Discrete Lusternik–Schnirelmann category
Nick Scoville1
joint with Seth Aaronson 2
1 Ursinus
2 Temple
Discrete Lusternik–Schnirelmann category
Brian Green1
College
University
Scoville, Aaronson, Green
Introduction
The Lusternik-Schnirelmann category of a topological space X ,
denoted cat(X ), is the least integer m such that there exists open
subsets {U0 , U1 , . . . , Um } covering X with each Ui contractible in
X.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Introduction
The Lusternik-Schnirelmann category of a topological space X ,
denoted cat(X ), is the least integer m such that there exists open
subsets {U0 , U1 , . . . , Um } covering X with each Ui contractible in
X.
Example
cat(S n ) = 1
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Introduction
The Lusternik-Schnirelmann category of a topological space X ,
denoted cat(X ), is the least integer m such that there exists open
subsets {U0 , U1 , . . . , Um } covering X with each Ui contractible in
X.
Example
cat(S n ) = 1
In general, cat(ΣX ) ≤ 1.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Introduction
The Lusternik-Schnirelmann category of a topological space X ,
denoted cat(X ), is the least integer m such that there exists open
subsets {U0 , U1 , . . . , Um } covering X with each Ui contractible in
X.
Example
cat(S n ) = 1
In general, cat(ΣX ) ≤ 1.
Theorem
(Lusternik, Schnirelmann 1934) If M is a smooth manifold
satisfying additional properties and f : M → R is smooth with m
critical points, then cat(M) + 1 ≤ m.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility
Recall that if a simplicial complex K contains a pair of simplicies
(σ, τ ) such that σ is a face of τ and σ has no other cofaces, then
K is said to collapse onto K − {σ, τ }. The complex K is said to
be collapsible if there exits a sequence of collapses
K → K1 → . . . → {v }.
Bing’s house with two rooms and the dunce cap are examples of
simplicial complexes which are not collpasible, but whose
geometric realization is contractible.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility Caveats
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility Caveats
• W. Lickorish and J. Martin have shown that collapsibility is
not invariant under barycentric subdivision.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility Caveats
• W. Lickorish and J. Martin have shown that collapsibility is
not invariant under barycentric subdivision.
• B. Benedetti and F. Lutz have shown the existence of a
(collapsible) 3-ball with 8 vertices, but that has a collapse
onto the dunce cap.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility Caveats
• W. Lickorish and J. Martin have shown that collapsibility is
not invariant under barycentric subdivision.
• B. Benedetti and F. Lutz have shown the existence of a
(collapsible) 3-ball with 8 vertices, but that has a collapse
onto the dunce cap.
• Allowing elementary expansions (i.e. working up to simple
homotopy type) would eliminate Bing’s house and dunce cap
as interesting examples (and “too close” to homotopy type).
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Collapsibility Caveats
• W. Lickorish and J. Martin have shown that collapsibility is
not invariant under barycentric subdivision.
• B. Benedetti and F. Lutz have shown the existence of a
(collapsible) 3-ball with 8 vertices, but that has a collapse
onto the dunce cap.
• Allowing elementary expansions (i.e. working up to simple
homotopy type) would eliminate Bing’s house and dunce cap
as interesting examples (and “too close” to homotopy type).
• If L ⊆ K and L & L0 is a collapse, we would like
dcatK (L) = dcatK (L0 ).
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS pre-category
Definition
Let L be a (closed) subcomplex of K . We say that L has discrete
g K (L) = m, if m is the least integer
pre-category m, denoted dcat
such that there exists m + 1 closed subcomplexes {U0 , U1 , . . . , Um }
m
S
Ui .
of K each of which is collapsible such that L ⊆
i=0
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS pre-category
Definition
Let L be a (closed) subcomplex of K . We say that L has discrete
g K (L) = m, if m is the least integer
pre-category m, denoted dcat
such that there exists m + 1 closed subcomplexes {U0 , U1 , . . . , Um }
m
S
Ui .
of K each of which is collapsible such that L ⊆
i=0
Example
Let B be the collapsible 3-ball of Benedetti and Lutz, D ⊆ B the
g B (D) = 0 while
dunce cap. Since B is collapsible, dcat
g
dcat(D)
= 1.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Let K =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Let K =
Then






,
















g ) ≤ 1.
is a collapsible cover of K . Hence dcat(K
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS category
g K (L) ≥ dcat
g K (L0 ). Can I show that
If L & L0 , then dcat
0
g K (L) ≤ dcat
g K (L )?
dcat
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS category
g K (L) ≥ dcat
g K (L0 ). Can I show that
If L & L0 , then dcat
0
g K (L) ≤ dcat
g K (L )?
dcat
Not yet...
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS category
g K (L) ≥ dcat
g K (L0 ). Can I show that
If L & L0 , then dcat
0
g K (L) ≤ dcat
g K (L )?
dcat
Not yet...
