On the cell structure of flag manifolds

TL;DR

This paper introduces a unified geometric cell structure for real, complex, and quaternionic flag manifolds, inspired by Milnor and Stasheff's approach for Grassmann manifolds.

Contribution

It provides a novel, unified geometric method to define cell structures across different types of flag manifolds, extending prior work on Grassmannians.

Findings

Unified cell structure for all three types of flag manifolds

Method inspired by Milnor and Stasheff's approach

Potential applications in topology and geometry

Abstract

We here define a cell structure for real, complex and quaternionic flag manifolds in a unified way. Our method is geometric in nature and is inspired from a method due to Milnor and Stasheff, which they used to define a cell structure for real Grassmann manifolds.

Equations

$$\mathring{D^n}-B(0,r_0)$$

$$S^{n-1}\subset D^n-B(0,r_0)=\overline{\mathring{D^n}-B(0,r_0)}.$$

$$f(S^{n-1})$$

$$f(\mathring{D^n})\cap f(S^{n-1})=e\cap f(S^{n-1})=\varnothing$$

$$\partial e$$

$$\partial (e_1\times e_2\times \dots e_q)=\bigcup _{i=1}^q{\bar{e}}_1\times {\bar{e}}_2\times \cdots \times {\bar{e}}_{i-1}\times \partial e_i\times {\bar{e}}_{i+1}\times \cdots \times {\bar{e}}_q$$

$$\dim e_1\times e_2\times \cdots \times e_q=\dim e_1+\dim e_2+\cdots +\dim e_q.$$

$$\begin{array}[]{rrll}\langle\cdot,\cdot\rangle:&H\times H&\to&K\\ &(x,y)&\mapsto&\langle x,y\rangle\end{array}$$

$$⟨x+y,z⟩$$

$$⟨\lambda x,y⟩$$

$$⟨y,x⟩=\overline{⟨x,y⟩}$$

$$⟨x,x⟩\ge 0\text{mbox}and⟨x,x⟩=0\Rightarrow x=0$$

$$||u||=\underset{||x||\le 1}{\sup }||u(x)||$$

$$⟨u(x),u(y)⟩=⟨x,y⟩\text{mbox}forallx,y\in F.$$

$$d(u.v)=||u-v||$$

$$V(N,P)\times V(M,N)\to V(M,P)$$

$$⟨x,y⟩=\sum _{n=1}^{\infty }{x}_n\overline{y_n}.$$

$$p(n,H):V(n,H)\to G(n,H)$$

$$p(n,H{)}^{-1}(p(n,H)(O))=\bigcup _{u\in V(n,n)}Ou$$

$$\begin{array}[]{ccc}V(n,H)\times V(n,n)&\longrightarrow&V(n,H)\times V(n,H)\\ (v,u)&\longmapsto&(v,vu)\end{array}$$

$$\sigma \le \tau ⟺\sigma (k)\le \tau (k),1\le k\le n$$

$$\begin{array}[]{lrrl}d:&S(n,m)&\to&\mathbb{N}\\ &\sigma&\mapsto&d(\sigma)=\sum_{k=1}^{n}\sigma(k)-k\end{array}$$

$$V_{\sigma }(n,H)=\{u\in V(n,H)|u(E_k)\subset H_{\sigma (k)}\text{mbox}and⟨u(e_k),h_{\sigma (k)}⟩>0,1\le k\le n\}$$

$$\overline{V_{\sigma }(n,H)}=\{u\in V(n,H)|u(E_k)\subset H_{\sigma (k)}\text{mbox}and⟨u(e_k),h_{\sigma (k)}⟩\ge 0,1\le k\le n\}$$

$$T(u,v)(x)=x-\frac{⟨x,u+v⟩}{1+⟨u,v⟩}(u+v)+2⟨x,u⟩v.$$

$$\begin{array}[]{rrcl}\omega_{X}^{W}:&\{0,1,\dots,m\}&\to&\{0,1,\dots,n\}\\ &k&\mapsto&\dim X\cap W_{k}\end{array}$$

$$\begin{array}[]{rrcl}\sigma_{X}^{W}:&\{1,2,\dots,n\}&\to&\{1,2,\dots,m\}\\ &k&\mapsto&\min(\omega_{X}^{W})^{-1}(k)\end{array}$$

$$\dim X\cap W_i\ge k⟺i\ge {\sigma }_X^W(k),1\le k\le n.$$

$$0=X_0\subset X_1\subset \cdots \subset X_n=X$$

$${\omega }_X^Y{\omega }_Y^Z(k)$$

$${\sigma }_X^Z(k)$$

Full text

††Moncef Ghazel††E-mail: moncef.ghazel@fst.utm.tn††Faculté des Sciences de Tunis, University of Tunis El Manar. Tunis, Tunisia.

Keywords: CW-complex, Stiefel manifold, Grassmann manifold, Flag manifold.

2010 Mathematics Subject Classification: 14M15, 57Q05.

1. Introduction

A cell structure on a Hausdorff topological space is a partition of it into open cells in a regular way. Such a structure yields, in many interesting cases, information about the topological data associated to the space using the means of algebraic topology. More precisely, a cell structure allows for computation of Betti numbers, Poincaré polynomial and Euler characteristic. Singular (co)homology of a CW-complex is readily computable using cellular homology. To compute an extraordinary (co)homology theory for a CW-complex, one has the Atiyah-Hirzebruch spectral sequence which is the analogue of cellular homology [1]. The Serre spectral sequence is defined using a CW-complex approximation of the base space of the fibration.

Perhaps one of the most important cell structures defined in the literature is that of the real and complex Grassmann manifolds given by Ehresmann [6], for it was essentially used in the theory of caracteristic classes developed by Pontrjagin and Chern [10, 4]. In a subsequent work [7], Ehresmann extended his construction of the cell structure to the case of real flag manifolds. His definitions have certain deficiencies in the sense that they do not allow for a comprehensive proof of the axioms of a CW-complex. This is no surprise, for his work preceded that Whitehead on his celebrated combinatorial homotopy paper, where he gave the definitive definition of a CW-complex. The purpose of the present work is two fold: firstly, to suggest a new definition of Ehresmann CW-complex structure of real fleg manifolds which allows one to give an exhaustive proof of the required axioms; and secondly, to extend this definition to the complex and quaternionic cases. Our method maybe viewed as a strict generalization of that of Milnor and Stasheff for the cell structure of real Grassmann manifolds [9, Chapter 6].

