### 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

Let $\mathbb{X}_n$ be a (possible noisy) finite sample of an unknown topological space $\mathcal X$.

### Topological inference

Let $\mathbb{X}_n$ be a (possible noisy) finite sample of an unknown topological space $\mathcal X$.

Goal: Describe (algorithmic) tools to infer topological properties of $\mathcal{X}$ from $\mathbb{X}_n$.

### Persistence

• Metric space $(\mathbb{X}_n,d)$

### Persistence

• Metric space $(\mathbb{X}_n,d)$
• Filtration of complexes $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$

### Persistence

• Metric space $(\mathbb{X}_n,d)$
• Filtration of complexes $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• Apply a topological invariant $F$ $F(K_1) \xrightarrow{(i_1)_*} F(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} F(K_N)$

### Persistence

• Metric space $(\mathbb{X}_n,d)$
• Filtration of complexes $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• Apply a topological invariant $F$ $F(K_1) \xrightarrow{(i_1)_*} F(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} F(K_N)$
• For every $i < j$, there exist a homomorphism $f_{i,j}:F(K_i) \to F(K_j)$ and the persistent invariant $i\to j$ is defined as $F^{i\rightarrow j} := \mathrm{Im}(f_{i,j})$

### Persistent homology

• Metric space $(\mathbb{X}_n,d)$
• Filtration of complexes $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• Apply $H_d(\bullet)$ $H_d(K_1) \xrightarrow{(i_1)_*} H_d(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} H_d(K_N)$
• For every $i < j$, there exist a homomorphism $f_{i,j}:H_d(K_i) \to H_d(K_j)$ and the persistent homology $i\to j$ is defined as $H_d^{i\rightarrow j} := \mathrm{Im}(f_{i,j})$

### Persistent homology

• Metric space $(\mathbb{X}_n,d)$
• Filtration of complexes $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• The invariant $F = H_d(\bullet, k)$ with $k$ a field induces a $k[x]$- graded module defined as $M_d = \bigoplus_{i\geq 0} H_d(K_i)$
• Theorem (Zomorodian & Carlsson, 2005). For every degree $d\geq 0$, there is a decomposition $$M_d = \left( \bigoplus_{s=1}^n \Sigma^{\alpha_s} k[x]\right) \oplus \left(\bigoplus_{l=1}^m \Sigma^{\gamma_l}k[x]/x^{n_l}k[x]\right)$$ where $\alpha_s, \gamma_j\in \mathbb{Z}$, $n_l< n_{l+1}$ y $\Sigma^{\alpha}$ denotes an $\alpha$-shift in the degree.

We can represent $M_d$ as a set of intervals $(\alpha_s, + \infty)$ and $(\gamma_l,\gamma_l+n_l)$, also known as barcode.

### Persistent homology

• Metric space $(\mathbb{X}_n,d)$

• Filtration of complexes $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• The invariant $F = H_d(\bullet, k)$ with $k$ a field induces a $k[x]$- graded module defined as $M_d = \bigoplus_{i\geq 0} H_d(K_i)$

We can represent $M_d$ as a set of intervals $(\alpha_s, + \infty)$ and $(\gamma_l,\gamma_l+n_l)$, also known as barcode.

• The algorithm to compute the interval decomposition of $M_d$ reduces to the computation of homology of $K_N$ seen as a filtered complex (based on matrix normal reduction).

• The complexity of the computation of the persistent homology at degree $d$ is $$O(M^3)$$ with $M$ the number of $(d+1)$-simplices in $K_N$.

