Mayer-Vietoris Sequences  for  the  Constrained  Persistent  Homology  of   Simplicial  Complexes

Shiquan Ren

The  notion  of  independence  hypergraphs  is  introduced    to   investigate  the  relations between  random hypergraphs  and  random  simplicial complexes [30].
With  the  help  of   the   differential  calculus  on  discrete  sets,  the  constrained  homology of  simplicial  complexes  as  well as the  constrained  cohomology  of  independence  hypergraphs  are  constructed  [27].   In  this  paper,  by  proving  the  functorialities  of  the  constrained  homology  of  simplicial  complexes  as well  as  the  constrained  cohomology  of  independence  hypergraphs,
we  study  the  constrained  persistent  homology  for  filtrations  of  simplicial complexes  as well as
the  constrained  persistent  cohomology  for  filtrations  of  independence  hypergraphs.
We  study  the  Mayer-Vietoris sequences  for the  constrained  (co)homology  as  well as  their  functorialities.
As  a  result,  we  prove  the  Mayer-Vietoris sequences   for the  persistent constrained  homology for
filtrations  of  simplicial complexes  and  the    persistent constrained  cohomology for
filtrations  of  independence  hypergraphs.

2010 Mathematics Subject Classification.  	Primary  55U10,  	55U15,  Secondary  	53A45,  08A50

Keywords and Phrases.   simplicial complexes,  hypergraphs,   (co-)chain complexes,  (co-)homology,  differential calculus

§ INTRODUCTION

Let   $V$  be  a  discrete  set  whose elements are  called  vertices.  Suppose  there  is  a  total  order  $\prec$  on  $V$.
Let  $\Delta[V]$  be  the  collection of all   the  non-empty  finite  subsets  of  $V$.
A   hypergraph    with its  vertices  from   $V$  is  an  arbitrary  subset   $\mathcal{H}\subseteq \Delta[V]$.
A  simplicial complex  $\mathcal{K}$  with  its  vertices  from  $V$
is  a hypergraph such  that  for  any  $\sigma\in \mathcal{K}$  and  any non-empty  subset
$\tau\subseteq \sigma$,  it  holds  $\tau\in \mathcal{K}$.
An  independence  hypergraph   $\mathcal{L}$  with  its  vertices  from  $V$
is  a hypergraph such  that  for  any  $\sigma\in \mathcal{L}$  and  any  finite   superset
$\tau\supseteq \sigma$,  it  holds  $\tau\in \mathcal{L}$.

Let  $V$  and  $V'$   be  two  discrete  sets.
Let  $\mathcal{H}$  be  a  hypergraph  on  $V$  and  let  $\mathcal{H}'$  be  a  hypergraph  on  $V'$.
A  morphism   of  hypergraphs  is  a  map  $\varphi:  V\longrightarrow  V'$  such  that  for  any  hyperedge  $\sigma$  of  $\mathcal{H}$,  the  image  $f(\sigma)$,  which   is  a  subset   of  $V'$    consisting   of  the  images    $f(v)\in  V'$  of  the  vertices   $v\in  \sigma$,  is
a  hyperedge of $\mathcal{H}'$.
Let  $\varphi:  \mathcal{H}\longrightarrow \mathcal{H}'$  be  a  morphism  of  hypergraphs.
Note that  if   $\mathcal{H}$  is  a  simplicial complex,  then  the  image $\varphi(\mathcal{H})$  must a simplicial  complex  with  its  vertices from  $V'$.   Moreover,  if  $\mathcal{H}$  is  an  independence  hypergraph  and  $\varphi$  is  induced
by  a  surjective map  $\varphi:  V\longrightarrow  V'$,  then  the  image $\varphi(\mathcal{H})$   must  be  an  independence  hypergraph  with  its  vertices  from  $V'$.

Given  a  simplicial complex  $\mathcal{K}$  with  its  vertices
from  $V$  and  a  simplicial  complex  $\mathcal{K}'$  with  its  vertices  from  $V'$,     a  morphism   of  hypergraphs   $\varphi: \mathcal{K}\longrightarrow \mathcal{K}'$
is  called  a simplicial  map.   Similarly,  given  an  independence  hypergraph   $\mathcal{L}$  with  its  vertices
from  $V$  and  an  independence  hypergraph   $\mathcal{L}'$  with  its  vertices  from  $V'$,    we  call   a  morphism  of hypergraphs  $\varphi: \mathcal{L}\longrightarrow \mathcal{L}'$  induced  by  a  surjective  map  $\varphi:  V\longrightarrow  V'$
a  morphism of  independence hypergraphs.   Note  that  the  surjectivity  of  $\varphi:  V\longrightarrow  V'$  does  not  imply  the  surjectivity  of
$\varphi: \mathcal{L}\longrightarrow  \mathcal{L}'$.

In  this  paper,   we  study  the  persistence  of  the  constrained  homology  for  simplicial  complexes  as well  as  the
persistence  of  the  constrained  cohomology  for  independence  hypergraphs.  We prove  the  Mayer-Vietoris sequences  for  the  constrained  persistent  homology  of  simplicial complexes  and  the  constrained  persistent cohomology  of  independence  hypergraphs.  This  paper  is  motivated  as  follows.

Motivation 1.   simplicial complexes  and  independence hypergraphs from  probabilistic   combinatorics.
Random  hypergraphs  and  random  simplicial  complexes   are  higher-dimensional generalizations  of  random graphs.  They  have  been  intensively  studied in  recent  decades  (for  example,   [9, 10, 21, 23, 24, 25, 29]).

Let   $\mathcal{H}$   be   a  hypergraph   with its  vertices  from   $V$.  The  associated  simplicial complex $\Delta\mathcal{H}$  of  $\mathcal{H}$  and  the  lower-associated  simplicial  complex  $\delta\mathcal{H}$  of  $\mathcal{H}$    are
the smallest  simplicial  complex    containing  $\mathcal{H}$  and  the  largest  simplicial  complex    contained in  $\mathcal{H}$  respectively.   The  associated  independence   hypergraph   $\bar\Delta\mathcal{H}$  of  $\mathcal{H}$  and  the  lower-associated  independence  hypergraph   $\bar\delta\mathcal{H}$  of  $\mathcal{H}$
are  the smallest   independence  hypergraph   containing  $\mathcal{H}$  and
the  largest  independence  hypergraph   contained in  $\mathcal{H}$  respectively.
In  addition,   if   $V$  is  a  finite set,  then
the  complement  hypergraph  of  $\mathcal{H}$  is  a  hypergraph   given  by  $
\gamma\mathcal{H}=\{\sigma\in\Delta[V]\mid  \sigma\notin \mathcal{H}\}
$     with  its  vertices  from  $V$.

For  any  simplicial complex  $\mathcal{K}$  with its  vertices  from  $V$,  an  external face of  $\mathcal{K}$  is  a  nonempty  subset $\sigma$  of $V$  such that $\sigma\notin \mathcal{K}$  and  $\tau\in \mathcal{K}$  for any proper  subset
$\tau$  of  $\sigma$.   Let  $E(\mathcal{K})$  be  the set  of  all the  external faces of $\mathcal{K}$.       For  any  independence  hypergraph   $\mathcal{L}$   with its  vertices  from  $V$,      a   co-external face  of  $\mathcal{L}$  is  a     subset $\sigma$  of $V$  such that $\sigma\notin \mathcal{L}$  and  $\tau\in \mathcal{L}$  for any proper  superset
$\tau$  of  $\sigma$  such  that  $\tau$ a  subset of  $V$.   Let  $\bar  E(\mathcal{L})$  be  the set  of  all the  co-external faces of $\mathcal{L}$.

Let  an  arbitrary  function $p:  \Delta[V]\longrightarrow  [0,1]$.
Consider  the  random  hypergraph  whose  probability   is  given by

\begin{eqnarray}\label{eq-3.a1}
\bar{\rm P}_{p}(\mathcal{H})=\prod_{\sigma\in\mathcal{H}}  p(\sigma)\prod_{\sigma\notin   \mathcal{H}}\big(1-p(\sigma)\big),
\end{eqnarray}

the       random  simplicial  complex  whose  probability   is  given by

\begin{eqnarray}\label{eq-3.a2}
{\rm P}_{p}(\mathcal{K})=\prod_{\sigma\in\mathcal{K}}  p(\sigma)\prod_{\sigma\in  E(\mathcal{K})}\big(1-p(\sigma)\big)
\end{eqnarray}

and  the      random  independence  hypergraph  whose  probability   is  given by

\begin{eqnarray}\label{eq-3.a3}
{\rm Q}_{p}(\mathcal{L})=\prod_{\sigma\in\mathcal{L}}  p(\sigma)\prod_{\sigma\in  \bar  E(\mathcal{L})}\big(1-p(\sigma)\big).
\end{eqnarray}

Here  in  (<ref>)  -  (<ref>),   $ \mathcal{H} $,  $\mathcal{K}$  and  $ \mathcal{L} $   respectively  are  any  hypergraph,  simplicial  complex  and  independence  hypergraph  with  their  vertices  from   $V$.   The  product  of  countably many  numbers  between  $0$  and  $1$  is  defined  to be  the limit  of the  products  of  finitely  many   numbers.

Suppose  $V$  is  a  finite  set.  Let  $\mathcal{H}\sim  \bar{\rm  P}_p$   be  a  randomly  generated  hypergraph  on  $V$.    We  have  the following  assertions  (cf.  <cit.>,   <cit.>, <cit.>):
the  lower-associated  simplicial  complex   of   $\mathcal{H}$   is   a  randomly generated
simplicial  complex   $\delta\mathcal{H} \sim  {\rm  P}_{p}$,
the  lower-associated  independence  hypergraph   of   $\mathcal{H}$     is   a   randomly generated
independence  hypergraph   $ \bar\delta\mathcal{H}\sim  {\rm  Q}_{p}$,
the  complement   of   $\mathcal{H}$  is  a randomly  generated  hypergraph  $\gamma\mathcal{H}\sim  \bar{\rm  P}_{1-p}$,
the  complement  of  the  associated  simplicial  complex  of   $\mathcal{H}$   is  a  randomly  generated
independence  hypergraph  $\gamma \Delta\mathcal{H} \sim  {\rm  Q}_{1-p}$,
the  complement  of  the  associated  independence  hypergraph   of   $\mathcal{H}$   is a randomly generated
simplicial  complex    $\gamma \bar\Delta\mathcal{H} \sim  {\rm   P}_{1-p}$.

Motivation 2.   differential  calculus  on  discrete  sets  and  the  constrained homology.   The  differential  calculus  on  discrete  sets  are  studied  by  A. Dimakis  and  F. Müller-Hoissen  [11, 12].
Let  $\mathbb{N}$  be  the  set  of  natural numbers  $0,1,2,\ldots$.
By    considering  the  exterior  algebra   ${\rm  Ext}_*(V)$  generated  by  the  operators  $\frac{\partial}{\partial  v}$,   $v\in  V$  (cf.  Eq.  (<ref>)),    and  taking  $\alpha\in {\rm  Ext}_{2t+1}(V)$ for  any    $t\in\mathbb{N}$,   we  constructed  the constrained  homology  $H_*(\mathcal{K},\alpha,m)$,   $m\in \mathbb{Z}$,   for   a  simplicial  complex  $\mathcal{K}$  in  [27]  as  a  generalization  of  the  weighted  homology  $H_*(\mathcal{K},w)$  of  $\mathcal{K}$  (cf.  [26, 31, 32, 33]).   In   [27],   by  considering  the  exterior  algebra   ${\rm  Ext}^*(V)$  generated  by  the  operators  $dv$,   $v\in  V$  (cf.  Eq. (<ref>)),      and  taking  $\omega\in {\rm  Ext}^{2t+1}(V)$ for  any  nonnegative  integer $t$,    we     constructed the  constrained  cohomology $H^*(\mathcal{L},\omega,m)$,   $m\in \mathbb{Z}$,    of  an   independence  hypergraph  $\mathcal{L}$      as  a  dual  analog     of  the  constrained  homology  of   simplicial  complexes.
Taking  $\beta\in  {\rm  Ext}_{2s}(V)$  and  $\mu\in   {\rm  Ext}^{2s}(V)$  for  any     $s\in\mathbb{N}$.
It is  proved  in [27]  that
$\beta$  induces a  homomorphism  from  $H_*(\mathcal{K},\alpha,m)$  to  $H_*(\mathcal{K},\alpha,m-2s)$  and
$\mu$  induces  a  homomorphism  from  $H^*(\mathcal{L},\omega,m)$  to $H^*(\mathcal{L},\omega,m+2s)$.

Motivation  3.  persistent  homology  of  networks and  topological  data analysis.   Persistent homology is a widely-used  method  in topological data analysis  (for  examples,  [8, 5, 34, 13]).  Given  a  simplicial complex  $\mathcal{K}$,  we  take   a  filtration  $\{\mathcal{K}_t\}_{t\in\mathbb{R}}$       such  that  $\mathcal{K}_t\subseteq \mathcal{K}_s$  for  any  $-\infty<t\leq  s<+\infty$       and  $  \cup_{t\in  \mathbb{R}} \mathcal{K}_t=\mathcal{K}$.  Applying the homology  functor,    we  will  obtain  a  persistent  module
$\{H_*(\mathcal{K}_t)\}_{t\in\mathbb{R}}$  which  is  called  the  persistent  homology  of  the  filtration  $\{\mathcal{K}_t\}_{t\in\mathbb{R}}$.    The persistent  homology  of  simplicial  complexes  captures the multi-scale  or time-evolving   topological  features of  complex networks  (cf.  [1, 2, 7, 22]).

