k=1

The cohomology of $C_2$-equivariant Grassmannians $\text{Gr}_k(\mathbb{R}^{p,q})$ when $k=1$

These are the projective spaces $\mathbb{P}(\text{Gr}_k\mathbb{R}^{p,q})$. Kronholm defines "twisted" projective space $\mathbb{RP}^n_{tw}=\mathbb{P}(\mathbb{R}^{n+1,\lfloor\frac {n+1}2\rfloor})$.

$k=1$	$q=1$	$q=2$	$q=3$	$q=4$	$q=5$	$q=6$
$p=2$	$\mathbb{RP}^1_{tw}$
$p=3$	$\mathbb{RP}^2_{tw}$
$p=4$	$\text{Gr}_1(\mathbb{R}^{4,1})$	$\mathbb{RP}^3_{tw}$
$p=5$	$\text{Gr}_1(\mathbb{R}^{5,1})$	$\mathbb{RP}^4_{tw}$
$p=6$	$\text{Gr}_1(\mathbb{R}^{6,1})$	$\text{Gr}_1(\mathbb{R}^{6,2})$	$\mathbb{RP}^5_{tw}$
$p=7$	$\text{Gr}_1(\mathbb{R}^{7,1})$	$\text{Gr}_1(\mathbb{R}^{7,2})$	$\mathbb{RP}^6_{tw}$
$p=8$	$\text{Gr}_1(\mathbb{R}^{8,1})$	$\text{Gr}_1(\mathbb{R}^{8,2})$	$\text{Gr}_1(\mathbb{R}^{8,3})$	$\mathbb{RP}^7_{tw}$
$p=9$	$\text{Gr}_1(\mathbb{R}^{9,1})$	$\text{Gr}_1(\mathbb{R}^{9,2})$	$\text{Gr}_1(\mathbb{R}^{9,3})$	$\mathbb{RP}^8_{tw}$
$p=10$	$\text{Gr}_1(\mathbb{R}^{10,1})$	$\text{Gr}_1(\mathbb{R}^{10,2})$	$\text{Gr}_1(\mathbb{R}^{10,3})$	$\text{Gr}_1(\mathbb{R}^{10,4})$	$\mathbb{RP}^9_{tw}$
$p=11$	$\text{Gr}_1(\mathbb{R}^{11,1})$	$\text{Gr}_1(\mathbb{R}^{11,2})$	$\text{Gr}_1(\mathbb{R}^{11,3})$	$\text{Gr}_1(\mathbb{R}^{11,4})$	$\mathbb{RP}^{10}_{tw}$
$p=12$	$\text{Gr}_1(\mathbb{R}^{12,1})$	$\text{Gr}_1(\mathbb{R}^{12,2})$	$\text{Gr}_1(\mathbb{R}^{12,3})$	$\text{Gr}_1(\mathbb{R}^{12,4})$	$\text{Gr}_1(\mathbb{R}^{12,5})$	$\mathbb{RP}^{11}_{tw}$

A pattern emerges here. Kronholm gives $$H^{\ast,\ast}(\mathbb{RP}^n_{tw})=\bigoplus_{k=0}^n\Sigma^{k,\lceil\frac k2\rceil}\mathbb M_2$$ and more generally we can select constructions whose filtration spectral sequences have no nonzero differentials, for example if for $q\le \frac p2$ we identify $V=\mathbb{R}^{p,q}\cong (\mathbb{R}^{+-})^{\oplus q}\oplus (\mathbb{R}^+)^{\oplus (p-2q)}$ then $$H^{\ast,\ast}(\mathbb{P}(\mathbb{R}^{p,q}))=E_1(V)=\bigoplus_{k=0}^p\Sigma^{k,\min(q,\lceil\frac k2\rceil)}\mathbb M_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{2,1})$

Also known as $S^{1,1}$ or as $\mathbb{RP}^1_{tw}$.
Poincaré polynomial: $$x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{2,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{3,1})$

