# On the mod $2$ cohomology algebra of oriented Grassmannians

Milica Jovanović  University of Belgrade, Faculty of mathematics, Studentski
trg 16, Belgrade, Serbia<EMAIL_ADDRESS>and  Branislav I.
Prvulović  University of Belgrade, Faculty of mathematics, Studentski trg 16,
Belgrade, Serbia<EMAIL_ADDRESS>
###### Abstract.

For $n\in\\{2^{t}-3,2^{t}-2,2^{t}-1\\}$ ($t\geq 3$) we study the cohomology
algebra $H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$ of the Grassmann manifold
$\widetilde{G}_{n,3}$ of oriented $3$-dimensional subspaces of
$\mathbb{R}^{n}$. A complete description of
$H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$ is given in the cases $n=2^{t}-3$
and $n=2^{t}-2$, while in the case $n=2^{t}-1$ we obtain a description
complete up to a coefficient from $\mathbb{Z}_{2}$.

###### Key words and phrases:

Grassmann manifolds, Stiefel–Whitney classes, Gröbner bases

###### 2020 Mathematics Subject Classification:

Primary 55R40; Secondary 13P10

The first author was partially supported by the Science Fund of the Republic
of Serbia, Grant No. 7749891, Graphical Languages – GWORDS. The second author
was partially supported by the Ministry of Science, Technological Development
and Innovations of the Republic of Serbia [contract no.
451-03-47/2023-01/200104].

##  1\. Introduction and statement of main results

Let $G_{n,k}$ be the Grassmann manifold of $k$-dimensional subspaces in
$\mathbb{R}^{n}$ and let $\widetilde{G}_{n,k}$ be the Grassmann manifold of
oriented $k$-dimensional subspaces in $\mathbb{R}^{n}$. Over both manifolds
there are canonical $k$-dimensional vector bundles, denoted by $\gamma_{n,k}$
and $\widetilde{\gamma}_{n,k}$ respectively. There are several ways to
describe the mod $2$ cohomology algebra $H^{*}(G_{n,k};\mathbb{Z}_{2})$, one
of them being due to Borel’s famous result [2], which establishes that every
element in $H^{*}(G_{n,k};\mathbb{Z}_{2})$ can be expressed as a polynomial in
Stiefel–Whitney classes $w_{i}(\gamma_{n,k})\in
H^{i}(G_{n,k};\mathbb{Z}_{2})$, $1\leq i\leq k$. On the other hand, the
situation in $H^{*}(\widetilde{G}_{n,k};\mathbb{Z}_{2})$ is more complicated,
although $\widetilde{G}_{n,k}$ is simply connected, and so
$w_{1}(\widetilde{\gamma}_{n,k})=0$. Namely, besides the Stiefel–Whitney
classes $w_{i}(\widetilde{\gamma}_{n,k})\in
H^{i}(\widetilde{G}_{n,k};\mathbb{Z}_{2})$, which we from now on abbreviate to
$\widetilde{w}_{i}$, $2\leq i\leq k$, there are some additional indecomposable
classes (a positive dimensional cohomology class is indecomposable if it
cannot be written as a polynomial in classes of smaller dimension). Put in
other words, the subalgebra of $H^{*}(\widetilde{G}_{n,k};\mathbb{Z}_{2})$
generated by $\widetilde{w}_{2},\ldots,\widetilde{w}_{k}$ does not coincide
with $H^{*}(\widetilde{G}_{n,k};\mathbb{Z}_{2})$. There is a description of
this subalgebra as a quotient of the polynomial algebra in
$\widetilde{w}_{2},\ldots,\widetilde{w}_{k}$ (analogous to the Borel
description of $H^{*}(G_{n,k};\mathbb{Z}_{2})$), but the whole algebra
$H^{*}(\widetilde{G}_{n,k};\mathbb{Z}_{2})$ is not completely described in
general.

The case $k=1$ is trivial, since $\widetilde{G}_{n,1}$ is the
$(n-1)$-dimensional sphere $S^{n-1}$. The algebra
$H^{*}(\widetilde{G}_{n,k};\mathbb{Z}_{2})$ is known in the case $k=2$ as
well. In [5] one can find a complete description of this algebra for $k=2$. In
the same paper, the additional particular case $(n,k)=(6,3)$ was also fully
calculated (it suffices to study $\widetilde{G}_{n,k}$ with $n\geq 2k$, since
$\widetilde{G}_{n,k}\cong\widetilde{G}_{n,n-k}$).

For $k=3$ and $n\in\\{2^{t}-3,2^{t}-2,2^{t}-1,2^{t}\\}$ ($t\geq 3$),
significant results in this regard were obtained by Basu and Chakraborty in
[1]. In that paper the authors showed that, besides Stiefel–Whitney classes
$\widetilde{w}_{2}$ and $\widetilde{w}_{3}$, there is exactly one (up to
addition of a polynomial in $\widetilde{w}_{2}$ and $\widetilde{w}_{3}$)
indecomposable class $\widetilde{a}$ in
$H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$, and they identified the algebra
$H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$ as the quotient of the polynomial
algebra $\mathbb{Z}_{2}[w_{2},w_{3},a]$ by a certain ideal. In the generating
set of that ideal some polynomials in $w_{2}$ and $w_{3}$ remained
undetermined. This description was completed in the case $n=2^{t}$ ($t\geq 3$)
in [3]. For $n\in\\{2^{t}-3,2^{t}-2,2^{t}-1\\}$ ($t\geq 3$), the description
is essentially the following ([1, Theorem B(2)]):

(1.1)
$H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})\cong\frac{\mathbb{Z}_{2}[w_{2},w_{3},a_{2^{t}-4}]}{\big{(}g_{n-2},g_{n-1},g_{n},a_{2^{t}-4}^{2}-P_{1}a_{2^{t}-4}-P_{2}\big{)}},$

where $P_{1}$ and $P_{2}$ are some (unknown) polynomials in $w_{2}$ and
$w_{3}$, with $P_{2}=0$ for $n\in\\{2^{t}-3,2^{t}-2\\}$. Via the isomorphism
(1.1), the cosets of $w_{2}$ and $w_{3}$ from the right-hand side correspond
to the Stiefel–Whitney classes $\widetilde{w}_{2}\in
H^{2}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$ and $\widetilde{w}_{3}\in
H^{3}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$ respectively, while the coset of
$a_{2^{t}-4}$ corresponds to the indecomposable class
$\widetilde{a}_{2^{t}-4}\in H^{2^{t}-4}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$.
The polynomials $g_{r}$, $r\geq 0$, appearing in (1.1), are well-known
polynomials in $w_{2}$ and $w_{3}$.

This paper deals with the problem of computing the polynomials $P_{1}$ and
$P_{2}$ from (1.1). For $n\in\\{2^{t}-3,2^{t}-2\\}$, there is actually only
$P_{1}$ to be determined, and we prove that $P_{1}=0$ in these cases (more
precisely, that $P_{1}$ can be chosen to be the zero-polynomial in order for
the isomorphism (1.1) to hold). This means that
$H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$ is isomorphic to the tensor
product of $\mathbb{Z}_{2}[w_{2},w_{3}]/(g_{n-2},g_{n-1},g_{n})$ and the
exterior algebra
$\Lambda(a_{2^{t}-4})\cong\mathbb{Z}_{2}[a_{2^{t}-4}]/(a_{2^{t}-4}^{2})$, that
is, $\widetilde{a}_{2^{t}-4}^{2}=0$ in
$H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$. Therefore, we obtain a complete
description of $H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$ in these cases. In
the case $n=2^{t}-1$ we prove that $\widetilde{a}_{2^{t}-4}^{2}\neq 0$ in
$H^{*}(\widetilde{G}_{n,3};\mathbb{Z}_{2})$, and establish that
$P_{1}=g_{2^{t}-4}$ and $P_{2}=\gamma w_{2}^{2^{t}-4}$ for some
$\gamma\in\mathbb{Z}_{2}$. So the main results of this paper are given in the
following theorem (it is well known that $g_{2^{t}-3}=0$, and so this
polynomial is redundant in (1.1)).

######  Theorem 1.1.

For all integers $t\geq 3$ we have the following isomorphisms of graded
algebras (where $|w_{i}|=i$, $i=2,3$, and $|a_{2^{t}-4}|=2^{t}-4$):

  * (a)

$\displaystyle
H^{*}(\widetilde{G}_{2^{t}-1,3};\mathbb{Z}_{2})\cong\frac{\mathbb{Z}_{2}[w_{2},w_{3},a_{2^{t}-4}]}{\big{(}g_{2^{t}-2},g_{2^{t}-1},a_{2^{t}-4}^{2}-g_{2^{t}-4}a_{2^{t}-4}-\gamma
w_{2}^{2^{t}-4}\big{)}}$,

for some $\gamma\in\mathbb{Z}_{2}$;

  * (b)

$\displaystyle
H^{*}(\widetilde{G}_{2^{t}-2,3};\mathbb{Z}_{2})\cong\frac{\mathbb{Z}_{2}[w_{2},w_{3}]}{(g_{2^{t}-4},g_{2^{t}-2})}\otimes\Lambda(a_{2^{t}-4})$;

  * (c)

$\displaystyle
H^{*}(\widetilde{G}_{2^{t}-3,3};\mathbb{Z}_{2})\cong\frac{\mathbb{Z}_{2}[w_{2},w_{3}]}{(g_{2^{t}-5},g_{2^{t}-4})}\otimes\Lambda(a_{2^{t}-4})$.

