructing a homotopy equivalence between $ {Fl} (\mathbb {C}^\infty )$ and $(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ . Here, we will be using the fact that Čech cohomology satisfies the homotopy axiom [Reference SpanierSp66, Theorem 6.5.6].

In general, suppose $X_n$ and $Y_n$ are two inverse systems, with respective limits ${X = \varprojlim X_n}$ and $Y=\varprojlim Y_n$ . If $f_n \colon X_n \to Y_n$ and $g_n \colon Y_n \to X_n$ are homotopy inverse maps of the systems with homotopies respecting the inverse systems, then the induced maps $f\colon X \to Y$ and $g\colon Y \to X$ are homotopy inverses.

To prove Theorem B, it suffices to construct homotopy equivalences between the partial flag manifold $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ and $(\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ , compatible with the projections.

Let $H_n = \operatorname {\mathrm {Hom}}(\mathbb {C}^n,\mathbb {C}^\infty )$ be the vector space of $\infty \times n$ matrices with finitely many non-zero entries. Define the subspaces $H^{\circ \circ }_n \subset H^\circ _n \subset H_n$ by

$$\begin{align*}H^{\circ\circ}_n = \{ \text{linear } f\colon \mathbb{C}^n \to \mathbb{C}^\infty \,|\, f(v) \neq 0 \text{ for all }v\neq 0\} \end{align*}$$

and

$$\begin{align*}H^\circ_n = \{ \text{linear } f\colon \mathbb{C}^n \to \mathbb{C}^\infty \,|\, f(\mathrm{e}_i) \neq 0 \text{ for }1\leq i\leq n\}, \end{align*}$$

where $(\mathrm {e}_{i})$ denotes the standard basis for $\mathbb {C}^{n}$ . So $H^{\circ \circ }_n$ is the space of full-rank matrices – that is, of linear embeddings of $\mathbb {C}^n$ in $\mathbb {C}^\infty $ – and $H^\circ _n$ is the space of matrices with all columns nonzero. These spaces form inverse systems, via maps $H_n \to H_{n'}$ for $n\geq n'$ , by projecting onto the first $n'$ columns.

The group $B_n$ of upper-triangular matrices in $GL_n$ acts freely on $H^{\circ \circ }_n$ by right multiplication, with quotient

$$\begin{align*}H^{\circ\circ}_n/B_n = {Fl} (1,\ldots,n;\mathbb{C}^\infty). \end{align*}$$

The diagonal torus $T_n = (\mathbb {C}^*)^n$ also acts, with quotient

$$\begin{align*}H^{\circ\circ}_n/T_n = {Fl} ^{\mathrm{split}}(1,\ldots,n;\mathbb{C}^\infty), \end{align*}$$

where $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty )$ is the “split” flag manifold, parametrizing n linearly independent lines $L_1$ , …, $L_n$ in $\mathbb {C}^\infty $ . We equip it with the quotient topology coming from the above identity. There is a natural projection $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty ) \to {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ , defined by $V_i = L_1\oplus \cdots \oplus L_i$ . Choose a Hermitian metric on $\mathbb {C}^\infty $ – say, by writing $\mathbb {C}^\infty $ as the union of $\mathbb {C}^m$ according to the standard bases, and equipping each $\mathbb {C}^m$ with the standard Hermitian metric. This defines a section $\sigma $ of the projection map, by splitting the chain $V_1 \subset \cdots \subset V_n$ orthogonally with respect to the metric. Using Gram–Schmidt orthonormalization, this makes $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ a deformation retract of $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty )$ . (To ensure that the deformation is compatible with the inverse system, one uses Gram–Schmidt to simultaneously move all the lines $L_i$ into orthogonal position.)

Let $ {Fl} (\mathbb {C}^\infty )$ be the inverse limit of the spaces $ {Fl} (1,\ldots ,n;\mathbb {C}^\infty )$ , and define $ {Fl} ^{\mathrm {split}}(\mathbb {C}^\infty )$ , H, $H^{\circ }$ , and $H^{\circ \circ }$ analogously. Note that $ {Fl} (\mathbb {C}^\infty )$ is the inverse limit of the system $H^{\circ \circ }_{n}/B_{n}$ , and $ {Fl} ^{\mathrm {split}}(\mathbb {C}^\infty )$ that of the system $H^{\circ \circ }_{n}/T_{n}$ .