We next describe the contents of the different sections of this paper. In Section 2, we define the notion of an open cell in a topological space and give some of its technical properties. These properties will be used in the proof of our main results. Section 3 is a preliminary one, it contains the definitions and the basic properties of Stiefel and Grassmann manifolds. In Section 4, we briefly review the method of Milnor and Stasheff [9, Chapter 6] for the cell structure of real Grassmann manifolds and extend it to the complex and quaternionic cases. The results of Section 4 are used as the basis for the reconstruction of the cell structure for flag manifolds which is done in the last two Sections.

To conclude this introduction, we observe that there is a cell structure defined for flag manifolds using Lie theoretic methods (See [2, page 193] and [3])

2. Cells and CW-complexes

For any topological space $X$ and any subset $A$ of $X$, let $\mathring{A}$ and $\bar{A}$ denote the interior and the closure of $A$ respectively. Let $D^{n}$ be the closed unit ball in the $n$-dimensional Euclidien space $\mathbb{R}^{n}$ and $S^{n-1}$ the unit sphere in $\mathbb{R}^{n}$.

Definition 2.1**.**

Let $X$ be a topological space.

1. (1)

A subset $e$ of $X$ is called an open cell in $X$ if there exists a continuous map $f:D^{n}\longrightarrow X$ which maps $\mathring{D^{n}}$ homeomorphically onto $e$. $f$ is then called a caracteristic map for $e$. The integer $n$ is unique, it is called the dimension of the open cell $e$ and is denoted by $\dim e$. 2. (2)

The boundary of an open cell $e$ in $X$ is the subset $\partial e=\bar{e}-e$.

The next lemma will play an important role in the sequel.

Lemma 2.2**.**

Let $e$ be an open cell in a Hausdorff space $X$ with caracteristic map $f:D^{n}\longrightarrow X$. Then

1. (1)

$f(D^{n})=\bar{e}$. 2. (2)

$f(\mathring{D^{n}})\cap f(S^{n-1})=\emptyset$. 3. (3)

$\partial e=f(S^{n-1})$; i.e. $\partial e=f(\partial\mathring{D^{n}})$.

In particular, $f(D^{n})$ and $f(S^{n-1})$ depend only on $e$ and not on the choice of the characteristic map $f$ of $e$.

Proof.

1. (1)

$e\subset f(D^{n})$, $f(D^{n})$ is compact in $X$ which is Hausdorff, therefore $f(D^{n})$ is closed, it follows that $\bar{e}\subset f(D^{n})$. In the other hand,

$f(D^{n})=f(\bar{\mathring{D^{n}}})\subset\overline{f(\mathring{D^{n}})}=\bar{e}$; i.e. $f(D^{n})\subset\bar{e}$. Thus $\bar{e}=f(D^{n})$. 2. (2)

Let $g:\mathring{D^{n}}\longrightarrow e$, be the homeomorphism induced by $f$. Let $y_{0}\in e$, $V$ an open neighborhood of $y_{0}$ in the subspace $e$ of $X$ with compact closure $\bar{V}$ in $e$. $g^{-1}(\bar{V})$ is compact in $\mathring{D^{n}}$, there exists $r_{0}\in(0,1)$ such that $g^{-1}(\bar{V})\subset B(0,r_{0})$ where $B(0,r_{0})$ is the open ball of center [math] and radius $r_{0}$. $g^{-1}(\bar{V})\subset B(0,r_{0})$, thus

[TABLE]

i.e. $\mathring{D^{n}}-B(0,r_{0})\subset g^{-1}(e-V)$, therefore $g(\mathring{D^{n}}-B(0,r_{0}))\subset e-V$.

[TABLE]

Therefore

[TABLE]

$y_{0}\notin\bar{e}-V$, thus $y_{0}\notin f(S^{n-1})$. This shows that

[TABLE] 3. (3)

[TABLE]

∎

Lemma 2.3**.**

Let $K$ be compact space, $X$ a Hausdorff space and $f:K\longrightarrow X$ continuous. Then the induced map $\tilde{f}:K\longrightarrow f(K)$ is a quotient map.

Proof.

$\tilde{f}$ is continuous. Let $C$ be a subset of $f(K)$ such that $\tilde{f}^{-1}(C)$ is closed, and is therefore $\tilde{f}^{-1}(C)$ is compact. It follows that $C=\tilde{f}(\tilde{f}^{-1}(C))$ is compact in $f(K)$ which is Hausdorff. Therefore $C$ is closed in $f(K)$. It follows that $\tilde{f}$ is a quotient map. ∎

Lemma 2.4**.**

Let $X\stackrel{{\scriptstyle q}}{{\longrightarrow}}Y$ be a quotient map and $O$ an open subset of $Y$. Then the induced map $\tilde{q}:q^{-1}(O)\longrightarrow O$ is a quotient map.

Proof.

Clear. ∎

Lemma 2.5**.**

Let $X$ be a Hausdorff space, $f:D^{n}\longrightarrow X$ a continuous map such that

1. (1)

$f_{|\mathring{D^{n}}}$* is injective.* 2. (2)

$f(\mathring{D^{n}})\cap f(S^{n-1})=\emptyset$.

Then $f(\mathring{D^{n}})$ is an open cell in $X$ with characteristic map $f$.

Proof.

By Lemma 2.3 the induced map $D^{n}\longrightarrow f(D^{n})$ is a quotient map, $f(\mathring{D^{n}})\cap f(S^{n-1})=\emptyset$. Thus $f(\mathring{D^{n}})=f({D^{n}})-f(S^{n-1})$. It follows that $f(\mathring{D^{n}})$ is open in $f(D^{n})$. By Lemma 2.4, the induced map $\mathring{D^{n}}\longrightarrow f(\mathring{D^{n}})$ is a quotient map which is injective. Therefore the induced map $\mathring{D^{n}}\longrightarrow f(\mathring{D^{n}})$ is a homeomorphism. It follows that $f(\mathring{D^{n}})$ is an open cell in $X$ with characteristic map $f$. ∎

Lemma 2.6**.**

Let $X,Y$ be two Hausdorff spaces, $f:X\longrightarrow Y$ a continuous map and $e$ an open cell in $X$. Assume that

1. (1)

$f_{|e}$* is injective.* 2. (2)

$f(e)\cap f(\partial e)=\emptyset$.