The technique that we use to prove this theorem combines the theory of Gröbner
bases with some classical methods of algebraic topology. In Section 2 of the
paper we establish a framework for the proof of Theorem 1.1 by recalling some
basic facts concerning the cohomology algebras $H^{*}(G_{n,k};\mathbb{Z}_{2})$
and $H^{*}(\widetilde{G}_{n,k};\mathbb{Z}_{2})$, in particular the Gysin
sequence connecting these two cohomologies. In Section 3 we use the Gröbner
basis for the ideal
$(g_{2^{t}-2},g_{2^{t}-1})\trianglelefteq\mathbb{Z}_{2}[w_{2},w_{3}]$ obtained
by Fukaya in [4], as well as an additive basis for
$H^{*}(\widetilde{G}_{2^{t}-1,3};\mathbb{Z}_{2})$ that it produces, to prove
part (a) of the theorem. Finally, Sections 4 and 5 prove parts (b) and (c) of
the theorem.

##  2\. Some preliminaries on cohomology of Grassmann manifolds

We will be working with $\mathbb{Z}_{2}$ coefficients exclusively, so in the
rest of the paper, these coefficients for cohomology are understood.

By Borel’s description [2], as a graded algebra

(2.1) $H^{*}(G_{n,k})\cong\mathbb{Z}_{2}[w_{1},w_{2},\ldots,w_{k}]/I_{n,k},$

where the coset of $w_{i}$ in the quotient on the right-hand side corresponds
to the Stiefel–Whitney class $w_{i}(\gamma_{n,k})\in H^{i}(G_{n,k})$ on the
left-hand side. So the grading on $\mathbb{Z}_{2}[w_{1},w_{2},\ldots,w_{k}]$
is such that the degree of $w_{i}$ is $i$, $1\leq i\leq k$. $I_{n,k}$ is the
graded ideal in $\mathbb{Z}_{2}[w_{1},w_{2},\ldots,w_{k}]$ generated by
polynomials $\overline{w}_{n-k+1},\ldots,\overline{w}_{n}$, obtained as
appropriate homogeneous parts of the power series
$1/(1+w_{1}+w_{2}+\cdots+w_{k})$. Therefore, they satisfy the relation

(2.2)
$(1+w_{1}+w_{2}+\cdots+w_{k})(\overline{w}_{0}+\overline{w}_{1}+\overline{w}_{2}+\cdots)=1.$

Let $p:\widetilde{G}_{n,k}\rightarrow G_{n,k}$ be the map that forgets the
orientation of the $k$-plane in $\mathbb{R}^{n}$. It is well known that $p$ is
a two-fold (universal) covering map, and that for the induced map
$p^{*}:H^{*}(G_{n,k})\rightarrow H^{*}(\widetilde{G}_{n,k})$ one has
$p^{*}(w_{i}(\gamma_{n,k}))=w_{i}(\widetilde{\gamma}_{n,k})=\widetilde{w}_{i}$,
$1\leq i\leq k$. Therefore, $\operatorname{im}p^{*}$ is a subalgebra of
$H^{*}(\widetilde{G}_{n,k})$ generated by
$\widetilde{w}_{2},\ldots,\widetilde{w}_{k}$ ($\widetilde{w}_{1}=0$ since
$\widetilde{G}_{n,k}$ is simply connected). The Gysin exact sequence
associated to $p$ has the following form:

${\cdots}$${H^{r}(G_{n,k})}$${H^{r}(\widetilde{G}_{n,k})}$${H^{r}(G_{n,k})}$${H^{r+1}(G_{n,k})}$${H^{r+1}(\widetilde{G}_{n,k})}$${\cdots,}$$\scriptstyle{p^{*}}$$\scriptstyle{w_{1}}$$\scriptstyle{p^{*}}$

where $H^{r}(G_{n,k})\stackrel{{\scriptstyle
w_{1}}}{{\longrightarrow}}H^{r+1}(G_{n,k})$ is the multiplication with
$w_{1}(\gamma_{n,k})$.

The inclusion $\mathbb{R}^{n}\hookrightarrow\mathbb{R}^{n+1}$ induces
embeddings $i:G_{n,k}\hookrightarrow G_{n+1,k}$ and $j:G_{n,k}\hookrightarrow
G_{n+1,k+1}$; and likewise in the oriented case, it induces
$\widetilde{i}:\widetilde{G}_{n,k}\hookrightarrow\widetilde{G}_{n+1,k}$ and
$\widetilde{j}:\widetilde{G}_{n,k}\hookrightarrow\widetilde{G}_{n+1,k+1}$.
Moreover, these embeddings commute with the double covering $p$, so we have
the following commutative diagrams of two-fold coverings ($S^{0}$-bundles).

${S^{0}}$${S^{0}}$${S^{0}}$${S^{0}}$${\widetilde{G}_{n,k}}$${\widetilde{G}_{n+1,k}}$${\widetilde{G}_{n,k}}$${\widetilde{G}_{n+1,k+1}}$${G_{n,k}}$${G_{n+1,k}}$${G_{n,k}}$${G_{n+1,k+1}}$$\scriptstyle{\widetilde{i}}$$\scriptstyle{p}$$\scriptstyle{p}$$\scriptstyle{\widetilde{j}}$$\scriptstyle{p}$$\scriptstyle{p}$$\scriptstyle{i}$$\scriptstyle{j}$

These diagrams in return induce morphisms between Gysin sequences.

(2.3)
${\cdots}$${H^{r}(G_{n+1,k})}$${H^{r}(\widetilde{G}_{n+1,k})}$${H^{r}(G_{n+1,k})}$${H^{r+1}(G_{n+1,k})}$${\cdots}$${\cdots}$${H^{r}(G_{n,k})}$${H^{r}(\widetilde{G}_{n,k})}$${H^{r}(G_{n,k})}$${H^{r+1}(G_{n,k})}$${\cdots}$$\scriptstyle{p^{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{\varphi}$$\scriptstyle{\widetilde{i}^{*}}$$\scriptstyle{w_{1}}$$\scriptstyle{i^{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{p^{*}}$$\scriptstyle{\varphi}$$\scriptstyle{w_{1}}$
(2.4) ${\cdots
H^{r}(G_{n+1,k+1})}$${H^{r}(\widetilde{G}_{n+1,k+1})}$${H^{r}(G_{n+1,k+1})}$${H^{r+1}(G_{n+1,k+1})}$${\cdots}$${\cdots
H^{r}(G_{n,k})}$${H^{r}(\widetilde{G}_{n,k})}$${H^{r}(G_{n,k})}$${H^{r+1}(G_{n,k})}$${\cdots}$$\scriptstyle{p^{*}}$$\scriptstyle{j^{*}}$$\scriptstyle{\varphi}$$\scriptstyle{\widetilde{j}^{*}}$$\scriptstyle{w_{1}}$$\scriptstyle{j^{*}}$$\scriptstyle{j^{*}}$$\scriptstyle{p^{*}}$$\scriptstyle{\varphi}$$\scriptstyle{w_{1}}$

The following lemma will be used repeatedly in the upcoming sections.

######  Lemma 2.1.

(a) Suppose that $r\geq 0$ is an integer with the property that in
$H^{r}(G_{n+1,k})$ one has $\ker w_{1}\cap\ker i^{*}=0$ (see (2.3)). Then for
an arbitrary class $x\in H^{r}(\widetilde{G}_{n+1,k})$ the following
implication holds:

$x\notin\operatorname{im}p^{*}\quad\Longrightarrow\quad\widetilde{i}^{*}(x)\notin\operatorname{im}p^{*}.$

(b) Similarly, if $r\geq 0$ is an integer with the property that $\ker
w_{1}\cap\ker j^{*}=0$ in $H^{r}(G_{n+1,k+1})$ (see (2.4)), then for a class
$x\in H^{r}(\widetilde{G}_{n+1,k+1})$ the following implication holds:

$x\notin\operatorname{im}p^{*}\quad\Longrightarrow\quad\widetilde{j}^{*}(x)\notin\operatorname{im}p^{*}.$

###### Proof.

The proof of the lemma comes down to elementary diagram chasing. For (a) we
use diagram (2.3). If $x\notin\operatorname{im}p^{*}=\ker\varphi$, then
$\varphi(x)\neq 0$ and $\varphi(x)\in\ker w_{1}$. From $\ker w_{1}\cap\ker
i^{*}=0$ we now conclude that $\varphi(x)\notin\ker i^{*}$, and so

$\varphi\big{(}\,\,\widetilde{i}^{*}(x)\big{)}=i^{*}\big{(}\varphi(x)\big{)}\neq
0,$

which means that
$\widetilde{i}^{*}(x)\notin\ker\varphi=\operatorname{im}p^{*}$.

The proof of part (b) is identical (one just uses diagram (2.4)). ∎

Hence, we will be interested in the kernels of the maps
$i^{*}:H^{*}(G_{n+1,k})\rightarrow H^{*}(G_{n,k})$ and
$j^{*}:H^{*}(G_{n+1,k+1})\rightarrow H^{*}(G_{n,k})$. The total space of
canonical vector bundle $\gamma_{n,k}$ over $G_{n,k}$ consists of pairs
$(V,v)$, where $V\in G_{n,k}$ and $v\in V$, so it is easy to see that the
embeddings $i:G_{n,k}\hookrightarrow G_{n+1,k}$ and $j:G_{n,k}\hookrightarrow
G_{n+1,k+1}$ are covered by bundle maps from $\gamma_{n,k}$ to
$\gamma_{n+1,k}$ and from $\gamma_{n,k}$ to $\gamma_{n+1,k+1}$ respectively.
Therefore, the next lemma is straightforward from the naturality of the
Stiefel–Whitney classes and the fact that the vector bundle $\gamma_{n,k}$ is
$k$-dimensional (see [6, p. 37]).