Poincaré polynomial: $$x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{3,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{4,1})$

Poincaré polynomial: $$x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{4,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{4,2})$

Poincaré polynomial: $$x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{4,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{5,1})$

Poincaré polynomial: $$x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{5,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{5,2})$

Poincaré polynomial: $$x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{5,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{6,1})$

Poincaré polynomial: $$x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{6,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{6,2})$

Poincaré polynomial: $$x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{6,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{6,3})$

Poincaré polynomial: $$x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{6,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{7,1})$

Poincaré polynomial: $$x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{7,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{7,2})$

Poincaré polynomial: $$x^{6} y^{2} + x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{7,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{7,3})$

Poincaré polynomial: $$x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{7,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{8,1})$

Poincaré polynomial: $$x^{7} y + x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{8,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{8,2})$

Poincaré polynomial: $$x^{7} y^{2} + x^{6} y^{2} + x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{8,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{8,3})$

Poincaré polynomial: $$x^{7} y^{3} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{8,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{8,4})$

Poincaré polynomial: $$x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{8,4}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{9,1})$

Poincaré polynomial: $$x^{8} y + x^{7} y + x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{9,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{8,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{9,2})$

Poincaré polynomial: $$x^{8} y^{2} + x^{7} y^{2} + x^{6} y^{2} + x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{9,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{9,3})$

Poincaré polynomial: $$x^{8} y^{3} + x^{7} y^{3} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{9,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,3}\mathbb{M}_2\oplus\Sigma^{8,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{9,4})$

Poincaré polynomial: $$x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{9,4}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{10,1})$

Poincaré polynomial: $$x^{9} y + x^{8} y + x^{7} y + x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{10,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{8,1}\mathbb{M}_2\oplus\Sigma^{9,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{10,2})$

Poincaré polynomial: $$x^{9} y^{2} + x^{8} y^{2} + x^{7} y^{2} + x^{6} y^{2} + x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{10,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{10,3})$

Poincaré polynomial: $$x^{9} y^{3} + x^{8} y^{3} + x^{7} y^{3} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{10,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,3}\mathbb{M}_2\oplus\Sigma^{8,3}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{10,4})$

Poincaré polynomial: $$x^{9} y^{4} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{10,4}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,4}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{10,5})$

Poincaré polynomial: $$x^{9} y^{5} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{10,5}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{11,1})$

Poincaré polynomial: $$x^{10} y + x^{9} y + x^{8} y + x^{7} y + x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{11,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{8,1}\mathbb{M}_2\oplus\Sigma^{9,1}\mathbb{M}_2\oplus\Sigma^{10,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{11,2})$

Poincaré polynomial: $$x^{10} y^{2} + x^{9} y^{2} + x^{8} y^{2} + x^{7} y^{2} + x^{6} y^{2} + x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{11,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{11,3})$

Poincaré polynomial: $$x^{10} y^{3} + x^{9} y^{3} + x^{8} y^{3} + x^{7} y^{3} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{11,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,3}\mathbb{M}_2\oplus\Sigma^{8,3}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2\oplus\Sigma^{10,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{11,4})$

Poincaré polynomial: $$x^{10} y^{4} + x^{9} y^{4} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{11,4}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,4}\mathbb{M}_2\oplus\Sigma^{10,4}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{11,5})$

Poincaré polynomial: $$x^{10} y^{5} + x^{9} y^{5} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{11,5}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2\oplus\Sigma^{10,5}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{12,1})$

Poincaré polynomial: $$x^{11} y + x^{10} y + x^{9} y + x^{8} y + x^{7} y + x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{12,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{8,1}\mathbb{M}_2\oplus\Sigma^{9,1}\mathbb{M}_2\oplus\Sigma^{10,1}\mathbb{M}_2\oplus\Sigma^{11,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{12,2})$