Then $f(e)$ is an open cell in $Y$, $\dim f(e)=\dim e$, $\overline{f(e)}=f(\bar{e})$ and $\partial f(e)=f(\partial e)$.

Proof.

Immediate consequence of Lemma 2.5 and Lemma 2.2. ∎

Definition 2.7**.**

A CW-complex is a Hausdorff space $X$ together with a partition of $X$ into open cells such that

1. (1)

For each open cell $e$ of of the partition of $X$, $\partial e$ is contained in the union of finitely many open cells (of the partition of $X$) of lower dimensions. 2. (2)

A subset $C$ of $X$ such that $\overline{e}\cap C$ is closed for every open cell $e$ of the partition of $X$ is itself closed.

Lemma 2.8**.**

Let $X$ be a Hausdorff space space such that

1. (1)

$X$* has a finite partition into open cells.* 2. (2)

If $e$ is an open cell in the partition of $X$ and $x\in\partial e$, then the unique open cell $e^{{}^{\prime}}$ containing $x$ is of lower dimension than $e$.

Then $X$ is a CW-complex.

Proof.

Immediate consequence of the previous definition. ∎

A CW-complex with finitely many open cells is said to be finite; a CW-complex is finite iff it is compact.

Lemma 2.9**.**

Let $X_{1},X_{2},\dots,X_{q}$ be Hausdorff topological spaces and $e_{1},e_{2},\dots,e_{q}$ open cells in $X_{1},X_{2},\dots,X_{q}$ respectively, then $e_{1}\times e_{2}\times\dots e_{q}$ is an open cell in $X_{1}\times X_{2}\dots\times X_{q}$ with boundary

[TABLE]

and

[TABLE]

Proof.

By induction and is straightforward. ∎

3. Stiefel and Grassmann manifolds

Throughout this paper, $K$ is assumed to be either the field $\mathbb{R}$ of real numbers, the field $\mathbb{C}$ of complex numbers or the skew field $\mathbb{H}$ of quaternions. All vector spaces in this paper are vector spaces over $K$.

An inner product on a vector space $H$ is a map $\langle\cdot,\cdot\rangle$

[TABLE]

that satisfies the following axioms:

For all $x,y,z\in H$ and $\lambda\in K$

1. (1)

linearity in the first argument:

[TABLE] 2. (2)

Conjugate symmetry:

[TABLE] 3. (3)

Positive definite:

[TABLE]

A vector space on which there is defined an inner product is called an inner product vector space. An inner product vector space $H$ has a norm $|\!|{\>}|\!|$ associated to its inner product given by $|\!|{x}|\!|=\langle x,x\rangle^{\frac{1}{2}}$, $x\in H$. Given two finite dimensional inner product vector spaces $M$ and $N$, then any linear map $M\stackrel{{\scriptstyle u}}{{\longrightarrow}}N$ is continuous and the vector space $L(M,N)$ of linear maps has a norm $|\!|{\>}|\!|$ defined by

[TABLE]

A linear map $M\stackrel{{\scriptstyle u}}{{\longrightarrow}}N$ is said to be inner product preserving if

[TABLE]

Such a map is a unit vector in the normed vector space $L(M,N)$. Define the Stiefel manifold $V(M,N)$ to be the topological subspace of $L(M,N)$ of inner product preserving linear maps from $M$ to $N$. $V(M,N)$ is a closed subset of the unit sphere of the finite dimensional normed vector space $L(M,N)$. It is therefore a compact metric space with distance $d$ given by

[TABLE]

Observe that if $P$ is another inner product vector space, then the composition induces a continuous map

[TABLE]

so that the finite dimensional inner product vector spaces form a topological category $V$, i.e. a category enriched over the cartesian monoidal category of topological spaces. Define an equivalence relation $\sim$ on the topological space $V(M,N)$ by $u\sim v$ if $u$ and $v$ have the same image.

Let $E$ be the vector space with basis $(e_{n})_{n\in\mathbb{N}^{*}}$. A vector $x\in E$ may be written as $x=\sum_{n=1}^{\infty}x_{n}e_{n}$ with $x_{n}=0$ for $n$ sufficiently large. Define an inner product on $E$ by

[TABLE]

Let $E_{0}$ be the zero subspace of $E$ and $E_{n}$ the inner product $K$-subspace of $E$ generated by $(e_{k})_{1\leq k\leq n}$, $n\in\mathbb{N}^{*}$. For $n\in\mathbb{N}$ and $H\in V$, let $V(n,H)=V(E_{n},H)$ and define $G(n,H)$ to be the quotient space of $V(n,H)$ by the equivalence relation $\sim$ defined above. $V(n,H)$ and $G(n,H)$ are the ordinary Stiefel manifold and Grassmann manifold over $K$. Let

[TABLE]

be the quotient map. Observe that any subspace of $H$ having dimension $n$ is the image of certain inner product preserving linear map from $E_{n}$ to $H$ and any two such maps are equivalent. We may therefore view $G(n,H)$ as the set of $n$-dimensional subspaces of $H$ and $p(n,H)$ as the map that takes a linear map $u\in V(n,H)$ to its image and think of $G(n,H)$ as having the final topology induced by the map $p(n,H)$. For $n,m\in\mathbb{N}$, define $V(n,m)=V(n,E_{m})$, $G(n,m)=G(n,E_{m})$ and $p(n,m)=p(n,E_{m})$. The next proposition is a classical result which gives a sufficient condition for the quotient space to be Hausdorff [5, Proposition 1.6 page 140].

Proposition 3.1**.**

Let $\mathscr{R}$ be an equivalence relation on a topological space $X$ and assume that

1. (1)

The quotient map $p:X\to X/\mathscr{R}$ is open . 2. (2)

The graph $\Gamma$ of the equivalence relation $\mathscr{R}$ is closed in $X\times X$.

Then the quotient space $X/\mathscr{R}$ is Hausdorff.

