The Bundle Theorem in fragments of projective Grassmann spaces

Mariusz Żynel

mariusz@math.uwb.edu.pl

University of Białystok Institute of Mathematics

Combinatorics 2014 Gaeta 2014

Motivations and references

L. R. Wilcox Modularity in the theory of lattices Ann, of Math. 40 (1939), 490–505

J. Kahn Locally projective-planar lattices which satisfy the Bundle Theorem Math. Z. 175 (1980), 219–247

A. Kreuzer Locally projective spaces which satisfy the Bundle Theorem J. Geom. 56 (1996), 87–98

K. Petelczyc, M. Żynel The complement of a point subset in a projective space and a Grassmann space J. Appl. Logic (to appear)

Objectives

Our goal is to recover a deleted hyperplane from a projective Grassmann space.

Mariusz Żynel  The Bundle Theorem...

Pencils

V – a vector space of dimension n with 3 ≤ n < ∞ Sub_{k}(V ) – the set of all k-dimensional subspaces of V Assume that 0 < k < n.

For H ∈ Sub_{k−1}(V ), B ∈ Sub_{k+1}(V ) with H ⊂ B a k-pencil is a set of the form

p(H, B) := {U ∈ Sub_{k}(V ): H ⊂ U ⊂ B}.

U_{3}  H U_{2}  U_{1} B

Grassmann spaces

The point-line structure

𝔐 = P_{k}(V ) = ⟨Sub_{k}(V ), P_{k}(V )⟩,

where P_{V}(k) is the family of all k-pencils, is a Grassmann space. If 0 < k < n, then 𝔐 is a partial linear space. If k = 1 or k = n − 1, then 𝔐 is a projective space.

Mariusz Żynel  The Bundle Theorem...

Basic properties and facts

Every strong subspace of 𝔐 is a projective space. There are two disjoint classes of maximal strong subspaces in 𝔐. Stars For H ∈ Sub_{k−1}(V ) a star is a set of the form

S(H) = [H)_{k} = {U ∈ Sub_{k}(V ): H ⊂ U}.

Tops For B ∈ Sub_{k+1}(V ) a top is a set of the form

T(B) = (B]_{k} = {U ∈ Sub_{k}(V ): U ⊂ B}.

Every line of 𝔐 can be uniquely extended to a star and a top.

Mariusz Żynel  The Bundle Theorem...

Basic properties and facts

Every strong subspace of 𝔐 is a projective space. There are two disjoint classes of maximal strong subspaces in 𝔐. Stars For H ∈ Sub_{k−1}(V ) a star is a set of the form

S(H) = [H)_{k} = {U ∈ Sub_{k}(V ): H ⊂ U}.

Tops For B ∈ Sub_{k+1}(V ) a top is a set of the form

T(B) = (B]_{k} = {U ∈ Sub_{k}(V ): U ⊂ B}.

Every line of 𝔐 can be uniquely extended to a star and a top.

Mariusz Żynel  The Bundle Theorem...

Basic properties and facts

Every strong subspace of 𝔐 is a projective space. There are two disjoint classes of maximal strong subspaces in 𝔐. Stars For H ∈ Sub_{k−1}(V ) a star is a set of the form

S(H) = [H)_{k} = {U ∈ Sub_{k}(V ): H ⊂ U}.

Tops For B ∈ Sub_{k+1}(V ) a top is a set of the form

T(B) = (B]_{k} = {U ∈ Sub_{k}(V ): U ⊂ B}.

Every line of 𝔐 can be uniquely extended to a star and a top.

Mariusz Żynel  The Bundle Theorem...

Spine spaces

W – a fixed subspace of V m – an integer such that k − codim(W ) ≤ m ≤ k, dim(W ) F_{k,m}(W ) := {U ∈ Sub_{k}(V ): dim(U ∩ W ) = m} G_{k,m}(W ) := {L ∩ F_{k,m}(W ): L ∈ P_{k}(V ) and |L ∩ F_{k,m}(W )| ≥ 2}

The point-line structure

A_{k,m}(V , W ) = ⟨F_{k,m}(W ), G_{k,m}(W )⟩

is called a spine space.