######  Lemma 2.2.

(a) For the map $i^{*}:H^{*}(G_{n+1,k})\rightarrow H^{*}(G_{n,k})$ we have

$i^{*}\big{(}w_{r}(\gamma_{n+1,k})\big{)}=w_{r}(\gamma_{n,k}),\quad 1\leq
r\leq k.$

(b) For the map $j^{*}:H^{*}(G_{n+1,k+1})\rightarrow H^{*}(G_{n,k})$ we have

$\displaystyle j^{*}\big{(}w_{r}(\gamma_{n+1,k+1})\big{)}$
$\displaystyle=w_{r}(\gamma_{n,k}),\quad 1\leq r\leq k,$ $\displaystyle
j^{*}\big{(}w_{k+1}(\gamma_{n+1,k+1})\big{)}$
$\displaystyle=w_{k+1}(\gamma_{n,k})=0.$

The same arguments work in the oriented case as well, and so for embeddings
$\widetilde{i}:\widetilde{G}_{n,k}\hookrightarrow\widetilde{G}_{n+1,k}$ and
$\widetilde{j}:\widetilde{G}_{n,k}\hookrightarrow\widetilde{G}_{n+1,k+1}$ one
has

(2.5) $\widetilde{i}^{*}(\widetilde{w}_{r})=\widetilde{w}_{r},\quad 2\leq
r\leq k,$

and

(2.6) $\widetilde{j}^{*}(\widetilde{w}_{r})=\widetilde{w}_{r},\quad 2\leq
r\leq k;\qquad\widetilde{j}^{*}(\widetilde{w}_{k+1})=0.$

In the case $k=3$, equality (2.2) leads to the recurrence relation

(2.7)
$\overline{w}_{r+3}=w_{1}\overline{w}_{r+2}+w_{2}\overline{w}_{r+1}+w_{3}\overline{w}_{r},\quad
r\geq 0,$

and also an explicit formula for calculating $\overline{w}_{r}$. Specifically,

(2.8)
$\overline{w}_{r}=\sum_{a+2b+3c=r}\binom{a+b+c}{a}\binom{b+c}{b}w_{1}^{a}w_{2}^{b}w_{3}^{c},\quad
r\geq 0,$

where the sum is taken over all triples $(a,b,c)$ of nonnegative integers such
that $a+2b+3c=r$ (both binomial coefficients in the sum are considered modulo
$2$).

It is well known (see e.g. [4] or [3]) that in the case $k=3$

(2.9) $\operatorname{im}p^{*}\cong\mathbb{Z}_{2}[w_{2},w_{3}]/J_{n,3},$

where $J_{n,3}$ is the graded ideal in $\mathbb{Z}_{2}[w_{2},w_{3}]$ generated
by the polynomials $g_{n-2}$, $g_{n-1}$ and $g_{n}$. These polynomials
$g_{r}\in\mathbb{Z}_{2}[w_{2},w_{3}]$, $r\geq 0$, are modulo $w_{1}$
reductions of the polynomials
$\overline{w}_{r}\in\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]$. From (2.8) we now get

(2.10) $g_{r}=\sum_{2b+3c=r}{b+c\choose b}\,w_{2}^{b}w_{3}^{c},\quad r\geq 0,$

and (2.7) becomes

(2.11) $g_{r+3}=w_{2}g_{r+1}+w_{3}g_{r},\quad r\geq 0.$

By definition, $J_{n,3}=(g_{n-2},g_{n-1},g_{n})$, but from (2.11) we see that
actually

(2.12) $g_{r}\in J_{n,3}\quad\mbox{ for all }r\geq n-2.$

For a (homogeneous) polynomial $f\in\mathbb{Z}_{2}[w_{2},w_{3}]$, we shall
denote by $\widetilde{f}\in\operatorname{im}p^{*}\subset
H^{*}(\widetilde{G}_{n,3})$ the cohomology class corresponding to (the coset
of) $f$ via the isomorphism (2.9). This is in accordance with the notation
$\widetilde{w}_{i}$, that we use for the Stiefel–Whitney class
$w_{i}(\widetilde{\gamma}_{n,3})$, $i=2,3$. For instance, the relation (2.12)
can now be rewritten as

(2.13) $\widetilde{g}_{r}=0\quad\mbox{ in
}H^{*}(\widetilde{G}_{n,3}),\quad\mbox{ for all }r\geq n-2.$

##  3\. The case $n=2^{t}-1$

Let $t\geq 3$ be a fixed integer. By [1, Theorem A(b)], in
$H^{*}(\widetilde{G}_{2^{t}-1,3})$, besides $\widetilde{w}_{2}$ and
$\widetilde{w}_{3}$, there is one more indecomposable class
$\widetilde{a}_{2^{t}-4}$ in cohomological dimension $2^{t}-4$. Indecomposable
means that it cannot be expressed as a polynomial in classes of (strictly)
smaller dimension. Since all classes of smaller dimension are polynomials in
$\widetilde{w}_{2}$ and $\widetilde{w}_{3}$, and the subalgebra
$\operatorname{im}p^{*}$ is generated by these two Stiefel–Whitney classes,
this is equivalent to the statement
$\widetilde{a}_{2^{t}-4}\notin\operatorname{im}p^{*}$. The uniqueness of
$\widetilde{a}_{2^{t}-4}$ is only up to addition of an element of
$\operatorname{im}p^{*}$. Put in other words, if $\widetilde{a}_{2^{t}-4}$ is
an indecomposable class in $H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$, then so
is $\widetilde{a}_{2^{t}-4}+\widetilde{w}$ for any $\widetilde{w}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})\cap\operatorname{im}p^{*}$. This means
that for $\widetilde{a}_{2^{t}-4}$ we can choose any class from
$H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})\setminus\operatorname{im}p^{*}$.

We are going to make this choice in the following way. By [1, Theorem A(a)] we
have an indecomposable class (outside $\operatorname{im}p^{*}$)
$\widetilde{a}_{2^{t}-1}\in H^{2^{t}-1}(\widetilde{G}_{2^{t},3})$. We will
prove that $\widetilde{i}^{*}(\widetilde{a}_{2^{t}-1})\in
H^{2^{t}-1}(\widetilde{G}_{2^{t}-1,3})\setminus\operatorname{im}p^{*}$, where
$\widetilde{i}:\widetilde{G}_{2^{t}-1,3}\hookrightarrow\widetilde{G}_{2^{t},3}$
is the embedding from the previous section. Since every class in
$H^{*}(\widetilde{G}_{2^{t}-1,3})$ is a polynomial in $\widetilde{w}_{2}$,
$\widetilde{w}_{3}$ and some (indecomposable) class in
$H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ (by [1, Theorem A(b)]), we see that
an odd-dimensional class must be divisible by $\widetilde{w}_{3}$. Hence,
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-1})$ must be of the form
$\widetilde{w}_{3}\sigma$, for some $\sigma\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$, and if we show
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-1})\notin\operatorname{im}p^{*}$, we
will have $\sigma\notin\operatorname{im}p^{*}$. Then we will be able to take
$\widetilde{a}_{2^{t}-4}:=\sigma$ and thus ensure the property:

(3.1)
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-1})=\widetilde{w}_{3}\widetilde{a}_{2^{t}-4}.$

Therefore, we are left to prove that
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-1})\notin\operatorname{im}p^{*}$. This
will be an immediate consequence of Lemma 2.1(a) (for $r=n=2^{t}-1$ and $k=3$)
as soon as we establish the following proposition.

######  Proposition 3.1.

Let $H^{2^{t}-1}(G_{2^{t},3})\stackrel{{\scriptstyle
w_{1}}}{{\longrightarrow}}H^{2^{t}}(G_{2^{t},3})$ be the multiplication with
the Stiefel–Whitney class $w_{1}(\gamma_{2^{t},3})$ and
$i^{*}:H^{2^{t}-1}(G_{2^{t},3})\rightarrow H^{2^{t}-1}(G_{2^{t}-1,3})$ the map
induced by the embedding $i:G_{2^{t}-1,3}\hookrightarrow G_{2^{t},3}$. Then
$\ker w_{1}\cap\ker i^{*}=0$.

###### Proof.

For a homogeneous polynomial $f\in\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]$ we will
denote by $[f]$ the cohomology class corresponding to $f$ via the isomorphism
(2.1). If $[f]\in\ker w_{1}\cap\ker i^{*}$, then by Lemma 2.2(a) we first have
$[f]=i^{*}[f]=0$ in $H^{2^{t}-1}(G_{2^{t}-1,3})$, so

$f=(\alpha w_{1}^{2}+\beta w_{2})\overline{w}_{2^{t}-3}+\gamma
w_{1}\overline{w}_{2^{t}-2}+\delta\overline{w}_{2^{t}-1},$

for some $\alpha,\beta,\gamma,\delta\in\mathbb{Z}_{2}$. Since
$\overline{w}_{2^{t}-2},\overline{w}_{2^{t}-1}\in J_{2^{t},3}$,

$[f]=[(\alpha w_{1}^{2}+\beta w_{2})\overline{w}_{2^{t}-3}+\gamma
w_{1}\overline{w}_{2^{t}-2}+\delta\overline{w}_{2^{t}-1}]=[(\alpha
w_{1}^{2}+\beta w_{2})\overline{w}_{2^{t}-3}],$

so we can choose $\gamma=\delta=0$.