Firstly,  we  prove  the  Mayer-Vietoris sequences  for  the  constrained  homology  of  simplicial complexes  and  the  constrained  cohomology  of  independence  hypergraphs.
Let $t \in  \mathbb{N}$.
Let  $\mathcal{K}_1$  and  $\mathcal{K}_2$   be
simplicial  complexes     and  $\mathcal{L}_1$  and  $\mathcal{L}_2$   be
independence  hypergraphs   with  their  vertices from  $V$.
For  any
$\alpha\in {\rm Ext}_{2t+1}(V)$,      we  have
a  long  exact  sequence  of  the  constrained  homology  groups

\begin{eqnarray}
\cdots \longrightarrow      H_n(\mathcal{K}_1 \cap  \mathcal{K}_2,\alpha, m)\longrightarrow
H_n (\mathcal{K}_1,\alpha, m) \oplus   H_n( \mathcal{K}_2,\alpha, m)\longrightarrow  \nonumber\\
H_n(\mathcal{K}_1 \cup  \mathcal{K}_2,\alpha, m)\longrightarrow H_{n-1}(\mathcal{K}_1 \cap  \mathcal{K}_2,\alpha, m)\longrightarrow \cdots
\label{eq-lex1a}
\end{eqnarray}

which   long  exact  sequence   is  functorial  with  respect
to  simplicial maps.
For      any
$\omega\in {\rm Ext}^{2t+1}(V)$,      we  have
a  long  exact  sequence  of  the  constrained  homology  groups

\begin{eqnarray}
\cdots \longrightarrow      H^n(\mathcal{L}_1 \cap  \mathcal{L}_2,\omega, m)\longrightarrow
H^n (\mathcal{L}_1,\omega, m) \oplus   H^n( \mathcal{L}_2,\omega, m)\longrightarrow  \nonumber \\
H^n(\mathcal{L}_1 \cup  \mathcal{L}_2,\omega, m)\longrightarrow H^{n+1}(\mathcal{L}_1 \cap  \mathcal{L}_2,\omega, m)\longrightarrow \cdots
\label{eq-lex2b}
\end{eqnarray}

which    is  functorial  with  respect
to  morphisms  of  independence hypergraphs  induced by  bijective  self-maps  on  the  vertex set  $V$.
We  call   (<ref>)    the Mayer-Vietoris  sequence  for the  constrained  homology  of  simplicial  complexes  on  $V$  and  denote  it  as
${\bf  MV}_*(\mathcal{K}_1,\mathcal{K}_2,\alpha,m)$.     We   call
(<ref>)   the Mayer-Vietoris  sequence  for the  constrained  cohomology  of  independence  hypergraphs  on  $V$   and  denote  it  as
${\bf  MV}^*(\mathcal{L}_1,\mathcal{L}_2,\omega,m)$.

Secondly,  we  prove the functorialities
of  the  constrained    homology  for  simplicial complexes,   the  constrained  cohomology  for  independence hypergraphs  and   the  Mayer-Vietoris  sequences.   We give  the  definitions
of  morphisms  of  Mayer-Vietoris  sequences  in  Section <ref>.
We  study the persistent constrained  homology  $H_*(\mathcal{K}_x,\alpha,m\mid  x\in \mathbb{R})$  for a  filtration  $\{\mathcal{K}_x\}_{x\in\mathbb{R}}$  of  simplicial complexes  and  the  persistent constrained  cohomology
$H^*(\mathcal{L}_x,\omega,m\mid  x\in \mathbb{R})$  for  a  filtration  $\{\mathcal{L}_x\}_{x\in\mathbb{R}}$  of   independence  hypergraphs.
Given  two  filtrations  $\{\mathcal{K}_{1,x}\}_{x\in\mathbb{R}}$
and  $\{\mathcal{K}_{2,x}\}_{x\in\mathbb{R}}$  of  simplicial complexes,  we  denote  the  persistent  Mayer-Vietoris  sequence  as              ${\bf   PMV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x},  \alpha,m\mid  x\in\mathbb{R})$.
Similarly,  given  two  filtrations  $\{\mathcal{L}_{1,x}\}_{x\in\mathbb{R}}$
and  $\{\mathcal{L}_{2,x}\}_{x\in\mathbb{R}}$  of  independence  hypergraphs,  we  denote  the  persistent  Mayer-Vietoris  sequence  as              ${\bf   PMV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x},  \omega,m\mid  x\in\mathbb{R})$.   We  give   the  definition  of  morphisms  of  persistent  Mayer-Vietoris  sequences  in  Section <ref>.

we  prove  the  next  theorem  for  the persistent Mayer-Vietoris sequences  of  the constrained  homology  of  simplicial complexes  and  the
constrained  cohomology  of  independence  hypergraphs.

* Let  $\{\mathcal{K}_{1,x}\}_{x\in\mathbb{R}}$
and  $\{\mathcal{K}_{2,x}\}_{x\in\mathbb{R}}$  be   two  filtrations    of  simplicial complexes    with their  vertices from  $V$.
Then  we  have a  diagram  of   persistent  Mayer-Vietoris  sequences

\begin{eqnarray*}
\xymatrix{
\cdots \ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}_6(V)} \ar[r]^-{{\rm  Ext}_2(V)}\ar[r]  &{\bf   PMV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x}, \alpha,m+2\mid  x\in\mathbb{R})\ar[r]^-{{\rm  Ext}_2(V)}\ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}
&{\bf   PMV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x}, \alpha,m\mid  x\in\mathbb{R})\ar[r]^-{{\rm  Ext}_2(V)}  &\cdots
\end{eqnarray*}
such that  for  any  positive  integer  $k$,    any  $s_1,s_2,\ldots, s_k\in \mathbb{N}$  and
any  $\beta_i\in  {\rm  Ext}_{2s_i}(V)$  for  $i=1,2,\ldots, k$,
the  morphism   $\beta_1\wedge\beta_2\wedge\cdots\wedge\beta_k\in  {\rm  Ext}_{2(s_1+s_2+\cdots +s_k)}(V)$   of  persistent  Mayer-Vietoris sequences
can  be  identified  with  the  composition  $\beta_1\circ \beta_2\circ\cdots \circ \beta_k$  of the  morphisms $\beta_1,\beta_2,\ldots,\beta_k$;

* Let  $\{\mathcal{L}_{1,x}\}_{x\in\mathbb{R}}$
and  $\{\mathcal{L}_{2,x}\}_{x\in\mathbb{R}}$  be   two  filtrations    of   independence   hypergraphs    with their  vertices from  $V$.
Then  we  have a  diagram  of   persistent   Mayer-Vietoris  sequences

\begin{eqnarray*}
\xymatrix{
\cdots \ar@/_3pc/[rr]^-{{\rm  Ext}^4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}^6(V)} \ar[r]^-{{\rm  Ext}^2(V)}&{\bf   PMV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x}, \omega,m-2\mid  x\in\mathbb{R})\ar[r]^-{{\rm  Ext}^2(V)}\ar@/_3pc/[rr]^-{{\rm  Ext}^4(V)}      &{\bf  PMV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x}, \omega,m\mid  x\in\mathbb{R})\ar[r]^-{{\rm  Ext}^2(V)}  & \cdots
\end{eqnarray*}
such that  for  any  positive  integer  $k$,    any  $s_1,s_2,\ldots, s_k\in \mathbb{N}$  and
any  $\mu_i\in  {\rm  Ext}^{2s_i}(V)$  for  $i=1,2,\ldots, k$,
the  morphism   $\mu_1\wedge\mu_2\wedge\cdots\wedge\mu_k\in  {\rm  Ext}^{2(s_1+s_2+\cdots +s_k)}(V)$   of  persistent  Mayer-Vietoris sequences
can  be  identified  with  the  composition  $\mu_1\circ \mu_2\circ\cdots \circ \mu_k$  of the  morphisms $\mu_1,\mu_2,\ldots,\mu_k$.

In  Theorem <ref>,  we  use

\begin{eqnarray*}
&{\rm  Ext}_{2s}(V):  ~~~{\bf   PMV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x}, \alpha,m+2\mid  x\in\mathbb{R})\longrightarrow  {\bf   PMV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x}, \alpha,m \mid  x\in\mathbb{R})
\end{eqnarray*}

to  denote  the  family  of   morphisms   of  persistent  Mayer-Vietoris  sequences

\begin{eqnarray*}
& \beta_*: ~~~{\bf   PMV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x}, \alpha,m+2\mid  x\in\mathbb{R})\longrightarrow  {\bf   PMV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x}, \alpha,m \mid  x\in\mathbb{R})
\end{eqnarray*}

for  all $ \beta\in {\rm  Ext}_{2s}(V) $
and   use

\begin{eqnarray*}
& {\rm  Ext}^{2s}(V):~~~{\bf   PMV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x}, \omega,m-2\mid  x\in\mathbb{R})\longrightarrow  {\bf   PMV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x}, \omega,m \mid  x\in\mathbb{R})
\end{eqnarray*}

to denote   the  family  of
morphisms    of  persistent  Mayer-Vietoris  sequences

\begin{eqnarray*}
& \mu_*:~~~ {\bf   PMV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x}, \omega,m-2\mid  x\in\mathbb{R})\longrightarrow  {\bf   PMV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x}, \omega,m \mid  x\in\mathbb{R})
\end{eqnarray*}

for  all $ \mu\in {\rm  Ext}^{2s}(V) $.
The  next  corollary  follows  from   Theorem <ref>   immediately.

Suppose $V$  is  a  finite  set.

* Let  $\{\mathcal{K}_{1,x}\}_{x\in\mathbb{R}}$
and  $\{\mathcal{K}_{2,x}\}_{x\in\mathbb{R}}$  be   two  filtrations    of  simplicial complexes    with their  vertices from  $V$.
Let  $\mathcal{L}_{1,x}=\Delta[V]\setminus  \mathcal{K}_{1,-x}$   and  $\mathcal{L}_{2,x}=\Delta[V]\setminus  \mathcal{K}_{2,-x}$   for  each  $x\in \mathbb{R}$.    Then
$\{\mathcal{L}_{1,x}\}_{x\in\mathbb{R}}$  and     $\{\mathcal{L}_{2,x}\}_{x\in\mathbb{R}}$
are  filtrations  of  independence hypergraphs  whose  constrained persistent  cohomology  satisfies  Theorem <ref> (2);

* Let  $\{\mathcal{L}_{1,x}\}_{x\in\mathbb{R}}$
and  $\{\mathcal{L}_{2,x}\}_{x\in\mathbb{R}}$  be   two  filtrations    of   independence   hypergraphs    with their  vertices from  $V$.     Let  $\mathcal{K}_{1,x}=\Delta[V]\setminus  \mathcal{L}_{1,-x}$   and  $\mathcal{K}_{2,x}=\Delta[V]\setminus  \mathcal{L}_{2,-x}$   for  each  $x\in \mathbb{R}$.
Then   $\{\mathcal{K}_{1,x}\}_{x\in\mathbb{R}}$  and     $\{\mathcal{K}_{2,x}\}_{x\in\mathbb{R}}$
are  filtrations  of  simplicial  complexes  whose  constrained  persistent homology satisfies   Theorem <ref> (1).

§ DIFFERENTIAL  CALCULUS  ON  DISCRETE  SETS

Let  $\mathbb{C}$  be  be  the  complex  number  field.

Let  ${\rm  Ext} [V]$  be  the  exterior  algebra  generated  by  $V$  over  $\mathbb{C}$.

${\rm  Ext}[V]=\bigoplus  _{p=0}^\infty   {\rm  Ext}^p[V]$

${\rm  Ext}^p[V]$   is  the  complex  vector  space  consisting  of  all  the   formal   (finitely  many)  linear    combinations  of  the   elements  of  the  form  $v_0\wedge  v_1\wedge  \cdots  \wedge  v_p$  with  $ v_0,v_1,\ldots,v_p\in  V $     distinct.

Let  $n\in\mathbb{N}$.
An elementary $n$-path     on $V$  is an  ordered sequence $v_0v_1\ldots v_n$ of  (not  necessarily  distinct)  $n+1$ vertices in  $V$   (cf.  <cit.>,  [15, 17, 18, 20, 19]).
A formal linear combination of elementary $n$-paths on $V$ with coefficients in  $\mathbb{C}$  is called an $n$-path  on $V$.
Denote by $\Lambda_n(V)$  the  vector  space  of all the  $n$-paths  on  $V$ (cf.  <cit.>,   [15, 17, 18, 20, 19]).     We have a graded  vector  space
\Lambda_*(V)=\bigoplus_{n=0}^\infty \Lambda_n(V)$.
We have a canonical  Hermitian  product
on $\Lambda(V)$  given  by

\begin{eqnarray*}
\langle u_0u_1\ldots u_n,v_0v_1\ldots v_m\rangle =
\begin{cases}
\prod_{i=0}^n  \delta(u_i,v_i),  &  n=m\\
0,  &n\neq  m
\end{cases}
\end{eqnarray*}

which  extends  by
$\langle  \xi_1+\xi_2,\eta\rangle =\langle \xi_1,\eta\rangle+\langle \xi_2,\eta\rangle$,
$\langle \xi,\eta_1+\eta_2\rangle =\langle \xi,\eta_1\rangle +\langle \xi,\eta_2\rangle$   and
$\langle  c  \xi, \eta\rangle= \langle  \xi,\bar  c\eta\rangle=c \langle  \xi,\eta\rangle$  for any  $\xi,\xi_1,\xi_2,\eta,\eta_1,\eta_2\in  \Lambda(V)$  and  any  $c\in\mathbb{C}$.   Here     for  any  vertices  $u,v\in V$,  we  use the notation  $\delta(u,v)=1$  if $u=v$  and  $\delta(u,v)=0$  if  $u\neq v$.
The  join   $*$   sending  a  pair  of elementary  paths  $(v_0v_1\ldots  v_n, u_0u_1\ldots  u_m)$  to
$v_0v_1\ldots  v_n  u_0u_1\ldots  u_m$  induces  a  complex  bilinear  map
$*:  \Lambda_*(V)\times \Lambda_*(V)\longrightarrow  \Lambda_*(V)$  which  decomposes  into  graded  complex  bilinear  maps
$*: \Lambda_n(V)\times  \Lambda_m(V)\longrightarrow  \Lambda_{n+m+1}(V)$  for  $n,m\in  \mathbb{N}$.

For any $v\in V$,   the partial derivative on  $\Lambda_*(V)$ with respect to $v$ is defined  to be  a sequence of   linear maps   (cf.  <cit.>)

\begin{eqnarray}\label{eq-1.x1}
\frac{\partial}{\partial v}: ~~~ \Lambda_n(V)\longrightarrow \Lambda_{n-1}(V),~~~
n\in \mathbb{N}
\end{eqnarray}

given by

\begin{eqnarray}\label{eq-1.1}
\frac{\partial}{\partial v} (v_0v_1 \ldots v_n)=\sum_{i=0}^n (-1)^i  \delta(v,v_i) v_0\ldots \widehat{v_i}\ldots v_n
\end{eqnarray}

and   the partial differentiation  $dv$  on   $\Lambda_*(V)$  with respect to $v$  is  defined  to be a sequence of  linear maps
(cf.  <cit.>)

\begin{eqnarray}\label{eq-1.x2}
d v:  ~~~  \Lambda_{n}(V)\longrightarrow \Lambda_{n+1}(V), ~~~n\in \mathbb{N}
\end{eqnarray}

given  by

\begin{eqnarray} \label{eq-2.x5}
dv (u_0u_1\ldots u_{n-1})=\sum_{i=0}^n (-1)^i u_0u_1\ldots u_{i-1} v u_i u_{i+1}\ldots u_{n-1}.
\end{eqnarray}

It  can  be  derived  from  <cit.>  that  $\langle  \frac{\partial}{\partial v}(\eta),\xi\rangle =\langle \eta, dv(\xi)\rangle
$  for any $\xi,\eta\in \Lambda_*(V)$,   i.e.  $dv$  is the adjoint linear map of $\frac{\partial}{\partial v}$.