The sections $\sigma $ described above are compatible with the inverse system maps: for $n\geq n'$ , the diagram

commutes. Taking inverse limits, it follows that the projection $ {Fl} ^{\mathrm {split}}(\mathbb {C}^\infty ) = H^{\circ \circ }/T \to {Fl} (\mathbb {C}^\infty ) = H^{\circ \circ }/B$ is a homotopy equivalence.

On the other hand, $T_n$ also acts on $H^\circ _n$ , with quotient

$$\begin{align*}H^\circ_n/T_n = (\mathbb{C}\mathbb{P}^\infty)^{\times n}. \end{align*}$$

For the inverse limit, we get $H^\circ /T=(\mathbb {C}\mathbb{P} ^\infty )^{\times \mathbb {N}}$ . The projections making $H^{\circ \circ }_n$ and $H^\circ _n$ inverse systems induce the projections of flag manifolds and products of projective spaces.

Now, it suffices to construct a homotopy equivalence between $ {Fl} ^{\mathrm {split}}(1,\ldots ,n;\mathbb {C}^\infty )$ and $(\mathbb {C}\mathbb{P} ^\infty )^{\times n}$ , compatible with projections. The following lemma completes the proof of Theorem B.

In fact, we will prove the lemma directly on the limit spaces $H^{\circ \circ }$ and $H^\circ $ , in such a way that the constructions evidently restrict compatibly to $H^{\circ \circ }_n$ and $H^{\circ }_n$ for finite n.

Proof We construct a map $\beta \colon H^\circ \hookrightarrow H^{\circ \circ }$ as follows. The idea is to send $f\colon \mathbb {C}^\infty \to \mathbb {C}^\infty $ (with $f(\mathrm {e}_i)\neq 0$ for all i) to an injective map $\hat {f}\colon \mathbb {C}^\infty \to \mathbb {C}^\infty \otimes \mathbb {C}^\infty $ , and then compose with an isomorphism $\mu \colon \mathbb {C}^\infty \otimes \mathbb {C}^\infty \to \mathbb {C}^\infty $ to obtain $\beta (f)$ .

First, given any $f\in H = \operatorname {\mathrm {Hom}}(\mathbb {C}^\infty ,\mathbb {C}^\infty )$ , let $\hat {f}\colon \mathbb {C}^\infty \to \mathbb {C}^\infty \otimes \mathbb {C}^\infty $ be defined by

$$\begin{align*}\hat{f}(v) = \sum_i v_i f(\mathrm{e}_i)\otimes \mathrm{e}_i. \end{align*}$$

Since each $f(\mathrm {e}_i)$ is nonzero, the vectors $\{ f(\mathrm {e}_i) \otimes \mathrm {e}_i \}_{i=1,2,\ldots }$ are linearly independent. So $\hat {f}$ is injective, as claimed.

Next, let $\mu \colon \mathbb {N} \times \mathbb {N} \to \mathbb {N}$ be the bijection defined by enumerating pairs in the order indicated by reading matrix coordinates below:

$$\begin{align*}\begin{array}{ccccc} 1 & 3 & 6 & 10 & \cdots \\ 2 & 5 & 9 & \cdots \\ 4 & 8 & \cdots \\ 7 & \cdots \\ \vdots \end{array} \end{align*}$$

That is, the enumeration of $\mathbb {N}\times \mathbb {N}$ proceeds as

$$\begin{align*}(1,1), \, (2,1), \, (1,2), \, (3,1), \, (2,2), \, (1,3),\, (4,1), \, (3,2), \, (2,3), \, (1,4),\, \ldots \end{align*}$$

Reusing notation, let $\mu \colon \mathbb {C}^\infty \otimes \mathbb {C}^\infty \xrightarrow {\sim } \mathbb {C}^\infty $ be the isomorphism defined by $\mathrm {e}_i \otimes \mathrm {e}_j \mapsto \mathrm {e}_{\mu (i,j)}$ .

