---
abstract: 'This note proves that, for $F = \Bbb{R,C}$ or $\Bbb{H}$, the bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with $k < n$ and having real dimension $d$, constitute a linearly independent set in the unoriented bordism group ${\frak{N}}_d$ regarded as a ${\Bbb{Z}} _2$-vector space.'
address: |
    Department of Mathematics, North-Eastern Hill University, Permanent Campus, Shillong 793 022, India\
    E-mail: akdas@nehu.ac.in
author:
- ASHISH KUMAR DAS
title: Cobordism independence of Grassmann manifolds
---

=msam10 at 10pt

\[theore\][**Theorem**]{} \[theore\][PROPOSITION]{} \[theore\][Lemma]{} \[theore\][DEFINITION]{} \[theore\][COROLLARY]{} \[theore\][Remark]{} \[theore\][Example]{} \[theore\][Fact]{}

Introduction {#S:intro}
============

This paper is a continuation of the ongoing study of cobordism of Grassmann manifolds. Let $F$ denote one of the division rings $\Bbb{R}$ of reals, $\Bbb{C}$ of complex numbers, or $\Bbb{H}$ of quaternions. Let $t = \rm{dim}_{\Bbb{R}}$$F$. Then the Grassmannian manifold $G_k(F^{n+k})$ is defined to be the set of all $k$-dimensional (left) subspaces of $F^{n+k}$. $G_k(F^{n+k})$ is a closed manifold of real dimension $nkt$. Using the orthogonal complement of a subspace one identifies $G_k(F^{n+k})$ with $G_n(F^{n+k}).$

In [@pS91], Sankaran has proved that, for $F = \Bbb{R,C}$ or $\Bbb{H},$ the Grassmannian manifold $G_k(F^{n+k})$ bounds if and only if $\nu(n+k) > \nu(k),$ where, given a positive integer $m$, $\nu(m)$ denotes the largest integer such that $2^{\nu(m)}$ divides $m.$

Given a positive integer $d$, let $\cal{G}$$(d)$ denote the set of bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$ having real dimension $d$ such that $k<n.$ The restriction $k<n$ is imposed because $G_k(F^{n+k}) \approx G_n(F^{n+k})$ and, for $k = n,$ $G_k(F^{n+k})$ bounds. Thus, ${\cal{G}}(d) = \{ [G_k(F^{n+k})] \in {\frak{N}}_* \mid nkt
= d, k < n,\  {\rm and}\  \nu(n+k) \le \nu(k) \} \subset {\frak{N}}_d.$

The purpose of this paper is to prove the following:

\[T:Gd\] $\cal{G}$$(d)$ is a linearly independent set in the ${\Bbb{Z}}
_2$-vector space ${\frak{N}}_d$.

Similar results for Dold and Milnor manifolds can be found in [@sK89] and [@sD01] respectively.

The real Grassmannians — a Brief review {#S:real}
=======================================

The real Grassmannian manifold $G_k({\Bbb{R}}^{n+k})$ is an $nk$-dimensional closed manifold of $k$-planes in ${\Bbb{R}}^{n+k}$. It is well-known (see [@hH80]) that the mod-2 cohomology of $G_k({\Bbb{R}}^{n+k})$ is given by $$H^*(G_k({\Bbb{R}}^{n+k});\ {\Bbb{Z}} _2) \cong {\Bbb{Z}} _2[w_1,
w_2,\dots , w_k, {\bar{w}}_1, {\bar{w}}_2, \dots , {\bar{w}}_n]/
\{w.\bar{w}=1\},$$ where $w = 1 + w_1 + w_2 + \dots + w_k$ and ${\bar{w}} = 1 + {\bar{w}}_1
+ {\bar{w}}_2 +\dots + {\bar{w}}_n$ are the total Stiefel–Whitney classes of the universal $k$-plane bundle ${\gamma}_k$ and the corresponding complementary bundle ${\gamma}_k^{\perp},$ both over $G_k({\Bbb{R}}^{n+k}),$ respectively.