Poincaré polynomial: $$x^{11} y^{2} + x^{10} y^{2} + x^{9} y^{2} + x^{8} y^{2} + x^{7} y^{2} + x^{6} y^{2} + x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{12,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2\oplus\Sigma^{11,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{12,3})$

Poincaré polynomial: $$x^{11} y^{3} + x^{10} y^{3} + x^{9} y^{3} + x^{8} y^{3} + x^{7} y^{3} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{12,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,3}\mathbb{M}_2\oplus\Sigma^{8,3}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2\oplus\Sigma^{10,3}\mathbb{M}_2\oplus\Sigma^{11,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{12,4})$

Poincaré polynomial: $$x^{11} y^{4} + x^{10} y^{4} + x^{9} y^{4} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{12,4}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,4}\mathbb{M}_2\oplus\Sigma^{10,4}\mathbb{M}_2\oplus\Sigma^{11,4}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{12,5})$

Poincaré polynomial: $$x^{11} y^{5} + x^{10} y^{5} + x^{9} y^{5} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{12,5}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2\oplus\Sigma^{10,5}\mathbb{M}_2\oplus\Sigma^{11,5}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{12,6})$

Poincaré polynomial: $$x^{11} y^{6} + x^{10} y^{5} + x^{9} y^{5} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{12,6}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2\oplus\Sigma^{10,5}\mathbb{M}_2\oplus\Sigma^{11,6}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{13,1})$

Poincaré polynomial: $$x^{12} y + x^{11} y + x^{10} y + x^{9} y + x^{8} y + x^{7} y + x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{13,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{8,1}\mathbb{M}_2\oplus\Sigma^{9,1}\mathbb{M}_2\oplus\Sigma^{10,1}\mathbb{M}_2\oplus\Sigma^{11,1}\mathbb{M}_2\oplus\Sigma^{12,1}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{13,2})$

Poincaré polynomial: $$x^{12} y^{2} + x^{11} y^{2} + x^{10} y^{2} + x^{9} y^{2} + x^{8} y^{2} + x^{7} y^{2} + x^{6} y^{2} + x^{5} y^{2} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{13,2}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2\oplus\Sigma^{11,2}\mathbb{M}_2\oplus\Sigma^{12,2}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{13,3})$

Poincaré polynomial: $$x^{12} y^{3} + x^{11} y^{3} + x^{10} y^{3} + x^{9} y^{3} + x^{8} y^{3} + x^{7} y^{3} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{13,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,3}\mathbb{M}_2\oplus\Sigma^{8,3}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2\oplus\Sigma^{10,3}\mathbb{M}_2\oplus\Sigma^{11,3}\mathbb{M}_2\oplus\Sigma^{12,3}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{13,4})$

Poincaré polynomial: $$x^{12} y^{4} + x^{11} y^{4} + x^{10} y^{4} + x^{9} y^{4} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{13,4}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,4}\mathbb{M}_2\oplus\Sigma^{10,4}\mathbb{M}_2\oplus\Sigma^{11,4}\mathbb{M}_2\oplus\Sigma^{12,4}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{13,5})$

Poincaré polynomial: $$x^{12} y^{5} + x^{11} y^{5} + x^{10} y^{5} + x^{9} y^{5} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{13,5}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2\oplus\Sigma^{10,5}\mathbb{M}_2\oplus\Sigma^{11,5}\mathbb{M}_2\oplus\Sigma^{12,5}\mathbb{M}_2.$$

The cohomology of $\text{Gr}_1(\mathbb{R}^{13,6})$

Poincaré polynomial: $$x^{12} y^{6} + x^{11} y^{6} + x^{10} y^{5} + x^{9} y^{5} + x^{8} y^{4} + x^{7} y^{4} + x^{6} y^{3} + x^{5} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y + x y + 1$$ Generator count grid: Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{13,6}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2\oplus\Sigma^{10,5}\mathbb{M}_2\oplus\Sigma^{11,6}\mathbb{M}_2\oplus\Sigma^{12,6}\mathbb{M}_2.$$