#### Example

$S^2\vee S^1\vee S^1$

• Point cloud $\mathbb{X}_n$

#### Example

$S^2\vee S^1\vee S^1$

• Point cloud $\mathbb{X}_n$
• Vietoris–Rips filtration $\mathrm{VR}_{\alpha}(\mathbb{X}_n)$

#### Example

$S^2\vee S^1\vee S^1$

• Point cloud $\mathbb{X}_n$
• Vietoris–Rips filtration $\mathrm{VR}_{\alpha}(\mathbb{X}_n)$
• Persistent Homology

#### Example

$S^1\times S^1$ vs $S^2\vee S^1\vee S^1$

#### Example

$S^2\vee S^1\vee S^1$ vs $S^1\times S^1$

Vietoris–Rips filtration

Persistent homology

### The persistent fundamental group

• Metric space $(\mathbb{X}_n,d)$
• Filtration of (connected) complexes $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• Persistent fundamental group $\pi_1(K_1) \xrightarrow{(i_1)_*} \pi_1(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} \pi_1(K_N)$
• Algorithm?

#### CW-complexes of dim 2

$K$

$\rightarrow$

#### Group presentations

$\mathcal{P}_K = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | \\~ ~ ~ ~ ~ ~ ~ ~ ~ ~x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, \\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$

##### The persistent fundamental group

• Metric space $(\mathbb{X}_n,d)$
• Filtration of complexes (connected with constant set of vertices) $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• Spanning tree $T$ on $K_1$ (and hence, on $K_i ~\forall i\geq 0$).
• Persistent group presentation $\mathcal{P}_{K_1} \xrightarrow{(i_1)_*} \mathcal{P}_{K_2} \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} \mathcal{P}_{K_N}$

##### The persistent fundamental group

• Persistent group presentation $\mathcal{P}_{K_1} \xrightarrow{(i_1)_*} \mathcal{P}_{K_2} \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} \mathcal{P}_{K_N}$
• where $\mathcal{P}_{K_N} = \langle x_1, x_2, \dots, x_n ~ | ~ r_1, r_2, \dots, r_m\rangle$ with $X_N = \{x_1, x_2, \dots x_n\}$ the set of generators induced by the 1-cells in $K_N$ not in $T$, and $R_N = \{r_1, r_2, \dots, r_m\}$ the set of relators associated to the attaching maps of the 2-cells in $K_N$.

For $0\leq j \leq N$, $\mathcal{P}_{K_j} = \langle x_{j_1}, x_{j_2}, \dots, x_{j_{n_j}} ~ | ~ r_{j_1}, r_{j_2}, \dots, r_{j_{m_j}}\rangle$ with $X_j = \{x_{j_1}, x_{j_2}, \dots, x_{j_{n_j}}\}$ the set of the 1-cells in $K_j$ not in $T$ and $R_j = \{r_{j_1}, r_{j_2}, \dots, r_{j_{m_j}}\}$ the set of relators associated to the attaching maps of the 2-cells in $K_j$.

The homomorphisms $(i_j)_*$ are induced by the inclusions $X_j\subseteq X_{j+1}$, $R_j\subseteq R_{j+1}$.

Problem: In practice, the presentations $\mathcal{P}_{K_j}$ may be huge.

### Discrete Morse Theory

$\bullet$ Ximena Fernandez, Combinatorial methods and algorithms in low-dimensional topology and the Andrews-Curtis conjecture. PhD thesis, University of Buenos Aires (2017)

$\bullet$ 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.
• $f:K\to \mathbb{R}$ a discrete Morse function.
• For every cell $e^n$ in $K$, $|\{e^n\succ e^{n-1}: f(e^n)\leq f(e^{n-1})\}\ + |\{e^n\prec e^{n+1} : f(e^n)\geq f(e^{n+1})\}|\leq 1.$

### Discrete Morse Theory

Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.

• $K$ a regular CW-complex.
• $f:K\to \mathbb{R}$ a discrete Morse function.
• $C$ the set of critical cells of $K$.
• An $n$-cell $e^n \in K$ is a critical cell of index $n$ if the values of $f$ in every face and coface of $e^n$ 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.
• $f:K\to \mathbb{R}$ a discrete Morse function.
• $C$ the set of critical cells of $K$.

