The persistent fundamental group
of point clouds
XIMENA FERNANDEZ
Durham University
UK CENTRE FOR TOPOLOGICAL DATA ANALYSIS
GETCO 2022
Joint work with Kevin Piterman
Topological inference
Topological inference
Let Xn be a (possible noisy) finite sample of an unknown topological space X.
Topological inference
Let Xn be a (possible noisy) finite sample of an unknown topological space X.
Goal: Describe (algorithmic) tools to infer topological properties of X from Xn.
Persistence
- Metric space (Xn,d)
Persistence
- Metric space (Xn,d)
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
Persistence
- Metric space (Xn,d)
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
- Apply a topological invariant F F(K1)(i1)∗→F(K2)(i2)∗→⋯(iN−1)∗→F(KN)
Persistence
- Metric space (Xn,d)
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
- Apply a topological invariant F F(K1)(i1)∗→F(K2)(i2)∗→⋯(iN−1)∗→F(KN)
- For every i<j, there exist a homomorphism fi,j:F(Ki)→F(Kj) and the persistent invariant i→j is defined as Fi→j:=Im(fi,j)
Persistent homology
- Metric space (Xn,d)
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
- Apply Hd(∙) Hd(K1)(i1)∗→Hd(K2)(i2)∗→⋯(iN−1)∗→Hd(KN)
- For every i<j, there exist a homomorphism fi,j:Hd(Ki)→Hd(Kj) and the persistent homology i→j is defined as Hi→jd:=Im(fi,j)
Persistent homology
- Metric space (Xn,d)
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
- The invariant F=Hd(∙,k) with k a field induces a k[x]- graded module defined as Md=⨁i≥0Hd(Ki)
Theorem (Zomorodian & Carlsson, 2005). For every degree d≥0, there is a decomposition Md=(n⨁s=1Σαsk[x])⊕(m⨁l=1Σγlk[x]/xnlk[x]) where αs,γj∈Z, nl<nl+1 y Σα denotes an α-shift in the degree.
We can represent Md as a set of intervals (αs,+∞) and (γl,γl+nl), also known as barcode.
Persistent homology
- Metric space (Xn,d)
- Filtration of complexes K1i1→K2i2→⋯iN−1→KN
- The invariant F=Hd(∙,k) with k a field induces a k[x]- graded module defined as
Md=⨁i≥0Hd(Ki)
We can represent Md as a set of intervals (αs,+∞) and (γl,γl+nl), also known as barcode.
- The algorithm to compute the interval decomposition of Md reduces to the computation of homology of KN seen as a filtered complex (based on matrix normal reduction).
- The complexity of the computation of the persistent homology at degree d is O(M3) with M the number of (d+1)-simplices in KN.
Example
S2∨S1∨S1
- Point cloud Xn
Example
S2∨S1∨S1
- Point cloud Xn
- Vietoris–Rips filtration VRα(Xn)
Example
S2∨S1∨S1
- Point cloud Xn
- Vietoris–Rips filtration VRα(Xn)
- Persistent Homology
Example
S1×S1 vs S2∨S1∨S1
Example
S2∨S1∨S1 vs S1×S1
Vietoris–Rips filtration
Persistent homology
The persistent fundamental group
The persistent fundamental group
- Metric space (Xn,d)
- Filtration of (connected) complexes K1i1→K2i2→⋯iN−1→KN
- Persistent fundamental group π1(K1)(i1)∗→π1(K2)(i2)∗→⋯(iN−1)∗→π1(KN)
- Algorithm?
CW-complexes of dim 2
K
Group presentations
PK=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16, x−110x9x4,x−19x8,x−18x7x6,x−17⟩
The persistent fundamental group
- Metric space (Xn,d)
- Filtration of complexes (connected with constant set of vertices) K1i1→K2i2→⋯iN−1→KN
- Spanning tree T on K1 (and hence, on Ki ∀i≥0).
- Persistent group presentation PK1(i1)∗→PK2(i2)∗→⋯(iN−1)∗→PKN
The persistent fundamental group
- Persistent group presentation PK1(i1)∗→PK2(i2)∗→⋯(iN−1)∗→PKN
where PKN=⟨x1,x2,…,xn | r1,r2,…,rm⟩ with XN={x1,x2,…xn} the set of generators induced by the 1-cells in KN not in T, and RN={r1,r2,…,rm} the set of relators associated to the attaching maps of the 2-cells in KN.
For 0≤j≤N, PKj=⟨xj1,xj2,…,xjnj | rj1,rj2,…,rjmj⟩ with Xj={xj1,xj2,…,xjnj} the set of the 1-cells in Kj not in T and Rj={rj1,rj2,…,rjmj} the set of relators associated to the attaching maps of the 2-cells in Kj.