On the other hand, since $[f]\in\ker w_{1}$, we have $[w_{1}f]=0$ in
$H^{2^{t}}(G_{2^{t},3})$, so

(3.2) $w_{1}f=(\lambda w_{1}^{2}+\mu w_{2})\overline{w}_{2^{t}-2}+\nu
w_{1}\overline{w}_{2^{t}-1}+\varepsilon\overline{w}_{2^{t}},$

for some $\lambda,\mu,\nu,\varepsilon\in\mathbb{Z}_{2}$. By (2.7),

$\overline{w}_{2^{t}}=w_{1}\overline{w}_{2^{t}-1}+w_{2}\overline{w}_{2^{t}-2}+w_{3}\overline{w}_{2^{t}-3},$

and after substituting $(\alpha w_{1}^{2}+\beta w_{2})\overline{w}_{2^{t}-3}$
for $f$, (3.2) becomes

(3.3) $(\alpha w_{1}^{3}+\beta w_{1}w_{2}+\varepsilon
w_{3})\overline{w}_{2^{t}-3}=(\lambda
w_{1}^{2}+(\mu+\varepsilon)w_{2})\overline{w}_{2^{t}-2}+(\nu+\varepsilon)w_{1}\overline{w}_{2^{t-1}}.$

By (2.8), the polynomial $\overline{w}_{2^{t}-2}$ contains the monomial
$w_{2}^{2^{t-1}-1}$ with nonzero coefficient, and it is obvious that
$(\mu+\varepsilon)w_{2}\overline{w}_{2^{t}-2}$ is the only summand in (3.3)
that can contain $w_{2}^{2^{t-1}}$. So we conclude $\mu+\varepsilon=0$, i.e.,
$\mu=\varepsilon$. Equation (3.3) now becomes

(3.4) $(\alpha w_{1}^{3}+\beta w_{1}w_{2}+\mu
w_{3})\overline{w}_{2^{t}-3}=\lambda
w_{1}^{2}\overline{w}_{2^{t}-2}+(\nu+\mu)w_{1}\overline{w}_{2^{t-1}}.$

We now use the same idea to show that $\alpha=\beta=0$. For $\alpha$ we
observe the monomial $w_{1}^{4}w_{2}^{2^{t-1}-2}$, and use (2.8) in order to
determine the coefficients of this monomial in each of the summands in (3.4).
These coefficients are listed in Table 1.

summand | $a$ | $b$ | $c$ | $\binom{a+b+c}{a}\binom{b+c}{b}$
---|---|---|---|---
$\alpha w_{1}^{3}\overline{w}_{2^{t}-3}$ | 1 | $2^{t-1}-2$ | 0 | 1
$\beta w_{1}w_{2}\overline{w}_{2^{t}-3}$ | 3 | $2^{t-1}-3$ | 0 | 0
$\mu w_{3}\overline{w}_{2^{t}-3}$ | - | - | - | -
$\lambda w_{1}^{2}\overline{w}_{2^{t}-2}$ | 2 | $2^{t-1}-2$ | 0 | 0
$(\nu+\mu)w_{1}\overline{w}_{2^{t-1}}$ | 3 | $2^{t-1}-2$ | 0 | 0
Table 1.  Coefficients of $w_{1}^{4}w_{2}^{2^{t-1}-2}$ in (3.4)

So $\alpha w_{1}^{3}\overline{w}_{2^{t}-3}$ is the only summand containing
$w_{1}^{4}w_{2}^{2^{t-1}-2}$, and its coefficient is $\alpha$. We conclude
that $\alpha=0$.

For $\beta$ we proceed in the same way. We now observe the monomial
$w_{1}^{3}w_{2}^{2^{t-1}-3}w_{3}$, and list its coefficients in Table 2, which
allows us to conclude that $\beta=0$.

summand | $a$ | $b$ | $c$ | $\binom{a+b+c}{a}\binom{b+c}{b}$
---|---|---|---|---
$\alpha w_{1}^{3}\overline{w}_{2^{t}-3}$ | 0 | $2^{t-1}-3$ | 1 | 0
$\beta w_{1}w_{2}\overline{w}_{2^{t}-3}$ | 2 | $2^{t-1}-4$ | 1 | 1
$\mu w_{3}\overline{w}_{2^{t}-3}$ | 3 | $2^{t-1}-3$ | 0 | 0
$\lambda w_{1}^{2}\overline{w}_{2^{t}-2}$ | 1 | $2^{t-1}-3$ | 1 | 0
$(\nu+\mu)w_{1}\overline{w}_{2^{t-1}}$ | 2 | $2^{t-1}-3$ | 1 | 0
Table 2.  Coefficients of $w_{1}^{3}w_{2}^{2^{t-1}-3}w_{3}$ in (3.4)

Finally, we have $[f]=[(\alpha w_{1}^{2}+\beta
w_{2})\overline{w}_{2^{t}-3}]=0$ in $H^{2^{t}-1}(G_{2^{t},3})$, and since
$[f]$ was an arbitrary element of $\ker w_{1}\cap\ker i^{*}$, this concludes
the proof. ∎

We have thus chosen indecomposable classes $\widetilde{a}_{2^{t}-1}\in
H^{2^{t}-1}(\widetilde{G}_{2^{t},3})$ and $\widetilde{a}_{2^{t}-4}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ such that (3.1) holds.

It is well known (see e.g. [1, Lemma 2.1(i)]) that $g_{2^{t}-3}=0$, so
$J_{2^{t}-1,3}=(g_{2^{t}-3},g_{2^{t}-2},g_{2^{t}-1})=(g_{2^{t}-2},g_{2^{t}-1})$.
Also, according to [1, Theorem B(2)] we have an isomorphism of graded algebras

(3.5)
$H^{*}(\widetilde{G}_{2^{t}-1,3})\cong\frac{\mathbb{Z}_{2}[w_{2},w_{3},a_{2^{t}-4}]}{\big{(}g_{2^{t}-2},g_{2^{t}-1},a_{2^{t}-4}^{2}-P_{1}a_{2^{t}-4}-P_{2}\big{)}},$

mapping the classes $\widetilde{w}_{2}$, $\widetilde{w}_{3}$ and
$\widetilde{a}_{2^{t}-4}$ to the cosets of $w_{2}$, $w_{3}$ and $a_{2^{t}-4}$
respectively, where the (unknown) polynomials $P_{1}$ and $P_{2}$ do not
contain the variable $a_{2^{t}-4}$.

In [4] Fukaya found a Gröbner basis for the ideal $J_{2^{t}-1,3}$, and in [3]
it was shown that $J_{2^{t}-1,3}=J_{2^{t},3}$ and that this Gröbner basis
consists of polynomials

(3.6) $f_{i}=g_{2^{t}-3+2^{i}}=\sum_{2b+3c=2^{t}-3+2^{i}}{b+c\choose
b}w_{2}^{b}w_{3}^{c},\quad\mbox{ for }0\leq i\leq t-1.$

These polynomials, of course, belong to $J_{2^{t}-1,3}$ (see (2.12)), and so
we have that $\widetilde{f}_{i}=0$ in $H^{*}(\widetilde{G}_{2^{t}-1,3})$ for
all $i$. By [4, p. 203] (or [3, Proposition 3.6]) we know that the leading
monomial (with respect to the lexicographic order with $w_{2}>w_{3}$) of
$f_{i}$ is the following:

(3.7) $\mathrm{LM}(f_{i})=w_{2}^{2^{t-1}-2^{i}}w_{3}^{2^{i}-1}.$

Moreover, [3, Proposition 3.6] gives us that

(3.8) $f_{t-1}=\mathrm{LM}(f_{t-1})=w_{3}^{2^{t-1}-1}.$

In the same way as in [3] we can use this Gröbner basis and the isomorphism
(3.5) to find an additive basis for the cohomology algebra
$H^{*}(\widetilde{G}_{2^{t}-1,3})$. First, if we choose the lexicographic
order with $a_{2^{t}-4}>w_{2}>w_{3}$ for the monomial order in
$\mathbb{Z}_{2}[w_{2},w_{3},a_{2^{t}-4}]$, then the set
$\\{f_{0},f_{1},\ldots,f_{t-1},a_{2^{t}-4}^{2}-P_{1}a_{2^{t}-4}-P_{2}\\}$
turns out to be a Gröbner basis for the ideal
$(g_{2^{t}-2},g_{2^{t}-1},a_{2^{t}-4}^{2}-P_{1}a_{2^{t}-4}-P_{2})$ in
$\mathbb{Z}_{2}[w_{2},w_{3},a_{2^{t}-4}]$ (whatever the polynomials $P_{1}$
and $P_{2}$ might be). This is essentially due to the fact that the leading
monomial of the polynomial $a_{2^{t}-4}^{2}-P_{1}a_{2^{t}-4}-P_{2}$ is
$a_{2^{t}-4}^{2}$ (since $P_{1}$ and $P_{2}$ are polynomials in $w_{2}$ and
$w_{3}$ only), and $a_{2^{t}-4}^{2}$ is coprime with every
$\mathrm{LM}(f_{i})$, $0\leq i\leq t-1$ (cf. [3, Theorem 3.9]). On the other
hand, if we observe the monomials not divisible by any of the leading
monomials from Gröbner basis, it is well known that their cosets form an
additive basis for the quotient. Therefore, the isomorphism (3.5) leads to the
conclusion that the set

(3.9)
$B=\left\\{\widetilde{a}_{2^{t}-4}^{r}\widetilde{w}_{2}^{b}\widetilde{w}_{3}^{c}\mid
r<2,\,\left(\forall i\in\\{0,1,\ldots,t-1\\}\right)\,b<2^{t-1}-2^{i}\vee
c<2^{i}-1\right\\}$

is an additive basis for $H^{*}(\widetilde{G}_{2^{t}-1,3})$ (cf. [3, Corollary
3.10]).