For computational convenience in this cohomology one uses the flag manifold ${\rm Flag}({\Bbb{R}}^{n+k})$ consisting of all ordered $(n+k)$-tuples $(V_1,
V_2,\ldots, V_{n+k})$ of mutually orthogonal one-dimensional subspaces of ${\Bbb{R}}^{n+k}$ with respect to the ‘standard’ inner product on ${\Bbb{R}}^{n+k}$. It is standard (see [@fH66]) that the mod-2 cohomology of ${\rm Flag}({\Bbb{R}}^{n+k})$ is given by $$H^*({\rm Flag}({\Bbb{R}}^{n+k});\ {\Bbb{Z}} _2) \cong {\Bbb{Z}} _2[e_1,
e_2, \dots , e_{n+k}] \bigg/ \left\{ \prod_{i=1}^{n+k} (1+e_i) = 1
\right\},$$ where $e_1, e_2,\dots , e_{n+k}$ are one-dimensional classes. In fact each $e_i$ is the first Stiefel–Whitney class of the line bundle ${\lambda}_i$ over ${\rm Flag}({\Bbb{R}}^{n+k})$ whose total space consists of pairs, a flag $(V_1, V_2, \ldots, V_{n+k})$ and a vector in $V_i$.

There is a map ${\pi}_{n+k}\ :\ {\rm Flag}({\Bbb{R}}^{n+k})
\longrightarrow G_k({\Bbb{R}}^{n+k})$ which assigns to $(V_1, V_2, \ldots,
V_{n+k})$, the $k$-dimensional subspace $V_1 \oplus V_2 \oplus \dots
\oplus V_k$. In the cohomology, ${\pi}_{n+k}^*\ :\
H^*(G_k({\Bbb{R}}^{n+k});$ ${\Bbb{Z}}_2) \longrightarrow
H^*({\rm Flag}({\Bbb{R}}^{n+k});\ {\Bbb{Z}}_2)$ is injective and is described by $${\pi}_{n+k}^*(w) = \underset{i=1}{\overset{k}{\prod}}(1+e_i),\quad  
{\pi}_{n+k}^*(\bar{w}) = \underset{i=k+1}{\overset{n+k}{\prod}}(1+e_i).$$

In [@rS82], Stong has observed, among others, the following facts:

\[F:val\] [The value of the class $u \in H^*(G_k({\Bbb{R}}^{n+k});\ {\Bbb{Z}} _2)$ on the fundamental class of $G_k({\Bbb{R}}^{n+k})$ is the same as the value of $${\pi}_{n+k}^*(u)e_1^{k-1}e_2^{k-2} \ldots
e_{k-1}e_{k+1}^{n-1}e_{k+2}^{n-2} \ldots e_{n+k-1}$$ on the fundamental class of ${\rm Flag}({\Bbb{R}}^{n+k})$.]{}

\[F:for\] [In $H^*({\rm Flag}({\Bbb{R}}^{n+k});\ {\Bbb{Z}}_2)$ one has $$e_{i_1}^{n+k-(r-1)}e_{i_2}^{n+k-(r-1)} \dots
e_{i_{r-1}}^{n+k-2}e_{i_r}^{n+k-1}\ =\ 0$$ if $1 \leq r \leq n+k$ and the set $\{i_1, i_2,\dots , i_r\} \subset
\{1, 2,\dots , n+k\}.$ In particular  $e_i^{n+k}\ =\ 0$  for each $i$ , $1 \leq i \leq n+k.$]{}

\[F:top\] [In the top dimensional cohomology of ${\rm Flag}({\Bbb{R}}^{n+k}),$ a monomial $e_1^{i_1}e_2^{i_2} \dots e_{n+k}^{i_{n+k}}$ represents the non-zero class if and only if the set $\{i_1 , i_2 , \dots , i_{n+k}\} = \{0 , 1
, \dots , n+k-1\}.$]{}

The tangent bundle $\tau$ over $G_k({\Bbb{R}}^{n+k})$ is given (see [@wH64]) by $$\tau \oplus {\gamma}_k \otimes {\gamma}_k \cong (n+k){\gamma}_k.$$ In particular, the total Stifel–Whitney class $W(G_k({\Bbb{R}}^{n+k}))$ of the tangent bundle over $G_k({\Bbb{R}}^{n+k})$ maps under ${\pi}_{n+k}^*$ to $$\underset{1 \leq i \leq k}{\prod}
(1+e_i)^{n+k}\  .\   \underset{1
\leq i < j \leq k}{\prod}(1+e_i+e_j)^{-2}.$$ Choosing a positive integer $\alpha$ such that $2^{\alpha} \geq
n+k,$ we have, using Fact \[F:for\], $${\pi}_{n+k}^*(W(G_k({\Bbb{R}}^{n+k}))) = \underset{1 \leq i \leq k}{\prod}
(1+e_i)^{n+k}\  .\   \underset{1
\leq i < j \leq k}{\prod}(1+e_i+e_j)^{2^{\alpha} -2}.$$