K. Prażmowski, M. Żynel

Automorphisms of spine spaces,

Abh. Math. Sem. Univ. Hamb. 72 (2002), 59–77.

Bundles and planes in partial linear spaces

The set [U] of all lines through a point U is called a bundle.

Given three lines L_{1}, L_{2}, L_{3} that form a triangle in a partial linear space, the plane Π(L_{1}, L_{2}, L_{3}) spanned by this triangle is the set of those points which lie on the lines that intersect two of those three lines L_{1}, L_{2}, L_{3} in two distinct points.

In linear spaces where the dimension function can be defined we can say that a plane is a subspace of dimension 2.

Two lines L_{1}, L_{2} ∈ L are said to be coplanar, which is written as L_{1} π L_{2}, iff there is a plane containing them.

The set [U] of all lines through a point U is called a bundle.

Given three lines L_{1}, L_{2}, L_{3} that form a triangle in a partial linear space, the plane Π(L_{1}, L_{2}, L_{3}) spanned by this triangle is the set of those points which lie on the lines that intersect two of those three lines L_{1}, L_{2}, L_{3} in two distinct points.

In linear spaces where the dimension function can be defined we can say that a plane is a subspace of dimension 2.

Two lines L_{1}, L_{2} ∈ L are said to be coplanar, which is written as L_{1} π L_{2}, iff there is a plane containing them.

The set [U] of all lines through a point U is called a bundle.

Given three lines L_{1}, L_{2}, L_{3} that form a triangle in a partial linear space, the plane Π(L_{1}, L_{2}, L_{3}) spanned by this triangle is the set of those points which lie on the lines that intersect two of those three lines L_{1}, L_{2}, L_{3} in two distinct points.

In linear spaces where the dimension function can be defined we can say that a plane is a subspace of dimension 2.

Two lines L_{1}, L_{2} ∈ L are said to be coplanar, which is written as L_{1} π L_{2}, iff there is a plane containing them.

The set [U] of all lines through a point U is called a bundle.

Given three lines L_{1}, L_{2}, L_{3} that form a triangle in a partial linear space, the plane Π(L_{1}, L_{2}, L_{3}) spanned by this triangle is the set of those points which lie on the lines that intersect two of those three lines L_{1}, L_{2}, L_{3} in two distinct points.

In linear spaces where the dimension function can be defined we can say that a plane is a subspace of dimension 2.

Two lines L_{1}, L_{2} ∈ L are said to be coplanar, which is written as L_{1} π L_{2}, iff there is a plane containing them.

Hyperplane complement in a Grassmann space

W – a fixed subspace of V with codim(W ) = k ℋ(W ) := {U ∈ Sub_{k}(V ): U ∩ W ≠ 0} – a hyperplane in 𝔐 S_{ℋ(W )} := {Sub_{k}(V ) \ ℋ(W )} L_{ℋ(W )} := {L ∩ S_{ℋ(W )} : L ∈ P_{k}(V ) and |L ∩ S_{ℋ(W )} | ≥ 2}

𝔇 = 𝔇_{𝔐}(ℋ(W )) = ⟨S_{ℋ(W )} , L_{ℋ(W )} ⟩

is the complement of the hyperplane ℋ(W ) in 𝔐.

Fact In every hyperplane of the form ℋ(W ) in 𝔐 there are points collinear with no point outside ℋ(W ).

𝔇 = ⟨F_{k,0}(W ), G_{k,0}(W )⟩ = A_{k,0}(V , W ) ℋ(W ) = F_{k,1}(W ) ∪ ⋯ ∪ F_{k,min(k,n−k)}(W )

Mariusz Żynel  The Bundle Theorem...