The elements of $B$ with $r=0$, i.e., the classes
$\widetilde{w}_{2}^{b}\widetilde{w}_{3}^{c}$ such that for all
$i\in\\{0,1,\ldots,t-1\\}$ either $b<2^{t-1}-2^{i}$ or $c<2^{i}-1$, form an
additive basis for $\operatorname{im}p^{*}$ (this could be obtained e.g. from
the isomorphism (2.9) by an analogous consideration). Now it is not hard to
see that the basis element from $\operatorname{im}p^{*}$ of the highest
cohomological dimension is
$\widetilde{w}_{2}^{2^{t-2}-1}\widetilde{w}_{3}^{2^{t-1}-2}\in
H^{2^{t+1}-8}(\widetilde{G}_{2^{t}-1,3})$. Indeed, for
$\widetilde{w}_{2}^{b}\widetilde{w}_{3}^{c}\in B$ one must have $c\leq
2^{t-1}-2$ (since $c\geq 2^{t-1}-1$ would imply $b<0$), and if
$i\in\\{0,1,\ldots,t-2\\}$ is such that $2^{i}-1\leq c\leq 2^{i+1}-2$, then
$b\leq 2^{t-1}-2^{i}-1$. So

(3.10) $2b+3c\leq 2(2^{t-1}-2^{i}-1)+3(2^{i+1}-2)=2^{t}+2^{i+2}-8\leq
2^{t+1}-8.$

This means that the following implication holds

(3.11) $j>2^{t+1}-8\quad\Longrightarrow\quad
H^{j}(\widetilde{G}_{2^{t}-1,3})\cap\operatorname{im}p^{*}=0.$

Note also that $\widetilde{w}_{2}^{2^{t-2}-1}\widetilde{w}_{3}^{2^{t-1}-2}$ is
the only element in $B\cap\operatorname{im}p^{*}$ of the cohomological
dimension $2^{t+1}-8$. Namely, in order to have equality in (3.10) one must
have $b=2^{t-1}-2^{i}-1$, $c=2^{i+1}-2$ and $i=t-2$. On the other hand, by [4,
Corollary 4.12] we know that $\widetilde{w}_{2}^{2^{t}-4}\neq 0$, and so we
conclude

(3.12)
$\widetilde{w}_{2}^{2^{t}-4}=\widetilde{w}_{2}^{2^{t-2}-1}\widetilde{w}_{3}^{2^{t-1}-2}\neq
0\quad\mbox{ in }H^{2^{t+1}-8}(\widetilde{G}_{2^{t}-1,3}).$

Similarly, the only element of $B$ in the top dimension $3\cdot 2^{t}-12$
(this is the dimension of the manifold $\widetilde{G}_{2^{t}-1,3}$) is
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-2}-1}\widetilde{w}_{3}^{2^{t-1}-2}$,
so we have

(3.13)
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4}=\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-2}-1}\widetilde{w}_{3}^{2^{t-1}-2}\neq
0\quad\mbox{ in }H^{3\cdot
2^{t}-12}(\widetilde{G}_{2^{t}-1,3})\cong\mathbb{Z}_{2}.$

Theorem 1.1(a) states that in (3.5) we can take $P_{1}=g_{2^{t}-4}$ and
$P_{2}=\gamma w_{2}^{2^{t}-4}$ for some $\gamma\in\mathbb{Z}_{2}$. In order to
prove this it suffices to show that in $H^{*}(\widetilde{G}_{2^{t}-1,3})$ we
have

(3.14)
$\widetilde{a}_{2^{t}-4}^{2}=\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}+\gamma\widetilde{w}_{2}^{2^{t}-4}$

(for some $\gamma\in\mathbb{Z}_{2}$).

We start off by representing $\widetilde{a}_{2^{t}-4}^{2}$ in the basis $B$
(see (3.9)). Since $\widetilde{a}_{2^{t}-4}^{2}\in
H^{2^{t+1}-8}(\widetilde{G}_{2^{t}-1,3})$, we need to list basis elements in
this cohomological dimension. As we have already noticed, from
$\operatorname{im}p^{*}$ we have
$\widetilde{w}_{2}^{2^{t-2}-1}\widetilde{w}_{3}^{2^{t-1}-2}$, and outside of
$\operatorname{im}p^{*}$ we have
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k}$
for every $k\geq 0$ such that $2^{t-1}-2-3k\geq 0$, i.e.,
$k\leq(2^{t-1}-2)/3$. Indeed,
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k}\in
B$ for all such $k$, since $2^{t-1}-2-3k\geq 2^{t-1}-2^{i}$ and $2k\geq
2^{i}-1$ for some $i\in\\{0,1,\ldots,t-1\\}$ would imply $2^{i}\geq 3k+2$ and
$2^{i}\leq 2k+1$, which is obviously impossible. Having in mind (3.12), we
thus have

(3.15)
$\widetilde{a}_{2^{t}-4}^{2}=\gamma\widetilde{w}_{2}^{2^{t}-4}+\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\lambda_{k}\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k},$

for some uniquely determined
$\gamma,\lambda_{0},\lambda_{1},\ldots\in\mathbb{Z}_{2}$. In order to obtain
(3.14), we see that we need to equate the sum on the right-hand side with
$\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}$. As the first step in that
direction, we prove the following proposition.

######  Proposition 3.2.

If $\lambda_{k}$, $0\leq k\leq\lfloor(2^{t-1}-2)/3\rfloor$, are the
coefficients from (3.15), then

$\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\lambda_{k}\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k}=\lambda_{0}\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}.$

###### Proof.

The main result of [3] is the equality $\widetilde{a}_{2^{t}-1}^{2}=0$, where
$\widetilde{a}_{2^{t}-1}\in H^{2^{t}-1}(\widetilde{G}_{2^{t},3})$ is the
indecomposable class. Therefore, by (3.1) we get

$\widetilde{w}_{3}^{2}\widetilde{a}_{2^{t}-4}^{2}=(\widetilde{w}_{3}\widetilde{a}_{2^{t}-4})^{2}=\big{(}\,\widetilde{i}^{*}(\widetilde{a}_{2^{t}-1})\big{)}^{2}=\widetilde{i}^{*}(\widetilde{a}_{2^{t}-1}^{2})=0.$

So, multiplying (3.15) by $\widetilde{w}_{3}^{2}$ leads to

(3.16)
$0=\widetilde{w}_{3}^{2}\widetilde{a}_{2^{t}-4}^{2}=\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\lambda_{k}\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k+2}$

($\gamma\widetilde{w}_{2}^{2^{t}-4}\widetilde{w}_{3}^{2}=0$ due to (3.11)).
The classes
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k+2}$
for $k\geq 1$ are elements of the basis $B$, since there is no
$i\in\\{0,1,\ldots,t-1\\}$ such that $2^{t-1}-2-3k\geq 2^{t-1}-2^{i}$ and
$2k+2\geq 2^{i}-1$ (if $k\geq 1$). However, for $k=0$ we have the class
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2}\widetilde{w}_{3}^{2}$,
which is not in $B$. We wish to express this class as a sum of basis elements.
In order to do so, note that the monomial
$a_{2^{t}-4}w_{2}^{2^{t-1}-2}w_{3}^{2}$ is divisible by
$\mathrm{LM}(f_{1})=w_{2}^{2^{t-1}-2}w_{3}$ (see (3.7)), and by (3.6) we have

(3.17)
$f_{1}=g_{2^{t}-1}=\sum_{2b+3c=2^{t}-1}\binom{b+c}{b}w_{2}^{b}w_{3}^{c}=\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\binom{2^{t-1}-1-k}{2k+1}w_{2}^{2^{t-1}-2-3k}w_{3}^{2k+1}.$

The last equality comes from the assertion that $c$ must be odd (since
$2b+3c=2^{t}-1$), and then one substitutes $2k+1$ for $c$. We know that
$\widetilde{f}_{1}=0$ in $H^{*}(\widetilde{G}_{2^{t}-1,3})$, and so

$\displaystyle 0$
$\displaystyle=\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\binom{2^{t-1}-1-k}{2k+1}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k+1}$
$\displaystyle=\widetilde{w}_{2}^{2^{t-1}-2}\widetilde{w}_{3}+\sum_{k=1}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\binom{2^{t-1}-1-k}{2k+1}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k+1}.$

Multiplying this equation with $\widetilde{a}_{2^{t}-4}\widetilde{w}_{3}$ we
get

(3.18)
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2}\widetilde{w}_{3}^{2}=\sum_{k=1}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\binom{2^{t-1}-1-k}{2k+1}\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k+2}$

Inserting (3.18) in (3.16) leads to

$0=\sum_{k=1}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\left(\lambda_{k}+\lambda_{0}\binom{2^{t-1}-1-k}{2k+1}\right)\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k+2}.$

As we have already indicated, the right-hand side is a linear combination of
basis elements, which means that all coefficients must vanish. Therefore,
$\lambda_{k}=\lambda_{0}\binom{2^{t-1}-1-k}{2k+1}$ for all $k$ such that
$1\leq k\leq\lfloor(2^{t-1}-2)/3\rfloor$. We conclude that

$\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\lambda_{k}\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k}=\lambda_{0}\widetilde{a}_{2^{t}-4}\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\binom{2^{t-1}-1-k}{2k+1}\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k}.$

It remains to prove that this sum on the right-hand side is equal to
$\widetilde{g}_{2^{t}-4}$. If

$S=\sum_{k=0}^{\left\lfloor\frac{2^{t-1}-2}{3}\right\rfloor}\binom{2^{t-1}-1-k}{2k+1}w_{2}^{2^{t-1}-2-3k}w_{3}^{2k},$

then using (3.17) and (2.11) we get that in $\mathbb{Z}_{2}[w_{2},w_{3}]$ the
following identity holds

$w_{3}S=g_{2^{t}-1}=w_{2}g_{2^{t}-3}+w_{3}g_{2^{t}-4}=w_{3}g_{2^{t}-4}$

(since $g_{2^{t}-3}=0$). Canceling out $w_{3}$ finishes the proof. ∎

So now (3.15) simplifies to

(3.19)
$\widetilde{a}_{2^{t}-4}^{2}=\lambda_{0}\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}+\gamma\widetilde{w}_{2}^{2^{t}-4}$

(for some $\lambda_{0},\gamma\in\mathbb{Z}_{2}$), and in order to obtain
(3.14) (and thus finish the proof of Theorem 1.1(a)) we are left to prove that
$\lambda_{0}=1$.