The homomorphisms (ij)∗ are induced by the inclusions Xj⊆Xj+1, Rj⊆Rj+1.
Problem: In practice, the presentations PKj may be huge.
Discrete Morse Theory
∙ Ximena Fernandez, Combinatorial methods and algorithms in low-dimensional topology and the Andrews-Curtis conjecture. PhD thesis, University of Buenos Aires (2017)
∙ Ximena Fernandez, Morse theory for group presentations (2019) arXiv:1912.00115
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
- f:K→R a discrete Morse function.
For every cell en in K, |{en≻en−1:f(en)≤f(en−1)} +|{en≺en+1:f(en)≥f(en+1)}|≤1.
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
- f:K→R a discrete Morse function.
- C the set of critical cells of K.
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.
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
- f:K→R a discrete Morse function.
- C the set of critical cells of K.
Theorem [Forman, '98]. K is homotopy equivalent to a CW-complex KM with exactly one cell of dimension k for every critical cell of index k. In particular, π1(K)≅π1(KM).
Discrete Morse Theory
Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.
- K a regular CW-complex.
- f:K→R a discrete Morse function.
- C the set of critical cells of K.
Theorem [Forman, '98]. K is homotopy equivalent to a CW-complex KM with exactly one cell of dimension k for every critical cell of index k. In particular, π1(K)≅π1(KM).
Discrete Morse Theory
Discrete Morse Theory & Internal collapses
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≃L∪˜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.
[Kozlov, D. Organized collapse: an introduction to discrete Morse theory. Graduate Studies in Mathematics, 207. Amer. Math. Soc., Providence, RI, 2020.]
Discrete Morse Theory & Internal collapses
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≃L∪˜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.
[Kozlov, D. Organized collapse: an introduction to discrete Morse theory. Graduate Studies in Mathematics, 207. Amer. Math. Soc., Providence, RI, 2020.]
Discrete Morse Theory & Internal collapses
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≃L∪˜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.
[Kozlov, D. Organized collapse: an introduction to discrete Morse theory. Graduate Studies in Mathematics, 207. Amer. Math. Soc., Providence, RI, 2020.]
Discrete Morse Theory & Internal collapses
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≃L∪˜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.
[Kozlov, D. Organized collapse: an introduction to discrete Morse theory. Graduate Studies in Mathematics, 207. Amer. Math. Soc., Providence, RI, 2020.]
Discrete Morse Theory & Internal collapses
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≃L∪˜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.
[Kozlov, D. Organized collapse: an introduction to discrete Morse theory. Graduate Studies in Mathematics, 207. Amer. Math. Soc., Providence, RI, 2020.]
Discrete Morse Theory & Internal collapses
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≃L∪˜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.
[Kozlov, D. Organized collapse: an introduction to discrete Morse theory. Graduate Studies in Mathematics, 207. Amer. Math. Soc., Providence, RI, 2020.]
Discrete Morse Theory & Internal collapses
Theorem: Let K be a regular CW-complex 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∪dj⋃i=1eji with {eji:0≤j≤N,1≤i≤dj} the set of critical cells of f. Moreover, K≃L0∪N⋃j=1dj⋃i=1˜eji=KM. Here, the attaching maps of the cells ˜eji can be explicitly reconstructed from the internal collapses.*
∗ X. Fernandez, Combinatorial methods and algorithms in low-dimensional topology and the Andrews-Curtis conjecture. PhD thesis, University of Buenos Aires (2017)
∗ X. Fernandez, Morse theory for group presentations (2019) arXiv:1912.00115
Discrete Morse Theory & Internal collapses
CW-complexes of dim 2
K
Group presentations
PK=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16, x−110x9x4,x−19x8,x−18x7x6,x−17⟩
K
PK=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16, x−110x9x4,x−19x8,x−18x7x6,x−17⟩
K+f:K→R Morse function
PKM=⟨x4,x6 | x6x4x−16x−14⟩
Morse theory for group presentations
Given K a regular CW-complex and f:K→R a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation PKM.
O(M2) with M = |2-cells of K|
ximenafernandez/Finite-Topological-Spaces (SAGE)
Morse theory for group presentations
Given K a regular CW-complex of dim 2 and f:K→R a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation PKM.
- Every Morse function determines a matching on the set of cells of K.
Morse theory for group presentations
Given K a regular CW-complex of dim 2 and f:K→R a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation PKM.
- Every Morse function determines a matching on the set of cells of K.
- A Morse function with a single critical 0-cell determines a spanning tree on K(1).
Morse theory for group presentations
Given K a regular CW-complex of dim 2 and f:K→R a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation PKM.