$\left.\right.$

Thus, the $m$th Stiefel–Whitney class $W_m = W_m
(G_k({\Bbb{R}}^{n+k}))$ maps under ${\pi}_{n+k}^*$ to the $m$th elementary symmetric polynomial in $e_i$, $1 \leq i \leq k,$ each with multiplicity $n+k$, and $e_i +e_j$, $1 \leq i < j \leq k,$ each with multiplicity$2^{\alpha} -2.$ Therefore, if $S_p( {\sigma}_1,{\sigma}_2, \dots ,
{\sigma}_p)$ denotes the expression of the power sum $\sum_{m=1}^q y_m^p$ as a polynomial in elementary symmetric polynomials $\sigma_m$’s in $q$ ‘unknowns’ $y_1,y_2, \dots , y_q$, $q \geq p$, we have (see [@pS91]) $$S_p ({\pi}_{n+k}^*(W_1),{\pi}_{n+k}^*(W_2), \dots , {\pi}_{n+k}^*(W_p))
= \underset{1 \leq i \leq k}{\sum} (n+k)e_i^p.$$ Thus we have a polynomial $$S_p(G_k({\Bbb{R}}^{n+k})) = S_p (W_1,W_2, \dots , W_p) \in
H^p(G_k({\Bbb{R}}^{n+k});\ {\Bbb{Z}} _2)$$ of Stiefel–Whitney classes of $G_k({\Bbb{R}}^{n+k})$ such that $$\label{E:sym}
{\pi}_{n+k}^*(S_p(G_k({\Bbb{R}}^{n+k}))) =
\begin{cases}
\underset{1 \leq i \leq k}{\sum} e_i^p,  
&\text{if $n+k$ is odd and\ $p<n+k$}\\
0,  &\text{otherwise.} 
\end{cases}$$

Proof of Theorem 1.1 {#S:proof}
====================

It is shown in [@eF71] that $$[G_{2k}({\Bbb{R}}^{2n+2k})] = [G_k({\Bbb{R}}^{n+k})]^4 \quad \text{in}\ 
{\frak{N}}_{4nk}.$$ From this, we have, in particular, $$[G_k(F^{n+k})]=[G_k({\Bbb{R}}^{n+k})]^t \quad \text{in}\  {\frak{N}}_{nkt}.$$ For this one has to simply observe that the mod-2 cohomology of the ${\Bbb F}$-Grassmannian is isomorphic as ring to that of the corresponding real Grassmannian by an obvious isomorphism that multiplies the degree by $t$. On the other hand, since $\frak{N}_*$ is a polynomial ring over the field ${\Bbb{Z}} _2$, we have the following:

\[R:ind\][A set $\{[M_1],[M_2], \dots , [M_m]\}$ is linearly independent in ${\frak{N}}_d$ if and only if the set $\{[M_1]^{2^\beta},[M_2]^{2^\beta}, \dots , [M_m]^{2^\beta}\}$ is linearly independent in ${\frak{N}}_{d.2^\beta}$, $\beta \geq 0.$]{}

Therefore, noting that $t = 1, 2$, or $4$, it is enough to prove Theorem \[T:Gd\] for real Grassmannians only. Thus, from now onwards, we shall take $${\cal{G}}(d) = \{[G_k({\Bbb{R}}^{n+k})]\  \mid \  nk = d,\ k<n,\ \text{and} \;
{\nu}(n+k) \leq {\nu}(k) \}.$$

If $G_k({\Bbb{R}}^{n+k})$ is an odd-dimensional real Grassmannian manifold then both $n$ and $k$ must be odd, and so ${\nu}(n+k) >
{\nu}(k).$ This means that $G_k({\Bbb{R}}^{n+k})$ bounds and so it follows that  ${\cal{G}}(d) = \emptyset$  if $d$ is odd. Therefore we assume that $d$ is even.