W – a fixed subspace of V with codim(W ) = k ℋ(W ) := {U ∈ Sub_{k}(V ): U ∩ W ≠ 0} – a hyperplane in 𝔐 S_{ℋ(W )} := {Sub_{k}(V ) \ ℋ(W )} L_{ℋ(W )} := {L ∩ S_{ℋ(W )} : L ∈ P_{k}(V ) and |L ∩ S_{ℋ(W )} | ≥ 2}

𝔇 = 𝔇_{𝔐}(ℋ(W )) = ⟨S_{ℋ(W )} , L_{ℋ(W )} ⟩

is the complement of the hyperplane ℋ(W ) in 𝔐.

Fact In every hyperplane of the form ℋ(W ) in 𝔐 there are points collinear with no point outside ℋ(W ).

𝔇 = ⟨F_{k,0}(W ), G_{k,0}(W )⟩ = A_{k,0}(V , W ) ℋ(W ) = F_{k,1}(W ) ∪ ⋯ ∪ F_{k,min(k,n−k)}(W )

Mariusz Żynel  The Bundle Theorem...

W – a fixed subspace of V with codim(W ) = k ℋ(W ) := {U ∈ Sub_{k}(V ): U ∩ W ≠ 0} – a hyperplane in 𝔐 S_{ℋ(W )} := {Sub_{k}(V ) \ ℋ(W )} L_{ℋ(W )} := {L ∩ S_{ℋ(W )} : L ∈ P_{k}(V ) and |L ∩ S_{ℋ(W )} | ≥ 2}

𝔇 = 𝔇_{𝔐}(ℋ(W )) = ⟨S_{ℋ(W )} , L_{ℋ(W )} ⟩

is the complement of the hyperplane ℋ(W ) in 𝔐.

Fact In every hyperplane of the form ℋ(W ) in 𝔐 there are points collinear with no point outside ℋ(W ).

𝔇 = ⟨F_{k,0}(W ), G_{k,0}(W )⟩ = A_{k,0}(V , W ) ℋ(W ) = F_{k,1}(W ) ∪ ⋯ ∪ F_{k,min(k,n−k)}(W )

Mariusz Żynel  The Bundle Theorem...

Key facts about the complement

Lemma (i) If L ∈ A_{k,1}(W ), then S(L), T(L) ⊆ ℋ(W ). (ii) If L ∈ L_{k,1}^{α}(W ), then T(L) ⊈ ℋ(W ) and S(L) ⊆ ℋ(W ). (iii) If L ∈ L_{k,1}^{ω}(W ), then S(L) ⊈ ℋ(W ) and T(L) ⊆ ℋ(W ).

In what follows we assume that 2 < k < n − 2.

Corollary For every point U ∈ F_{k,1}(W ) there are U_{1}, U_{2}, U_{3} ∈ F_{k,0}(W ) such that U, U_{1}, U_{2}, U_{3} form a tetrahedron in 𝔐.

Mariusz Żynel  The Bundle Theorem...

Key facts about the complement

Lemma (i) If L ∈ A_{k,1}(W ), then S(L), T(L) ⊆ ℋ(W ). (ii) If L ∈ L_{k,1}^{α}(W ), then T(L) ⊈ ℋ(W ) and S(L) ⊆ ℋ(W ). (iii) If L ∈ L_{k,1}^{ω}(W ), then S(L) ⊈ ℋ(W ) and T(L) ⊆ ℋ(W ).

In what follows we assume that 2 < k < n − 2.

Corollary For every point U ∈ F_{k,1}(W ) there are U_{1}, U_{2}, U_{3} ∈ F_{k,0}(W ) such that U, U_{1}, U_{2}, U_{3} form a tetrahedron in 𝔐.

Mariusz Żynel  The Bundle Theorem...

Bundle Theorem

Both in the Grassmann space 𝔐 and in the hyperplane complement 𝔇 the following is true.

Theorem (Bundle Theorem) Let L_{1}, L_{2}, L_{3}, L_{4} be lines such that no three of them are coplanar. If five of the six pairs {L_{i}, L_{j}}, 1 ≤ i < j ≤ 4 are coplanar, then so is the sixth pair.