- Every Morse function determines a matching on the set of cells of K.
- A Morse function with a single critical 0-cell determines a spanning tree on K(1).
- The internal collapses induced by every pair of matched cells (x,e) of dim 1 and 2 determines a rewriting in the attaching map of every 2-cell.
They can be performed in any order.
Example
Q0=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
ω0(x13,x−113x12x−14)=x12x−14
Example
Q0=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
ω0(x13,x−113x12x−14)=x12x−14
Q1=⟨x4,x6,x7,x8,x9,x10,x11,x12 | x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
ω1(x12,x−112x12)=x11
Q2=⟨x4,x6,x7,x8,x9,x10,x11 | x11x−14,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Example
Q0=⟨x4,x6,x7,x8,x9,x10,x11,x12,x13 | x13,x−113x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q1=⟨x4,x6,x7,x8,x9,x10,x11,x12 | x12x−14,x−112x11,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q2=⟨x4,x6,x7,x8,x9,x10,x11 | x11x−14,x−111x10x−16,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q3=⟨x4,x6,x7,x8,x9,x10 | x10x−16x−14,x−110x9x4,x−19x8,x−18x7x6,x−17⟩
Q4=⟨x4,x6,x7,x8,x9 | x9x4x−16x−14,x−19x8,x−18x7x6,x−17⟩
Q5=⟨x4,x6,x7,x8 | x8x4x−16x−14,x−18x7x6,x−17⟩
Q6=⟨x4,x6,x7 | x7x6x4x−16x−14,x−17⟩
Q7=⟨x4,x6 | x6x4x−16x−14⟩
Morse theory for group presentations
Definition [F.2017] Let K be a regular CW-complex. Let f:K→R be a Morse function
with only one critical cell of dimension 0.
Label M(2)={(x1,e1),…,(xm,em)} the subset of induced matched pairs of cells of dimension 1 and 2.
The Morse presentation QK,f is the presentation Qm defined by the following iterative procedure:
∙ Q0 is the standard presentation PK constructed using
the spanning tree T induced by M(1), the induced matched pairs of cells of dimension 0 and 1.
∙ For 0≤i<m, let Qi+1 be the presentation obtained from Qi after removing the relator ri associated to ei and the generator xi, and applying the rewriting rule* ω(xi,ri) on every occurrence of the generator xi in the rest of the relators.
* Given a relator r=w1xϵw2 and a generator x that appears neither in
w1 nor in w2 and ϵ=±1, the equivalent expression ω(x,r) of x induced by r
is defined as (w−11w−12)ϵ.
Theorem [F.2017] QK,f is the standard presentation PKM.
The persistent fundamental group (cont.)
∙ Ximena Fernandez and Kevin Piterman, The persistent fundamental group of point clouds (2022) (In preparation)
The persistent fundamental group
Morse reduction. Given f:KN→R a Morse function (K1)M?→(K2)M?→⋯?→(KN)M
π1((K1)M)?→π1((K2)M)?→⋯?→π1((KN)M)
The persistent fundamental group
- Metric space (Xn,d)
- Filtration (connected with constant set of vertices) K1i1→K2i2→⋯iN−1→KN
- f:KN→R a Morse function (with a single critical 0-cell)
- Persistent fundamental group
We developed an algorithm with complexity O(M2) with M = |2-cells of KN| to compute presentations of π1((Kj)M) and homomorphisms gj:π1((Kj)M)→π1((Kj+1)M) for all j s.t. π1(K1)(i1)∗→π1(K2)(i2)∗→⋯(iN−1)∗→π1(KN) ≅ ↺ ~~~~~~~~~~\pi_1((K_1)_\mathcal{M})\xrightarrow{g_1} \pi_1((K_2)_\mathcal{M}) \xrightarrow{g_2} \dots \xrightarrow{g_{N-1}}\pi_1((K_N)_\mathcal{M})
Example
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle~~~~~~P_{(K_2)_\mathcal M} =\langle x_4, x_6, x_7~ | ~ x_7x_6 x_4x_6^{-1}x_4^{-1}\rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
Example
~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle~~~~~~P_{(K_2)_\mathcal M} =\langle x_4, x_6, x_7~ | ~ x_7x_6 x_4x_6^{-1}x_4^{-1}\rangle ~~~~~~ P_{(K_3)_\mathcal M} = \langle x_4, x_6~ | ~ x_6 x_4x_6^{-1}x_4^{-1}\rangle
\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle
\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle
\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle
The persistent fundamental group
Theorem [F. Piterman 2022]. Let K be a regular CW-complex and let \phi:K\to \mathbb{R} be a filtration such that K_{\min \phi} is connected and contains all the vertices of K.