Let  ${\rm  Ext}_*(V)$  be  the  exterior  algebra  generated  by  $\frac{\partial}{\partial  v}$,   $v\in  V$,   over  $\mathbb{C}$
and  let   ${\rm  Ext}^*(V)$  be  the  exterior  algebra  generated  by  $ dv $,   $v\in  V$,   over  $\mathbb{C}$.
Then  both  ${\rm  Ext}_*(V)$  and  ${\rm  Ext}^*(V)$  are  isomorphic  to  ${\rm  Ext}[V]$.
It  is  proved  in  <cit.>  that
$\frac{\partial}{\partial  v}\circ  \frac{\partial}{\partial  u}=-\frac{\partial}{\partial  u}\circ  \frac{\partial}{\partial  v}$
and   in  <cit.>  that
$dv \circ du=-du\circ  dv$  for  any  $u,v\in  V$.
Thus  both ${\rm  Ext}_*(V)$  and  ${\rm  Ext}^*(V)$   act  on  $\Lambda_*(V)$  such that the
compositions  of  maps are  represented  by
exterior  products.
Let  $k\in  \mathbb{N}$.  Let  $\alpha\in  {\rm  Ext}_k(V)$  and  $\omega\in  {\rm  Ext}^k(V)$.
We  say that $\alpha$ and $\omega$ are adjoint to each other      if $\langle \alpha(\eta),\xi \rangle= \langle \eta,  \omega(\xi)\rangle
$ for any $n\in \mathbb{N}$,  any  $\xi\in \Lambda_n(V)$  and any $\eta\in \Lambda_{n+k}(V)$.
Similar  to  <cit.>,   it  is  direct    that
$\alpha$  and  $\omega$  are  adjoint  if  and  only  if

\begin{eqnarray*}
\alpha=\sum_{v_1,v_2,\ldots, v_k\in V}  z_{v_1,v_2,\ldots, v_k} \frac{\partial}{\partial v_1}\wedge  \frac{\partial}{\partial  v_2}\wedge \cdots \wedge\frac{\partial}{\partial  v_k},~~~ z_{v_1,v_2,\ldots, v_k}\in\mathbb{C}.
\end{eqnarray*}

\begin{eqnarray*}
\omega={\rm sgn}(k)\sum_{v_1,v_2,\ldots, v_k\in V} \bar {z}_{v_1,v_2,\ldots, v_k} d v_1\wedge  d  v_2\wedge \cdots \wedge  d v_k,~~~ \bar z_{v_1,v_2,\ldots, v_k}\in\mathbb{C}
\end{eqnarray*}

where ${\rm sgn}(k)=1$  if $k\equiv 0,1$  modulo  $4$  and ${\rm sgn}(k)=-1$  if $k\equiv 2,3$  modulo $4$.

Let  $\tilde  \Lambda_n(V)$  be  the   subspace  of  $\Lambda_n(V)$
spanned  by  the  elementary  $n$-paths  $v_0v_1\ldots  v_n$   such  that   $v_0,v_1,\ldots,  v_n$  are  distinct,  which are taken  as the  canonical basis.
An   elementary  $n$-path $v_0 v_1\ldots v_n$  on $V$  is  called   non-simplicial    if there exist integers  $0\leq i<j\leq n$  such that  either  $v_j\prec v_i$  or  $v_j=v_i$  (cf.  <cit.>).
Let $\mathcal{O}_n(V)$  be   the  complex  vector space spanned by all the  non-simplicial   elementary  $n$-paths   on $V$  (cf.  <cit.>).
$\Lambda_n(V)= \tilde  \Lambda_n(V)  \oplus \mathcal{O}_n(V)$   is  an  orthogonal  sum   for  $n\in\mathbb{N}$.   Let  $\tilde \Lambda_*(V)=\bigoplus _{n=0}^\infty   \tilde \Lambda_n(V)$  and    $\mathcal{O}_*(V)=\bigoplus _{n=0}^\infty  \mathcal{O}_n(V)$.
The  restriction  of  (<ref>)   gives  a  graded  linear  map

\begin{eqnarray*}
{\frac{\partial}{\partial v}}: ~~~ \tilde \Lambda_n(V)\longrightarrow \tilde \Lambda_{n-1}(V),~~~
n\in \mathbb{N}.
\end{eqnarray*}

By composing with the  canonical  projection $\pi:  \Lambda_*(V)\longrightarrow \tilde \Lambda_*(V)$
sending   $\mathcal{O}_*(V)$  to zero  and  sending  $\tilde \Lambda_*(V)$  identically to itself,   (<ref>)  gives  a graded  linear map

\begin{eqnarray*}%\label{eq-1.x2}
{d v}:  ~~~  \tilde \Lambda_{n}(V)\overset{dv}{\longrightarrow}  \Lambda_{n+1}(V)\overset{\pi}{\longrightarrow}\tilde  \Lambda_{n+1}(V), ~~~n\in \mathbb{N}.
\end{eqnarray*}

An  element  $\sigma\in \Delta[V]$  consisting  of  $n+1$  vertices in  $V$   is  an  $n$-hyperedge  on  $V$,
denoted as  $\sigma^{(n)}$.
Let  $S_{n+1}$  be the  symmetric  group  on  $(n+1)$-letters.  Then
$S_{n+1}$  acts  on  the  set  of  all  the  elementary  $n$-paths on  $V$.
With the  help of    this   group  action,
$\sigma^{(n)}$   can  be  expressed  as   an  orbit

\begin{eqnarray*}
\sigma^{(n)}= S_{n+1}  (v_0 v_1\ldots  v_n)= \{v_{s(0)} v_{s(1)}\ldots  v_{s(n)}\mid  s\in  S_{n+1}\}
\end{eqnarray*}

for  some  distinct  $v_0, v_1,\ldots,  v_n\in  V$.   Without  loss  of  generality,  we  choose  the  representative    $  v_0v_1\ldots   v_n$  of  $\sigma^{(n)}$  such that    $v_0\prec  v_1\prec \cdots\prec v_n$  and  write
$\sigma^{(n)}=  \{v_0,v_1,\ldots,v_n\}$.

\begin{eqnarray*}
\mathbb{C}_n(\Delta[V] )={\rm Span}_{\mathbb{C}}\{\sigma^{(n)}\in \Delta[V]\}
\end{eqnarray*}

be  the vector space  consisting of all the linear combinations of the $n$-hyperedges on $V$.   Consider   the  direct  sum

\begin{eqnarray*}
\mathbb{C}_*(\Delta[V] )=\bigoplus_{n\geq 0} \mathbb{C}_n(\Delta[V] ).
\end{eqnarray*}

Then  $\mathbb{C}_n(\Delta[V] )$   is  a    quotient  vector space  of   $\tilde \Lambda_n(V)$  by  moduling   the action  of  $S_{n+1}$  on  the  canonical basis   of  $\tilde \Lambda_n(V)$     for  each   $n\in \mathbb{N}$.
Choosing    the elementary  $n$-path  $v_0v_1\ldots  v_n$  on  $V$ satisfying
$v_0\prec  v_1\prec \cdots  \prec  v_n$  as   an  representative of $\sigma^{(n)}$,
we  have  that    $\mathbb{C}_*(\Delta[V] )$   is  a  graded  subspace  of
$\tilde  \Lambda_*(V)$  hence  is   also  a  graded  subspace  of  $  \Lambda_*(V)$.   It  is  direct
to verify  that   the  restriction of  $\frac{\partial}{\partial v}$  to  $\mathbb{C}_*(\Delta[V] )$  is  a  well-defined  map

\begin{eqnarray*}
\frac{\partial}{\partial  v}: ~~~ \mathbb{C}_{n }(\Delta[V] )\longrightarrow  \mathbb{C}_{n-1}(\Delta[V] ),  ~~~  n\in \mathbb{N}.
\end{eqnarray*}

Thus   for each   $\alpha\in {\rm  Ext}_k(V)$,  $k\in\mathbb{N}$,     the  restriction  of   $\alpha: \Lambda_n(V)\longrightarrow \Lambda_{n-k}(V)$   to  $\mathbb{C}_n(\Delta[V] )$  induces    a
graded  linear  map

\begin{eqnarray*}
\alpha:~~~ \mathbb{C}_n(\Delta[V] )\longrightarrow    \mathbb{C}_{n-k}(\Delta[V] ),~~~  n\in\mathbb{N}.
\end{eqnarray*}

On  the  other  hand,  it  is  direct
to verify  that   the  restriction  of   $dv$  to  $\mathcal{O}_*(V)$   is  a
well-defined  map

\begin{eqnarray*}
dv:  ~~~\mathcal{O}_n(V)\longrightarrow  \mathcal{O}_{n+1}(V),~~~  n\in\mathbb{N}.
\end{eqnarray*}

Thus  for each   $\omega\in {\rm  Ext}^k(V)$,  $k\in\mathbb{N}$,   the  restriction  of   $\omega: \Lambda_n(V)\longrightarrow \Lambda_{n+k}(V)$   to   $\mathcal{O}_{n}(V)$  induces  a  well-defined  graded  linear  map   $\omega: \mathcal{O}_n(V)\longrightarrow    \mathcal{O}_{n+k}(V)$.     Therefore,   the  quotient  map  gives  a  well-defined  graded  linear  map
$\omega:  \tilde  \Lambda_n(V)\longrightarrow \tilde \Lambda_{n+k}(V)$  which  consequently  induces
a graded  linear  map

\begin{eqnarray*}
\omega:~~~ \mathbb{C}_n(\Delta[V] )\longrightarrow    \mathbb{C}_{n+k}(\Delta[V] ),~~~  n\in\mathbb{N}.
\end{eqnarray*}

§ THE  CONSTRAINED   HOMOLOGY  OF  SIMPLICIAL  COMPLEXES  AND  THE  CONSTRAINED  COHOMOLOGY
OF  INDEPENDENCE  HYPERGRAPHS

Let  $t\in  \mathbb{N}$.
Let $\alpha\in {\rm Ext}_{2t+1}(V)$ and $\omega\in {\rm Ext}^{2t+1}(V)$.  Then for any  integer  $0\leq q\leq 2t$,    we  have a chain complex

\begin{eqnarray*}
\xymatrix{
\cdots\ar[r]^-{\alpha}
\cdots\ar[r]^-{\alpha}
\end{eqnarray*}

denoted  by  $\Lambda_*(V,\alpha,q)$,  and  a co-chain complex

\begin{eqnarray*}
\xymatrix{
\cdots
&  \ar[l]_-{\omega}\\
\cdots&\Lambda_{(2t+1)+q}(V)\ar[l]_-{\omega}
&\ar[l]_-\omega  0,
\end{eqnarray*}

denoted  by $\Lambda^*(V,\omega,q)$.
The  chain complex    $\Lambda_*(V,\alpha,q)$  has  a  sub-chain  complex

\begin{eqnarray*}
\xymatrix{
\cdots\ar[r]^-{\alpha}
&\tilde \Lambda_{n(2t+1)+q}(V)\ar[r]^-{\alpha}
&\tilde \Lambda_{(n-1)(2t+1)+q}(V)\ar[r]^-{\alpha}
\cdots\ar[r]^-{\alpha}
&\tilde \Lambda_{(2t+1)+q}(V)\ar[r]^-\alpha
&\tilde \Lambda_{q}(V)\ar[r]^-\alpha
\end{eqnarray*}

denoted  by  $\tilde \Lambda_*(V,\alpha,q)$.
The   co-chain  complex  $\Lambda^*(V,\omega,q)$
has  a  sub-co-chain  complex

\begin{eqnarray*}
\xymatrix{
\cdots
&  \ar[l]_-{\omega}\\
\cdots&\mathcal{O}_{(2t+1)+q}(V)\ar[l]_-{\omega}
&\ar[l]_-\omega  0,
\end{eqnarray*}

denoted  by $\mathcal{O}^*(V,\omega,q)$.
Moduling   $\mathcal{O}^*(V,\omega,q)$  in    $\Lambda^*(V,\omega,q)$,    we  obtain   a   quotient  co-chain  complex

\begin{eqnarray*}
\xymatrix{
\cdots
&\tilde \Lambda_{n(2t+1)+q}(V)\ar[l]_-{\omega}
&\tilde \Lambda_{(n-1)(2t+1)+q}(V)\ar[l]_-{\omega}
&  \ar[l]_-{\omega}\\
\cdots&\tilde \Lambda_{(2t+1)+q}(V)\ar[l]_-{\omega}
&\tilde \Lambda_{q}(V)\ar[l]_-\omega
&\ar[l]_-\omega  0,
\end{eqnarray*}

denoted  by $\tilde \Lambda^*(V,\omega,q)$.
Moduling  the  action of  the  symmetric  groups  on  the  canonical basis  of  $\tilde \Lambda_*(V,\alpha,q)$,  we  obtain    an  induced   chain  complex

\begin{eqnarray*}
\xymatrix{
\cdots\ar[r]^-{\alpha}
\cdots\ar[r]^-{\alpha}
\end{eqnarray*}

denoted  by  $C_*(\Delta[V],\alpha,q)$.    Moduling  the  action of  the  symmetric  groups  on  the  canonical  basis  of   $\tilde \Lambda^*(V,\alpha,q)$,  we  obtain    an  induced   co-chain  complex

\begin{eqnarray*}
\xymatrix{
\cdots
&  \ar[l]_-{\omega}\\
\cdots&\mathbb{C}_{(2t+1)+q}(\Delta[V])\ar[l]_-{\omega}
&\ar[l]_-\omega  0,
\end{eqnarray*}

denoted  by  $C^*(\Delta[V],\omega,q)$.