In locally-projective linear spaces (cf. [Kreuzer, 1996]) this theorem is used to prove that two lines determine a bundle uniquely.

Maximal π-clique and bundles

Let K be a maximal π-clique in 𝔇. There are two possibilities: (i) there is a point U in 𝔐 such that U ∈ \overline{L} for all L ∈ K, or (ii) there is a plane Π in 𝔇 such that L ⊂ Π for all L ∈ K.

They can be distinguished by requirement that:

(∗) there are three non-coplanar lines in K. The π-clique K is of type (i) iff it satisfies (∗).

There is a strong subspace in 𝔇 containing K.

Every bundle [U] breaks up into π-cliques.

Let K be a maximal π-clique in 𝔇. There are two possibilities: (i) there is a point U in 𝔐 such that U ∈ \overline{L} for all L ∈ K, or (ii) there is a plane Π in 𝔇 such that L ⊂ Π for all L ∈ K.

They can be distinguished by requirement that:

(∗) there are three non-coplanar lines in K. The π-clique K is of type (i) iff it satisfies (∗).

There is a strong subspace in 𝔇 containing K.

Every bundle [U] breaks up into π-cliques.

Let K be a maximal π-clique in 𝔇. There are two possibilities: (i) there is a point U in 𝔐 such that U ∈ \overline{L} for all L ∈ K, or (ii) there is a plane Π in 𝔇 such that L ⊂ Π for all L ∈ K.

They can be distinguished by requirement that:

(∗) there are three non-coplanar lines in K. The π-clique K is of type (i) iff it satisfies (∗).

There is a strong subspace in 𝔇 containing K.

Every bundle [U] breaks up into π-cliques.

Let K be a maximal π-clique in 𝔇. There are two possibilities: (i) there is a point U in 𝔐 such that U ∈ \overline{L} for all L ∈ K, or (ii) there is a plane Π in 𝔇 such that L ⊂ Π for all L ∈ K.

They can be distinguished by requirement that:

(∗) there are three non-coplanar lines in K. The π-clique K is of type (i) iff it satisfies (∗).

There is a strong subspace in 𝔇 containing K.

Every bundle [U] breaks up into π-cliques.

Gluing π-cliques together

X_{1}, X_{2} – strong subspaces of different types in 𝔇

K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X_{2}}(U_{2}) : ⟺ U_{1}, U_{2} ∈ X_{1} ∩ X_{2} ∧

(∃ L_{1} ∈ K_{π}^{X_{1}}(U_{1}))(∃ L_{2} ∈ K_{π}^{X_{2}}(U_{2}))[ L_{1}, L_{2} ≠ X_{1} ∩ X_{2} ∧ T(L_{1})∩S(L_{2}) or S(L_{1})∩T(L_{2}) is a proper line distinct from X_{1}∩X_{2} ] X_{1}, X_{2} – strong subspaces of the same type in 𝔇

K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X_{2}}(U_{2}) : ⟺ (∃ U ∈ F_{k,0}(W ) ∪ F_{k,1}(W )) (∃ a maximal strong subspace X of the other type than X_{1}, X_{2})

[ U ∈ X ∧ K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X} (U) σ K_{π}^{X_{2}}(U_{2}) ]

σ is an equivalence relation; its equivalence classes are bundles. To every point U ∈ F_{k,0}(W ) ∪ F_{k,1}(W ) we can associate a unique bundle [U]_{𝔇} of lines of 𝔇.

Mariusz Żynel  The Bundle Theorem...