######  Proposition 3.3.

Let $H^{2^{t}-4}(G_{2^{t}-1,3})\stackrel{{\scriptstyle
w_{1}}}{{\longrightarrow}}H^{2^{t}-3}(G_{2^{t}-1,3})$ be the multiplication
map and $j^{*}:H^{2^{t}-4}(G_{2^{t}-1,3})\rightarrow
H^{2^{t}-4}(G_{2^{t}-2,2})$ the map induced by the embedding
$j:G_{2^{t}-2,2}\hookrightarrow G_{2^{t}-1,3}$ (see Section 2). Then $\ker
w_{1}\cap\ker j^{*}=0$.

###### Proof.

As in the proof of Proposition 3.1, let
$f\in\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]$ be such that for the corresponding
cohomology class in $H^{2^{t}-4}(G_{2^{t}-1,3})$ one has $[f]\in\ker
w_{1}\cap\ker j^{*}$. We want to prove that $[f]=0$ in
$H^{2^{t}-4}(G_{2^{t}-1,3})$.

By Lemma 2.2(b), $j^{*}[f]=[\rho(f)]$, where
$\rho:\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]\rightarrow\mathbb{Z}_{2}[w_{1},w_{2}]$
is the reduction modulo $w_{3}$. Since $j^{*}[f]=0$, we first get
$[\rho(f)]=0$, but actually $\rho(f)=0$, because the degree of $\rho(f)$ in
$\mathbb{Z}_{2}[w_{1},w_{2}]$ is $2^{t}-4$, and
$H^{*}(G_{2^{t}-2,2})\cong\mathbb{Z}_{2}[w_{1},w_{2}]/(\overline{w}_{2^{t}-3},\overline{w}_{2^{t}-2})$.
We conclude that $f=w_{3}h$, for some $h\in\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]$.

On the other hand, $[f]\in\ker w_{1}$ means that $[w_{1}f]=0$ in
$H^{2^{t}-3}(G_{2^{t}-1,3})$, and since
$H^{*}(G_{2^{t}-1,3})\cong\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]/(\overline{w}_{2^{t}-3},\overline{w}_{2^{t}-2},\overline{w}_{2^{t}-1})$,
we conclude that $w_{1}f=\alpha\overline{w}_{2^{t}-3}$ for some
$\alpha\in\mathbb{Z}_{2}$. When we combine these two results, we get

$w_{1}w_{3}h=\alpha\overline{w}_{2^{t}-3}.$

If $\alpha$ were nonzero, then the polynomial $\overline{w}_{2^{t}-3}$ would
be divisible by $w_{3}$. However, it is obvious that the monomial
$w_{1}^{2^{t}-3}$ appears in $\overline{w}_{2^{t}-3}$ with nonzero coefficient
(see (2.8)). Therefore, $\alpha=0$, which implies
$w_{1}f=\alpha\overline{w}_{2^{t}-3}=0$, and so $f=0$. ∎

We know that $\widetilde{a}_{2^{t}-4}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})\setminus\operatorname{im}p^{*}$, and so
Proposition 3.3 and Lemma 2.1(b) (for $r=2^{t}-4$, $n=2^{t}-2$ and $k=2$)
imply

(3.20) $\widetilde{j}^{*}(\widetilde{a}_{2^{t}-4})\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-2,2})\setminus\operatorname{im}p^{*}$

(here, $\operatorname{im}p^{*}$ is the subalgebra of
$H^{*}(\widetilde{G}_{2^{t}-2,2})$ generated by $\widetilde{w}_{2}$). A
complete description of $H^{*}(\widetilde{G}_{2^{t}-2,2})$ is given in [5,
Theorem 2.1(b)] and reads as follows:

(3.21)
$H^{*}(\widetilde{G}_{2^{t}-2,2})\cong\frac{\mathbb{Z}_{2}[w_{2},b_{2^{t}-4}]}{\big{(}w_{2}^{2^{t-1}-1},b_{2^{t}-4}^{2}-w_{2}^{2^{t-1}-2}b_{2^{t}-4}\big{)}}.$

If $\widetilde{b}_{2^{t}-4}\in H^{2^{t}-4}(\widetilde{G}_{2^{t}-2,2})$ is the
class corresponding to (the coset of) $b_{2^{t}-4}$ via the isomorphism
(3.21), then (3.20) implies

(3.22)
$\widetilde{j}^{*}(\widetilde{a}_{2^{t}-4})=\widetilde{b}_{2^{t}-4}+\mu\widetilde{w}_{2}^{2^{t-1}-2},$

for some $\mu\in\mathbb{Z}_{2}$. Since $\widetilde{w}_{2}^{2^{t}-4}=0$ in
$H^{*}(\widetilde{G}_{2^{t}-2,2})$, squaring the equation (3.22) leads to

(3.23)
$\widetilde{j}^{*}(\widetilde{a}_{2^{t}-4}^{2})=\widetilde{b}_{2^{t}-4}^{2}\neq
0\quad\mbox{ in }H^{2^{t+1}-8}(\widetilde{G}_{2^{t}-2,2}).$

On the other hand, by (3.19) and (2.6)

$\widetilde{j}^{*}(\widetilde{a}_{2^{t}-4}^{2})=\lambda_{0}\widetilde{j}^{*}(\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4})+\gamma\widetilde{w}_{2}^{2^{t}-4}=\lambda_{0}\widetilde{j}^{*}(\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}),$

leading to the conclusion $\lambda_{0}=1$. This completes the proof of Theorem
1.1(a).

Note that we have proved that $\widetilde{a}_{2^{t}-4}^{2}\neq 0$ in
$H^{2^{t+1}-8}(\widetilde{G}_{2^{t}-1,3})$. This follows from (3.23), and it
also follows from the description of $H^{*}(\widetilde{G}_{2^{t}-1,3})$ (given
in Theorem 1.1(a)) itself.

We were not able to compute the coefficient $\gamma\in\mathbb{Z}_{2}$ from
this description, and likewise we could not deduce whether the top dimensional
cohomology class $\widetilde{a}_{2^{t}-4}^{3}\in H^{3\cdot
2^{t}-12}(\widetilde{G}_{2^{t}-1,3})$ vanishes or not. Nevertheless, in the
upcoming Proposition 3.5 we prove that these two problems are equivalent.

We first show that $\widetilde{w}_{2}^{2^{t-1}-2}$ is the only basis element
in $\operatorname{im}p^{*}\cap H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ whose
square is nonzero. Actually, we already know that
$(\widetilde{w}_{2}^{2^{t-1}-2})^{2}=\widetilde{w}_{2}^{2^{t}-4}\neq 0$ (see
(3.12)), so we now verify that the square of
$\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k}$ vanishes if $k>0$.

######  Lemma 3.4.

If $k>0$ and $2^{t}-4-6k\geq 0$, then

$\widetilde{w}_{2}^{2^{t}-4-6k}\widetilde{w}_{3}^{4k}=0\quad\mbox{ in
}H^{*}(\widetilde{G}_{2^{t}-1,3}).$

###### Proof.

The proof is by reverse induction on $k$. If $k\geq 2^{t-3}$, i.e., $4k\geq
2^{t-1}$, then

$\widetilde{w}_{2}^{2^{t}-4-6k}\widetilde{w}_{3}^{4k}=\widetilde{w}_{2}^{2^{t}-4-6k}\widetilde{w}_{3}^{4k-2^{t-1}+1}\widetilde{w}_{3}^{2^{t-1}-1}=\widetilde{w}_{2}^{2^{t}-4-6k}\widetilde{w}_{3}^{4k-2^{t-1}+1}\widetilde{f}_{t-1}=0,$

by (3.8).

Now let $0<k<2^{t-3}$ and suppose that
$\widetilde{w}_{2}^{2^{t}-4-6j}\widetilde{w}_{3}^{4j}=0$ for all integers
$j>k$ (such that $2^{t}-4-6j\geq 0$). There exists a unique integer
$i\in\\{2,\ldots,t-2\\}$ with the property $2^{i}\leq 4k\leq 2^{i+1}-4$. Note
that then

$12k\leq 3\cdot 2^{i+1}-12=2^{i+2}+2^{i+1}-12\leq 2^{t}+2^{i+1}-12,$

i.e., $6k\leq 2^{t-1}+2^{i}-6$, implying $2^{t}-4-6k>2^{t-1}-2^{i}$. This
means that we can write

(3.24)
$\widetilde{w}_{2}^{2^{t}-4-6k}\widetilde{w}_{3}^{4k}=\widetilde{w}_{2}^{2^{t}-4-6k-2^{t-1}+2^{i}}\widetilde{w}_{3}^{4k-2^{i}+1}\widetilde{w}_{2}^{2^{t-1}-2^{i}}\widetilde{w}_{3}^{2^{i}-1}.$

On the other hand, according to (3.7) and (3.6),

$w_{2}^{2^{t-1}-2^{i}}w_{3}^{2^{i}-1}=\mathrm{LM}(f_{i})=f_{i}+\\!\\!\\!\sum_{\begin{subarray}{c}2b+3c=2^{t}-3+2^{i}\\\
(b,c)\neq(2^{t-1}-2^{i},2^{i}-1)\end{subarray}}{b+c\choose
b}w_{2}^{b}w_{3}^{c}.$

Since $\widetilde{f}_{i}=0$, by (3.24) in cohomology we have:

$\displaystyle\widetilde{w}_{2}^{2^{t}-4-6k}\widetilde{w}_{3}^{4k}$
$\displaystyle=\widetilde{w}_{2}^{2^{t-1}+2^{i}-4-6k}\widetilde{w}_{3}^{4k-2^{i}+1}\widetilde{w}_{2}^{2^{t-1}-2^{i}}\widetilde{w}_{3}^{2^{i}-1}$
$\displaystyle=\widetilde{w}_{2}^{2^{t-1}+2^{i}-4-6k}\widetilde{w}_{3}^{4k-2^{i}+1}\\!\\!\\!\sum_{\begin{subarray}{c}2b+3c=2^{t}-3+2^{i}\\\
(b,c)\neq(2^{t-1}-2^{i},2^{i}-1)\end{subarray}}{b+c\choose
b}\widetilde{w}_{2}^{b}\widetilde{w}_{3}^{c}$
$\displaystyle=\\!\\!\\!\sum_{\begin{subarray}{c}2b+3c=2^{t}-3+2^{i}\\\
(b,c)\neq(2^{t-1}-2^{i},2^{i}-1)\end{subarray}}{b+c\choose
b}\widetilde{w}_{2}^{b+2^{t-1}+2^{i}-4-6k}\widetilde{w}_{3}^{c+4k-2^{i}+1}$

To complete the induction step, it now suffices to prove that for a summand
with ${b+c\choose b}\equiv 1\pmod{2}$ one has $c+4k-2^{i}+1=4j$ for some
$j>k$.