Theorem [Forman, '98]. $K$ is homotopy equivalent to a CW-complex $K_\mathcal{M}$ with exactly one cell of dimension $k$ for every critical cell of index $k$. In particular, $\pi_1(K) \cong \pi_1(K_\mathcal{M}).$

### Discrete Morse Theory

Goal: 'Simplify' the cell decomposition of a CW-complex while preserving its homotopy type.

• $K$ a regular CW-complex.
• $f:K\to \mathbb{R}$ a discrete Morse function.
• $C$ the set of critical cells of $K$.

Theorem [Forman, '98]. $K$ is homotopy equivalent to a CW-complex $K_\mathcal{M}$ with exactly one cell of dimension $k$ for every critical cell of index $k$. In particular, $\pi_1(K) \cong \pi_1(K_\mathcal{M}).$

### Discrete Morse Theory

##### Discrete Morse Theory & Internal collapses

Lemma (Internal collapse): Let $K$ be a CW-complex of dimension $\leq n$. Let $\varphi:\partial D^n\to K$ be the attaching map of an $n$-cell $e^n$. If $K \searrow L$, then $K\cup e^n \simeq L\cup \widetilde{e}^n$ where the attaching map $\widetilde{\varphi}\colon \partial D^ n\to L$ of $\widetilde{e}^n$ is defined as $\widetilde{\varphi}=r \varphi$ with $r:K\to L$ the canonical strong deformation retract induced by the collapse $K \searrow L$.

[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 $\leq n$. Let $\varphi:\partial D^n\to K$ be the attaching map of an $n$-cell $e^n$. If $K \searrow L$, then $K\cup e^n \simeq L\cup \widetilde{e}^n$ where the attaching map $\widetilde{\varphi}\colon \partial D^ n\to L$ of $\widetilde{e}^n$ is defined as $\widetilde{\varphi}=r \varphi$ with $r:K\to L$ the canonical strong deformation retract induced by the collapse $K \searrow L$.

[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 $\leq n$. Let $\varphi:\partial D^n\to K$ be the attaching map of an $n$-cell $e^n$. If $K \searrow L$, then $K\cup e^n \simeq L\cup \widetilde{e}^n$ where the attaching map $\widetilde{\varphi}\colon \partial D^ n\to L$ of $\widetilde{e}^n$ is defined as $\widetilde{\varphi}=r \varphi$ with $r:K\to L$ the canonical strong deformation retract induced by the collapse $K \searrow L$.

[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 $\leq n$. Let $\varphi:\partial D^n\to K$ be the attaching map of an $n$-cell $e^n$. If $K \searrow L$, then $K\cup e^n \simeq L\cup \widetilde{e}^n$ where the attaching map $\widetilde{\varphi}\colon \partial D^ n\to L$ of $\widetilde{e}^n$ is defined as $\widetilde{\varphi}=r \varphi$ with $r:K\to L$ the canonical strong deformation retract induced by the collapse $K \searrow L$.

[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 $\leq n$. Let $\varphi:\partial D^n\to K$ be the attaching map of an $n$-cell $e^n$. If $K \searrow L$, then $K\cup e^n \simeq L\cup \widetilde{e}^n$ where the attaching map $\widetilde{\varphi}\colon \partial D^ n\to L$ of $\widetilde{e}^n$ is defined as $\widetilde{\varphi}=r \varphi$ with $r:K\to L$ the canonical strong deformation retract induced by the collapse $K \searrow L$.

[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 $\leq n$. Let $\varphi:\partial D^n\to K$ be the attaching map of an $n$-cell $e^n$. If $K \searrow L$, then $K\cup e^n \simeq L\cup \widetilde{e}^n$ where the attaching map $\widetilde{\varphi}\colon \partial D^ n\to L$ of $\widetilde{e}^n$ is defined as $\widetilde{\varphi}=r \varphi$ with $r:K\to L$ the canonical strong deformation retract induced by the collapse $K \searrow L$.