\[L:Flag\] In $H^*({\rm Flag}({\Bbb{R}}^{n+k});\ {\Bbb{Z}}_2)$ one has[,]{} for $1 \leq j
\leq k,$ $$\begin{aligned}
&\left(\underset{1 \leq i \leq k}{\sum} e_i^{n+k-(2j-1)} \right) \cdot
e_1^{k-1}e_2^{k-2} \dots e_{k-j}^j \cdot e_{k-(j-1)}^{j-1} \cdot
e_{k-(j-2)}^{n+k-(j-1)} \dots e_k^{n+k-1} \\[.2pc]
&= e_1^{k-1}e_2^{k-2} \dots e_{k-j}^j \cdot e_{k-(j-1)}^{n+k-j}\cdot
e_{k-(j-2)}^{n+k-(j-1)} \dots e_k^{n+k-1}.\end{aligned}$$

Note that

1.  if $i \ne k-(j-1)$ then the exponent of $e_i$ in the product $$\hskip -.5cm e_1^{k-1}e_2^{k-2} \dots
    e_{k-j}^j \cdot e_{k-(j-1)}^{j-1} \cdot e_{k-(j-2)}^{n+k-(j-1)} \dots
    e_k^{n+k-1}$$ is greater than or equal to $j$, and

2.  $\{n+k-(2j-1)\}+j =n+k-(j-1).$

Therefore, invoking Fact \[F:for\], the lemma follows.$\Box$

\[P:odd\]$\left.\right.$

Let ${\cal{O}}(d) = \{[G_k({\Bbb{R}}^{n+k})] \in {\cal{G}}(d)\
\mid \ n+k$ is odd $\}$. Then ${\cal{O}}(d)$ is linearly independent in ${\frak{N}}_d$.

Arrange the members of ${\cal{O}}(d)$ in descending order of the values of $n+k$, so that $${\cal{O}}(d) =
\{[G_{k_1}({\Bbb{R}}^{n_1+k_1})],\ [G_{k_2}({\Bbb{R}}^{n_2+k_2})],\ \dots ,
\ [G_{k_s}({\Bbb{R}}^{n_s+k_s})]\},$$ where $n_1+k_1 > n_2+k_2 > \dots > n_s+k_s$. Note that $n_1 =d$ and $k_1 = 1$.

For a $d$-dimensional Grassmannian manifold $G_k({\Bbb{R}}^{n+k})$, consider the polynomials $${f_{\ell}}(G_k({\Bbb{R}}^{n+k})) \; = 
\; \underset{1 \leq j \leq k_{\ell}}{\prod}
S_{n_{\ell} +k_{\ell} -(2j-1)} (G_k({\Bbb{R}}^{n+k}))\; \in
H^d(G_k({\Bbb{R}}^{n+k}); {\Bbb{Z}} _2)$$

$\left.\right.$

of Stiefel–Whitney classes of $G_k({\Bbb{R}}^{n+k}),$ where $1
\leq \ell \leq s.$ Then, for each $\ell ,$ $1 \leq \ell \leq s$, we have, using , $$\begin{aligned}
&{\pi}_{n_{\ell}+k_{\ell}}^*(f_{\ell}
(G_{k_{\ell}}({\Bbb{R}}^{n_{\ell}+k_{\ell}})))e_1^{k_{\ell}-
1}e_2^{k_{\ell}-2} \dots
e_{k_{\ell}-1}e_{k_{\ell}+1}^{n_{\ell}-1}e_{k_{\ell}+2}^{n_{\ell}-2}
\dots e_{n_{\ell}+k_{\ell}-1} \\[.2pc]
&= \left(\underset{1 \leq j \leq k_{\ell}}{\prod} \left(\underset{1 \leq
i \leq k_{\ell}}{\sum} e_i^{n_{\ell}+k_{\ell}-(2j-1)}
\right)\right)e_1^{k_{\ell}-1}e_2^{k_{\ell}-2} \dots
e_{k_{\ell}-1}e_{k_{\ell}+1}^{n_{\ell}-1}e_{k_{\ell}+2}^{n_{\ell}-2}\\[.2pc]
&\quad\ \dots e_{n_{\ell}+k_{\ell}-1} \\[.2pc]
&= e_1^{n_{\ell}}e_2^{n_{\ell}+1}
\dots e_{k_{\ell}}^{n_{\ell}+k_{\ell}-1}e_{k_{\ell}+1}^{n_{\ell}-1}
e_{k_{\ell}+2}^{n_{\ell}-2} \dots e_{n_{\ell}+k_{\ell}-1}, \end{aligned}$$ applying Lemma \[L:Flag\] repeatedly for successive values of $j$.