Definition
The discrete category of L in K is defined by
g K (L0 ) : L collapses to L0 in K }.
dcatK (L) = min{dcat
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Smooth vs Discrete LS category
Since a collapse corresponds to a deformation retraction, we have
that a collapsible complex is contractible. Hence
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Smooth vs Discrete LS category
Since a collapse corresponds to a deformation retraction, we have
that a collapsible complex is contractible. Hence
Proposition
cat(|K |) ≤ dcat(K ).
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Smooth vs Discrete LS category
Since a collapse corresponds to a deformation retraction, we have
that a collapsible complex is contractible. Hence
Proposition
cat(|K |) ≤ dcat(K ).
Example
dcat(S n ) = 1
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Smooth vs Discrete LS category
Since a collapse corresponds to a deformation retraction, we have
that a collapsible complex is contractible. Hence
Proposition
cat(|K |) ≤ dcat(K ).
Example
dcat(S n ) = 1
dcat(ΣK ) ≤ 1
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Smooth vs Discrete LS category
Since a collapse corresponds to a deformation retraction, we have
that a collapsible complex is contractible. Hence
Proposition
cat(|K |) ≤ dcat(K ).
Example
dcat(S n ) = 1
dcat(ΣK ) ≤ 1
dcat(CK ) = 0.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
dcat(K ) 6≤ dim(K )
In the smooth case, it is well-known that cat(X ) ≤ dim(X ). This
is not true in the discrete case, even for a 1-dimensional complex.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
dcat(K ) 6≤ dim(K )
In the smooth case, it is well-known that cat(X ) ≤ dim(X ). This
is not true in the discrete case, even for a 1-dimensional complex.
Proposition
Let K be a 1-dimensional complex with v vertices and e edges.
e
Then d v −1
e − 1 ≤ dcat(K ).
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
dcat(K ) 6≤ dim(K )
In the smooth case, it is well-known that cat(X ) ≤ dim(X ). This
is not true in the discrete case, even for a 1-dimensional complex.
Proposition
Let K be a 1-dimensional complex with v vertices and e edges.
e
Then d v −1
e − 1 ≤ dcat(K ).
This is a special case of a more general combinatorial lower bound
which we will discuss later.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
dcat(K ) 6≤ dim(K )
In the smooth case, it is well-known that cat(X ) ≤ dim(X ). This
is not true in the discrete case, even for a 1-dimensional complex.
Proposition
Let K be a 1-dimensional complex with v vertices and e edges.
e
Then d v −1
e − 1 ≤ dcat(K ).
This is a special case of a more general combinatorial lower bound
which we will discuss later.
Example
Let K = Kn , the complete graph on n vertices. Then
e
d v −1
e − 1 = d n2 e − 1 ≤ dcat(K ).
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
dcat(K ) 6≤ dim(K )
In the smooth case, it is well-known that cat(X ) ≤ dim(X ). This
is not true in the discrete case, even for a 1-dimensional complex.
Proposition
Let K be a 1-dimensional complex with v vertices and e edges.
e
Then d v −1
e − 1 ≤ dcat(K ).
This is a special case of a more general combinatorial lower bound
which we will discuss later.
Example
Let K = Kn , the complete graph on n vertices. Then
e
d v −1
e − 1 = d n2 e − 1 ≤ dcat(K ). A collapsible cover of size d n2 e
can be constructed for Kn . Thus
dcat(K ) = d n2 e − 1 6≤ 1 = dim(K ).
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete Morse theory
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
3
2
5
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
3
2
5
Discrete Lusternik–Schnirelmann category
2
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
3
2
4
5
5
Discrete Lusternik–Schnirelmann category
2
2
7
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
3
2
4
5
2
2
5
7
2
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
10
8
2
2
5
2
11
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
10
8
2
2
5
2
11
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
10
8
2
2
5
2
11
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
10
8
2
2
5
2
11
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
10
8
2
2
5
2
11
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Discrete Morse theory
Let K =
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
10
8
2
2
5
2
11
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
Discrete Lusternik–Schnirelmann category
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (0) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (1) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (3) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (6) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (7) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (8) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (9) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (14) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Building a complex
The critical values are 0, 1, 3, 6, 7, 8, 9, 14, and 15.
0
4
3
2
4
5
8
2
2
5
2
11
10
7
1
9
6
15
12
14
13
K (15) =
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS Theorem
Theorem
Let f : K → R be a discrete Morse function with m critical points.
Then dcat(K ) + 1 ≤ m.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS Theorem
Theorem
Let f : K → R be a discrete Morse function with m critical points.
Then dcat(K ) + 1 ≤ m.
Proof.
Let cn := min{a ∈ R : ∃L(a) ⊆ K s.t. dcatK (L(a)) ≥ n − 1}.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS Theorem
Theorem
Let f : K → R be a discrete Morse function with m critical points.
Then dcat(K ) + 1 ≤ m.
Proof.
Let cn := min{a ∈ R : ∃L(a) ⊆ K s.t. dcatK (L(a)) ≥ n − 1}.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS Theorem
Theorem
Let f : K → R be a discrete Morse function with m critical points.