First, since $\mathrm{LM}(f_{i})=w_{2}^{2^{t-1}-2^{i}}w_{3}^{2^{i}-1}$, for a
nonzero summand in this sum, one must have $b<2^{t-1}-2^{i}$, and consequently
$c>2^{i}-1$ (because $2b+3c$ is the same for all monomials in $f_{i}$). This
implies $c+4k-2^{i}+1>4k$, and so we only need to prove that $c+4k-2^{i}+1$ is
divisible by $4$.

Since $i\geq 2$ and $2b+3c=2^{t}-3+2^{i}$, we first conclude that $c$ is odd,
and therefore $c+4k-2^{i}+1$ is even, which means that it remains to rule out
the possibility $c+4k-2^{i}+1\equiv 2\pmod{4}$, i.e., $c\equiv 1\pmod{4}$. But
$c\equiv 1\pmod{4}$ and $2b+3c=2^{t}-3+2^{i}$ would imply $2b\equiv
2\pmod{4}$, and we would have that $b$ is odd, contradicting the fact
${b+c\choose b}\equiv 1\pmod{2}$. ∎

This lemma ensures that

(3.25) $\widetilde{g}_{2^{t}-4}^{2}=\widetilde{w}_{2}^{2^{t}-4}.$

Namely, the leading monomial of $g_{2^{t}-4}$ is obviously
$w_{2}^{2^{t-1}-2}$, and all other monomials are of the form
$w_{2}^{2^{t-1}-2-3k}w_{3}^{2k}$ for some $k>0$.

As we have already indicated, the indecomposable class in
$H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ is not unique. If
$\widetilde{a}_{2^{t}-4}$ is such a class, then so is
$\widetilde{a}_{2^{t}-4}+\widetilde{w}$ for any
$\widetilde{w}\in\operatorname{im}p^{*}\cap
H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ ($\widetilde{a}_{2^{t}-4}$ is unique
only up to addition of such a $\widetilde{w}$). As another consequence of
Lemma 3.4, we note that taking some other indecomposable class from
$H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ has no effect on equality (3.14).
Indeed,

$(\widetilde{a}_{2^{t}-4}+\widetilde{w})^{2}=\widetilde{a}_{2^{t}-4}^{2}+\widetilde{w}^{2}=\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}+\gamma\widetilde{w}_{2}^{2^{t}-4}+\widetilde{w}^{2}=\widetilde{g}_{2^{t}-4}(\widetilde{a}_{2^{t}-4}+\widetilde{w})+\gamma\widetilde{w}_{2}^{2^{t}-4},$

where the last equality comes from Lemma 3.4. Namely, one can fairly easy show
that $\widetilde{w}^{2}=\widetilde{w}\widetilde{g}_{2^{t}-4}$ using the lemma
and the following facts: $\widetilde{w}$ is a sum of monomials of the form
$\widetilde{w}_{2}^{2^{t-1}-2-3k}\widetilde{w}_{3}^{2k}$ for some $k\geq 0$;
$\widetilde{w}_{3}\widetilde{g}_{2^{t}-4}=\widetilde{g}_{2^{t}-1}=0$ in
$H^{*}(\widetilde{G}_{2^{t}-1,3})$ (see (2.13)); and for every monomial
$\widetilde{w}_{2}^{b}\widetilde{w}_{3}^{c}$ with nonzero coefficient in
$\widetilde{g}_{2^{t}-4}$ the exponent $c$ is divisible by $4$ (see the proof
of Proposition 3.2, $g_{2^{t}-4}=S$ in that proof).

######  Proposition 3.5.

For an indecomposable class $\widetilde{a}_{2^{t}-4}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ and the coefficient
$\gamma\in\mathbb{Z}_{2}$ from Theorem 1.1(a) the following equivalence holds:

$\widetilde{a}_{2^{t}-4}^{3}\neq 0\quad\Longleftrightarrow\quad\gamma=0.$

###### Proof.

By (3.14) and (3.25) we have

$\displaystyle\widetilde{a}_{2^{t}-4}^{3}$
$\displaystyle=\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}^{2}+\gamma\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4}=\widetilde{g}_{2^{t}-4}^{2}\widetilde{a}_{2^{t}-4}+\gamma\widetilde{g}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4}+\gamma\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4}$
$\displaystyle=\widetilde{g}_{2^{t}-4}^{2}\widetilde{a}_{2^{t}-4}+\gamma\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4}=(1+\gamma)\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4},$

since $\widetilde{g}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4}\in H^{3\cdot
2^{t}-12}(\widetilde{G}_{2^{t}-1,3})\cap\operatorname{im}p^{*}=0$ (by (3.11)).
On the other hand, by (3.13) we know that
$\widetilde{a}_{2^{t}-4}\widetilde{w}_{2}^{2^{t}-4}$ is the nonzero element in
$H^{3\cdot 2^{t}-12}(\widetilde{G}_{2^{t}-1,3})$, and so this equality proves
the proposition. ∎

##  4\. The case $n=2^{t}-2$

By [1, Theorem A(b)] the cohomology algebra $H^{*}(\widetilde{G}_{2^{t}-2,3})$
contains an indecomposable class in dimension $2^{t}-4$. Similarly as in the
case $n=2^{t}-1$, for this class we can choose any $\widetilde{a}_{2^{t}-4}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-2,3})\setminus\operatorname{im}p^{*}$. If
$\widetilde{i}:\widetilde{G}_{2^{t}-2,3}\hookrightarrow\widetilde{G}_{2^{t}-1,3}$
is the embedding and $\widetilde{a}_{2^{t}-4}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-1,3})$ the indecomposable class from the
previous section, we will prove that
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4})\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-2,3})\setminus\operatorname{im}p^{*}$, and
then make a choice:

(4.1) $\widetilde{a}_{2^{t}-4}:=\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4}).$

Similarly as before, the relation
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4})\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-2,3})\setminus\operatorname{im}p^{*}$ will be
a direct consequence of Lemma 2.1(a) and the following proposition.

######  Proposition 4.1.

Let $H^{2^{t}-4}(G_{2^{t}-1,3})\stackrel{{\scriptstyle
w_{1}}}{{\longrightarrow}}H^{2^{t}-3}(G_{2^{t}-1,3})$ be the multiplication
map and $i^{*}:H^{2^{t}-4}(G_{2^{t}-1,3})\rightarrow
H^{2^{t}-4}(G_{2^{t}-2,3})$ the map induced by the embedding
$i:G_{2^{t}-2,3}\hookrightarrow G_{2^{t}-1,3}$ (see Section 2). Then $\ker
w_{1}\cap\ker i^{*}=0$.

###### Proof.

The proof is similar to those of Propositions 3.1 and 3.3. Take a polynomial
$f\in\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]$ such that $[f]\in\ker w_{1}\cap\ker
i^{*}$. By Lemma 2.2(a), $[f]=i^{*}[f]=0$ in $H^{2^{t}-4}(G_{2^{t}-2,3})$, and
so $f=\alpha\overline{w}_{2^{t}-4}$ for some $\alpha\in\mathbb{Z}_{2}$. On the
other hand, since $[w_{1}f]=0$ in $H^{2^{t}-3}(G_{2^{t}-1,3})$, we have
$w_{1}f=\beta\overline{w}_{2^{t}-3}$ for some $\beta\in\mathbb{Z}_{2}$.
Combining these two equations, we get

(4.2) $\alpha w_{1}\overline{w}_{2^{t}-4}=\beta\overline{w}_{2^{t}-3}.$

If $\alpha$ were nonzero, then the monomial $w_{1}^{3}w_{2}^{2^{t-1}-3}$ would
appear on the left-hand side of (4.2) with nonzero coefficient (since
$w_{1}^{2}w_{2}^{2^{t-1}-3}$ appears in $\overline{w}_{2^{t}-4}$ with nonzero
coefficient, by (2.8)). However, the coefficient of
$w_{1}^{3}w_{2}^{2^{t-1}-3}$ in $\overline{w}_{2^{t}-3}$ is
$\binom{2^{t-1}}{3}=0$. We conclude that $\alpha=0$, and so
$f=\alpha\overline{w}_{2^{t}-4}=0$. ∎

So now we have the indecomposable class $\widetilde{a}_{2^{t}-4}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-2,3})$, chosen by (4.1). According to [1,
Theorem B(2)] there exists an algebra isomorphism

(4.3)
$H^{*}(\widetilde{G}_{2^{t}-2,3})\cong\frac{\mathbb{Z}_{2}[w_{2},w_{3},a_{2^{t}-4}]}{\big{(}g_{2^{t}-4},g_{2^{t}-2},a_{2^{t}-4}^{2}-P_{1}a_{2^{t}-4}\big{)}},$

which maps the classes $\widetilde{w}_{2}$, $\widetilde{w}_{3}$ and
$\widetilde{a}_{2^{t}-4}$ to the cosets of $w_{2}$, $w_{3}$ and $a_{2^{t}-4}$
respectively, and $P_{1}$ is some polynomial in variables $w_{2}$ and $w_{3}$
only. In order to prove part (b) of Theorem 1.1, we need to show that for
$P_{1}$ we can take the zero polynomial, which amounts to showing that
$\widetilde{a}_{2^{t}-4}^{2}=0$ in $H^{*}(\widetilde{G}_{2^{t}-2,3})$. We
shall need the following lemma.