Thus, in view of Facts \[F:val\] and \[F:top\], the Stiefel–Whitney number $$\langle {f_{\ell}}(G_{k_{\ell}}({\Bbb{R}}^{n_{\ell}+k_{\ell}})),
[G_{k_{\ell}}({\Bbb{R}}^{n_{\ell}+k_{\ell}})] \rangle \; \ne \; 0$$ for each $\ell$, $1 \le \ell \le s.$ On the other hand, using , it is clear that $$\langle {f_{\ell}}(G_{k_h}({\Bbb{R}}^{n_h+k_h})),
[G_{k_h}({\Bbb{R}}^{n_h+k_h})] \rangle \; = \; 0$$ for each $h > \ell$, since $n_{\ell}+k_{\ell}-1 \; \ge \; n_h+k_h.$ Therefore, it follows that the $s \times s$ matrix $$[\langle {f_{\ell}}(G_{k_h}({\Bbb{R}}^{n_h+k_h})),
[G_{k_h}({\Bbb{R}}^{n_h+k_h})] \rangle ]_{1 \le {\ell}\le s,\; 1 \le h
\le s}$$ is non-singular; being lower triangular with 1’s in the diagonal. This completes theproof.$\Box$

Now we shall complete the proof of Theorem \[T:Gd\] using induction on $d$. First note that $$\begin{aligned}
{\cal G}(2) &= \{[G_1({\Bbb{R}}^{2+1})]\} = \{[{\Bbb{R}}P^2]\},\\[.2pc]
{\cal G}(4) &= \{[G_1({\Bbb{R}}^{4+1})]\} = \{[{\Bbb{R}}P^4]\},\end{aligned}$$ and so both are linearly independent in ${\frak{N}}_2,$ ${\frak{N}}_4$ respectively. Assume that the theorem holds for all dimensions less than $d$.

We have ${\cal G}(d) \; = \; {\cal E}(d) \bigcup {\cal O}(d) $, where $${\cal E}(d) \; = \; \{[G_k({\Bbb{R}}^{n+k})] \in {\cal G}(d) | n+k\ \ 
\text{is even}\}$$ and $${\cal O}(d) \; = \; \{[G_k({\Bbb{R}}^{n+k})] \in
{\cal G}(d) | n+k\ \ \text{is odd}\}.$$ Observe that if $[G_k({\Bbb{R}}^{n+k})] \in {\cal E}(d)$ then both $n$ and $k$ are even with $\nu(k) \ne \nu(n).$ On the other hand, $[G_2({\Bbb{R}}^{\frac{d}{2} +2})] \ \in \ {\cal E}(d)$  if $ d \equiv
0 \pmod{8}.$ Thus, ${\cal E}(d) \ne \emptyset $ if and only if $ d
\equiv 0 \pmod{8}.$

In view of Proposition \[P:odd\], we may assume without any loss that ${\cal E}(d) \; \ne \; \emptyset .$ Then, by the above observation and by Theorem 2.2 of [@pS91] every member of ${\cal E}(d)$ is of the form $[G_{\frac{k}{2}}({\Bbb{R}}^{\frac{n}{2} +\frac{k}{2}})]^4$, where $[G_{\frac{k}{2}}({\Bbb{R}}^{\frac{n}{2} +\frac{k}{2}})]$ $\in$ $
{\cal G}(\frac{d}{4}).$ By induction hypothesis, $ {\cal
G}(\frac{d}{4})$ is linearly independent in ${\frak{N}}_{\frac{d}{4}}$.