Proposition 3.2**.**

The space $G(n,H)$ is compact Hausdorff, $n\in\mathbb{N}$ and $H\in V$.

Proof.

$G(n,H)$ is the image by the continuous map $p(n,H)$ of the compact space $V(n,H)$, thus $G(n,H)$ is compact. Let $O$ be open in $V(n,H)$.

[TABLE]

is open, therefore the quotient map $p(n,H)$ is open. Let $\Gamma$ be the graph of the equivalence relation $\sim$. $\Gamma$ is the image of the following continuous map

[TABLE]

$V(n,H)\times V(n,n)$ is compact, therefore $\Gamma$ is a compact subset of the Hausdorff space $V(n,H)\times V(n,H)$. It follows that $\Gamma$ is closed. By Proposition 3.1, $G(n,H)$ is Hausdorff. ∎

4. The cell structure for Grassmann manifolds

In this section, we show that the method of Milnor and Stasheff for the construction of the cell structure for real Grassman manifolds [9, Chapter 6] extends easily to the complex and quaternionic cases.

Define an elementary Schubert symbol of type $(n,m)$ to be a strictly increasing function $\{1,2,\dots,n\}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}\{1,2,\dots,m\}$. The set of elementary Schubert symbols of type $(n,m)$ is denoted by $S(n,m)$. There is an order relation on $S(n,m)$ given by

[TABLE]

Define a dimension function

[TABLE]

Let $\sigma,\tau\in S(n,m)$ with $\sigma<\tau$, then $d(\sigma)<d(\tau)$ so that $d$ is a strictly increasing function.

Let $H$ be finite dimentional inner product vector space of dimension $m$, $(h_{k})_{1\leq k\leq m}$ an orthonormal basis of $H$, $H_{k}$ the subspace of $H$ generated by $(h_{i})_{1\leq i\leq k}$, $1\leq k\leq m$. Define

[TABLE]

and $V_{\sigma}(n,m)=V_{\sigma}(n,E_{m})$. The closure $\overline{V_{\sigma}(n,H)}$ is given by

[TABLE]

Let $\dim_{\mathbb{R}}K$ be the dimension of $K$ as an $\mathbb{R}$-vector space.

Lemma 4.1**.**

$V_{\sigma}(n,H)$* is an open cell in $V(n,H)$ of dimension $\dim_{\mathbb{R}}K.d(\sigma)$, $\sigma\in S(n,m)$.*

Proof.

First, observe that the inner product preserving map $f:E_{m}\longrightarrow H$ given $f(e_{k})=h_{k}$ is a isomorphism in the category $V$, therefore it induces a homeomorphism $V(n,m)\longrightarrow V(n,H)$ which maps $V_{\sigma}(n,m)$ onto $V_{\sigma}(n,H)$, we may then assume without loss of generalities that $H=E_{m}$. In [9, Lemma 6.2.], it is shown that for $K=\mathbb{R}$, $V_{\sigma}(n,m)$ is an open cell in $V(n,m)$ of dimension $d(\sigma)$. The method used there extends readily to the other cases of $K$. The only minor modification is due to the fact that the inner product is not symmetric when $K=\mathbb{C}$ or $K=\mathbb{H}$ and the formula of the rotation $T(u,v)$ defined in [9, page 77] should be

[TABLE]

where $u$ and $v$ are two unit vectors in $E_{m}$ with $\langle u,v\rangle>0$; ($\langle u,v\rangle$ is a positive real number). ∎

Definition 4.2**.**

Let $W$ be a finite dimensional vector space of dimension $m$.

1. (1)

A flag in $W$ is a strictly increasing sequence $0=W_{0}\subset W_{1}\subset W_{2}\subset\dots\subset W_{q}=W$ of subspaces of $W$. If we write $d_{k}=\dim W_{k}$, then $0=d_{0}\prec d_{1}\prec d_{2}\prec\dots\prec d_{q}=m$. The sequence $(d_{0},d_{1},\dots,d_{q})$ is called the signature of the flag $(W_{k})_{0\leq k\leq q}$. 2. (2)

A flag $(W_{k})_{0\leq k\leq q}$ in $W$ is said to be complete if $q=m$, in which case $\dim W_{k}=k$, $0\leq k\leq m$ and the flag has signature $(0,1,\dots,m)$. 3. (3)

A vector space that is assigned a fixed flag (resp. a fixed complete flag) is said to be flagged (resp. completely flagged).

Let $W$ be a completely flagged finite dimensional vector space with dimension $m$. $(W_{k})_{0\leq k\leq m}$ its fixed complete flag and $X$ an $n$-dimensional vector subspace of $W$. Define the Schubert function of $X$ to be the function

[TABLE]

Then

• •

$0=\omega_{X}^{W}(0)\leq\omega_{X}^{W}(1)\leq\dots\leq\omega_{X}^{W}(m)=n$, so that $\omega_{X}^{W}$ is increasing.

• •

$\omega_{X}^{W}(k+1)\leq\omega_{X}^{W}(k)+1,\quad 0\leq k\leq m-1$.

• •

$\omega_{X}^{W}$is surjective.

Define the Schubert symbol of $X$ to be the elementary Schubert symbol

[TABLE]

$\sigma_{X}^{W}$ is characterized by the following property

[TABLE]

In particular, if we set $X_{0}=0$ and $X_{k}=X\cap W_{\sigma_{X}^{W}(k)}$, $1\leq k\leq n$, then

[TABLE]

is a complete flag in $X$ called the induced complete flag, so that a subspace of a completely flagged finite dimensional vector space is naturally a completely flagged vector space.

Lemma 4.3**.**

Let $Z$ be a finite dimensional completely flagged vector space, $Y$ a subspace of $Z$ and $X$ a nonzero subspace of $Y$. Then with the above notation, we have

1. (1)

$\omega_{X}^{Z}=\omega_{X}^{Y}\omega_{Y}^{Z}.$** 2. (2)

$\sigma_{X}^{Z}=\sigma_{Y}^{Z}\sigma_{X}^{Y}.$**

Proof.

Let $p=\dim X$, $q=\dim Y$ and $r=\dim Z$.