Then dcat(K ) + 1 ≤ m.
Proof.
Let cn := min{a ∈ R : ∃L(a) ⊆ K s.t. dcatK (L(a)) ≥ n − 1}. If cn
is a regular value of f , then there is an > 0 such that
K (cn + ) & K (cn − ) in K by R. Forman.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS Theorem
Theorem
Let f : K → R be a discrete Morse function with m critical points.
Then dcat(K ) + 1 ≤ m.
Proof.
Let cn := min{a ∈ R : ∃L(a) ⊆ K s.t. dcatK (L(a)) ≥ n − 1}. If cn
is a regular value of f , then there is an > 0 such that
K (cn + ) & K (cn − ) in K by R. Forman. Hence
dcatK (K (cn + )) = dcatK (K (cn − )) ≥ n − 1 so that
cn > cn − ∈ {a ∈ R : ∃L(a) ⊆ K s.t dcatK (L(a)) ≥ n − 1}
contradicting the fact that cn is minimum. Thus each cn is a
critical value of f .
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Discrete LS Theorem
Theorem
Let f : K → R be a discrete Morse function with m critical points.
Then dcat(K ) + 1 ≤ m.
Proof.
Let cn := min{a ∈ R : ∃L(a) ⊆ K s.t. dcatK (L(a)) ≥ n − 1}. If cn
is a regular value of f , then there is an > 0 such that
K (cn + ) & K (cn − ) in K by R. Forman. Hence
dcatK (K (cn + )) = dcatK (K (cn − )) ≥ n − 1 so that
cn > cn − ∈ {a ∈ R : ∃L(a) ⊆ K s.t dcatK (L(a)) ≥ n − 1}
contradicting the fact that cn is minimum. Thus each cn is a
critical value of f . A straightforward induction argument now
shows that if c1 < c2 < . . . < cdcat(K )+1 are the critical values,
then K (cdcat(K )+1 ) ⊆ K contains at least dcat(K ) + 1 critical
simplicies. Thus dcat(K ) + 1 ≤ m.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Lower bound
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Lower bound
Let K be a simplicial complex of dimension n or n + 1, and let ciK
denote the number of simplicies of K of dimension i. Let
K .
E (c K ) := c0K + c2K + . . . + cnK and O(c K ) := c1K + c3K + . . . + cn±1
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Lower bound
Let K be a simplicial complex of dimension n or n + 1, and let ciK
denote the number of simplicies of K of dimension i. Let
K .
E (c K ) := c0K + c2K + . . . + cnK and O(c K ) := c1K + c3K + . . . + cn±1
Proposition
Let K be a simplicial complex of dimension n with ci the number
K
K
of
l simplicies
m of K of dimension i. If E (c ) − 1 ≥ O(c ), then
K
E (c )−1
− 1 ≤ dcat(K ). If E (c K ) − 1 ≤ O(c K ), then
K
l O(c K ) m
O(c )
− 1 ≤ dcat(K ).
E (c K )−1
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Computation
Let K be a simplicial complex.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Computation
Let K be a simplicial complex.
1
Initialize V := K and U := ∅.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Computation
Let K be a simplicial complex.
1
Initialize V := K and U := ∅.
2
Let U = ∅. Pick random top dimensional subcomplex ∆ and
add to U, remove from V .
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Computation
Let K be a simplicial complex.
1
Initialize V := K and U := ∅.
2
Let U = ∅. Pick random top dimensional subcomplex ∆ and
add to U, remove from V .
3
Expand along 0-simplicies, then 1-simplicies, 2-simplicies, etc.
Add to U, remove from V .
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Computation
Let K be a simplicial complex.
1
Initialize V := K and U := ∅.
2
Let U = ∅. Pick random top dimensional subcomplex ∆ and
add to U, remove from V .
3
Expand along 0-simplicies, then 1-simplicies, 2-simplicies, etc.
Add to U, remove from V .
4
Add U to U. Go to step 2.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Computation
Let K be a simplicial complex.
1
Initialize V := K and U := ∅.
2
Let U = ∅. Pick random top dimensional subcomplex ∆ and
add to U, remove from V .
3
Expand along 0-simplicies, then 1-simplicies, 2-simplicies, etc.
Add to U, remove from V .
4
Add U to U. Go to step 2.
End when V = ∅.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Computation
Let K be a simplicial complex.
1
Initialize V := K and U := ∅.
2
Let U = ∅. Pick random top dimensional subcomplex ∆ and
add to U, remove from V .
3
Expand along 0-simplicies, then 1-simplicies, 2-simplicies, etc.
Add to U, remove from V .
4
Add U to U. Go to step 2.
End when V = ∅.
The set U obtained in the above algorithm is a collapsible cover of
K so that dcat(K ) ≤ |U| − 1.
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
Scoville, Aaronson, Green
Example
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Discrete Lusternik–Schnirelmann category
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Scoville, Aaronson, Green