XIMENA FERNANDEZ
City St George's University of London
Conference on Algebraic Topology - Santiago de Compostela
24 April 2025
'Every mathematician has a secret weapon.
Mine is Morse theory.'
Raoul Bott.
Let K be a finite CW-complex.
Q: Is there any algorithm to "determine" its homotopy type?
A: NO
Let K be a finite CW-complex.
Q: Is there any algorithm to find a complex L such that K≃L,
with L "simpler"?
A: YES.
Collapses & Discrete Morse Theory
Strong Collapses & Strong Discrete Morse Theory
J.H.C. Whitehead ~1950
Let K,L be CW-complexes.
Let K,L be CW-complexes.
Let K,L be CW-complexes.
Let K,L be CW-complexes.
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
For every cell en in K,
#{en≻en−1:f(en)≤f(en−1)}≤1 and
#{en≺en+1:f(en)≥f(en+1)}≤1.
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
An n-cell en∈K is a critical cell of index n if the values of f in every face and coface of en increase with dimension.
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
Theorem [Forman, '98]. K is homotopy equivalent to a CW-complex coref(K) with exactly one cell of dimension k for every critical cell of index k.
Lemma [Forman, '98]. Let f:K→R be a discrete Morse function.
A. If f−1(α,β] contains no critical cells, then K(β)↘K(α).
B. If f−1(α,β] contains exactly one critical k-cell, then K(β)≃K(α)∪ek.
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪en where the attaching map φ:∂Dn→L of en is defined as φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪en where the attaching map φ:∂Dn→L of en is defined as φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪en where the attaching map φ:∂Dn→L of en is defined as φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪en where the attaching map φ:∂Dn→L of en is defined as φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪en where the attaching map φ:∂Dn→L of en is defined as φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Lemma (Internal collapse): Let K be a CW-complex of dimension ≤n. Let φ:∂Dn→K be the attaching map of an n-cell en. If K↘L, then K∪en↗↘n+1L∪en where the attaching map φ:∂Dn→L of en is defined as φ=rφ with r:K→L the canonical strong deformation retract induced by the collapse K↘L.
Theorem [F. 2024]: Let K be a regular CW-complex of dim n and let f:K→R be discrete Morse function.
Then, f induces a sequence of internal collapses given by a filtration of K
∅=K−1⊆L0⊆K0⊆L1⊆K1⋯⊆LN⊆KN=K
such that:
Kj↘Lj for all 1≤j≤N and
Lj=Kj−1∪i=1⋃djeij with {eij:0≤j≤N,1≤i≤dj} the set of critical cells of f.
Moreover,
K↗↘n+1L0∪⋃j=1N⋃i=1djeij=coref(K) the internal core of K.
* Here, the attaching maps of the cells eij can be explicitly reconstructed from the internal collapses.
Conjecture [Andrews & Curtis, 1965].
Any finite balanced presentation P=⟨x1,…,xn ∣ r1,…,rn⟩ of the trivial group
can be transformed into the empty presentation ⟨ ∣ ⟩ by a
finite sequence of the following operations:
Theorem [Tietze, 1908].
Any finite presentation P=⟨x1,…,xn ∣ r1,…,rm⟩ of a group G
can be transformed into any other presentation of the same group by a
finite sequence of the following operations:
Potential counterexamples.
Theorem [Bridson, 2015] There exist balanced presentations of the trivial group that satisfy the AC-conjecture for which the minimun length of any simplification sequence is superexponential in the total length of the relators.
Remark [Anonymous Referee]. AC-transformations of group presentations are in correspondence with 3-deformations of 2-complexes.
Conjecture [Andrews & Curtis, 1965].
For any contractible CW-complex of dimension 2, it 3-deforms to a point.