1. (1)

For $k\in\{1,2,\dots,r\}$

[TABLE]

We then have $\omega_{X}^{Z}=\omega_{X}^{Y}\omega_{Y}^{Z}.$ 2. (2)

For $k\in\{1,2,\dots,p\}$

[TABLE]

We then have $\sigma_{X}^{Z}=\sigma_{Y}^{Z}\sigma_{X}^{Y}.$

∎

Let $H$ be a finite dimensional vector space with orthonormal basis $(h_{k})_{1\leq k\leq m}$, $H_{0}=0$ and $H_{k}$ the subspace of $H$ generated by $(h_{i})_{1\leq i\leq k}$, $1\leq k\leq m$. $(H_{k})_{0\leq k\leq m}$ is a complete flag in $H$. Define

[TABLE]

The following lemma is clearly an immediate consequence of this last definition.

Lemma 4.4**.**

The family $(G_{\sigma}(n,H))_{\sigma\in S(n,m)}$ is a partition of $G(n,H)$.

Lemma 4.5**.**

Let $\sigma\in S(n,m)$, then

1. (1)

$p(n,H)$* induces a bijection $p(n,H)_{/}:V_{\sigma}(n,H)\longrightarrow G_{\sigma}(n,H)$.* 2. (2)

Let $v\in\partial V_{\sigma}(n,H)$ and $Y=p(n,H)(v)$, then $\sigma_{Y}\prec\sigma$ where $\sigma_{Y}$ is the Schubert symbol of $Y$. In particular

[TABLE]

Proof.

1. (1)

• •

Let $u\in V_{\sigma}(n,H)$, $X=p(n,H)(u)$ and $k\succ 0.$

$X\cap H_{\sigma(k)}\subset\mbox{Im }u$ and $u$ is injective. Thus

[TABLE]

while

[TABLE]

It follows that $\sigma$ is the Schubert symbol of $X$ and

[TABLE]

• •

Let $X\in G_{\sigma}(n,H)$ and assume that $u\in V_{\sigma}(n,H)$ is such that

$p(n,H)(u)=X$. Let $k\in\{1,2,\dots,n\}$

[TABLE]

[TABLE]

It follows that $u(e_{k})$ is the unique unit vector $x_{k}$ in $(X\cap H_{\sigma(k)})\cap(X\cap H_{\sigma(k)-1})^{\bot}$ such that $\langle x_{k},h_{\sigma(k)}\rangle\succ 0$. This shows that the map

[TABLE]

is injective. To show that it is surjective, we only need to construct an antecedent $u$ for a given element $X\in G_{\sigma}(n,H)$ as suggested by the previous argument. 2. (2)

• •

Recall that

[TABLE]

[TABLE]

$v\in\partial V_{\sigma}(n,H)=\overline{V_{\sigma}(n,H)}-V_{\sigma}(n,H)$. Therefore

[TABLE]

It follows that

[TABLE]

and

[TABLE]

by (7)

[TABLE]

i.e. $\sigma_{Y}<\sigma$

• •

If $X\in p(n,H)(V_{\sigma}(n,H))$, then $\sigma_{X}=\sigma>\sigma_{Y}$. Therefore $X\neq Y$ and $p(n,H)(V_{\sigma}(n,H))\cap p(n,H)(\partial V_{\sigma}(n,H))=\emptyset$.

∎

Lemma 4.6**.**

Let $\sigma\in S(n,m)$, then

1. (1)

$G_{\sigma}(n,H)$* is an open cell in $G(n,H)$ with dimension $\dim_{\mathbb{R}}K.d(\sigma)$.* 2. (2)

$\partial G_{\sigma}(n,H)=p(n,H)(\partial V_{\sigma}(n,H))$. 3. (3)

Let $Y\in\partial G_{\sigma}(n,H)$ and $\sigma_{Y}$ the Schubert symbol of $Y$, then $\sigma_{Y}\prec\sigma$. In particular $d(\sigma_{Y})\prec d(\sigma)$.

Proof.

Immediate consequence of Lemma 2.6, Lemma 4.1 and lemma 4.5. ∎

To sumarize, one has the following

Theorem 4.7**.**

$G(n,H)$* has a finite CW-complex structure with open cells $G_{\sigma}(n,H)$, $\sigma\in S(n,m).$*

Proof.

Immediate consequence of Lemma 2.8, Lemma 4.4 and lemma 4.6.

∎

Corollary 4.8**.**

$G(n,H)$* is a topological manifold of dimension $n(m-n).\dim_{\mathbb{R}}K$.*

The proof is similar but not identical to the one given in [8, page 31].

Proof.

$G(n,H)$ is Hausdorff. Let $X_{0}\in G(n,H)$ and choose $(h_{1},h_{2},\dots,h_{m})$ to be a basis of $H$ such that $(h_{m-n+1},\dots,h_{m-1},h_{m})$ is a basis of $X_{0}$. Define $u_{0}\in V(n,H)$ by $u_{0}(e_{k})=h_{m-n+k},1\leq k\leq n$. $p(n,H)(u_{0})=X_{0}$, $u_{0}\in V_{\sigma}(n,H)$ where $\sigma\in S(n,m)$ is given by $\sigma(k)=m-n+k$. By Lemma 4.5, $X_{0}\in G_{\sigma}(n,H)$. $G_{\sigma}(n,H)$ is an open cell in $G(n,H)$ of top dimension, thus $G_{\sigma}(n,H)$ is an open neighborhood of $X_{0}$.

[TABLE]

It follows that $G(n,H)$ is a topological manifold of dimension $\dim_{\mathbb{R}}K.n(m-n)$. ∎

5. Schubert cells in a generalized Stiefel manifold

Let $H$ be a finite dimensional inner product vector space of dimension $n$. $(n_{k})_{1\leq k\leq q}$ a sequence of integers such that $1\leq n_{1}\prec n_{2}\dots\prec n_{q}\prec n$. Define the generalized Stiefel manifold $V(n_{1},\dots,n_{q},H)$ to be the subspace of $\prod^{q}_{k=1}V(n_{k},H)$ given by

[TABLE]

$V(n_{1},\dots,n_{q},H)$ is a subspace of a Haudorff space, it is then Hausdorff. Let

[TABLE]

$\psi(n_{1},\dots,n_{q},H)$ is continuous injective map with image $V(n_{1},\dots,n_{q},H)$.

