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Discrete Morse Theoryand applications
Combinatorics seminar
Sven Persson
2003-06-13
Outline
• 1 Introduction
• 2 Preliminaries
• 3 Discrete morse theory
• 4 Generalized shellings and discrete
morse functions
• 5 References
1 Introduction
• Discrete version of morse theory [Forman]
• Shellings [Björner, Wachs]
• Discrete morse theory and shellings [Chari]
1 Introduction
• Principle idea: construct a more efficient complex, while retaining topological properties
• For a given generalized shelling of a regular cell comples, there is a canonical discrete morse function
2 Preliminaries
finite regular cell complex• P() face poset wrt containment
order, cover , boundary subcomplex, = ∪
≅ Bdim, ≅ Sdim-1, dim = max dim
• M() maximal cells of pure if dim = d ∀ M()
2 Preliminaries
• Shelling – simplicial complex version:
pure d-dimensional simplicial complex
F1, ..., Fn , ordering of M(), shelling of if ∀i>1:
Fi ∩(∪j<i Fj) nonempty union of (d-1)-faces of Fj
2 Preliminaries
• Nonpure shelling [Björner, Wachs]:
1, ..., m, ordering of M(), shelling if
dim = 0 or
(i) ∃ ordering of M(1) which is shelling
(ii) j ∩ (∪1k j-1 k ) is pure and (dim j-1)-dimensional, for 2 j m
(iii) ∃ ordering of M(j) which is shelling where
M(j ∩ (∪1k j-1 k ) ) appears first, for 2 j m
2 Preliminaries
• Theorem 2.1 [Björner, Wachs]
shellable ⇒ ≃ ∨ Sd
2 Preliminaries
1
2
3
a
c d
b e
f
2 Preliminaries• Generalized shelling [Chari]:
1, ..., m, ordering of cells of , generalized shelling if
dim = 0 or
(i) ∃ ordering of M(1) which is shelling
(ii) j ∩ (∪1k j-1 k ) is pure and (dim j-1)-dimensional, for 2 j m
(iii) ∃ ordering of M(j) which is shelling where
M(j ∩ (∪1k j-1 k ) ) appears first, for 2 j m
(iv) M() ⊆ {1, ..., m}
(v) i j ⇒ i < j
2 Preliminaries
(iv) M() = {1, ..., m} ⇒
generalized shelling =
nonpure shelling
2 Preliminaries
• Proposition 2.1
simplicial complex, F1, ..., Fm ordered subset of faces:
F1, ..., Fm generalized shelling⇔
∃ G1, ..., Gm, Gi ⊆ Fi, { [Gi, Fi]} 1im
partitions (S-partition) and ∪1ik [Gi, Fi] simplicial complex for k= 1, .. , m
3 Discrete morse theory
• f discrete morse function iff : → R and
B () 1 and C () 1 ∀ where B () = | { B(); f () ≥ f ()} |
C () = | { C(); f () f ()} |
ie almost increasing wrt dimensionnon interesting example: f = dim
3 Discrete morse theory
1
2
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a
c d
b e
f
0 0 0
00
0
1
1
1
1
1
1 1
1
22
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a
c d
b e
f
1 3 5
75
3
2
2
4
7
4
6 6
8
67
3 Discrete morse theory
• σ critical cell if B () = C () = 0
• C(f) = {σ ; B () = C () = 0 }
1
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a
c d
b e
f
1 3 5
75
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a
c d
b e
f
0 0 0
00
0
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1 1
1
2 2
An efficient morse function has few critical cells
3 Discrete morse theory
• Theorem 3.1 [Forman]
(i) ≃ C(f)
(ii) j mj
where mj = | {σ C(f); dim σ = j} |
3 Discrete morse theory
• Lemma 3.2
B () C () = 0
3 Discrete morse theory
• G () hasse diagram of directed acyclic
• ∃ M(f) avoids exactly C(f), ie cover relations where f in noincreasing wrt dimension
3 Discrete morse theory
abcd
ab bd cd acbe
de ef df
def
a b c d e fGM () = G () reversing the edges in M
3 Discrete morse theory
abcd
ab bd cd acbe
de ef df
def
a b c d e fGM () = G () reversing the edges in M
3 Discrete morse theory
• Proposition 3.3
C ⊆ C = C(f) ∃ f ⇔ ∃ M on G () : GM () acyclic
C avoids exactly M
4 Generalized shellings and morse functions
• d-pseudomanifold
pure d-dimensional regular cell complex
(i) every (d-1)-cell is contained in at most two d-cells
(ii) for any d-cells ∃ sequence of d-cells m : i, i+1 share (d-1)-cell
4 Generalized shellings and morse functions
• σj bounded if j ∩ (∪1k j k ) = j
σ1σ2
σ3
σ4
σ5
σ5 bounded
4 Generalized shellings and morse functions
• Proposition 4.1
d-pseudomanifold, m shelling, v
⇒ admits morse function f:
(i) d-sphere ⇒ C(f) = { v, m}
d-ball ⇒ C(f) = { v }
(ii) restricted to ∪1k j k C(f) = { v }, j<m
4 Generalized shellings and morse functions
• Theorem 4.2
m generalized shelling, v
⇒ ∃ f : v critical
critical⇔ bounded
4 Generalized shellings and morse functions
• Corollary 4.3
d-dimensional mj = | j-dimensional bounded cells in generalized shelling |
⇒(i) ≃ cell complex with m0+1 points and
mj j-dimensional cells for j=1, .., d
(ii) the bounded cells appear in non-increasing order of dimension ⇒ ≃ ∨0jd Sm
j
References
• Chari (2000) On discrete morse functions and combinatorial decompositions Discrete Mathematics 217 101-113
• Forman (2002) A user’s guide to discrete morse theory Séminaire Lotharingien 48
• Forman (accepted) How many equilibria are there? An introduction to morse theory