Let  $\mathbb{Z}$  be  the ring  of  integers.
For any  $m\in \mathbb{Z}$,  there is a unique  $\lambda\in \mathbb{Z}$   and a unique  integer  $0\leq q\leq 2t$  such that $m=\lambda(2t+1)+q$.
Denote the chain complex

\begin{eqnarray*}
\xymatrix{
\cdots\ar[r]^-{\alpha}
\cdots\ar[r]^-{\alpha}
\end{eqnarray*}

$\Lambda_*(V,\alpha,m) $
the  co-chain complex

\begin{eqnarray*}
\xymatrix{
\cdots
&  \ar[l]_-{\omega}\\
\cdots&\Lambda_{(1+\lambda)(2t+1)+q}(V)\ar[l]_-{\omega}
&\ar[l]_-\omega  0
\end{eqnarray*}

\Lambda^*(V,\omega,m)$.
Here   we  use the notation $\Lambda_k(V)=0$  for $k<0$.
Similar  notations  for $\tilde \Lambda_*(V,\alpha,m) $,   $
\tilde \Lambda^*(V,\omega,m)$,
$\mathbb{C}_*(\Delta[V],\alpha,m) $,  $\mathbb{C}^*(\Delta[V],\omega,m) $  and  $\mathcal{O}^*(V, \omega,m)$.
Let $s\in \mathbb{N}$.  Let  $\beta\in {\rm Ext}_{2s}(V)$  and $\mu\in {\rm Ext}^{2s}(V)$.  Then $\beta$  gives  a  chain map

\begin{eqnarray}\label{eq-8a}
\beta:~~~  \Lambda_*(V,\alpha,m)\longrightarrow \Lambda_*(V,\alpha,m-2s)
\end{eqnarray}

and $\mu$  gives a   co-chain map

\begin{eqnarray}\label{eq-8b}
\mu:~~~  \Lambda^*(V,\omega,m)\longrightarrow \Lambda^*(V,\omega,m+2s).
\end{eqnarray}

The  chain map (<ref>)  induces  a  chain  map

\begin{eqnarray*}
\beta:~~~  \tilde \Lambda_*(V,\alpha,m)\longrightarrow \tilde  \Lambda_*(V,\alpha,m-2s)
\end{eqnarray*}

and  consequently induces a  chain map

\begin{eqnarray*}
\beta:~~~  \mathbb{C}_*(\Delta[V],\alpha,m)\longrightarrow \mathbb{C}_*(\Delta[V],\alpha,m-2s).
\end{eqnarray*}

Similarly,   the  co-chain map
(<ref>)  induces  a  co-chain  map

\begin{eqnarray*}
\mu:~~~\mathcal{O}^*(V,\omega,m)\longrightarrow \mathcal{O}^*(V,\omega,m+2s),
\end{eqnarray*}

which,  together  with  (<ref>),   induces  a  quotient co-chain  map

\begin{eqnarray*}\label{eq-1.8.b}
\mu:~~~  \tilde \Lambda^*(V,\omega,m)\longrightarrow \tilde\Lambda^*(V,\omega,m+2s)
\end{eqnarray*}

and  consequently  induces  a  co-chain map

\begin{eqnarray*}
\mu:~~~  \mathbb{C}^*(\Delta[V],\omega,m)\longrightarrow \mathbb{C}^*(\Delta[V],\omega,m+2s).
\end{eqnarray*}

Let  $s_1,s_2\in \mathbb{N}$.

* Let $\beta_1\in    {\rm Ext}_{2s_1}(V) $   and   $\beta_2\in    {\rm Ext}_{2s_2}(V) $.   Then  the  diagram  commutes

\begin{eqnarray*}
\xymatrix{
\mathbb{C}_*(\Delta[V],\alpha,m)\ar[r]^{\beta_1} \ar[rd] _{\beta_1\wedge \beta_2}&     \mathbb{C}_*(\Delta[V],\alpha,m-2s_1)\ar[d]^{\beta_2}\\
&      \mathbb{C}_*(\Delta[V],\alpha,m-2(s_1+s_2));
\end{eqnarray*}

* Let $\mu_1\in    {\rm Ext}^{2s_1}(V) $   and   $\mu_2\in    {\rm Ext}^{2s_2}(V) $.   Then  the  diagram  commutes

\begin{eqnarray*}
\xymatrix{
\mathbb{C}^*(\Delta[V],\omega,m)\ar[r]^{\mu_1} \ar[rd] _{\mu_1\wedge \mu_2}&     \mathbb{C}^*(\Delta[V],\omega,m+2s_1)\ar[d]^{\mu_2}\\
&    \mathbb{C}^*(\Delta[V],\omega,m+2(s_1+s_2)).
\end{eqnarray*}

Since  $\beta_1\wedge \beta_2=\beta_2\wedge\beta_1=\beta_2\circ \beta_1$  is  the  composition  of  $\beta_2$  and  $\beta_1$  (or  equivalently,  the  composition  of  $\beta_1$  and  $\beta_2$),  we  have (1).   Similarly,  since  $\mu_1\wedge \mu_2=\mu_2\wedge\mu_1=\mu_2\circ \mu_1$  is  the  composition  of  $\mu_2$  and  $\mu_1$,   we  have  (2).

Let $\mathcal{K}$  be a simplicial  complex  with its vertices from  $V$.
Let  $\mathbb{C}_n(\mathcal{K} )$  be the  complex  vector space spanned  by the $n$-simplices in $\mathcal{K}$.
Let  $t,s\in  \mathbb{N}$.   Let  $m\in \mathbb{Z}$.    Suppose $m=\lambda(2t+1)+q$  where $\lambda\in \mathbb{Z}$  and  $0\leq q\leq 2t$.  It  can  be  derived  from  <cit.>  that
for any $\alpha\in {\rm Ext}_{2t+1}(V)$,
the graded vector space
\mathbb{C}_{(n+\lambda)(2t+1)+q}(\mathcal{K})\}_{  n\in  \mathbb{N}}
equipped with the  chain  map  $\alpha$  gives a sub-chain  complex   of $C_*(\Delta[V],\alpha, m)$,  which will be  denoted  as $C_*(\mathcal{K},\alpha, m)$.   Moreover,
for any $\beta\in {\rm Ext}_{2s}(V)$,  there is  an  induced   chain map

\begin{eqnarray*}%\label{eq-4.89a}
\beta:~~~  C_*(\mathcal{K},\alpha,m)\longrightarrow  C_*(\mathcal{K},\alpha,m-2s).
\end{eqnarray*}

The  $n$-th  constrained homology group $H_n(\mathcal{K},\alpha,m)$ of $\mathcal{K}$  with respect to $\alpha$ and $m$   is  defined  to be the  $n$-th   homology group  (cf.  <cit.>)

\begin{eqnarray*}
= \frac{{\rm Ker}\Big(\alpha:C_{(n+\lambda)(2t+1)+q}(\mathcal{K};\mathbb{R})\longrightarrow C_{(n-1+\lambda)(2t+1)+q}(\mathcal{K};\mathbb{R})\Big)}{ {\rm Im}\Big(\alpha:C_{(n+1+\lambda)(2t+1)+q}(\mathcal{K};\mathbb{R})\longrightarrow C_{(n+\lambda)(2t+1)+q}(\mathcal{K};\mathbb{R})\Big) }
\end{eqnarray*}

of the chain complex $C_*(\mathcal{K},\alpha, m)$.
It   can  be   derived  from  <cit.>  that   for any $\alpha\in {\rm Ext}_{2t+1}(V)$  and $\beta\in {\rm Ext}_{2s}(V)$,  there is an  induced homomorphism

\begin{eqnarray*}
\beta_*: ~~~H_n(\mathcal{K},\alpha,m)\longrightarrow  H_n(\mathcal{K},\alpha,m-2s), ~~~~~~  n\in  \mathbb{N}
\end{eqnarray*}

of  the  constrained  homology groups.

Let $\mathcal{L}$  be a  independent hypergraph  with its vertices in $V$.
Let  $\mathbb{C}_n(\mathcal{L} )$  be the  complex  vector space spanned  by the $n$-hyperedges
in $\mathcal{L}$.
It  can  be  derived  from  <cit.>  that   for any $\omega\in {\rm Ext}^{2t+1}(V)$,
the graded vector space
C_{(n+\lambda)(2t+1)+q}(\mathcal{L};\mathbb{R})\}_{  n\in  \mathbb{N}}$
equipped with  the  co-boundary  map  $\omega$  gives a sub-co-chain  complex   of $C^*(\Delta[V],\omega, m)$,  which will be denoted  as $C^*(\mathcal{L},\omega, m)$.
Moreover,  for any $\mu\in {\rm Ext}^{2s}(V)$,  there is  an  induced  co-chain map

\begin{eqnarray*}
\mu:~~~  C^*(\mathcal{L},\omega,m)\longrightarrow  C^*(\mathcal{L},\omega,m+2s).
\end{eqnarray*}

The  $n$-th  constrained  cohomology group  $H^n(\mathcal{L},\omega, m)$  of $\mathcal{L}$  with respect to $\omega$ and $m$   is  defined  to be the  cohomology group  (cf.  <cit.>)

\begin{eqnarray*}
H^n(\mathcal{L},\omega, m)    =  \frac{{\rm Ker}\Big(\omega:  C_{(n+\lambda)(2t+1)+q}(\mathcal{L};\mathbb{R})\longrightarrow C_{(n+1+\lambda)(2t+1)+q}(\mathcal{L};\mathbb{R})\Big)}{ {\rm Im}\Big(\omega: C_{(n-1+\lambda)(2t+1)+q}(\mathcal{L};\mathbb{R})\longrightarrow C_{(n+\lambda)(2t+1)+q}(\mathcal{L};\mathbb{R})\Big) }
\end{eqnarray*}

of the  co-chain complex $C^*(\mathcal{L},\omega, m)$.
It   can  be   derived  from  <cit.>  that
for any $\omega\in {\rm Ext}^{2t+1}(V)$  and $\mu\in {\rm Ext}^{2s}(V)$,  there is an  induced homomorphism

\begin{eqnarray*}
\mu_*: ~~~H^n(\mathcal{L},\omega,m)\longrightarrow  H^n(\mathcal{L},\omega,m+2s), ~~~~~~  n\in  \mathbb{N}
\end{eqnarray*}

of  the  constrained  cohomology groups.

Let  $s_1,s_2\in \mathbb{N}$.

* Let $\beta_1\in    {\rm Ext}_{2s_1}(V) $   and   $\beta_2\in    {\rm Ext}_{2s_2}(V) $.   Then  the  diagram  commutes

\begin{eqnarray*}
& \xymatrix{
\mathbb{C}_*(\mathcal{K},\alpha,m)\ar[r]^{\beta_1} \ar[rd] _{\beta_1\wedge \beta_2}&     \mathbb{C}_*(\mathcal{K},\alpha,m-2s_1)\ar[d]^{\beta_2}\\
&      \mathbb{C}_*(\mathcal{K},\alpha,m-2(s_1+s_2));
\end{eqnarray*}

* Let $\mu_1\in    {\rm Ext}^{2s_1}(V) $   and   $\mu_2\in    {\rm Ext}^{2s_2}(V) $.   Then  the  diagram  commutes

\begin{eqnarray*}
& \xymatrix{
\mathbb{C}^*(\mathcal{L},\omega,m)\ar[r]^{\mu_1} \ar[rd] _{\mu_1\wedge \mu_2}&     \mathbb{C}^*(\mathcal{L},\omega,m+2s_1)\ar[d]^{\mu_2}\\
&    \mathbb{C}^*(\mathcal{L},\omega,m+2(s_1+s_2)).
\end{eqnarray*}

The restriction of  the  chain maps in  the diagram in  Proposition <ref> (1)  to the  corresponding  sub-chain complexes
gives  the  commutative  diagram in  (1).   The restriction of   the  co-chain maps in the diagram in  Proposition <ref> (2)  to the  corresponding  sub-co-chain complexes
gives  the  commutative  diagram in  (2).

Let  $s_1,s_2\in \mathbb{N}$.

* Let $\beta_1\in    {\rm Ext}_{2s_1}(V) $   and   $\beta_2\in    {\rm Ext}_{2s_2}(V) $.   Then  the  diagram  commutes

\begin{eqnarray*}
\xymatrix{
H_*(\mathcal{K},\alpha,m)\ar[r]^{\beta_1} \ar[rd] _{\beta_1\wedge \beta_2}&    H_*(\mathcal{K},\alpha,m-2s_1)\ar[d]^{\beta_2}\\
&      H_*(\mathcal{K},\alpha,m-2(s_1+s_2));
\end{eqnarray*}

* Let $\mu_1\in    {\rm Ext}^{2s_1}(V) $   and   $\mu_2\in    {\rm Ext}^{2s_2}(V) $.   Then  the  diagram  commutes

\begin{eqnarray*}
\xymatrix{
H^*(\mathcal{L},\omega,m)\ar[r]^{\mu_1} \ar[rd] _{\mu_1\wedge \mu_2}&   H^*(\mathcal{L},\omega,m+2s_1)\ar[d]^{\mu_2}\\
&    H^*(\mathcal{L},\omega,m+2(s_1+s_2)).
\end{eqnarray*}

Taking  the constrained  homology  to  the  diagram in  Lemma <ref> (1) and by the functoriality  of  the  homology  of  chain  complexes,  we  obtain  (1).
Similarly,  taking the constrained  cohomology  to  the  diagram  in  Lemma <ref> (2)   and by the functoriality  of  the  cohomology  of  co-chain  complexes,  we  obtain  (2).

Let $t \in  \mathbb{N}$.

* Let  $\mathcal{K}_1$  and  $\mathcal{K}_2$   be
simplicial  complexes   with  their  vertices in $V$.
For  any
$\alpha\in {\rm Ext}_{2t+1}(V)$,      we  have
a  long  exact  sequence  (<ref>).
Moreover,     the  long  exact  sequence   is  functorial  with  respect
to  simplicial maps;

* Let  $\mathcal{L}_1$  and  $\mathcal{L}_2$   be
independence  hypergraphs   with  their  vertices in $V$.
For      any
$\omega\in {\rm Ext}^{2t+1}(V)$,      we  have
a  long  exact  sequence   (<ref>).
Moreover,     the  long  exact  sequence   is  functorial  with  respect
to  morphisms  of  independence hypergraphs  induced by  bijective  self-maps  on  the  vertex set  $V$.

(1).  Let  $\mathcal{K}_1$  and  $\mathcal{K}_2$   be
simplicial  complexes   with  their  vertices from  $V$.
Then  both
$\mathcal{K}_1\cap\mathcal{K}_2$   and
$\mathcal{K}_1\cup \mathcal{K}_2$  are  simplicial  complexes  with  their  vertices  from  $V$.
We  have a  short  exact  sequence  of  chain  complexes

\begin{eqnarray*}
&0\longrightarrow       C_*(\mathcal{K}_1 \cap  \mathcal{K}_2,\alpha, m)\longrightarrow  C_*(\mathcal{K}_1,\alpha, m) \oplus   C_*( \mathcal{K}_2,\alpha, m)\longrightarrow  C_*(\mathcal{K}_1 \cup  \mathcal{K}_2,\alpha, m)\longrightarrow 0
\end{eqnarray*}

which  is  functorial  with  respect to  simplicial maps.
This  induces  a  long  exact sequence  (<ref>)  of the constrained  homology  groups  which  is  functorial  with  respect  to  simplicial  maps.