$\prod^{q-1}_{k=1}V(n_{k},n_{k+1})\times V(n_{q},H)$ is compact, therefore $V(n_{1},\dots,n_{q},H)$ is compact Hausdorff and $\psi(n_{1},\dots,n_{q},H)$ induces a homeomorphism of

$\prod^{q-1}_{k=1}V(n_{k},n_{k+1})\times V(n_{q},H)$ onto $V(n_{1},\dots,n_{q},H)$.

Define

[TABLE]

An element of $S(n_{1},n_{2},\dots,n_{q},n)$ is called a general Schubert symbol of type $(n_{1},n_{2},\dots,n_{q},n)$. It is then a finite sequence $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})$ of elementary Schubert symbols. The order relation $\leq$ on the elementary Schubert symbols defined by (2) induces an order relation $\leq$ on $S(n_{1},n_{2},\dots,n_{q},n)$ given by

[TABLE]

Let

[TABLE]

then

[TABLE]

If $\sigma<\tau$, then $d(\sigma)<d(\tau)$ so that $d$ is a strictly increasing function.

Lemma 5.1**.**

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$ then

[TABLE]

Proof.

By Induction, for $q=2$

[TABLE]

A similar argument shows that the formula is true for $q+1$ if it holds for $q$. ∎

Let $(h_{i})_{1\leq i\leq n}$ an orthonormal basis of $H$. For $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$, define $V_{\sigma}(n_{1},\dots,n_{q},H)$ to be the subset of $V(n_{1},\dots,n_{q},H)$ given by

[TABLE]

Lemma 5.2**.**

*Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(\sigma_{1},\sigma_{2},\dots,\sigma_{q},n)$, then $V_{\sigma}(n_{1},\dots,n_{q},H)$ is an open cell in $V(n_{1},\dots,n_{q},H)$ of dimension $\dim_{\mathbb{R}}K.d(\sigma)$ and boundary

$\partial V_{\sigma}(n_{1},\dots,n_{q},H)$ is equal to*

[TABLE]

Proof.

Immediate consequence of Lemma 2.9 and the fact that

[TABLE]

is a homeomorphism. ∎

Lemma 5.3**.**

Let $m_{1},m_{2},...,m_{r},m$ be positive integers with $m_{1}\prec m_{2}\prec...\prec m_{r}\prec m$, $F$ an $m$-dimensional inner product vector space, $(f_{i})_{1\leq i\leq m}$ an orthonormal basis of $F$, $\sigma=(\sigma_{1},\sigma_{2},...,\sigma_{r})\in S(m_{1},m_{2},...,m_{r},m)$ and

[TABLE]

the composition map. Then

1. (1)

$\phi(m_{1},\dots,m_{r},F)(\prod^{r-1}_{k=1}V_{\sigma_{k}}(m_{k},m_{k+1})\times V_{\sigma_{r}}(m_{r},F))\subset V_{\sigma_{r}\dots\sigma_{2}\sigma_{1}}(m_{1},F).$** 2. (2)

$\phi(m_{1},\dots,m_{r},F)(\partial V_{\sigma_{1}}(m_{1},m_{2})\times\prod^{r-1}_{k=2}V_{\sigma_{k}}(m_{k},m_{k+1})\times V_{\sigma_{r}}(m_{r},F))\\ \subset\partial V_{\sigma_{r}\dots\sigma_{2}\sigma_{1}}(m_{1},F).$**

Proof.

The first assertion is proved by induction on $r$ and is straightforward. Similarly, the second assertion is proved by induction on $r$. It uses the first assertion and (4). ∎

Lemma 5.4**.**

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$, then

1. (1)

$V_{\sigma}(n_{1},n_{2},\dots,n_{q},H)=V(n_{1},n_{2},\dots,n_{q},H)\cap\prod^{q}_{k=1}V_{\sigma_{q}\dots\sigma_{k+1}\sigma_{k}}(n_{k},H).$** 2. (2)

$\partial V_{\sigma}(n_{1},n_{2},\dots,n_{q},H)\subset\cup_{k=1}^{q}V(n_{1},H)\times\dots\times V(n_{k-1},H)\times\\ \partial V_{\sigma_{q}\dots\sigma_{k+1}\sigma_{k}}(n_{k},H)\times V(n_{k+1},H)\times\dots\times V(n_{q},H).$**

Proof.

1. (1)

By Lemma 5.3

[TABLE]

We prove by induction on $q$ that

[TABLE]

For $q=2$, let $(v_{1},v_{2})\in V(n_{1},n_{2},H)\cap V_{\sigma_{2}\sigma_{1}}(n_{1},H)\times V_{\sigma_{2}}(n_{2},H)$, there exists $(u_{1},u_{2})\in V(n_{1},n_{2})\times V(n_{2},H)$ such that $(v_{1},v_{2})=(u_{2}u_{1},u_{2})$. $u_{2}=v_{2}\in V_{\sigma_{2}}(n_{2},H)$. To show that $u_{1}\in V_{\sigma_{1}}(n_{1},n_{2})$, let $k\in\{1,2,\dots,n_{1}\}$, $u_{2}u_{1}(E_{k})=v_{1}(E_{k})\subset H_{\sigma_{2}\sigma_{1}(k)}$, thus $u_{1}(E_{k})\subset u_{2}^{-1}(H_{\sigma_{2}\sigma_{1}(k)})\subset E_{\sigma_{1}(k)}.$ Write $u_{1}(e_{k})=\lambda e_{\sigma_{1}(k)}+x$ where $x\in E_{\sigma_{1}(k)-1}$.

[TABLE]

In the other hand

[TABLE]

or $\langle u_{2}(e_{\sigma_{1}(k)}),e_{\sigma_{2}\sigma_{1}(k)}\rangle\succ 0$ therefore $\lambda\succ 0$, i.e. $\langle u_{1}(e_{k}),e_{\sigma_{1}(k)}\rangle\succ 0$, $1\leq k\leq n_{1}$. Thus $u_{1}\in V_{\sigma_{1}}(n_{1},n_{2})$ and $(v_{1},v_{2})\in V_{(\sigma_{1},\sigma_{2})}(n_{1},n_{2},H)$. Assume that the property is true for $q-1$, $q\geq 3$ and let

[TABLE]

for some $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$. Let

[TABLE]

be such that $v_{k}=u_{q}\dots u_{k+1}u_{k}$, $1\leq k\leq q.$

[TABLE]

By the induction assumption

[TABLE]

i.e.