Gluing π-cliques together

X_{1}, X_{2} – strong subspaces of different types in 𝔇

K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X_{2}}(U_{2}) : ⟺ U_{1}, U_{2} ∈ X_{1} ∩ X_{2} ∧

(∃ L_{1} ∈ K_{π}^{X_{1}}(U_{1}))(∃ L_{2} ∈ K_{π}^{X_{2}}(U_{2}))[ L_{1}, L_{2} ≠ X_{1} ∩ X_{2} ∧ T(L_{1})∩S(L_{2}) or S(L_{1})∩T(L_{2}) is a proper line distinct from X_{1}∩X_{2} ] X_{1}, X_{2} – strong subspaces of the same type in 𝔇

K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X_{2}}(U_{2}) : ⟺ (∃ U ∈ F_{k,0}(W ) ∪ F_{k,1}(W )) (∃ a maximal strong subspace X of the other type than X_{1}, X_{2})

[ U ∈ X ∧ K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X} (U) σ K_{π}^{X_{2}}(U_{2}) ]

σ is an equivalence relation; its equivalence classes are bundles. To every point U ∈ F_{k,0}(W ) ∪ F_{k,1}(W ) we can associate a unique bundle [U]_{𝔇} of lines of 𝔇.

Mariusz Żynel  The Bundle Theorem...

Gluing π-cliques together

X_{1}, X_{2} – strong subspaces of different types in 𝔇

K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X_{2}}(U_{2}) : ⟺ U_{1}, U_{2} ∈ X_{1} ∩ X_{2} ∧

(∃ L_{1} ∈ K_{π}^{X_{1}}(U_{1}))(∃ L_{2} ∈ K_{π}^{X_{2}}(U_{2}))[ L_{1}, L_{2} ≠ X_{1} ∩ X_{2} ∧ T(L_{1})∩S(L_{2}) or S(L_{1})∩T(L_{2}) is a proper line distinct from X_{1}∩X_{2} ] X_{1}, X_{2} – strong subspaces of the same type in 𝔇

K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X_{2}}(U_{2}) : ⟺ (∃ U ∈ F_{k,0}(W ) ∪ F_{k,1}(W )) (∃ a maximal strong subspace X of the other type than X_{1}, X_{2})

[ U ∈ X ∧ K_{π}^{X_{1}}(U_{1}) σ K_{π}^{X} (U) σ K_{π}^{X_{2}}(U_{2}) ]

σ is an equivalence relation; its equivalence classes are bundles. To every point U ∈ F_{k,0}(W ) ∪ F_{k,1}(W ) we can associate a unique bundle [U]_{𝔇} of lines of 𝔇.

Mariusz Żynel  The Bundle Theorem...

Adjacency of bundles

We will write ℬ(𝔇) for the set of all bundles in 𝔇. If a ∈ ℬ(𝔇) we write ā for the bundle of 𝔐 with a ⊆ ā. For a point U of 𝔇 and a bundle a ∈ ℬ(𝔇) we write U ∼ a if there is a line L ∈ a such that U ∈ L, we write L = \overline{U, a}. Frankly, two bundles a, b ∈ ℬ(𝔇) are adjacent iff they share a line. a ∼ b : ⟺ (∃ U ∈ F_{k,0}(W ))[ U ∼ a ∧ U ∼ b ∧ \overline{U, a} π \overline{U, b}]

a  U b

Lemma For two distinct bundles a, b ∈ ℬ(𝔇) we have a ∼ b iff there is a proper line L or L ∈ L_{k,1}^{α}(W ) ∪ L_{k,1}^{ω}(W ) such that L ∈ ā ∩ b̄.

Lines of bundles

Given two distinct adjacent bundles a, b ∈ ℬ(𝔇) we can define the new line through a, b:

[a, b] := {c ∈ ℬ(𝔇): for all points U of 𝔇 with [U]_{𝔇} ≠ a, b, c if U ∼ a, b, then U ∼ c and \overline{U, a}, \overline{U, b}, \overline{U, c} are coplanar\}.

a  c U b

Lemma If a, b ∈ ℬ(𝔇), a ≠ b, a ∼ b, and L ∈ ā ∩ b̄, then

[a, b] = {c ∈ ℬ(𝔇): L ∈ c̄}.

Mariusz Żynel  The Bundle Theorem...