(2).  Let  $\mathcal{L}_1$  and  $\mathcal{L}_2$   be
independence  hypergraphs   with  their  vertices from  $V$.
Then  both
$\mathcal{L}_1\cap\mathcal{L}_2$   and
$\mathcal{L}_1\cup \mathcal{L}_2$  are  independence  hypergraphs  with  their  vertices  from  $V$.
We  have a  short  exact  sequence  of  co-chain  complexes

\begin{eqnarray*}
&0\longrightarrow       C^*(\mathcal{L}_1 \cap  \mathcal{L}_2,\omega, m)\longrightarrow  C^*(\mathcal{L}_1,\omega, m) \oplus   C^*( \mathcal{L}_2,\omega, m)\longrightarrow  C^*(\mathcal{L}_1 \cup  \mathcal{L}_2,\omega, m)\longrightarrow 0
\end{eqnarray*}

which  is  functorial  with  respect to  morphisms  of  independence hypergraphs  induced by  bijective  self-maps  on  the  vertex  set   $V$.
This  induces  a  long  exact sequence  (<ref>)  of the constrained  cohomology  groups  which  is  functorial  with  respect  to morphisms  of  independence hypergraphs  induced by  bijective  self-maps  on    $V$.

§ THE  FUNCTORIALITIES

Let  $V$  and  $V'$  be     two  discrete  sets.
Suppose  $V$  as  well as   $V'$  has  a  total  order.
Let  $f:  V\longrightarrow  V'$  be  a  map.   We  have  an  induced  graded  linear  map  $\Lambda_*(f):  \Lambda_*(V)\longrightarrow  \Lambda_*(V')$  given  by
$ \Lambda_n(f)(v_0\ldots  v_n)=f(v_0)\ldots  f(v_n)
$   for  each  $n\in \mathbb{N}$ and  extending   linearly  over  $\mathbb{C}$.

Suppose  in addition  that  $f$  is  injective  from  $V$  to  $V'$.
Then   $\langle  \Lambda_*(f)(\eta), \Lambda_*(f)(\xi)\rangle=\langle \eta,\xi\rangle$  for any $\xi,\eta\in \Lambda_*(V)$.
We  have  induced  homomorphisms  of  exterior  algebras
${\rm  Ext}_*(f):  {\rm  Ext}_*(V)\longrightarrow {\rm  Ext}_*(V')$  sending  $\frac{\partial}{\partial  v}$  to $\frac{\partial}{\partial  f(v)}$      for  each  $v\in  V$
and   ${\rm  Ext}^*(f):  {\rm  Ext}^*(V)\longrightarrow {\rm  Ext}^*(V')$     sending
$dv$  to
$df(v)$     for  each  $v\in  V$.
For  any   $k\in  \mathbb{N}$,  if $\alpha\in  {\rm  Ext}_k(V)$  and  $\omega\in  {\rm  Ext}^k(V)$
are  adjoint,  then
${\rm  Ext}_*(f)(\alpha)$  and  ${\rm  Ext}^*(f)(\omega)$  are  adjoint  as  well.

The   restriction  of  $\Lambda_*(f)$  to  $\tilde \Lambda_*(f)$
induces  a  graded  linear map  $\tilde \Lambda_*(f): \tilde\Lambda_*(V)\longrightarrow \tilde \Lambda_*(V') $.
The  restriction  of   $\tilde \Lambda_*(f)$  to  $C_*(\Delta[V])$
induces  a   graded  linear  map
$\mathbb{C}_*(f):  \mathbb{C}_*(\Delta[V])\longrightarrow  \mathbb{C}_*(\Delta[V'])$.

For  any  $t\in  \mathbb{N}$,  any  integer    $0\leq q\leq 2t$,    any   $\alpha\in {\rm Ext}_{2t+1}(V)$ and   any  $\omega\in {\rm Ext}^{2t+1}(V)$,    we  have  an  induced  chain  map

\begin{eqnarray}\label{eq-3.1a}
f_\#: ~~~ \Lambda_*(V,\alpha,q)\longrightarrow  \Lambda_*(V', {\rm  Ext}_*(f)(\alpha),q)
\end{eqnarray}

and  an  induced   co-chain  map

\begin{eqnarray}\label{eq-3.1b}
f^\#: ~~~ \Lambda^*(V,\omega,q)\longrightarrow  \Lambda^*(V', {\rm  Ext}^*(f)(\omega),q).
\end{eqnarray}

Moreover,  for  any  $m\in \mathbb{Z}$,  any  $s\in \mathbb{N}$,  any  $\beta\in {\rm Ext}_{2s}(V)$  and  any  $\mu\in {\rm Ext}^{2s}(V)$,   we   have  a  commutative  diagram  whose  arrows  are   chain  maps

\begin{eqnarray}\label{diag-3.1.a}
\xymatrix{
\Lambda_*(V,\alpha,m)\ar[d]_{f_\#}\ar[rr]^-{\beta} && \Lambda_*(V,\alpha,m-2s)\ar[d]^{f_\#}\\
\Lambda_*(V',{\rm  Ext}_*(f)(\alpha),m)\ar[rr]^-{{\rm  Ext}_*(f)(\beta)} &&  \Lambda_*(V', {\rm  Ext}_*(f)(\alpha),m-2s)
\end{eqnarray}

and  a commutative  diagram    whose  arrows  are    co-chain   maps

\begin{eqnarray}\label{diag-3.1.b}
\xymatrix{
\Lambda^*(V,\omega,m)\ar[d]_{f^\#}\ar[rr]^-{\mu} && \Lambda^*(V,\omega,m+2s)\ar[d]^{f^\#}\\
\Lambda^*(V',{\rm  Ext}^*(f)(\omega),m)\ar[rr]^-{{\rm  Ext}^*(f)(\mu)} &&  \Lambda^*(V', {\rm  Ext}^*(f)(\omega),m+2s).
\end{eqnarray}

We   notice  that  the injectivity  of  $f$  is  essential  for   the  well-definedness ${\rm  Ext}_*(f)$,  ${\rm  Ext}^*(f)$,  $f_\#$  and  $f^\#$.

By restricting  (<ref>)   to  the  sub-chain complex  $\tilde\Lambda_*(V)$,  $f_\#$  induces  a  chain  map

\begin{eqnarray}\label{eq-3.2.kk}
f_\#: ~~~ \tilde \Lambda_*(V,\alpha,q)\longrightarrow  \tilde  \Lambda_*(V', {\rm  Ext}_*(f)(\alpha),q).
\end{eqnarray}

By  restricting  (<ref>)  to
the  sub-chain  complex  $\mathbb{C}_*(\Delta[V],\alpha,q)$,   we  obtain   a  chain  map

\begin{eqnarray*}%\label{eq-3.2a}
f_\# : ~~~ \mathbb{C}_*(\Delta[V],\alpha,q)\longrightarrow  \mathbb{C}_*(\Delta[V'], {\rm  Ext}_*(f)(\alpha),q).
\end{eqnarray*}

Similarly,  by  restricting  (<ref>)   to  the  sub-co-chain  complex    $\mathcal{O}^*(V,\omega,q)$,    $f^\#$  induces
a  co-chain  map

\begin{eqnarray}\label{eq-3.2bc}
f^\# : ~~~ \mathcal{O}^*(V,\omega,q)\longrightarrow  \mathcal{O}_*(V', {\rm  Ext}^*(f)(\omega),q).
\end{eqnarray}

By   (<ref>)   and (<ref>)  we  have  an  induced   co-chain  map

\begin{eqnarray}\label{eq-xyz123}
f^\#: ~~~ \tilde \Lambda^*(V,\omega,q)\longrightarrow \tilde \Lambda_*(V', {\rm  Ext}^*(f)(\omega),q).
\end{eqnarray}

By  restricting  (<ref>)  to  the  sub-co-chain  complex  $\mathbb{C}^*(\Delta[V],\omega,q)$,
we   obtain   a  co-chain map

\begin{eqnarray*}
f^\#:  ~~~  \mathbb{C}^*(\Delta[V],\omega,q)\longrightarrow  \mathbb{C}^*(\Delta[V'], {\rm  Ext}^*(f)(\omega),q).
\end{eqnarray*}

For  any  $m\in \mathbb{Z}$,  any  $s\in \mathbb{N}$,  any  $\beta\in {\rm Ext}_{2s}(V)$  and  any  $\mu\in {\rm Ext}^{2s}(V)$,   by  restricting  (<ref>)  to  the  corresponding   sub-chain complexes,   we   have  a  commutative  diagram  whose  arrows   are    chain  maps

\begin{eqnarray}\label{eq-diag.a0}
\xymatrix{
\mathbb{C}_*(\Delta[V],\alpha,m)\ar[d]_{ f_\# }\ar[rr]^-{\beta} && \mathbb{C}_*(\Delta[V],\alpha,m-2s)\ar[d]^{ f_\# }\\
\mathbb{C}_*(\Delta[V'],{\rm  Ext}_*(f)(\alpha),m)\ar[rr]^-{{\rm  Ext}_*(f)(\beta)} &&  \mathbb{C}_*(\Delta[V'], {\rm  Ext}_*(f)(\alpha),m-2s).
\end{eqnarray}

Similarly,   by  restricting  (<ref>)  to  the  corresponding   sub-co-chain complexes,   we   have  a  commutative  diagram   whose  arrows  are    co-chain  maps

\begin{eqnarray} \label{eq-diag.b0}
\xymatrix{
\mathbb{C}^*(\Delta[V],\omega,m)\ar[d]_{f^\#}\ar[rr]^-{\mu} && \mathbb{C}^*(\Delta[V],\omega,m+2s)\ar[d]^{f^\#}\\
\mathbb{C}^*(\Delta[V'],{\rm  Ext}^*(f)(\omega),m)\ar[rr]^-{{\rm  Ext}^*(f)(\mu)} &&  \mathbb{C}^*(\Delta[V'], {\rm  Ext}^*(f)(\omega),m+2s).
\end{eqnarray}

Let $\mathcal{K}$    and   $\mathcal{K}'$   be   simplicial  complexes   with their  vertices  from   $V$   and   $V'$  respectively.   Suppose  there  is  a    simplicial  map  $f:  \mathcal{K}\longrightarrow  \mathcal{K}'$
which  is  given  by an  injective  map
$f:  V\longrightarrow  V'$.    Then  the    diagram  (<ref>)  induces     a  commutative  diagram  whose
arrows  are   chain maps

\begin{eqnarray}\label{eq-diag.a1}
\xymatrix{
\mathbb{C}_*(\mathcal{K},\alpha,m)\ar[d]_{ f_\# }\ar[rr]^-{\beta} && \mathbb{C}_*(\mathcal{K},\alpha,m-2s)\ar[d]^{ f_\# }\\
\mathbb{C}_*(\mathcal{K}',{\rm  Ext}_*(f)(\alpha),m)\ar[rr]^-{{\rm  Ext}_*(f)(\beta)} &&  \mathbb{C}_*(\mathcal{K}', {\rm  Ext}_*(f)(\alpha),m-2s).
\end{eqnarray}

Each  chain  complex   in  (<ref>)   is   a  sub-chain complex  of  the  corresponding  chain  complex   in  (<ref>).
Applying the homology functor,  we  have a  commutative diagram  of  constrained homology groups

\begin{eqnarray}\label{eq-diag.a2}
\xymatrix{
H_*(\mathcal{K},\alpha,m)\ar[d]_{ f_* }\ar[rr]^-{\beta} && H_*(\mathcal{K},\alpha,m-2s)\ar[d]^{ f_* }\\
H_*(\mathcal{K}',{\rm  Ext}_*(f)(\alpha),m)\ar[rr]^-{{\rm  Ext}_*(f)(\beta)} &&  H_*(\mathcal{K}', {\rm  Ext}_*(f)(\alpha),m-2s).
\end{eqnarray}

Let $\mathcal{L}$    and   $\mathcal{L}'$   be  independence  hypergraphs   with their  vertices in $V$   and   $V'$  respectively.   Suppose  there  is  a    morphism   of   independence  hypergraphs  $f:  \mathcal{L}\longrightarrow  \mathcal{L}'$
which  is  given  by an  bijective  map
$f:  V\longrightarrow  V'$,  i.e.  $V=V'$  and  $f$  is  a  bijective  self-map  on  $V$.    Then  the    diagram  (<ref>)  induces     a  commutative  diagram   whose  arrows  are   chain maps

\begin{eqnarray} \label{eq-diag.b1}
\xymatrix{
\mathbb{C}^*(\mathcal{L},\omega,m)\ar[d]_{f^\#}\ar[rr]^-{\mu} && \mathbb{C}^*(\mathcal{L},\omega,m+2s)\ar[d]^{f^\#}\\
\mathbb{C}^*(\mathcal{L}',{\rm  Ext}^*(f)(\omega),m)\ar[rr]^-{{\rm  Ext}^*(f)(\mu)} &&  \mathbb{C}^*(\mathcal{L}', {\rm  Ext}^*(f)(\omega),m+2s).
\end{eqnarray}

Each  entry  in  (<ref>)   is   a  sub-co-chain complex  of  the  corresponding  entry  in  (<ref>).
We  have a  commutative diagram  of  constrained cohomology groups

\begin{eqnarray} \label{eq-diag.b2}
\xymatrix{
H^*(\mathcal{L},\omega,m)\ar[d]_{f^\#}\ar[rr]^-{\mu} && H^*(\mathcal{L},\omega,m+2s)\ar[d]^{f^\#}\\
H^*(\mathcal{L}',{\rm  Ext}^*(f)(\omega),m)\ar[rr]^-{{\rm  Ext}^*(f)(\mu)} &&  H^*(\mathcal{L}', {\rm  Ext}^*(f)(\omega),m+2s).
\end{eqnarray}

A  morphism  between  two  Mayer-Vietoris  sequences  of  the  constrained  homology  of  simplicial  complexes  is  a  sequence  of  maps  (expressed  as  vertical arrows)  such that the diagram

\begin{eqnarray*}
\xymatrix{
\cdots \ar[r]  &  H_n(\mathcal{K}_1 \cap  \mathcal{K}_2,\alpha, m)\ar[r]\ar[d]
& H_n (\mathcal{K}_1,\alpha, m) \oplus   H_n( \mathcal{K}_2,\alpha, m)\ar[r]\ar[d] &
\\
\cdots\ar[r]  &  H_n(\mathcal{K}'_1 \cap  \mathcal{K}'_2,\alpha', m')\ar[r]
& H_n(\mathcal{K}'_1,\alpha', m') \oplus   H_n( \mathcal{K}'_2,\alpha', m')\ar[r] &
\xymatrix{
\ar[r]
&  H_n(\mathcal{K}_1 \cap  \mathcal{K}_2,\alpha, m)\ar[r] \ar[d] & H_{n-1}(\mathcal{K}_1 \cap  \mathcal{K}_2,\alpha, m)\ar[r]\ar[d]  &\cdots\\
\ar[r]
&  H_n(\mathcal{K}'_1 \cap  \mathcal{K}'_2,\alpha', m')\ar[r]  & H_{n-1}(\mathcal{K}'_1 \cap  \mathcal{K}'_2,\alpha', m')\ar[r]  &\cdots
\end{eqnarray*}

commutes.   We  denote  such   a  morphism  as  ${\bf  MV}_*(\mathcal{K}_1,\mathcal{K}_2,\alpha,m)\longrightarrow  {\bf  MV}_*(\mathcal{K}'_1,\mathcal{K}'_2,\alpha',m')$.