[TABLE]

Furthermore, $(v_{1},v_{2})\in V(n_{1},n_{2},H)\cap V_{\sigma_{q}\dots\sigma_{2}\sigma_{1}}(n_{1},H)\times V_{\sigma_{q}\dots\sigma_{3}\sigma_{2}}(n_{2},H)$. By the induction assumption

[TABLE]

i.e.

[TABLE]

It follows that

[TABLE]

and

[TABLE]

as desired. 2. (2)

Let $(v_{1},v_{2},\dots,v_{q})\in\partial V_{\sigma}(n_{1},n_{2},\dots,n_{q},H)$. By Lemma 5.2, there exists $k\in\{1,2,...,q\}$ and

[TABLE]

such that

[TABLE]

Define

[TABLE]

Then $(u_{l},u_{l+1},\dots,u_{q})\in\partial V_{\sigma_{l}}(n_{l},n_{l+1})\times V_{\sigma_{l+1}(n_{l+1},n_{l+2}})\times\dots\times V_{\sigma_{q}}(n_{q},H)$. By Lemma 5.3. $v_{l}=u_{q}\dots u_{l+1}u_{l}\in\partial V_{\sigma_{q}\dots\sigma_{l+1}\sigma_{l}}(n_{l},H)$, therefore

$(v_{1},\dots,v_{l},\dots,v_{q})$ is in

[TABLE]

as desired.

∎

6. The cell structure for flag manifolds

Let $H$ be a finite dimensional inner product vector space of dimension $n$. $(n_{k})_{1\leq k\leq q}$ a sequence of integers such that $1\leq n_{1}\prec n_{2}\dots\prec n_{q}\prec n$. Define the flag manifold $G(n_{1},\dots,n_{q},H)$ to be the subspace of $\prod^{q}_{k=1}G(n_{k},H)$ given by

[TABLE]

$G(n_{1},\dots,n_{q},H)$ is a subspace of a Haudorff space, it is then Hausdorff. Let

[TABLE]

be the product of the quotient maps defined by (1). Then

[TABLE]

$V(n_{1},\dots,n_{q},H)$ is compact, then so is $G(n_{1},\dots,n_{q},H)$. Therefore the map $\prod^{q}_{k=1}p(n_{k},H)$ induces a quotient map

[TABLE]

Let $(h_{i})_{1\leq i\leq n}$ be an orthonormal basis of $H$, $H_{0}$ the zero subspace of $H$ and $H_{k}$ the subspace spanned by $(h_{i})_{1\leq i\leq k}$, $1\leq k\leq n$. $(H_{k})_{0\leq k\leq n}$ is a complete flag in $H$. Let $(X^{1},X^{2},\dots,X^{q})\in G(n_{1},\dots,n_{q},H)$, as explained in (8), the complete flag $(H_{k})_{0\leq k\leq n}$ in $H$ induces complete flags

[TABLE]

$X^{q}\in G(n_{q},H)$, let $\sigma_{X^{q}}$ be its Schubert symbol as defined in (6). Similarly for $1\leq i\prec q$, $X^{i}\in G(n_{i},X^{i+1})$, let $\sigma_{X^{i}}$ be its Schubert symbol and define the Schubert symbol $\sigma_{(X^{1},X^{2},\dots,X^{q})}$ of the flag $(X^{1},X^{2},\dots,X^{q})\in G(n_{1},\dots,n_{q},H)$ to be the general Schubert symbol

[TABLE]

For $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$, define

[TABLE]

The following lemma is an immediate consequence of the previous definition.

Lemma 6.1**.**

The subsets $G_{\sigma}(n_{1},\dots,n_{q},H)$, $\sigma\in S(n_{1},n_{2},\dots,n_{q},n)$ form a partition of $G(n_{1},\dots,n_{q},H)$.

Lemma 6.2**.**

Let $\sigma=(\sigma_{1},\sigma_{2},...,\sigma_{q})\in S(n_{1},n_{2},...,n_{q},n)$, then

1. (1)

$p(n_{1},n_{2},...,n_{q},H)_{/}:V_{\sigma}(n_{1},n_{2},...,n_{q},H)\to G(n_{1},n_{2},...,n_{q},H)$* is injective.* 2. (2)

$p(n_{1},n_{2},...,n_{q},H)(V_{\sigma}(n_{1},n_{2},...,n_{q},H))\\ \cap p(n_{1},n_{2},...,n_{q},H)(\partial V_{\sigma}(n_{1},n_{2},...,n_{q},H))=\emptyset.$**

Proof.

The first claim is a consequence of the first assertion of lemma 4.5 and the first assertion of Lemma 5.4 while the second claim is a consequence of the second assertions of the same Lemmas. ∎

Lemma 6.3**.**

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$ then

[TABLE]

Proof.

By Lemma 4.3

[TABLE]

therefore by Lemma 6.1, the subsets

[TABLE]

cover $G(n_{1},n_{2},\dots,n_{q},H)$. By lemma 4.4 the subsets

[TABLE]

are disjoint, therefore

[TABLE]

∎

Lemma 6.4**.**

Let $\sigma=(\sigma_{1},\sigma_{2},...,\sigma_{q})\in S(n_{1},n_{2},...,n_{q},n)$, then

[TABLE]

Proof.

By Lemma 5.4

[TABLE]

thus

[TABLE]

Let $(X^{1},X^{2},\dots,X^{q})\in G_{\sigma}(n_{1},\dots,n_{q},H)$. By Lemma 6.3, $(X^{1},X^{2},\dots,X^{q})$ is in $G(n_{1},n_{2},\dots,n_{q},H)\cap\prod^{q}_{k=1}G_{\sigma_{q}\dots\sigma_{k+1}\sigma_{k}}(n_{k},H)$ and by Lemma 4.5, there exists

[TABLE]

such that

[TABLE]

$(X^{1},X^{2},\dots,X^{q})\in G(n_{1},\dots,n_{q},H)$, thus $X^{1}\subset X^{2}\subset\dots\subset X^{q}$. By (9)

[TABLE]

thus

[TABLE]

By Lemma 5.4, $(v_{1},v_{2},\dots,v_{q})\in V_{\sigma}(n_{1},n_{2},...,n_{q},H)$ and

[TABLE]

∎

Lemma 6.5**.**

Let $\sigma\in S(n_{1},n_{2},\dots,n_{q},H)$, then $G_{\sigma}(n_{1},\dots,n_{q},H)$ is an open cell in $G(n_{1},\dots,n_{q},H)$ of dimension $\dim_{\mathbb{R}}K.d(\sigma)$. Furthermore

[TABLE]

Proof.