Bundle space

The set of all new lines will be written as

𝔏(𝔇) := {[a, b]: a, b ∈ ℬ(𝔇), a ∼ b and a ≠ b}. The structure

𝔅(𝔇) := ⟨ℬ(𝔇), 𝔏(𝔇)⟩

is called the bundle space over 𝔇. It is a partial linear space. Proposition The bundle space 𝔅(𝔇) is definable in terms of 𝔇. The bundle space 𝔅(𝔇) is isomorphic to

𝔇′ = ⟨F_{k,0}(W ) ∪ F_{k,1}(W ), G_{k,0}(W ) ∪ L_{k,1}^{α}(W ) ∪ L_{k,1}^{ω}(W )⟩. This is a new proof of a know fact that 𝔇′ is definable in 𝔇.

Mariusz Żynel  The Bundle Theorem...

Bundle space

The set of all new lines will be written as

𝔏(𝔇) := {[a, b]: a, b ∈ ℬ(𝔇), a ∼ b and a ≠ b}. The structure

𝔅(𝔇) := ⟨ℬ(𝔇), 𝔏(𝔇)⟩

is called the bundle space over 𝔇. It is a partial linear space. Proposition The bundle space 𝔅(𝔇) is definable in terms of 𝔇. The bundle space 𝔅(𝔇) is isomorphic to

𝔇′ = ⟨F_{k,0}(W ) ∪ F_{k,1}(W ), G_{k,0}(W ) ∪ L_{k,1}^{α}(W ) ∪ L_{k,1}^{ω}(W )⟩. This is a new proof of a know fact that 𝔇′ is definable in 𝔇.

Mariusz Żynel  The Bundle Theorem...

Bundle space

The set of all new lines will be written as

𝔏(𝔇) := {[a, b]: a, b ∈ ℬ(𝔇), a ∼ b and a ≠ b}. The structure

𝔅(𝔇) := ⟨ℬ(𝔇), 𝔏(𝔇)⟩

is called the bundle space over 𝔇. It is a partial linear space. Proposition The bundle space 𝔅(𝔇) is definable in terms of 𝔇. The bundle space 𝔅(𝔇) is isomorphic to

𝔇′ = ⟨F_{k,0}(W ) ∪ F_{k,1}(W ), G_{k,0}(W ) ∪ L_{k,1}^{α}(W ) ∪ L_{k,1}^{ω}(W )⟩. This is a new proof of a know fact that 𝔇′ is definable in 𝔇.

Mariusz Żynel  The Bundle Theorem...

Affine lines on the horizon

The affine lines A_{k,1}(W ) are not yet recovered. Let Π be a plane in 𝔇′. There are two possibilities:

(i) Π is a strong subspace of 𝔇′, then Π is a proper plane in 𝔇 or it is a projective plane in A_{k,1}(V , W ),

(ii) Π contains collinear and non-collinear points, then Π is a semi-affine plane in A_{k,1}(V , W ). Collinearity w.r.t. affine lines A_{k,1}(W ) can be defined as follows:

L^{τ} (a, b, c) : ⟺ (∃ a semi-affine plane Π in 𝔇′) [ a, b, c ∈ Π ∧ ≁(a, b, c) ].

Proposition (Prażmowski and Żynel, 2002)

For points a, b, c in 𝔇′ we have L^{τ} (a, b, c) iff there is a line L ∈ A_{k,1}(W ) such that a, b, c ∈ L.

The affine lines A_{k,1}(W ) are not yet recovered. Let Π be a plane in 𝔇′. There are two possibilities:

(i) Π is a strong subspace of 𝔇′, then Π is a proper plane in 𝔇 or it is a projective plane in A_{k,1}(V , W ),

(ii) Π contains collinear and non-collinear points, then Π is a semi-affine plane in A_{k,1}(V , W ). Collinearity w.r.t. affine lines A_{k,1}(W ) can be defined as follows:

L^{τ} (a, b, c) : ⟺ (∃ a semi-affine plane Π in 𝔇′) [ a, b, c ∈ Π ∧ ≁(a, b, c) ].

