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 $k$.

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

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 \[K\cup e^n \nearrow\hspace{-1.5 pt} \searrow\hspace{-8pt}^{^{n+1}}L\cup \widetilde{e}^n\] 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 $K \searrow L$.

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 \[K\cup e^n \nearrow\hspace{-1.5 pt} \searrow\hspace{-8pt}^{^{n+1}}L\cup \widetilde{e}^n\] 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 $K \searrow L$.

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 \[K\cup e^n \nearrow\hspace{-1.5 pt} \searrow\hspace{-8pt}^{^{n+1}}L\cup \widetilde{e}^n\] 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 $K \searrow L$.

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 \[K\cup e^n \nearrow\hspace{-1.5 pt} \searrow\hspace{-8pt}^{^{n+1}}L\cup \widetilde{e}^n\] 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 $K \searrow L$.

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 \[K\cup e^n \nearrow\hspace{-1.5 pt} \searrow\hspace{-8pt}^{^{n+1}}L\cup \widetilde{e}^n\] 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 $K \searrow L$.

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 \[K\cup e^n \nearrow\hspace{-1.5 pt} \searrow\hspace{-8pt}^{^{n+1}}L\cup \widetilde{e}^n\] 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 $K \searrow L$.

Proposition [F. 2017]: Let $ \displaystyle K \cup \bigcup_{i=1}^d e_i$ be a CW-complex where $\dim (K) \leq \dim (e_{i})\leq \dim (e_{i+1})\leq n$ for all $i=1, 2, \dots, d$. Let $\displaystyle \varphi_j:\partial D_j\to K\cup \bigcup_{i < j} e_i$ be the attaching map of $e_j$. If $K\searrow L$, then there exist CW-complexes $Z_1\leq Z_2\leq \dots \leq Z_d$ of dim $\leq n+1$ such that for every $j=1, 2, \dots, d,$ \[K\cup \bigcup_{i=1}^j e_i \nearrow Z_j \searrow L\cup \bigcup_{i=1}^j \widetilde e_i\] where the attaching map $\displaystyle \widetilde \varphi_j:\partial D_j\to L\cup \bigcup_{i< j} \widetilde e_i$ of the cell $\widetilde e_j$ is defined inductively as:
$\bullet ~~\displaystyle \widetilde \varphi_1=r_0\varphi_1$ with $r_0:K\to L$ the canonical strong deformation retract and for $j>1$, $\bullet ~~\widetilde \varphi_{j}=\widetilde r_{j-1}\imath_{j-1}\varphi_{j}$ where $ \widetilde r_{j-1}:Z_{j-1}\to L\cup \bigcup_{i< j} \widetilde e_i$ is the strong deformation retract and $\imath_{j-1}: K\cup \bigcup_{i< j} e_i \to Z_{j-1}$ is the inclusion.

Theorem [F. 2017]: Let $K$ be a regular CW-complex of dim $n$ 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, \[ K \nearrow\hspace{-1.5 pt} \searrow\hspace{-8pt}^{^{n+1}} L_0\cup \bigcup_{j=1}^{N} \bigcup_{i=1}^{d_j}\widetilde e_i^j = K_\mathcal{M}.\]

* Here, the attaching maps of the cells $\widetilde e_i^j$ can be explicitly reconstructed from the internal collapses.

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, I proposed an algorithm to compute the Morse presentation $\mathcal{P}_{K_{\mathcal M}}$.

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, I proposed an algorithm to compute the Morse presentation $\mathcal{P}_{K_{\mathcal M}}$.

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, I proposed an algorithm to compute the Morse presentation $\mathcal{P}_{K_{\mathcal M}}$.

Key points:

• Every Morse function is equivalent to an acyclic 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 determine a rewriting in the attaching map of every 2-cell.
  They can be performed in any order.

  Given $w_1x^{\epsilon}w_2$ the attaching map of the 2-cell $e$ (here $x$ appears neither in $w_1$ nor in $w_2$ and $\epsilon=\pm 1$), the internal collapse associated to $(x,e)$ induces the replacement of $x$ by $(w_1^{-1}w_2^{-1})^{\epsilon}$ in the rest of the 2-cells.

$\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}_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 $

$\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 $

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 $\mathcal P_{K_\mathcal M}$.

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, I proposed 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)

Posets-Package (GAP) (joint with K. Piterman & I. Sadofschi Costa).

Conjecture [Andrews & Curtis, 1965]. Any finite balanced presentation $\mathcal{P}=\langle x_1,\dots,x_n ~|~ r_1,\dots,r_n\rangle $ of the trivial group can be transformed into the empty presentation $\langle ~|~\rangle$ by a finite sequence of the following operations:

1. replace some relator $r_i$ by $r_i^{-1}$
2. replace some relator $r_i$ by $r_i r_j$ for some $j \neq i$
3. replace some relator $r_i$ by a conjugate $wr_i w^{-1}$ for some $w$ in the free group $F(x_1, x_2, \dots, x_n)$
4. add a generator $x_{n+1}$ and a relator $r_{n+1}$ that coincides with $x_{n+1}$, or the inverse of this operation

Remark [Referee]. $Q^{**}$-transformations of group presentations are in correspondence with 3-deformations of 2-complexes.