[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 $f:K\to \mathbb{R}$ be discrete Morse function. Then, $f$ induces a sequence of internal collapses given by a filtration of $K$ $\varnothing = K_{-1} \subseteq L_0 \subseteq K_0\subseteq L_1\subseteq K_1 \dots \subseteq L_{N}\subseteq K_{N}=K$ such that $K_j\searrow L_{j}$ for all $1\leq j\leq N$ and $\displaystyle L_{j}=K_{j-1}\cup \bigcup_{i=1}^{d_j} e_i^j$ with $\{e_i^j:0\leq j\leq N, 1\leq i \leq d_j\}$ the set of critical cells of $f$. Moreover, $K \simeq L_0\cup \bigcup_{j=1}^{N} \bigcup_{i=1}^{d_j}\widetilde e_i^j = K_\mathcal{M}.$ Here, the attaching maps of the cells $\widetilde e_i^j$ 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

#### CW-complexes of dim 2

$K$

$\rightarrow$

#### Group presentations

$\mathcal{P}_K = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | \\~ ~ ~ ~ ~ ~ ~ ~ ~ ~x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, \\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$

$K$

$\mathcal{P}_K = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | \\~ ~ ~ ~ ~ ~ ~ ~ ~ ~x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, \\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$

$K + f\colon K\to \mathbb{R}$ Morse function

$\mathcal{P}_{K_{\mathcal{M}}} = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle$

#### Morse theory for group presentations

Given $K$ a regular CW-complex and $f\colon K\to \mathbb{R}$ a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation $\mathcal{P}_{K_{\mathcal M}}$.

$~O(M^2)$ 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 $f\colon K\to \mathbb{R}$ a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation $\mathcal{P}_{K_{\mathcal M}}$.

• 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 $f\colon K\to \mathbb{R}$ a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation $\mathcal{P}_{K_{\mathcal M}}$.

• 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 $f\colon K\to \mathbb{R}$ a discrete Morse function with a single critical 0-cell, we developed an algorithm to compute the Morse presentation $\mathcal{P}_{K_{\mathcal M}}$.

• 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

$\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$

$\omega_0 (x_{13}, x_{13}^{-1}x_{12}x_4^{-1}) = x_{12}x_4^{-1}$

##### Example

$\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$

$\omega_0 (x_{13}, x_{13}^{-1}x_{12}x_4^{-1}) = x_{12}x_4^{-1}$

$\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$

$\omega_1 (x_{12}, x_{12}^{-1}x_{12}) = x_{11}$

$\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$

##### Example

$\mathcal{Q}_0 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13}~ | ~ x_{13}, x_{13}^{-1}x_{12}x_4^{-1},x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$
$\mathcal{Q}_1 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ~ | ~ x_{12}x_4^{-1}, x_{12}^{-1}x_{11}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$
$\mathcal{Q}_2 = \langle x_4, x_6, x_7, x_8, x_9, x_{10}, x_{11} ~ | ~ x_{11}x_4^{-1}, x_{11}^{-1}x_{10}x_6^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$
$\mathcal{Q}_3 = \langle x_4, x_6, x_7, x_8, x_9, x_{10} ~ | ~ x_{10}x_6^{-1}x_4^{-1}, x_{10}^{-1}x_9x_4, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$
$\mathcal{Q}_4 = \langle x_4, x_6, x_7, x_8, x_9 ~ | ~ x_{9}x_4x_6^{-1}x_4^{-1}, x_9^{-1}x_8, x_8^{-1}x_7x_6, x_7^{-1}\rangle$
$\mathcal{Q}_5 = \langle x_4, x_6, x_7, x_8~ | ~ x_{8}x_4x_6^{-1}x_4^{-1}, x_8^{-1}x_7x_6, x_7^{-1}\rangle$
$\mathcal{Q}_6 = \langle x_4, x_6, x_7~ | ~ x_7x_6x_4x_6^{-1}x_4^{-1}, x_7^{-1}\rangle$
$\mathcal{Q}_7 = \langle x_4, x_6~ | ~ x_6x_4x_6^{-1}x_4^{-1}\rangle$

#### Morse theory for group presentations

Definition [F.2017] Let $K$ be a regular CW-complex. Let $f:K\to \mathbb{R}$ be a Morse function with only one critical cell of dimension 0. Label $M^{(2)}=\{(x_1, e_1), \dots, (x_m, e_m)\}$ the subset of induced matched pairs of cells of dimension 1 and 2.