Let f:K\to \mathbb{R} be a Morse function with a single critical cell of dimension 0 such that the induced spanning tree is included in K_{\min \phi}.
Define a total ordering \{(x_1, e_1), (x_2, e_2), \dots, (x_m, e_m)\} on the set M^{(2)} of induced matched pairs of cells of dimension 1 and 2, such that if \phi(e_i)\leq \phi(e_j) then i\leq j.
Let \mathcal Q_0, \mathcal Q_1, \dots, \mathcal Q_m be the family of presentations associated to (K,f), with generators X(\mathcal Q_i) and relators R(\mathcal Q_i) for 1\leq i \leq m.
\bullet~~~ Given
\alpha\in \mathbb{R}, define
i(\alpha) := \begin{cases}
0 & \text{if } \alpha <\phi(e_i)\text{ for all }1\leq i \leq n\\
\max\{1\leq i\leq n \colon\phi(e_i)\leq \alpha\}&\text{otherwise.}\\\end{cases}
Then, \mathcal P_{({K_\alpha})_{\mathcal M}} = \langle X(\mathcal Q_{i(\alpha)})\cap K_\alpha ~ | ~ R(\mathcal Q_{i(\alpha)})\cap K_\alpha \rangle.
The persistent fundamental group
Theorem [F. Piterman 2022] (cont.)
For 1\leq i \leq m, denote by \Omega_i\colon G(\mathcal Q_{i-1}) \to G(\mathcal Q_{i}) the homomorphism induced by the rewriting rule associated to x_i and r_i, defined on the generators as
\Omega_i(x_j) := \begin{cases}
\omega(x_j, r_j) & \text{if }j =i \\
x_j & \text{if }j \neq i \end{cases}
for all 1\leq j \leq m.
For every
\bullet~~~ Given \alpha \leq \alpha', the homomorphism
\widetilde g_{\alpha, \alpha'}\colon F\left(X(\mathcal Q_{i(\alpha)})\cap K_\alpha\right)
\to F\left(X(\mathcal Q_{i(\alpha')})\cap K_{\alpha'}\right)
defined by
\widetilde g_{\alpha, \alpha'}(x) := \Omega_{i(\alpha')}\circ \dots \circ \Omega_{i(\alpha)+1}\circ \Omega_{i(\alpha)}(x)
for all x\in X(\mathcal Q_{i(\alpha)})\cap K_\alpha
induces a homomorphism g_{\alpha, \alpha'}\colon \pi_1((K_\alpha)_{\mathcal M}) \to \pi_1((K_{\alpha'})_{\mathcal M}) .
Moreover, the following diagram commutes
~~~~~~~~~~\pi_1(K_{\alpha}) \xrightarrow{f_{\alpha, \alpha'}} \pi_1(K_{\alpha'})
~~~~~~~~~\varphi_{\alpha}\downarrow ~~~~~\circlearrowleft~~~~~\downarrow \varphi_{\alpha'}
~~~~~~~~~~\pi_1((K_{\alpha})_\mathcal{M})\xrightarrow{g_{\alpha, \alpha'}} \pi_1((K_{\alpha'})_\mathcal{M})
where \varphi_{\alpha}:\mathcal P_{K_\alpha} \to \mathcal P_{(K_\alpha)_{\mathcal M}} is the isomorphism defined as \varphi_{\alpha}(x) := \Omega_m\circ \dots \circ \Omega_2\circ \Omega_1(x) on the set of generators x\in X(K_\alpha), for all \alpha \in \mathbb{R}.
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Persistent fundamental group
For \alpha \in [1, \sqrt{2}) and \alpha' \in [\sqrt{2}, 2), \varphi_{\alpha, \alpha'}(x_i) = \begin{cases} x_{16} & \text{if }i = 13,14,15,16\\ x_{17} & \text{if }i = 4,8,12,17\\ 1 & \text{otherwise}.\end{cases}
Example
S^2\vee S^1\vee S^1 vs S^1\times S^1
Persistent fundamental group
Persistent homology
Future work
- Canonical Morse functions for Vietoris-Rips filtrations of point clouds.
- Stability to small perturbations.
- Barcode decompositions (interaction with Persistent Homology).
- Persistence of group properties.
- Implementation in GAP.
- Applications in real data.
References
- X. Fernandez, Combinatorial methods and algorithms in low-dimensional topology and the Andrews-Curtis conjecture. PhD thesis, University of Buenos Aires (2017)
- X. Fernandez, Morse theory for group presentations (2019) arXiv:1912.00115
- X. Fernandez and K. Piterman, The persistent fundamental group of point clouds (2022) (In preparation)
Thanks!