Given $f\in H^\circ $ and $v = (v_1,v_2,\ldots ) \in \mathbb {C}^\infty $ , define $\beta (f)$ by

$$\begin{align*}\beta(f)(v) = \mu\bigl(\hat{f}(v)\bigr) = \mu\big( v_1 f(\mathrm{e}_1)\otimes \mathrm{e}_1 + v_2 f(\mathrm{e}_2) \otimes \mathrm{e}_2 + \cdots \big). \end{align*}$$

In terms of matrices, $\beta $ sends

(3.1) $$ \begin{align} f= \left[ \begin{array}{cccc} f_{11} & f_{12} & f_{13} & \cdots \\ f_{21} & f_{22} & f_{23} & \cdots \\ f_{31} & f_{32} & f_{33} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array} \right] \qquad \text{to} \qquad \beta(f)= \left[ \begin{array}{cccc} f_{11} & 0 & 0 & \cdots \\ f_{21} & 0 & 0 & \cdots \\ 0 & f_{12} & 0 & \cdots \\ f_{31} & 0 & 0 & \cdots \\ 0 & f_{22} & 0 & \cdots \\ 0 & 0 & f_{13} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array} \right]. \end{align} $$

Note that each column has only finitely many nonzero entries. Since no column of f is zero, the columns of $\beta (f)$ are evidently linearly independent.

Both $\alpha $ and $\beta $ are torus-equivariant, since they preserve the column-indices of entries. And they are evidently compatible with the inverse system – in fact, the above description applies verbatim to maps $\beta \colon H_n^\circ \hookrightarrow H_n^{\circ \circ }$ , defined analogously for all n.

Finally, we construct homotopies $\alpha \circ \beta \simeq \mathrm {id}_{H^\circ }$ and $\beta \circ \alpha \simeq \mathrm {id}_{H^{\circ \circ }}$ . Since $\alpha $ is just a natural inclusion, we suppress it from the notation. (So both $\alpha \circ \beta $ and $\beta \circ \alpha $ are given by the same formula that defines $\beta $ ; in matrices, this is (3.1).) We use the straight line homotopy

$$\begin{align*}(f,t) \mapsto t f + (1-t) \beta(f). \end{align*}$$

When $f \in H^\circ $ , so each column is nonzero, it is easy to see that the same is true of $t f + (1-t) \beta (f)$ for each $t\in [0,1]$ . (For $t=1$ , this is true by assumption; for $t=0,$ it holds because the entries of the columns of $\beta (f)$ are the same as those of f, but with $0$ ’s inserted. For $0<t<1$ , the first nonzero entry in column j of f appears strictly above the corresponding entry of $\beta (f)$ , so it remains nonzero in $t f + (1-t)\beta (f)$ – except if $j=1$ and the first nonzero entry is $f_{11}$ or $f_{21}$ , but those entries are constant in t for all $tf + (1-t)\beta (f)$ .) So we have shown $\alpha \circ \beta \simeq \mathrm {id}_{H^\circ }$ .

For the other composition, we factor the linear maps $\beta (f)$ and f as follows. From the definition, $\beta $ factors as $\beta (f) = \mu \circ \hat {f}$ . Let $\phi \colon \mathbb {C}^\infty \otimes \mathbb {C}^\infty \to \mathbb {C}^\infty $ be defined by $\phi (\mathrm {e}_i\otimes \mathrm {e}_j) = \mathrm {e}_i$ , so $f = \phi \circ \hat {f}$ . The two factorizations are summarized by the diagram

The homotopy between f and $\beta (f)$ factors as

$$\begin{align*}tf + (1-t)\beta(f) = (t\phi + (1-t)\mu)\circ\hat{f}. \end{align*}$$

To show $\beta \circ \alpha \simeq \mathrm {id}_{H^{\circ \circ }}$ , we must show that $tf + (1-t)\beta (f)$ is injective whenever f is; for this, it suffices to show that $t\phi + (1-t)\mu $ is injective for all $t\neq 1$ . We do this by direct computation.

Given $v = \sum _{i,j} a_{ij} \mathrm {e}_i\otimes \mathrm {e}_j$ , suppose $(t\phi + (1-t)\mu )(v)=0$ for some $t\neq 1$ . Expanding the coefficients, we obtain equations

$$ \begin{align*} a_{11} &= -t(a_{12} + a_{13} + \cdots ), \\ a_{21} &= -t(a_{22} + a_{23} + \cdots ), \end{align*} $$

and

$$ \begin{align*} a_{ij} &= \frac{t}{t-1}\sum_{\ell\geq1} a_{\mu(i,j),\ell} \end{align*} $$

for all other $(i,j)$ . Since $\mu (i,j)>i$ for all $(i,j)$ not equal to $(1,1)$ or $(2,1)$ , these linear equations express $a_{ij}$ as a combination of $a_{k,\ell }$ for pairs $(k,\ell )$ greater than $(i,j)$ in lexicographic order. Since all but finitely many coefficients $a_{k,\ell }$ are $0$ , this implies ${v=0}$ , completing the proof of the lemma.