The Morse presentation $\mathcal{Q}_{K,f}$ is the presentation $\mathcal Q_m$ defined by the following iterative procedure:

$\bullet$ $\mathcal Q_0$ is the standard presentation $\mathcal P_K$ constructed using the spanning tree $T$ induced by $M^{(1)}$, the induced matched pairs of cells of dimension 0 and 1.

$\bullet$ For $0\leq i < m$, let $\mathcal Q_{i+1}$ be the presentation obtained from $\mathcal Q_{i}$ after removing the relator $r_i$ associated to $e_i$ and the generator $x_i$, and applying the rewriting rule* $\omega(x_i, r_i)$ on every occurrence of the generator $x_i$ in the rest of the relators.
* Given a relator $r=w_1x^{\epsilon}w_2$ and a generator $x$ that appears neither in $w_1$ nor in $w_2$ and $\epsilon=\pm 1$, the equivalent expression $\omega(x,r)$ of $x$ induced by $r$ is defined as $(w_1^{-1}w_2^{-1})^{\epsilon}$.

Theorem [F.2017] $\mathcal Q_{K,f}$ is the standard presentation $\mathcal P_{K_\mathcal M}$.

### The persistent fundamental group (cont.)

$\bullet$ Ximena Fernandez and Kevin Piterman, The persistent fundamental group of point clouds (2022) (In preparation)

### The persistent fundamental group

• Metric space $(\mathbb{X}_n,d)$

• Filtration of (connected) complexes$K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$

• Persistent fundamental group
• $\pi_1(K_1) \xrightarrow{(i_1)_*} \pi_1(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} \pi_1(K_N)$

Morse reduction. Given $f\colon K_N\to \mathbb{R}$ a Morse function $(K_1)_\mathcal{M} \xrightarrow{?} (K_2)_\mathcal{M} \xrightarrow{?} \dots \xrightarrow{?} (K_N)_\mathcal{M}$

$\pi_1((K_1)_\mathcal{M})\xrightarrow{?} \pi_1((K_2)_\mathcal{M}) \xrightarrow{?} \dots \xrightarrow{?}\pi_1((K_N)_\mathcal{M})$

### The persistent fundamental group

• Metric space $(\mathbb{X}_n,d)$
• Filtration (connected with constant set of vertices) $K_1 \xrightarrow{i_1} K_2 \xrightarrow{i_2} \dots \xrightarrow{i_{N-1}} K_N$
• $f\colon K_N\to \mathbb{R}$ a Morse function (with a single critical 0-cell)

• Persistent fundamental group
We developed an algorithm with complexity $O(M^2)$ with $M$ = |2-cells of $K_N$| to compute presentations of $\pi_1((K_j)_\mathcal{M})$ and homomorphisms $g_j: \pi_1((K_j)_\mathcal{M})\to\pi_1((K_{j+1})_\mathcal{M})$ for all $j$ s.t. $~~~~~~~~~~\pi_1(K_1) \xrightarrow{(i_1)_*} \pi_1(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} \pi_1(K_N)$ $~~~~~~~~~~\cong ~~~~~\circlearrowleft~~~~~\cong ~~~~~\circlearrowleft~~~~~~~~\circlearrowleft~~~~~\cong$ $~~~~~~~~~~\pi_1((K_1)_\mathcal{M})\xrightarrow{g_1} \pi_1((K_2)_\mathcal{M}) \xrightarrow{g_2} \dots \xrightarrow{g_{N-1}}\pi_1((K_N)_\mathcal{M})$
##### 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)