A  morphism  between  two  Mayer-Vietoris  sequences  of  the  constrained  cohomology  of  independence  hypergraphs  is  a  sequence  of  maps  (expressed  as  vertical arrows)  such that the diagram

\begin{eqnarray*}
\xymatrix{
\cdots \ar[r]  &  H^n(\mathcal{L}_1 \cap  \mathcal{L}_2,\omega, m)\ar[r]\ar[d]
& H^n (\mathcal{L}_1,\omega, m) \oplus   H^n( \mathcal{L}_2,\omega, m)\ar[r]\ar[d] &
\\
\cdots\ar[r]  &  H^n(\mathcal{L}'_1 \cap  \mathcal{L}'_2,\omega', m')\ar[r]
& H^n(\mathcal{L}'_1,\omega', m') \oplus   H^n( \mathcal{L}'_2,\omega', m')\ar[r] &
\xymatrix{
\ar[r]
&  H^n(\mathcal{L}_1 \cap  \mathcal{L}_2,\omega, m)\ar[r] \ar[d] & H^{n+1}(\mathcal{L}_1 \cap  \mathcal{L}_2,\omega, m)\ar[r]\ar[d]  &\cdots\\
\ar[r]
&  H^n(\mathcal{L}'_1 \cap  \mathcal{L}'_2,\omega', m')\ar[r]  & H^{n+1}(\mathcal{L}'_1 \cap  \mathcal{L}'_2,\omega', m')\ar[r]  &\cdots
\end{eqnarray*}

commutes.   We  denote  such   a  morphism  as  ${\bf  MV}^*(\mathcal{L}_1,\mathcal{L}_2,\omega,m)\longrightarrow  {\bf  MV}^*(\mathcal{L}'_1,\mathcal{L}'_2,\omega',m')$.

* Let  $\mathcal{K}_1$  and  $\mathcal{K}_2$  be  two  simplicial  complexes  with their  vertices from  $V$.
Then  we  have a  diagram  of   Mayer-Vietoris  sequences  and  morphisms

\begin{eqnarray*}
\xymatrix{
\cdots \ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}_6(V)} \ar[r]^-{{\rm  Ext}_2(V)}\ar@/_7pc/[rrrr]^-{{\rm  Ext}_8(V)}\ar[r]  &{\bf  MV}_*(\mathcal{K}_1,\mathcal{K}_2, \alpha,m+2)\ar[r]^-{{\rm  Ext}_2(V)}\ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}_6(V)}
&{\bf  MV}_*(\mathcal{K}_1,\mathcal{K}_2, \alpha,m)\ar[r]^-{{\rm  Ext}_2(V)} \ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}  &{\bf  MV}_*(\mathcal{K}_1,\mathcal{K}_2, \alpha,m-2)\ar[r]^-{{\rm  Ext}_2(V)}&\cdots
\end{eqnarray*}
such that  for  any  positive  integer  $k$,    any  $s_1,s_2,\ldots, s_k\in \mathbb{N}$  and
any  $\beta_i\in  {\rm  Ext}_{2s_i}(V)$  for  $i=1,2,\ldots, k$,
the  morphism   $\beta_1\wedge\beta_2\wedge\cdots\wedge\beta_k\in  {\rm  Ext}_{2(s_1+s_2+\cdots +s_k)}(V)$   of  Mayer-Vietoris sequences
can  be  identified  with  the  composition  $\beta_1\circ \beta_2\circ\cdots \circ \beta_k$  of the  morphisms $\beta_1,\beta_2,\ldots,\beta_k$.   Moreover,     the  diagram  is  functorial  with  respect
to  simplicial maps;

* Let  $\mathcal{L}_1$  and  $\mathcal{L}_2$  be  two  independence  hypergraphs  with their  vertices from  $V$.
Then  we  have a  diagram  of   Mayer-Vietoris  sequences  and  morphisms

\begin{eqnarray*}
\xymatrix{
\cdots \ar@/_3pc/[rr]^-{{\rm  Ext}^4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}^6(V)} \ar[r]^-{{\rm  Ext}^2(V)}\ar@/_7pc/[rrrr]^-{{\rm  Ext}^8(V)}\ar[r]  &{\bf  MV}^*(\mathcal{L}_1,\mathcal{L}_2, \omega,m-2)\ar[r]^-{{\rm  Ext}^2(V)}\ar@/_3pc/[rr]^-{{\rm  Ext}^4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}^6(V)}
&{\bf  MV}^*(\mathcal{L}_1,\mathcal{L}_2, \omega,m)\ar[r]^-{{\rm  Ext}^2(V)}  \ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)} &{\bf  MV}^*(\mathcal{L}_1,\mathcal{L}_2, \omega,m+2)\ar[r]^-{{\rm  Ext}^2(V)}&\cdots
\end{eqnarray*}
such that  for  any  positive  integer  $k$,    any  $s_1,s_2,\ldots, s_k\in \mathbb{N}$  and
any  $\mu_i\in  {\rm  Ext}^{2s_i}(V)$  for  $i=1,2,\ldots, k$,
the  morphism   $\mu_1\wedge\mu_2\wedge\cdots\wedge\mu_k\in  {\rm  Ext}^{2(s_1+s_2+\cdots +s_k)}(V)$   of  Mayer-Vietoris sequences
can  be  identified  with  the  composition  $\mu_1\circ \mu_2\circ\cdots \circ \mu_k$  of the  morphisms $\mu_1,\mu_2,\ldots,\mu_k$.   Moreover,     the  diagram  is  functorial  with  respect
to  morphisms  of  independence hypergraphs  induced by  bijective  maps  between the  vertex  sets.

(1).   For  each  $i=1,2,\ldots,k$,  we  have  a   commutative  diagram

\begin{eqnarray*}
\xymatrix{
\cdots \ar[r]  &   H _n(\mathcal{K}_{1 } \cap  \mathcal{K}_{2 },\alpha, m )\ar[r]\ar[d]_{(\beta_i)_*}
&   H _n (\mathcal{K}_{1 },\alpha, m ) \oplus     H _n( \mathcal{K}_{2},\alpha, m )\ar[r]\ar[d]_{(\beta_i)_*}&
\\
\cdots\ar[r]  &   H _n(\mathcal{K}_{1 } \cap  \mathcal{K}_{2 },\alpha, m-2s_i )\ar[r]
&    H_n(\mathcal{K}_{1 },\alpha, m-2s_i ) \oplus    H_n( \mathcal{K}_{2 },\alpha, m-2s_i )\ar[r] &
\xymatrix{
\ar[r]
&    H _n(\mathcal{K}_{1 } \cap  \mathcal{K}_{2 },\alpha, m )\ar[r] \ar[d]_{(\beta_i)_*}&  H _{n-1}(\mathcal{K}_{1 } \cap  \mathcal{K}_{2 },\alpha, m )\ar[r]\ar[d]_{(\beta_i)_*} &\cdots\\
\ar[r]
&   H _n(\mathcal{K}_{1 } \cap  \mathcal{K}_{2 },\alpha, m-2s_i )\ar[r]  &    H _{n-1}(\mathcal{K}_{1 } \cap  \mathcal{K}_{2 },\alpha, m-2s_i )\ar[r]  &\cdots
\end{eqnarray*}

such  that  each  row  is  a  Mayer-Vietoris sequence  of  the  constrained  homology
of  simplicial  complexes.  Thus  $\beta_i$  induces a     morphism  $(\beta_i)_*$  between  two  Mayer-Vietoris  sequences
of  the  constrained  homology
of  simplicial  complexes.
By  Proposition <ref> (1),    the  morphism  $(\beta_1\wedge\beta_2\wedge\cdots\wedge\beta_k)_*$
induced  by  $\beta_1\wedge\beta_2\wedge\cdots\wedge\beta_k\in  {\rm  Ext}_{2(s_1+s_2+\cdots +s_k)}(V)$   of  Mayer-Vietoris sequences
can  be  identified  with  the  composition  $(\beta_1)_*\circ (\beta_2)_*\circ\cdots \circ (\beta_k)_*$.
We  obtain  the  diagram  in  (1).   By  (<ref>)  and  Lemma <ref> (1),
the  diagram in (1)  is  functorial  with  respect
to  simplicial maps.

(2).  For  each  $i=1,2,\ldots,k$,  we  have  a   commutative  diagram

\begin{eqnarray*}
\xymatrix{
\cdots \ar[r]  &    H ^n(\mathcal{L}_{1 } \cap  \mathcal{L}_{2 },\omega, m )\ar[r]\ar[d]_{(\mu_i)*}
&   H ^n (\mathcal{L}_{1 },\omega, m ) \oplus    H ^n( \mathcal{L}_{2 },\omega, m )\ar[r]\ar[d]_{(\mu_i)_*}&
\\
\cdots\ar[r]  &   H^n(\mathcal{L}_{1 } \cap  \mathcal{L}_{2 },\omega, m+2s_i )\ar[r]
&  H ^n(\mathcal{L}_{1 },\omega, m+2s_i ) \oplus      H ^n( \mathcal{L}_{2 },\omega, m+2s_i )\ar[r] &
\xymatrix{
\ar[r]
&    H ^n(\mathcal{L}_{1 } \cap  \mathcal{L}_{2 },\omega, m )\ar[r] \ar[d]_{(\mu_i)_*}&   H ^{n+1}(\mathcal{L}_{1 } \cap  \mathcal{L}_{2 },\omega, m )\ar[r]\ar[d]_{(\mu_i)_*} &\cdots\\
\ar[r]
&   H ^n(\mathcal{L}_{1 } \cap  \mathcal{L}_{2 },\omega, m+2s_i )\ar[r]  &    H ^{n+1}(\mathcal{L}_{1 } \cap  \mathcal{L}_{2 },\omega, m+2s_i )\ar[r]  &\cdots
\end{eqnarray*}

such  that  each  row  is  a  Mayer-Vietoris sequence  of  the  constrained  cohomology
of   independence  hypergraphs. Thus  $(\mu_i)_*$  induces a     morphism  $(\mu_i)_*$  between  two  Mayer-Vietoris  sequences
of  the  constrained  cohomology
of   independent  hypergraphs.    By  Proposition <ref> (2),    the  morphism  $(\mu_1\wedge\mu_2\wedge\cdots\wedge\mu_k)_*$
induced  by  $\mu_1\wedge\mu_2\wedge\cdots\wedge\mu_k\in  {\rm  Ext}^{2(s_1+s_2+\cdots +s_k)}(V)$   of  Mayer-Vietoris sequences
can  be  identified  with  the  composition  $(\mu_1)_*\circ (\mu_2)_*\circ\cdots \circ (\mu_k)_*$.
We  obtain  the  diagram  in  (2).   By  (<ref>)  and  Lemma <ref> (2),
the  diagram in (1)  is  functorial  with  respect
to    morphisms  of  independence hypergraphs  induced by  bijective  maps  between the  vertex  sets.

In  Proposition <ref>,  we  use

\begin{eqnarray*}
&{\rm  Ext}_{2s}(V):  ~~~{\bf   MV}_*(\mathcal{K}_{1},\mathcal{K}_{2}, \alpha,m+2)\longrightarrow  {\bf   MV}_*(\mathcal{K}_{1},\mathcal{K}_{2}, \alpha,m )
\end{eqnarray*}

to  denote  the  family  of   morphisms     of   Mayer-Vietoris  sequences

\begin{eqnarray*}
& \beta_*: ~~~{\bf   MV}_*(\mathcal{K}_{1},\mathcal{K}_{2}, \alpha,m+2)\longrightarrow  {\bf   MV}_*(\mathcal{K}_{1,x},\mathcal{K}_{2,x}, \alpha,m)
\end{eqnarray*}

for  all $ \beta\in {\rm  Ext}_{2s}(V) $
and   use

\begin{eqnarray*}
& {\rm  Ext}^{2s}(V):~~~{\bf   MV}^*(\mathcal{L}_{1},\mathcal{L}_{2}, \omega,m-2)\longrightarrow  {\bf   MV}^*(\mathcal{L}_{1,x},\mathcal{L}_{2,x}, \omega,m )
\end{eqnarray*}

to denote   the  family  of
morphisms    of     Mayer-Vietoris  sequences

\begin{eqnarray*}
& \mu_*:~~~ {\bf   MV}^*(\mathcal{L}_{1},\mathcal{L}_{2}, \omega,m-2)\longrightarrow  {\bf   MV}^*(\mathcal{L}_{1},\mathcal{L}_{2}, \omega,m )
\end{eqnarray*}

for  all $ \mu\in {\rm  Ext}^{2s}(V) $.

§ THE  CONSTRAINED    PERSISTENT  (CO)HOMOLOGY

Let   $V$  be   a   discrete  set.
Let  $\{\mathcal{K}_x\}_{x\in\mathbb{R}}$  be  a  filtration  of  simplicial  complexes  such  that

* for  any   $x\in  \mathbb{R}$,   $\mathcal{K}_x$  is  a  simplicial complex  with its  vertices from  $V$;

* for  any  $-\infty<x\leq  y<+\infty$,  there  is  a  canonical  inclusion  $\iota_x^y:   \mathcal{K}_x\longrightarrow \mathcal{K}_y$.