######  Lemma 4.2.

We have $\widetilde{w}_{2}^{2^{t}-4}=0$ in
$H^{2^{t+1}-8}(\widetilde{G}_{2^{t}-2,3})$.

###### Proof.

Assume to the contrary that $\widetilde{w}_{2}^{2^{t}-4}\neq 0$. The dimension
of the manifold $\widetilde{G}_{2^{t}-2,3}$ is $3\cdot 2^{t}-15$, and so by
the Poincaré duality, there exists a class $\sigma\in H^{3\cdot
2^{t}-15-(2^{t+1}-8)}(\widetilde{G}_{2^{t}-2,3})$ such that
$\sigma\widetilde{w}_{2}^{2^{t}-4}\neq 0$ in the top dimensional cohomology
group $H^{3\cdot 2^{t}-15}(\widetilde{G}_{2^{t}-2,3})$. Since $3\cdot
2^{t}-15-(2^{t+1}-8)=2^{t}-7<2^{t}-4$, $\sigma$ is a polynomial in
$\widetilde{w}_{2}$ and $\widetilde{w}_{3}$, i.e.,
$\sigma\in\operatorname{im}p^{*}$. This means that the nonzero top dimensional
class $\sigma\widetilde{w}_{2}^{2^{t}-4}$ belongs to $\operatorname{im}p^{*}$.
However, the relevant part of the Gysin sequence shows that this is not the
case.

${\cdots}$${H^{3\cdot 2^{t}-15}(G_{2^{t}-2,3})}$${H^{3\cdot
2^{t}-15}(\widetilde{G}_{2^{t}-2,3})}$${H^{3\cdot
2^{t}-15}(G_{2^{t}-2,3})}$${0}$$\scriptstyle{p^{*}}$$\scriptstyle{\varphi}$

Namely, $H^{3\cdot 2^{t}-15}(\widetilde{G}_{2^{t}-2,3})\cong H^{3\cdot
2^{t}-15}(G_{2^{t}-2,3})\cong\mathbb{Z}_{2}$, $\varphi$ is onto, and hence
isomorphism. Therefore, $p^{*}=0$. This contradiction concludes the proof. ∎

Now, according to (4.1), (3.14) and (2.5)

$\widetilde{a}_{2^{t}-4}^{2}=\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4}^{2})=\widetilde{i}^{*}(\widetilde{g}_{2^{t}-4})\cdot\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4})+\gamma\widetilde{i}^{*}(\widetilde{w}_{2}^{2^{t}-4})=\widetilde{g}_{2^{t}-4}\widetilde{a}_{2^{t}-4}+\gamma\widetilde{w}_{2}^{2^{t}-4}.$

By Lemma 4.2, $\widetilde{w}_{2}^{2^{t}-4}=0$, and by (2.13),
$\widetilde{g}_{2^{t}-4}=0$ in $H^{*}(\widetilde{G}_{2^{t}-2,3})$. So,
$\widetilde{a}_{2^{t}-4}^{2}=0$, and the proof of Theorem 1.1(b) is complete.

##  5\. The case $n=2^{t}-3$

In this section we prove part (c) of Theorem 1.1, and the proof goes along the
same lines as the proof of part (b) (given in the previous section). We choose
the indecomposable class $\widetilde{a}_{2^{t}-4}\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-3,3})$ to be
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4})$, where
$\widetilde{a}_{2^{t}-4}\in H^{2^{t}-4}(\widetilde{G}_{2^{t}-2,3})$ is the
indecomposable class chosen in the previous section, and
$\widetilde{i}:\widetilde{G}_{2^{t}-3,3}\hookrightarrow\widetilde{G}_{2^{t}-2,3}$
is the embedding from Section 2. When we do so, then it is immediate that
$\widetilde{a}_{2^{t}-4}^{2}=\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4}^{2})=0$,
and Theorem 1.1(c) follows.

So the only thing to be proved is the claim that we can take
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4})$ as the indecomposable class in
$H^{2^{t}-4}(\widetilde{G}_{2^{t}-3,3})$, that is,
$\widetilde{i}^{*}(\widetilde{a}_{2^{t}-4})\in
H^{2^{t}-4}(\widetilde{G}_{2^{t}-3,3})\setminus\operatorname{im}p^{*}$. As in
the previous cases, Lemma 2.1(a) ensures that it is enough to establish the
following proposition.

######  Proposition 5.1.

For the multiplication map $H^{2^{t}-4}(G_{2^{t}-2,3})\stackrel{{\scriptstyle
w_{1}}}{{\longrightarrow}}H^{2^{t}-3}(G_{2^{t}-2,3})$ and the map
$i^{*}:H^{2^{t}-4}(G_{2^{t}-2,3})\rightarrow H^{2^{t}-4}(G_{2^{t}-3,3})$
induced by the embedding $i:G_{2^{t}-3,3}\hookrightarrow G_{2^{t}-2,3}$, we
have $\ker w_{1}\cap\ker i^{*}=0$.

###### Proof.

For a polynomial $f\in\mathbb{Z}_{2}[w_{1},w_{2},w_{3}]$ such that $[f]\in\ker
w_{1}\cap\ker i^{*}$ we have $[f]=i^{*}[f]=0$ in $H^{2^{t}-4}(G_{2^{t}-3,3})$,
and so $f=\alpha w_{1}\overline{w}_{2^{t}-5}+\beta\overline{w}_{2^{t}-4}$ for
some $\alpha,\beta\in\mathbb{Z}_{2}$. But $\overline{w}_{2^{t}-4}\in
J_{2^{t}-2,3}$, which means that in $H^{2^{t}-4}(G_{2^{t}-2,3})$

$[f]=[\alpha w_{1}\overline{w}_{2^{t}-5}+\beta\overline{w}_{2^{t}-4}]=[\alpha
w_{1}\overline{w}_{2^{t}-5}],$

and so we can take $\beta=0$. We wish to prove that $\alpha=0$ as well.

We also know that $[w_{1}f]=0$ in $H^{2^{t}-3}(G_{2^{t}-2,3})$, and conclude
$w_{1}f=\lambda w_{1}\overline{w}_{2^{t}-4}+\mu\overline{w}_{2^{t}-3}$, for
some $\lambda,\mu\in\mathbb{Z}_{2}$. Combining these equations, we obtain

(5.1) $\alpha w_{1}^{2}\overline{w}_{2^{t}-5}=\lambda
w_{1}\overline{w}_{2^{t}-4}+\mu\overline{w}_{2^{t}-3}.$

Observe the monomial $w_{1}^{2^{t}-5}w_{2}$. We use (2.8) and in Table 3 list
its coefficients in each of the summands from (5.1).

summand | $a$ | $b$ | $c$ | $\binom{a+b+c}{a}\binom{b+c}{b}$
---|---|---|---|---
$\alpha w_{1}^{2}\overline{w}_{2^{t}-5}$ | $2^{t}-7$ | 1 | 0 | 0
$\lambda w_{1}\overline{w}_{2^{t}-4}$ | $2^{t}-6$ | 1 | 0 | 1
$\mu\overline{w}_{2^{t}-3}$ | $2^{t}-5$ | 1 | 0 | 0
Table 3.  Coefficients of $w_{1}^{2^{t}-5}w_{2}$ in (5.1)

From the table we conclude that $\lambda=0$. This means equation (5.1) now
becomes

(5.2) $\alpha w_{1}^{2}\overline{w}_{2^{t}-5}=\mu\overline{w}_{2^{t}-3}.$

The monomial $w_{1}w_{2}^{2^{t-1}-2}$ appears on the right-hand side of (5.2)
with the coefficient $\mu\binom{2^{t-1}-1}{1}\binom{2^{t-1}-2}{0}=\mu$, and it
is obvious that it cannot appear on the left-hand side. We conclude that
$\mu=0$, and consequently $\alpha=0$. This finishes the proof. ∎

## References

  * [1] S. Basu and P. Chakraborty, On the cohomology ring and upper characteristic rank of Grassmannian of oriented $3$-planes, J. Homotopy Relat. Struct. 15 (2020) 27–-60.
  * [2] A. Borel, La cohomologie mod 2 de certains espaces homogènes, Comm. Math. Helv. 27 (1953) 165–197.
  * [3] U. A. Colović and B. I. Prvulović, Gröbner bases in the mod $2$ cohomology of oriented Grassmann manifolds $\widetilde{G}_{2^{t},3}$, to appear in Math. Slovaca (arXiv:2306.05618 [math.AT]).
  * [4] T. Fukaya, Gröbner bases of oriented Grassmann manifolds, Homol. Homotopy Appl. 10:2 (2008) 195–209.
  * [5] J. Korbaš and T. Rusin, A note on the $\mathbb{Z}_{2}$-cohomology algebra of oriented Grassmann manifolds, Rend. Circ. Mat. Palermo, II. Ser 65 (2016) 507–517.
  * [6] J. W. Milnor, J. D. Stasheff, Characteristic Classes, Ann. of Math. Studies 76, Princeton University Press, New Jersey (1974).