Immediate consequence of Lemma 6.2, Lemma 6.4 and lemma 2.5. ∎

Lemma 6.6**.**

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$, $Y=(Y_{1},Y_{2},\dots,Y_{q})\in\partial G_{\sigma}(n_{1},n_{2},\dots,n_{q},H)$ and $\tau=(\tau_{1},\tau_{2},\dots,\tau_{q})$ the Schubert symbol of $(Y_{1},Y_{2},\dots,Y_{q})$ then

1. (1)

$\tau_{q}\dots\tau_{k+1}\tau_{k}\leq\sigma_{q}\dots\sigma_{k+1}\sigma_{k},1\leq k\leq q$. 2. (2)

$\tau_{q}\dots\tau_{k_{0}+1}\tau_{k_{0}}<\sigma_{q}\dots\sigma_{k_{0}+1}\sigma_{k_{0}}$* for some $k_{0}\in\{1,2,\dots,q\}$.* 3. (3)

$d(\tau)<d(\sigma)$.

Proof.

1. (1)

[TABLE]

thus $Y_{k}\in\overline{G_{\sigma_{q}\dots\sigma_{k+1}\sigma_{k}}(n_{k},H)},1\leq k\leq q.$

[TABLE]

It follows that

[TABLE]

By Lemma 4.6, $\tau_{q}\dots\tau_{k+1}\tau_{k}\leq\sigma_{q}\dots\sigma_{k+1}\sigma_{k}$, $1\leq k\leq q$. 2. (2)

$Y\in\partial G_{\sigma}(n_{1},n_{2},\dots,n_{q},H)$, thus $Y\notin G_{\sigma}(n_{1},n_{2},\dots,n_{q},H)$. There exists $k_{0}\in\{1,\dots,q\}$ such that $Y_{k_{0}}\notin G_{\sigma_{q}\dots\sigma_{k_{0}+1}\sigma_{k_{0}}}(n_{k_{0}},H),$ thus $Y_{k_{0}}\in\partial G_{\sigma_{q}\dots\sigma_{k_{0}+1}\sigma_{k_{0}}}(n_{k_{0}},H)$. By Lemma 4.5 $\tau_{q}\dots\tau_{k_{0}+1}\tau_{k_{0}}<\sigma_{q}\dots\sigma_{k_{0}+1}\sigma_{k_{0}}$. 3. (3)

The proof of the fact that $d(\tau)<d(\sigma)$ is by induction on $q$ and uses the previous two properties.

∎

Theorem 6.7**.**

There is a CW-complex structure on $G(n_{1},n_{2},\dots,n_{q},H)$ with open cells $G_{\sigma}(n_{1},n_{2},\dots,n_{q},H),\sigma\in S(n_{1},n_{2},\dots,n_{q},H)$.

Proof.

immediate consequence of lemmas 2.8, 6.1, 6.5 and 6.6. ∎

Corollary 6.8**.**

$G(n_{1},n_{2},\dots,n_{q},H)$* is a topological manifold of dimension*

[TABLE]

Proof.

$G(n_{1},n_{2},\dots,n_{q},H)$ is a Hausdorff. Let $X_{0}=(X_{0}^{1},X_{0}^{2},\dots,X_{0}^{q})\in G(n_{1},n_{2},\dots,n_{q},H)$ and construct by induction an orthonormal basis for $H$ such that $(h_{n-n_{k}+1},\dots,h_{n-1},h_{n})$ is a basis for $X_{0}^{k}$, $1\leq k\leq q$. Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{q})\in S(n_{1},n_{2},\dots,n_{q},n)$ be the general Schubert symbol given by

[TABLE]

It is easily checked that $X_{0}\in G_{\sigma}(n_{1},n_{2},\dots,n_{q},H).$

$G_{\sigma}(n_{1},n_{2},\dots,n_{q},H)$ is an open cell in $G(n_{1},n_{2},\dots,n_{q},H)$ of top dimension. It is therefore an open neighborhood of $X_{0}$ which is homeomorphic to an Euclidean space. $G(n_{1},n_{2},\dots,n_{q},H)$ is a topological manifold of dimension

[TABLE]

∎

Remark 6.9**.**

1. (1)

For $K=\mathbb{C}$ or $K=\mathbb{H}$, all cells in $G(n_{1},n_{2},\dots,n_{q},H)$ are even dimensional. Consequently, they are all cycles and they form a basis for the integral homology groups of $G(n_{1},n_{2},\dots,n_{q},H)$. 2. (2)

Let $H_{\mathbb{C}}$ (resp. $H_{\mathbb{H}}$) be an $n$-dimensional $\mathbb{C}$ vector space (resp. $H_{\mathbb{H}}$ vector space). There is a one-to-one correspondence between cells of $G(n_{1},n_{2},\dots,n_{q},H_{\mathbb{C}})$ of dimension $2i$ and cells of $G(n_{1},n_{2},\dots,n_{q},H_{\mathbb{H}})$ of dimension $4i$. In particular $P_{\mathbb{H}}(t)=P_{\mathbb{C}}(t^{2})$ where $P_{\mathbb{C}}$ (resp. $P_{\mathbb{H}}$) is the Poincaré polynomial of $G(n_{1},n_{2},\dots,n_{q},H_{\mathbb{C}})$ (resp. $G(n_{1},n_{2},\dots,n_{q},H_{\mathbb{H}})$. It follows that $P_{\mathbb{H}}(1)=P_{\mathbb{C}}(1)$ and the two manifolds $G(n_{1},n_{2},\dots,n_{q},H_{\mathbb{C}})$ and $G(n_{1},n_{2},\dots,n_{q},H_{\mathbb{H}})$ have the same Euler characteristic.