Let  $t,s,n\in  \mathbb{N}$.
Let $\alpha\in {\rm Ext}_{2t+1}(V)$  and  $\beta\in {\rm Ext}_{2s}(V)$.     Let  $m\in \mathbb{Z}$.      We  have  a  family  of  constrained  homology  groups

\begin{eqnarray}\label{eq-9.876}
\{H_n(\mathcal{K}_x,\alpha,m)\}_{x\in\mathbb{R}}
\end{eqnarray}

and  a  family  of  homomorphisms  of  homology  groups

\begin{eqnarray}\label{eq-9.878}
\{(\iota_x^y)_*:  H_n(\mathcal{K}_x,\alpha,m)\longrightarrow  H_n(\mathcal{K}_y,\alpha,m)\}_{-\infty<x\leq  y<+\infty}.
\end{eqnarray}

We  call  (<ref>)  together  with  (<ref>)  the $n$-th  constrained  persistent homology   of
$\{\mathcal{K}_x\}_{x\in\mathbb{R}}$  with  respect   to   $\alpha$  and  $m$  and  denote  it  as
${\bf  H}_n(\mathcal{K}_x,\alpha,m\mid  x\in\mathbb{R})$.   For  any  $-\infty<x\leq  y<+\infty$,   we  have  a  commutative diagram

\begin{eqnarray*}
\xymatrix{
H_n(\mathcal{K}_x,\alpha,m) \ar[r]  ^-{\beta_*}  \ar[d]_{(\iota_x^y)_*}& H_n(\mathcal{K}_x,\alpha,m-2s) \ar[d]^{(\iota_x^y)_*}\\
H_n(\mathcal{K}_y,\alpha,m) \ar[r]  ^-{\beta_*}  & H_n(\mathcal{K}_y,\alpha,m-2s).
\end{eqnarray*}

Thus  we  have  an  induced   morphism  of  persistent complex vector  spaces    (i.e.  a  homomorphism  of  persistent  complex  vector  spaces  defined  in <cit.>)

\begin{eqnarray*}
\beta_*:  ~~~{\bf  H}_n(\mathcal{K}_x,\alpha,m\mid  x\in\mathbb{R})\longrightarrow  {\bf  H}_n(\mathcal{K}_x,\alpha,m-2s\mid  x\in\mathbb{R}).
\end{eqnarray*}

Similarly,   let  $\{\mathcal{L}_x\}_{x\in\mathbb{R}}$  be  a  filtration  of   independence  hypergraphs  such  that

* for  any   $x\in  \mathbb{R}$,   $\mathcal{L}_x$  is  an  independence  hypergraph  with its  vertices from  $V$;

* for  any  $-\infty<x\leq  y<+\infty$,  there  is  a  canonical  inclusion  $\iota_x^y:   \mathcal{L}_x\longrightarrow \mathcal{L}_y$.

Let $\omega\in {\rm Ext}^{2t+1}(V)$  and  $\mu\in {\rm Ext}^{2s}(V)$.     We  have  a  family  of  constrained   cohomology  groups

\begin{eqnarray}\label{eq-9.976}
\{H^n(\mathcal{L}_x,\omega,m)\}_{x\in\mathbb{R}}
\end{eqnarray}

and  a  family  of  homomorphisms  of  homology   groups

\begin{eqnarray}\label{eq-9.978}
\{(\iota_x^y)_*:  H^n(\mathcal{L}_x,\omega,m)\longrightarrow  H^n(\mathcal{L}_y,\omega,m)\}_{-\infty<x\leq  y<+\infty}.
\end{eqnarray}

We  call  (<ref>)  together  with  (<ref>)  the $n$-th  constrained  persistent  cohomology   of
$\{\mathcal{L}_x\}_{x\in\mathbb{R}}$  with  respect   to   $\omega$  and  $m$  and  denote  it  as
${\bf  H}^n(\mathcal{L}_x,\omega,m\mid  x\in\mathbb{R})$.   For  any  $-\infty<x\leq  y<+\infty$,   we  have  a  commutative diagram

\begin{eqnarray*}
\xymatrix{
H^n(\mathcal{L}_x,\omega,m) \ar[r]  ^-{\mu_*}  \ar[d]_{(\iota_x^y)_*}& H^n(\mathcal{L}_x,\omega,m+2s) \ar[d]^{(\iota_x^y)_*}\\
H^n(\mathcal{L}_y,\omega,m) \ar[r]  ^-{\mu_*}  & H^n(\mathcal{L}_y,\omega,m+2s).
\end{eqnarray*}

Thus  we  have  an  induced   morphism  of  persistent complex vector  spaces

\begin{eqnarray*}
\mu_*: ~~~ {\bf  H}^n(\mathcal{L}_x,\omega,m\mid  x\in\mathbb{R})\longrightarrow  {\bf  H}^n(\mathcal{L}_x,\omega,m+2s\mid  x\in\mathbb{R}).
\end{eqnarray*}

* We  have  a  diagram  of  persistent complex vector  spaces

\begin{eqnarray*}
\xymatrix{
\cdots  \ar[r]^-{{\rm  Ext}_2(V)}   \ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}_6(V)}
\ar@/_7pc/[rrrr]^-{{\rm  Ext}_8(V)}& {\bf  H}_n(\mathcal{K}_x,\alpha,m+2\mid  x\in\mathbb{R}) \ar[r]^-{ {\rm  Ext}_2(V)}  \ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}  \ar@/_5pc/[rrr] ^-{{\rm  Ext}_6(V)} &  {\bf  H}_n(\mathcal{K}_x,\alpha,m\mid  x\in\mathbb{R})  \ar[r]^-{{\rm  Ext}_2(V)}   \ar@/_3pc/[rr]^-{{\rm  Ext}_4(V)}  & {\bf  H}_n(\mathcal{K}_x,\alpha,m-2\mid  x\in\mathbb{R})
\ar[r]^-{{\rm  Ext}_2(V)}
\end{eqnarray*}
such that  for  any  positive  integer  $k$,    any  $s_1,s_2,\ldots, s_k\in \mathbb{N}$  and
any  $\beta_i\in  {\rm  Ext}_{2s_i}(V)$  for  $i=1,2,\ldots, k$,  the   morphism  of  persistent   complex  vector  spaces
$\beta_1\wedge \beta_2\wedge\cdots \wedge \beta_k\in  {\rm  Ext}_{2(s_1+s_2+\cdots +s_k)}(V)$
can  be  identified  with  the  composition  $\beta_1\circ \beta_2\circ\cdots \circ \beta_k$  of the   morphisms  of  persistent  complex  vector  spaces   $\beta_1,\beta_2,\ldots,\beta_k$;

* We  have  a  diagram  of  persistent complex vector  spaces

\begin{eqnarray*}
\xymatrix{
\cdots  \ar[r]^-{{\rm  Ext}^2(V)}   \ar@/_3pc/[rr]^-{{\rm  Ext}^4(V)}\ar@/_5pc/[rrr]^-{{\rm  Ext}^6(V)}
\ar@/_7pc/[rrrr]^-{{\rm  Ext}^8(V)}& {\bf  H}^n(\mathcal{L}_x,\omega,m-2\mid  x\in\mathbb{R}) \ar[r]^-{ {\rm  Ext}^2(V)}  \ar@/_3pc/[rr]^-{{\rm  Ext}^4(V)}  \ar@/_5pc/[rrr] ^-{{\rm  Ext}^6(V)} &  {\bf  H}^n(\mathcal{L}_x,\omega,m\mid  x\in\mathbb{R})  \ar[r]^-{{\rm  Ext}^2(V)}   \ar@/_3pc/[rr]^-{{\rm  Ext}^4(V)}  & {\bf  H}^n(\mathcal{L}_x,\omega,m+2\mid  x\in\mathbb{R})
\ar[r]^-{{\rm  Ext}_2(V)}
\end{eqnarray*}
such that  for  any  positive  integer  $k$,    any  $s_1,s_2,\ldots, s_k\in \mathbb{N}$  and
any  $\mu_i\in  {\rm  Ext}^{2s_i}(V)$  for  $i=1,2,\ldots, k$,  the   morphism  of  persistent  complex  vector  spaces
$\mu_1\wedge \mu_2\wedge\cdots \wedge \mu_k\in  {\rm  Ext}^{2(s_1+s_2+\cdots +s_k)}(V)$
can  be  identified  with  the  composition  $\mu_1\circ \mu_2\circ\cdots \circ \mu_k$  of the   morphisms  of  persistent  complex  vector spaces   $\mu_1,\mu_2,\ldots,\mu_k$.

(1).   For  each  $i=1,2,\ldots,k$,  we  have  that  $\beta_i: {\bf  H}_n(\mathcal{K}_x,\alpha,m\mid  x\in\mathbb{R})\longrightarrow  {\bf  H}_n(\mathcal{K}_x,\alpha,m-2s_i\mid  x\in\mathbb{R})$   is   a   morphism  of  persistent complex vector  spaces  .   By  the  definition  of  ${\rm  Ext}_*(V)$,
$\beta_1\wedge  \beta_2\wedge\cdots \wedge \beta_k$  is  the  composition of
We  obtain  (1).

(2).      For  each  $i=1,2,\ldots,k$,  we  have  that  $\mu_i: {\bf  H}^n(\mathcal{L}_x,\alpha,m\mid  x\in\mathbb{R})\longrightarrow  {\bf  H}^n(\mathcal{L}_x,\alpha,m+2s_i\mid  x\in\mathbb{R})$   is   a   morphism  of  persistent complex vector  spaces  .   By  the  definition  of  ${\rm  Ext}^*(V)$,
$\mu_1\wedge  \mu_2\wedge\cdots \wedge \mu_k$  is  the  composition of
We  obtain  (2).

In  Proposition <ref>,    we  use  we  use

\begin{eqnarray*}
&{\rm  Ext}_{2s}(V):  ~~~{\bf   H}_*(\mathcal{K}_{x}, \alpha,m+2\mid  x\in\mathbb{R})\longrightarrow  {\bf   H}_*(\mathcal{K}_{x}, \alpha,m \mid  x\in\mathbb{R})
\end{eqnarray*}

to  denote  the  family  of    morphisms   of  persistent complex vector  spaces

\begin{eqnarray*}
& \beta_*: ~~~{\bf   H}_*(\mathcal{K}_{x}, \alpha,m+2\mid  x\in\mathbb{R})\longrightarrow  {\bf   H}_*(\mathcal{K}_{x}, \alpha,m \mid  x\in\mathbb{R})
\end{eqnarray*}

for  all $ \beta\in {\rm  Ext}_{2s}(V) $
and   use

\begin{eqnarray*}
& {\rm  Ext}^{2s}(V):~~~{\bf   H}^*(\mathcal{L}_{x}, \omega,m-2\mid  x\in\mathbb{R})\longrightarrow  {\bf   H}^*(\mathcal{L}_{x}, \omega,m \mid  x\in\mathbb{R})
\end{eqnarray*}

to denote   the  family  of
morphisms   of  persistent complex vector  spaces

\begin{eqnarray*}
& \mu_*:~~~ {\bf   H}^*(\mathcal{L}_{x}, \omega,m-2\mid  x\in\mathbb{R})\longrightarrow  {\bf   H}^*(\mathcal{L}_{x}, \omega,m \mid  x\in\mathbb{R})
\end{eqnarray*}

for  all $ \mu\in {\rm  Ext}^{2s}(V) $.

$\{\mathcal{K}_x\}_{x\in\mathbb{R}}$  and  $\{\mathcal{K}'_x\}_{x\in\mathbb{R}}$
be  two  filtrations  of  simplicial complexes  with  their  vertices  from  $V$.
Then  both  $\{\mathcal{K}_x\cap \mathcal{K}'_x\}_{x\in\mathbb{R}}$  and
$\{\mathcal{K}_x\cup \mathcal{K}'_x\}_{x\in\mathbb{R}}$
are  filtrations  of  simplicial  complexes   with  vertices from  $V$.
For  any  $-\infty<x\leq  y<+\infty$,  the  canonical  inclusions
$\iota_x^y: \mathcal{K}_x\longrightarrow  \mathcal{K}_y$  and
${\iota'}_x^y: \mathcal{K}'_x\longrightarrow  \mathcal{K}'_y$
induces  a  morphism  of  the Mayer-Vietoris
sequence of the constrained  homology

\begin{eqnarray}\label{eq-1oiu}
(I_x^y)_*: ~~~  {\bf  MV}_*(\mathcal{K}_x,\mathcal{K}'_x,  \alpha,m)\longrightarrow {\bf  MV}_*(\mathcal{K}_y,\mathcal{K}'_y,\alpha,m).
\end{eqnarray}

We  call the  family  $\{{\bf  MV}_*(\mathcal{K}_x,\mathcal{K}'_x,\alpha,m)\}_{x\in\mathbb{R}}$   together with the  family  of  morphisms     given by  (<ref>)  the
persistent  Mayer-Vietoris  sequence  of  the  constrained  persistent homology  for the filtrations $\{\mathcal{K}_x\}_{x\in\mathbb{R}}$ and  $\{\mathcal{K}'_x\}_{x\in\mathbb{R}}$ of  simplicial complexes  and  denote it  as

\begin{eqnarray*}
{\bf   PMV}_*(\mathcal{K}_x,\mathcal{K}'_x,  \alpha,m\mid  x\in\mathbb{R}).
\end{eqnarray*}

Similarly,  let
$\{\mathcal{L}_x\}_{x\in\mathbb{R}}$  and  $\{\mathcal{L}'_x\}_{x\in\mathbb{R}}$
be  two  filtrations  of  independence hypergraphs  with  their  vertices  from  $V$.
Then  both  $\{\mathcal{L}_x\cap \mathcal{L}'_x\}_{x\in\mathbb{R}}$  and
$\{\mathcal{L}_x\cup \mathcal{L}'_x\}_{x\in\mathbb{R}}$
are  filtrations  of  independence  hypergraphs   with  vertices from  $V$.
For  any  $-\infty<x\leq  y<+\infty$,  the  canonical  inclusions
$\iota_x^y: \mathcal{L}_x\longrightarrow  \mathcal{L}_y$  and
${\iota'}_x^y: \mathcal{L}'_x\longrightarrow  \mathcal{L}'_y$
induces  a  morphism  of  the Mayer-Vietoris
sequence of the constrained  homology

\begin{eqnarray}\label{eq-2oiu}
(I_x^y)_*: ~~~  {\bf  MV}^*(\mathcal{L}_x,\mathcal{L}'_x,  \omega,m)\longrightarrow {\bf  MV}^*(\mathcal{L}_y,\mathcal{L}'_y,\omega,m).
\end{eqnarray}

We  call the  family  $\{{\bf  MV}^*(\mathcal{L}_x,\mathcal{L}'_x,\alpha,m)\}_{x\in\mathbb{R}}$   together with the  family  of  morphisms     given by  (<ref>)  the
persistent  Mayer-Vietoris  sequence  of  the  constrained  persistent  cohomology  for the filtrations $\{\mathcal{L}_x\}_{x\in\mathbb{R}}$ and  $\{\mathcal{L}'_x\}_{x\in\mathbb{R}}$ of  independence  hypergraphs  and  denote it  as

\begin{eqnarray*}
{\bf   PMV}^*(\mathcal{L}_x,\mathcal{L}'_x,  \omega,m\mid  x\in\mathbb{R}).
\end{eqnarray*}

Take  the  maps  on the  set  of  vertices as the  identity  map ${\rm  id}_V: V\longrightarrow  V$.  Consider  the  filtrations  of  simplicial  complexes  and  independence  hypergraphs  with  respect  to  ${\rm  id}_V$.
The  diagram in Proposition <ref> (1)  is  functorial with respect to  simplicial maps.
By taking the persistence of the constrained homology,    (1) follows from  Proposition <ref> (1).
The  diagram in Proposition <ref> (2)  is  functorial with respect to  morphisms  of  independence hypergraphs  induced by  bijective  maps  between the  vertex  sets.
By taking the persistence of the constrained cohomology,    (2) follows from  Proposition <ref> (2).