Corollary [F. 2017]. Let $\mathcal{P}$ be a finite presentation of a finitely presented group. Let $K'_{\mathcal{P}}$ be the barycentric subdivision of the standard complex $K_{\mathcal{P}}$ and let $f:K'_{\mathcal{P}}\to \mathbb{R}$ be a discrete Morse function.
Then $\mathcal{P}\sim_{Q^{**}}\mathcal{P}_{(K'_{\mathcal{P}})_{\mathcal M}}$.

Theorem [F. 2017-2021]. The following balanced presentations of the trivial group satisfies the Andrews-Curtis conjecture:
$\bullet ~~\mathcal{P}=\langle x, y~|~ xyx = yxy,~ x^2 = y^{3}\rangle$* [Akbulut & Kirby, 1985]
$\bullet ~~\mathcal{P}= \langle x,y~|~x^{-1}y^3 x = y^{4}, ~x = y^{-1}xyx^{-1}\rangle$ [Miller & Schupp, 1999]
$\bullet ~~\mathcal{P}=\langle x,y~| ~x=[x^{-1},y^{-1}], y=[y^{-1},x^q]\rangle, \forall q \in \mathbb{N}$ [Gordon, 1984]

* First proved by Miasnikov in 2003 using genetic algorithms.

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

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 $\mathbb{X}_n$.

• Metric space $(\mathbb{X}_n,d)$
  

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

• 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 $F$ \[F(K_1) \xrightarrow{(i_1)_*} F(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} F(K_N)\]

• 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 $H_d(\bullet, k)$ with $k$ a field \[H_d(K_1) \xrightarrow{(i_1)_*} H_d(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} H_d(K_N)\]

Theorem (Zomorodian & Carlsson, 2005). For every degree $d\geq 0$, there is a decomposition of the $k[x]$- graded module $M_d := \bigoplus_{i\geq 0} H_d(K_i)$ as $$M_d= \left( \bigoplus_{s=1}^n \Sigma^{\alpha_s} k[x]\right) \oplus \left(\bigoplus_{l=1}^m \Sigma^{\gamma_l}k[x]/x^{n_l}k[x]\right)$$ 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.

• 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 $K_N$.

• Point cloud $\mathbb{X}_n$
  

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

• Point cloud $\mathbb{X}_n$

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

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

• Point cloud $\mathbb{X}_n$

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

• Persistent Homology

• 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?

• 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 \[\mathcal{P}_{K_1} \xrightarrow{(i_1)_*} \mathcal{P}_{K_2} \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} \mathcal{P}_{K_N}\]

• 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.

• 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 have 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. \[~~~~~~~~~~\pi_1(K_1) \xrightarrow{(i_1)_*} \pi_1(K_2) \xrightarrow{(i_2)_*} \dots \xrightarrow{(i_{N-1})_*} \pi_1(K_N)\] \[~~~~~~~~~~\cong ~~~~~\circlearrowleft~~~~~\cong ~~~~~\circlearrowleft~~~~~~~~\circlearrowleft~~~~~\cong \] \[~~~~~~~~~~\pi_1((K_1)_\mathcal{M})\xrightarrow{g_1} \pi_1((K_2)_\mathcal{M}) \xrightarrow{g_2} \dots \xrightarrow{g_{N-1}}\pi_1((K_N)_\mathcal{M})\]

$~O(M^2)$ with $M$ = |2-cells of $K_N$|

$\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 $

$\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 $

$~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle$

$\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 $

$~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle$

$\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 $

$~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle~~~~~~P_{(K_2)_\mathcal M} =\langle x_4, x_6, x_7~ | ~ x_7x_6 x_4x_6^{-1}x_4^{-1}\rangle$

$\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 $

$~~~~~~~~~P_{(K_1)_\mathcal M} = \langle x_4, x_6, x_{10}, x_{11}, x_{12}~ | ~ \rangle~~~~~~P_{(K_2)_\mathcal M} =\langle x_4, x_6, x_7~ | ~ x_7x_6 x_4x_6^{-1}x_4^{-1}\rangle ~~~~~~ P_{(K_3)_\mathcal M} = \langle x_4, x_6~ | ~ x_6 x_4x_6^{-1}x_4^{-1}\rangle$

$\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 $

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, \[\mathcal P_{({K_\alpha})_{\mathcal M}} = \langle X(\mathcal Q_{i(\alpha)})\cap K_\alpha ~ | ~ R(\mathcal Q_{i(\alpha)})\cap K_\alpha \rangle.\]

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 $\alpha \in \mathbb{R}.$

Persistent fundamental group

• 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 & SAGE.
• Applications in real data.