KP
⟵
P=⟨x,y ∣ xyx−1y−1⟩
K
⟶
PK=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 ∣ x13,x13−1x12x4−1,x12−1x11,x11−1x10x6−1,x10−1x9x4,x9−1x8,x8−1x7x6,x7−1⟩
K
PK=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 ∣ x13,x13−1x12x4−1,x12−1x11,x11−1x10x6−1, x10−1x9x4,x9−1x8,x8−1x7x6,x7−1⟩
K+f:K→R Morse function
Pcoref(K)=⟨x4,x6 ∣ x6x4x6−1x4−1⟩
Theorem [F.]: Given K a regular CW-complex of dim 2 and f:K→R a discrete Morse function with a single critical 0-cell, there exist an algorithm to compute the attaching maps of coref(K).
O(M2) with M = |2-cells of K|.
ximenafernandez/Finite-Topological-Spaces (SAGE)
Posets-Package (GAP) (joint with K. Piterman & I. Sadofschi Costa).
Corollary [F.]: Given P a group presentation, and a discrete Morse function on KP′, there exist an algorithm to compute P~=Pcoref(KP′). In particular, P∼ACP~.
O(R2) with R the total length relator of P.
(c.f. Brendel, Ellis, Juda, Mrozek. Fundamental group algorithm for low dimensional tessellated CW-complexes, 2015. arXiv:1507.03396)
Theorem [F.]. The following balanced presentations of the trivial group satisfies the Andrews-Curtis conjecture:
∙ P=⟨x,y ∣ xyx=yxy, x2=y3⟩* [Akbulut & Kirby, 1985]
∙ P=⟨x,y ∣ x−1y3x=y4, x=y−1xyx−1⟩ [Miller & Schupp, 1999]
∙ P=⟨x,y ∣ x=[x−1,y−1],y=[y−1,xq]⟩,∀q∈N [Gordon, 1984]
* First proved by Miasnikov in 2003 using genetic algorithms.
Rmk: If K↘L, then there exists f:K→R discrete Morse function s.t. L=coref(K).
Goal: Understand the attaching maps of coref(K).
Rmk: If K↘L, then there exists f:K→R discrete Morse function s.t. L=coref(K).
Goal: Understand the attaching maps of coref(K).
Goal: Understand when the attaching maps of coref(K) are understandable.
Let K,L be a simplicial complexes.
Let K,L be a simplicial complexes.
Remark: K↘↘L implies K↘L BUT the converse is not true.
Let K be a simplicial complex.
Let g:V(K)→R be a real-valued function.
Define Kα={σ∈K:f(v)≤α ∀v∈V(σ)} sublevel complex.
Let K be a simplicial complex.
Let g:V(K)→R be a real-valued function.
Define Kα={σ∈K:f(v)≤α ∀v∈V(σ)} sublevel complex.
Theorem [F. 2025]. K is homotopy equivalent to a CW-complex coreg(K) with exactly one cell of dimension k for every k-simplex in st(v,Kf(v)) with v strong critical vertex.
Moreover, coreg(K) is regular.
Lemma [F., 2025]. Let f:V(K)→R be a real-valued function.
A. If f−1(α,β] has no strong critical vertices, then Kβ↘↘Kα.
B. If f−1(α,β] has exactly one strong critical vertex v, then Kβ≃Kα∪⋃ek with ek in corrspondence with simplices in st(v,Kf(v)).
Lemma [F., 2025]. Let f:V(K)→R be a real-valued function.
A. If f−1(α,β] has no strong critical vertices, then K(β)↘↘K(α).
B. If f−1(α,β] has exactly one strong critical vertex v, then K(β)≃Kα∪⋃ek with ek in corrspondence with simplices in st(v,Kf(v)).
Proposition [F., 2025]. Let K be a finite simplicial complex.
Given g:V(K)→R map, the face poset of coreg(K) determines it.
Rmk: The face poset of coreg(K) is the poset of incidence of the critical simplices.
Given g:V(K)→R map, the face poset of coreg(K) determines it.
Rmk: The face poset of coreg(K) is the poset of incidence of the critical simplices.
Library of triangulations by Benedetti & Lutz