Suppose  $V$  is  a  finite  set.   Then  a   filtration  $\{\mathcal{K}_x\}_{x\in\mathbb{R}}$    of  simplicial  complexes  gives a  filtration
$\{\mathcal{L}_x\}_{x\in\mathbb{R}}$,  where
$\mathcal{L}_x=\Delta[V]\setminus \mathcal{K}_{-x}$,  of  independence  hypergraphs.
Conversely,  a  filtration   $\{\mathcal{L}_x\}_{x\in\mathbb{R}}$  of  independence  hypergraphs
also  gives  a  filtration   $\{\mathcal{K}_x\}_{x\in\mathbb{R}}$,
where  $\mathcal{K}_x=\Delta[V]\setminus \mathcal{L}_{-x}$,  of  simplicial  complexes.
The  corollary  follows.

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Shiquan Ren

Address:   School of Mathematics and Statistics,  Henan University,  Kaifeng  475004,  China.

E-mail<EMAIL_ADDRESS>


March 15, 2022

# Theory of Muon g-2

Bogdan Malaescu1 1LPNHE1LPNHE Sorbonne Université Sorbonne Université
Université Paris Cité Université Paris Cité CNRS/IN2P3 CNRS/IN2P3 Paris Paris
France France<EMAIL_ADDRESS>
###### Abstract

The longstanding discrepancy between the measured and the predicted values of
the anomalous magnetic moment of the muon, $a_{\mu}=(g-2)/2$, is one of the
most intriguing potential hints of new physics in particle physics. After a
brief introduction, the status of the theoretical prediction of g-2 is
presented, with some focus on the contributions yielding the dominant
uncertainties. The status of the comparison with the experimental measurement
is then discussed.

muon, g-2, prediction

##  1 Introduction

A longstanding discrepancy has been observed between the measured and the
predicted values of the anomalous magnetic moment of the muon,
$a_{\mu}=(g-2)/2$. This represents a potential hint of new physics in particle
physics and calls for an improvement of the theoretical prediction, in view of
the significant gain in precision foreseen for the Fermilab measurement. A
series of workshops organised by the Muon g-2 Theory Initiative has allowed
for detailed discussions on the various approaches that are being considered
for computing the different contributions to $a_{\mu}$. A consensus has been
reached for a conservative evaluation of the theoretical prediction and its
uncertainty. The results presented here are based mainly on the White Paper of
the Muon g-2 Theory Initiative [1] and on the inputs used therein from Refs.
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].

While $a_{\mu}$ is sensitive to quantum fluctuations of fields, it has to be
computed very precisely, which is a highly-nontrivial exercise. This
evaluation involves QED contributions, currently known up to O($\alpha^{5}$)
[33, 34], which corresponds to a precision of $0.001$ ppm. Then there are
electroweak contributions, currently known with a precision of $0.01$ ppm. The
dominant uncertainties originate from non-perturbative contributions, namely
the hadronic vacuum polarisation (HVP) and the hadronic light-by-light (HLbL),
with a precision of $0.34$ ppm and $0.15$ ppm respectively. All these
contributions have to be well under control before one can make any claim
about the observation of potential contributions from new physics, beyond the
Standard Model.

##  2 Status of the theoretical prediction

The dominant uncertainty for the theoretical prediction originates from
lowest-order HVP contribution ($a_{\mu}^{\text{HVP, LO}}$), which cannot be
calculated based on perturbative QCD because of the low mass scale. Instead,
one can use experimental data on hadronic production cross section (
$\sigma\left(\mathrm{e^{+}e^{-}}\rightarrow{\rm hadrons}\right)$ ). Indeed,
the optical theorem relates the imaginary part of the two-point correlator to
the hadronic cross section and a dispersion integral allows to compute
$a_{\mu}^{\text{HVP, LO}}$. The kernel of this integral strongly enhances the
low energy region and therefore precise measurements of the
$\sigma\left(\mathrm{e^{+}e^{-}}\rightarrow{\rm hadrons}\right)$ in this
region are very important. We do not use anymore data from hadronic $\tau$
decays, because they are less precise and are also impacted by theory
uncertainties [6].

Figure 1: The top plots show data from CMD-2 [37, 38] (left) and SND [39]
(right) on $\mathrm{e^{+}e^{-}}\rightarrow\pi^{+}\pi^{-}$ in the $\rho$
region. Bottom-left: the plot, adapted from Ref. [42], shows the KLOE data
sets on $\mathrm{e^{+}e^{-}}\rightarrow\pi^{+}\pi^{-}$ in the $\rho$ region
obtained in the three experimental configurations described in the text [40,
41, 42]. Bottom-right: the plot, reprinted from Ref. [43], displays the
results from BABAR [43, 44] using the large-angle ISR method:
$\mathrm{e^{+}e^{-}}\rightarrow\mu^{+}\mu^{-}$ compared to NLO QED (top frame)
and $\mathrm{e^{+}e^{-}}\rightarrow\pi^{+}\pi^{-}$ from threshold to
$3\mathrm{Ge\kern-1.00006ptV}$ using the $\pi\pi/\mu\mu$ ratio (bottom frame).
The insert shows the $\rho$ region.

Fig. 1 shows examples of experimental spectra for the dominant
$\pi^{+}\pi^{-}$ channel. At CMD-2 [37, 38] and SND [39] the measurements are
performed through a scan of the energy in the centre-of-mass of the collider.
KLOE used the ISR technique, selecting events with a hard photon emitted from
the initial state, which allows to cover a broad range of masses for the
hadronic final state system [40, 41, 42]. The measurement performed at BABAR
using the ISR approach selects events with the hard photon reconstructed in
the detector, allowing to cover the full energy range of interest.
Furthermore, the ISR luminosity is evaluated in-situ based on
$\mathrm{e^{+}e^{-}}\rightarrow\mu^{+}\mu^{-}$ events, which allows to reduce
several systematic uncertainties, and in both the $\pi\pi$ and $\mu\mu$
channels events with extra photons are considered (performing an ”NLO
measurement”). In the $\mu\mu$ channel, the comparison of the measured
spectrum with the NLO QED-based prediction provides an excellent test of the
procedure [43, 44]. While the KLOE ${}^{\prime}08$ [40] and KLOE
${}^{\prime}12$ [42] measurements consider events with the hard ISR photon
along the beam, the KLOE ${}^{\prime}10$ measurement uses large-angle ISR
photons [41]. The KLOE ${}^{\prime}12$ measurement also uses the muon spectrum
for the normalisation [42]. Still, for the time being, only the BABAR study
uses the reconstructed extra photons, necessary at the sub-percent level
precision aimed for here.

Figure 2: Left: Cross section for $\mathrm{e^{+}e^{-}}\to\pi^{+}\pi^{-}$
annihilation measured by the different experiments for the entire energy
range. The error bars contain both statistical and systematic uncertainties,
added in quadrature. The shaded (green) band represents the average of all the
measurements obtained by HVPTools, which is used for the numerical integration
following the procedure discussed in Ref. [6]. Right: Rescaling factor
accounting for inconsistencies among experiments versus $\sqrt{s}$. Reprinted
from Ref. [6].

Computing the $a_{\mu}^{\text{HVP, LO}}$ contribution requires combining the
experimental spectra measured by various experiments, as performed e.g. by the
DHMZ [2, 6] (see Fig. 2 left) and KNT [3, 7] teams respectively, and summing
the contributions of the various exclusive channels. While the channel-by-
channel combinations are performed taking into account the information on the
uncertainties and their correlations between bins/points and between
experiments, the DHMZ approach (implemented in the HVPTools software) also
accounts for correlations between different channels. This induces a necessary
enhancement of the total uncertainty.

Furthermore, a method for taking into account the local tensions between the
measurements (based on the computation of a $\chi^{2}/{\rm ndof}$ in fine
energy ranges, used for the rescaling of the uncertainties) has been
implemented by DHMZ in Refs. [45, 46] and is still being used [2, 6] (see Fig.
2 right). Such approach has also been used in the more recent KNT studies [3,
7]. Still, this local rescaling of the uncertainties is not sufficient in
presence of systematic tensions, as the ones between the BABAR and the KLOE
measurements in the $\pi\pi$ channel. These tensions are taken into account in
the most recent DHMZ study [6] by treating half of the difference between
integrals without either BABAR or KLOE as an extra uncertainty. This yields
the dominant uncertainty in the study, amounting to $2.8\cdot 10^{-10}$.

Figure 3: Left: The theoretical predictions based on various determinations of
$a_{\mu}^{\text{HVP, LO}}$ are compared with the BNL experimental result [48],
indicated by the yellow vertical band. The result of the Theory Initiative
White Paper [1] (WP20), as well as the DHMZ19 [6] and KNT19 [7] inputs, are
indicated below the dashed horizontal line. The uncertainty goal for the
Fermilab measurement is also displayed. Right: Comparison of the values for a
”Window” moment defined in Ref. [11], for various Lattice QCD studies.

Fig. 3 left shows the comparison between the theoretical predictions based on
various determinations of $a_{\mu}^{\text{HVP, LO}}$ and the BNL experimental
result [48]. The result of the Theory Initiative White Paper [1] is obtained
through the merging of model independent results, as obtained by DHMZ [6] and
KNT [7] (and CHHKS for the $\pi^{+}\pi^{-}$ [4] and $\pi^{+}\pi^{-}\pi^{0}$
[5] channels). The corresponding central value is obtained from the simple
average of the input integrals. The largest of the DHMZ and KNT uncertainties
is used in each channel. The treatment of the BABAR-KLOE tension in the
$\pi\pi$ channel and of the correlations between the various channels is based
on the DHMZ result. The importance of properly accounting for these two
aspects, as done in the DHMZ study, is indeed visible when comparing the
uncertainties of the DHMZ and KNT results (see Fig. 3 left).

During the last few years there has been excellent progress on the Lattice QCD
(+QED) calculations [9, 10, 11, 12, 13, 14, 15, 16, 17, 47]. In particular,
while the BMW20 [47] result is still to be cross-checked by other lattice
groups, its precision became similar to the one of dispersive approaches.
There are indeed ongoing cross-checks using Euclidean time windows (related to
the HVP with suppression of very low and high energies [11]) for which various
groups achieved similar precision (see Fig. 3 right). If the BMW20 result is
confirmed, the difference with respect to the dispersive results will have to
be understood.

Figure 4: Summary of the $a_{\mu}^{\text{HLbL}}$ values obtained through
various approaches: hadronic model + perturbative QCD (yellow diamond),
lattice QCD+QED (blue circles) and data-driven (red square). The result of the
Theory Initiative White Paper [1] (WP20) indicated by the vertical grey band
uses only the inputs marked with filled symbols.

A summary of the $a_{\mu}^{\text{HLbL}}$ values obtained through various
approaches, as well as the result of the Theory Initiative White Paper [1],
are indicated in Fig. 4. One can notice that there is indeed good progress on
systematically improvable approaches. Furthermore, the g-2 Theory Initiative
provides an adequate environment for cross-checks among these results. The
uncertainty of $a_{\mu}^{\text{HLbL}}$ is currently controlled with a
precision of $0.15$ ppm. The result obtained by the Mainz group [49], with a
somewhat better precision than the result of the Theory Initiative White Paper
[1], became publicly available after the latter and is to be considered in
future iterations.

Figure 5: The theoretical predictions based on various data-driven
determinations of $a_{\mu}^{\text{HVP, LO}}$ are compared with the Fermilab
[50] and BNL [48] combined experimental result, indicated by the blue vertical
band. The result of the Theory Initiative White Paper [1] is indicated by the
red empty circle and the gray vertical band. The DHMZ19 [6] and KNT19 [7]
results, updated to use the $a_{\mu}^{\text{HLbL}}$ from the White Paper, are
indicated by the filed black circles.

Putting together all the contributions to the theoretical prediction, the
dominant uncertainty originates from $a_{\mu}^{\text{HVP, LO}}$, based on the
merging of model-independent methods, while the $a_{\mu}^{\text{HLbL}}$ is
impacted by an important uncertainty too [1]. At the same time, Lattice QCD
results become more and more interesting. In Fig. 5 the theoretical
predictions based on various data-driven determinations of
$a_{\mu}^{\text{HVP, LO}}$are compared with the Fermilab [50] and BNL [48]
combined experimental result. A tension between the BNL measurement and the
reference SM prediction, at the level of $3.7~{}\sigma$ is observed. The
tension increases to $4.2~{}\sigma$ when including the FNAL result. The
tension is significantly smaller when using BMW20 for the $a_{\mu}^{\text{HVP,
LO}}$, a result which is still to be confirmed by other lattice groups.

##  3 Conclusion

While the tension between the nominal Standard Model prediction and the
experimental measurement of $a_{\mu}$ has reached the level of $4.2$ standard
deviations, it is worth phrasing some words of caution about this
significance. Indeed, while the experimental measurement is statistics-
dominated, the uncertainty of the prediction is limited by non-Gaussian
systematic effects that do not have a rigorous statistical interpretation (see
also the discussion in Ref. [51]). Nevertheless, one can clearly state that
there is a large discrepancy between measurement and reference Standard Model
prediction. The latter needs to be significantly improved in view of the
forthcoming updates of the Fermilab measurement. One can therefore conclude
that we have an interesting, long standing, multifaceted problem to solve.

## Acknowledgements

I kindly thank the organisers of the SPIN 2021 Symposium for the invitation to
give this talk. I thank my colleagues and friends Michel DAVIER, Andreas
HOECKER and Zhiqing ZHANG for our fruitful collaboration, as well as the
members of the g-2 Theory Initiative for the numerous interesting discussions.

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