So, by Remark \[R:ind\], $$\label{E:ind}
{\cal E}(d) \ {\rm is\ linearly\ independent\ in} \ {\frak{N}}_d.$$

Again note that if $[G_k({\Bbb{R}}^{n+k})] \; \in \; {\cal E}(d),$ then, by (\[E:sym\]), the polynomial $S_p(G_k({\Bbb{R}}^{n+k}))$ $=$ $0, \;
\forall \; p \ge 1$. So, for each of the polynomials $f_{\ell},$ $1
\le \ell \le s,$ considered in Proposition \[P:odd\], we have $$\langle {f_{\ell}}(G_{k}({\Bbb{R}}^{n+k})), [G_{k}({\Bbb{R}}^{n+k})]
\rangle \; = \; 0.$$ Therefore, writing $$\begin{aligned}
\hskip -4pc {\cal E}(d) = \{ [G_{k_{s+1}}({\Bbb{R}}^{n_{s+1}+k_{s+1}})],
[G_{k_{s+2}}({\Bbb{R}}^{n_{s+2}+k_{s+2}})],\dots,
[G_{k_{s+q}}({\Bbb{R}}^{n_{s+q}+k_{s+q}})]\},\end{aligned}$$ where $n_{s+1}+k_{s+1} > n_{s+2}+k_{s+2} > \dots > n_{s+q}+k_{s+q},$ we see that the $s \times (s+q)$ matrix $$[\langle {f_{\ell}}(G_{k_h}({\Bbb{R}}^{n_h+k_h})),
[G_{k_h}({\Bbb{R}}^{n_h+k_h})] \rangle ]_{1 \le {\ell}\le s,\; 1 \le h
\le s+q}$$ is of the form $$\label{M:od}
\underset{{\cal O}(d) \qquad \qquad \qquad \qquad {\cal E}(d)}{
\left[\begin{matrix}
1 & 0 & 0 & \cdots & 0 \quad & 0 & 0 & 0 &\cdots & 0\\
 \star & 1 & 0 & \cdots & 0 \quad & 0 & 0 & 0 &\cdots & 0\\
\star & \star & 1 & \cdots & 0 \quad & 0 & 0 & 0 &\cdots & 0\\
 - & - & - & \cdots & - \quad & - & - & - &\cdots & -\\
 - & - & - & \cdots & - \quad & - & - & - &\cdots & -\\
\star & \star & \star & \cdots & 1 \quad & 0 & 0 & 0 &\cdots & 0
\end{matrix}
\right]}.$$ Thus, no non-trivial linear combination of members of ${\cal O}(d)$ can be expressed as a linear combination of the members of ${\cal E}(d).$ This, together with (\[E:ind\]) and Proposition \[P:odd\], proves that the set ${\cal G}(d) \; = \; {\cal E}(d) \bigcup {\cal O}(d) $ is linearly independent in ${\frak{N}}_d$. Hence, by induction, Theorem \[T:Gd\] is completely proved.

[Using the decomposition of the members of ${\cal E}(d)$, and the polynomials $f_{\ell}$, in the lower dimensions together with the [*doubling homomorphism*]{} defined by Milnor [@jM65], one can obtain a set of polynomials of Stiefel–Whitney classes which yield, as in Proposition \[P:odd\], a lower triangular matrix for ${\cal E}(d)$ with $1$’s in the diagonal. Thus using (\[M:od\]) we have a lower triangular matrix, with 1’s in the diagonal, for the whole set ${\cal
G}(d).$]{}

Acknowledgement {#acknowledgement .unnumbered}
===============

Part of this work was done under a DST project

[9]{} Dutta S and Khare S S, Independence of bordism classes of Milnor manifolds, [*J. Indian Math. Soc.*]{} [**68(1–4)**]{} (2001) 1–16

Floyd E E, Steifel–Whitney numbers of quaternionic and related manifolds, [*Trans. Am. Math. Soc.*]{} [**155**]{} (1971) 77–94

Hiller H L, On the cohomology of real Grassmannians, [*Trans. Am. Math. Soc.*]{} [**257**]{} (1980) 521–533

Hirzebruch F, Topological Methods in Algebraic Geometry (New York: Springer-Verlag) (1966)

Hsiang W-C and Szczarba R H, On the tangent bundle for the Grassmann manifold, [*Am. J. Math.*]{} [**86**]{} (1964) 698–704

Khare S S, On Dold manifolds, [*Topology Appl.*]{} [**33**]{} (1989) 297–307

Milnor J W, On the Stiefel–Whitney numbers of complex manifolds and of spin manifolds, [*Topology*]{} [**3**]{} (1965) 223–230

Sankaran P, Determination of Grassmann manifolds which are boundaries, [*Canad. Math. Bull.*]{} [**34**]{} (1991) 119–122

Stong R E, Cup products in Grassmannians, [*Topology Appl.*]{} [**13**]{} (1982) 103–113