Proposition (Prażmowski and Żynel, 2002)

For points a, b, c in 𝔇′ we have L^{τ} (a, b, c) iff there is a line L ∈ A_{k,1}(W ) such that a, b, c ∈ L.

The affine lines A_{k,1}(W ) are not yet recovered. Let Π be a plane in 𝔇′. There are two possibilities:

(i) Π is a strong subspace of 𝔇′, then Π is a proper plane in 𝔇 or it is a projective plane in A_{k,1}(V , W ),

(ii) Π contains collinear and non-collinear points, then Π is a semi-affine plane in A_{k,1}(V , W ). Collinearity w.r.t. affine lines A_{k,1}(W ) can be defined as follows:

L^{τ} (a, b, c) : ⟺ (∃ a semi-affine plane Π in 𝔇′) [ a, b, c ∈ Π ∧ ≁(a, b, c) ].

Proposition (Prażmowski and Żynel, 2002)

For points a, b, c in 𝔇′ we have L^{τ} (a, b, c) iff there is a line L ∈ A_{k,1}(W ) such that a, b, c ∈ L.

Recovery of the hyperplane ℋ(W )

Proposition (Prażmowski and Żynel, 2002) A_{k,1}(V , W ) is definable in terms of A_{k,0}(V , W ).

Continuing this procedure we can recover A_{k,2}(V , W ), then A_{k,3}(V , W ), and so on. Finally, A_{k,min(k,n−k)}(V , W ) is recoverable in 𝔇 = A_{k,0}(V , W ).

Theorem (Prażmowski and Żynel, 2002) The hyperplane ℋ(W ) and the ambient space 𝔐 are recoverable from the complement 𝔇.

Proposition (Prażmowski and Żynel, 2002) A_{k,1}(V , W ) is definable in terms of A_{k,0}(V , W ).

Continuing this procedure we can recover A_{k,2}(V , W ), then A_{k,3}(V , W ), and so on. Finally, A_{k,min(k,n−k)}(V , W ) is recoverable in 𝔇 = A_{k,0}(V , W ).

Theorem (Prażmowski and Żynel, 2002) The hyperplane ℋ(W ) and the ambient space 𝔐 are recoverable from the complement 𝔇.

Proposition (Prażmowski and Żynel, 2002) A_{k,1}(V , W ) is definable in terms of A_{k,0}(V , W ).

Continuing this procedure we can recover A_{k,2}(V , W ), then A_{k,3}(V , W ), and so on. Finally, A_{k,min(k,n−k)}(V , W ) is recoverable in 𝔇 = A_{k,0}(V , W ).

Theorem (Prażmowski and Żynel, 2002) The hyperplane ℋ(W ) and the ambient space 𝔐 are recoverable from the complement 𝔇.

Proposition (Prażmowski and Żynel, 2002) A_{k,1}(V , W ) is definable in terms of A_{k,0}(V , W ).

Continuing this procedure we can recover A_{k,2}(V , W ), then A_{k,3}(V , W ), and so on. Finally, A_{k,min(k,n−k)}(V , W ) is recoverable in 𝔇 = A_{k,0}(V , W ).

Theorem (Prażmowski and Żynel, 2002) The hyperplane ℋ(W ) and the ambient space 𝔐 are recoverable from the complement 𝔇.

Summary and final remarks

DONE: We have successfully recovered the ambient projective Grassmann space form its hyperplane complement.

TO DO: Express the relation σ in terms of the points, lines and planes of the complement 𝔇.

Delete two hyperplanes from a Grassmann space and check how to recover the ambient space from their complement.

Summary and final remarks

DONE: We have successfully recovered the ambient projective Grassmann space form its hyperplane complement.

TO DO: Express the relation σ in terms of the points, lines and planes of the complement 𝔇.

Delete two hyperplanes from a Grassmann space and check how to recover the ambient space from their complement.

Thank you for your attention