# On the sequential topological complexity of group homomorphisms

Nursultan Kuanyshov  Nursultan Kuanyshov, Department of Mathematics,
University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA and
Institute of Mathematics and Mathematical Modeling 125 Pushkin str., 050010
Almaty, Kazakhstan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We define and develop a homotopy invariant notion for the sequential
topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that
interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott’s
topological complexity map $TC(f)$ interacts with $TC(X)$ and $TC(Y).$
Furthermore, we apply $TC_{r}(f)$ to studying group homomorphisms $\phi:G\to
H.$

In addition, we prove that the sequential topological complexity of any
nonzero homomorphism of a torsion group cannot be finite. Also, we give the
characterisation of cohomological dimension of group homomorphisms.

###### Key words and phrases:

The sequential topological complexity, cohomological dimension, Schwartz
genus, Lusternik-Schnirelmann category, group homomorphism

###### 2020 Mathematics Subject Classification:

Primary 50M30; Secondary 20J06, 20K45, 20K30, 20K10

##  1\. Introduction

Let $r\geq 2$ be an integer. The sequential topological complexity $TC_{r}(X)$
of a path-connected space X was introduced by Rudyak [Rud], generalising
Farber’s topological complexity [Fa1]. The motivation for these numerical
invariants arises from robotics (see [Fa1, Fa2, Fa3, KL, La]). They provide a
measure of complexity for the motion planning problem in the configuration
space X with prescribed initial and final states, as well as $r-2$ consecutive
intermediate states. More precisely, consider the path-fibration $p_{r}:PX\to
X^{r}$ that maps a path $\gamma:[0,1]\to X$ to the tuple
$\big{(}\gamma(0),\gamma(\frac{1}{r-1}),\cdots,\gamma(\frac{r-2}{r-1}),\gamma(1)\big{)}$.
Then $TC_{r}(X)$ is defined as the minimal integer $n$ for which $X^{r}$ can
be covered by $n+1$ many open subsets $U_{0},\cdots,U_{k}$ such that $p_{r}$
admits a local section over each $U_{i}.$ If no such $n$ exists, we set
$TC_{r}(X)=\infty.$ Note that $TC_{2}(X)$ recovers Farber’s topological
complexity.

When the mechanical system is a robotic arm, the $TC_{r}$ input on the
navigation problem is not quite satisfactory. In the case of robotic arm one
has a map $f:X\to Y$ of the configuration space X to the work space Y.
Defining such a notion has gone through many forms. It started when
Dranishnikov suggested to study the topological complexity of maps $TC(f)$ in
his lectures at the workshop on Applied Algebraic Topology in Castro Urdiales
in 2014. He defined it as the minimal number $k$ such that $X\times Y$ can be
covered by open sets $U_{0},\cdots,U_{k}$ such that over each $U_{i}$ there is
a section to the map $q:PX\to X\times Y$ defined as
$q(\gamma)=(\gamma(0),f\gamma(1)).$ The drawback of his definition is that it
can only apply to maps that have the path lifting property. In [Pa1, Pa2]
Pavesic modified above the definition to cover general maps, but his
definition lacks the homotopy invariant feature. In 2020, Murillo and Wu [MW]
gave their notion, and J. Scott [Sc] independently gave his definition and
proved that his definition is equivalent to Murillo’ and Wu’s definition. Note
that all of these definitions naturally generalizes to the sequential
topological complexity of maps. In 2022, Rudyak and Soumen [RS] gave their
notions using relative version of LS-category of map and they conjectured
their notion is equivalent to Scott’s and Murillo and Wu’s notions. Last year,
C. Zapata and J. Gonzalez [ZG] gave the generalized definitions that includes
all previous notions. Further, they proved that the notions introduced by [RS,
Sc, MW] are equivalent.

We extend Scott’s definition for several reasons: 1) His definition makes it
easy to prove basic inequalities, and we can prove that almost all known
results from spaces to maps in this paper. 2) Very recently, probabilistic
variants of LS-category and the sequential topological complexity of a metric
space are given independently in [DJ, Ju] and [KW]. One can use the techniques
from this paper to extend their notions from metric spaces to maps between
metric spaces. These are relevant notions because they gave lower bounds for
LS-category and the sequential topological complexity respectively.

Our objective in this paper is to compute the sequential complexity of group
homomorphisms. Let us recall that given a discrete group $\Gamma$, a
classifying space (Eilenberg-MacLane space) $B\Gamma=K(\Gamma,1)$ is defined
to be a path-connected space such that $\pi_{1}(B\Gamma)=\Gamma$ and
$\pi_{k}(B\Gamma)=0$ for all $k\neq 1.$ Since $B\Gamma$ is the unique up to
homotopy equivalence, we define that LS-category and the sequential
topological complexity of a discrete group to be LS-category and the
sequential topological complexity of its classifying space respectively, i.e.
$\operatorname{cat}(\Gamma):=\operatorname{cat}(B\Gamma)$ and
$TC_{r}(\Gamma):=TC_{r}(B\Gamma).$

Similarly, we can define the LS-category and the sequential topological
complexity of group homomorphism $\phi:\Gamma\to\Lambda,$
$\operatorname{cat}(\phi):=\operatorname{cat}(B\phi)$ and
$TC_{r}(\phi):=TC_{r}(B\phi)$ where $B\phi:B\Gamma\to B\Lambda$ is the
classifying map between classifying spaces.

The main results of the paper is the following theorems:

######  Theorem 1.1 (Theorem 6.5).

Let $\phi:\Gamma\to\Lambda$ be an epimorphism of finitely generated abelian
groups. Then

$TC_{r}(\phi)=(r-1)\operatorname{cd}(\phi).$

In particular, if $\Gamma$ is free abelain groups, then
$TC_{r}(\phi)=(r-1)(\operatorname{rank}(\Lambda)+k(T(\Lambda)))$ where
$k(T(\lambda))$ is the Smith Normal number for given finite abelian group
$T(\Lambda).$

######  Theorem 1.2 (Theorem 6.7).

Let $\phi:\Gamma\to\Lambda$ be a nonzero epimorphism of free groups. Then

$TC_{r}(\phi)=\begin{cases}r-1&\mbox{if}\,\,\,\Lambda\cong{\mathbb{Z}}\\\
r&\mbox{if}\,\,\,\Lambda\cong F_{m},m>1\end{cases}$

######  Theorem 1.3 (Proposition 4.2).

Let $\phi:\Gamma\to\Lambda$ be an nonzero epimorphism of finitely generated
groups $\Gamma,\Lambda$. Then $\operatorname{cd}(\phi)=0$ if and only if
$\Gamma$ is the trivial group.

######  Theorem 1.4 (Theorem 4.5).

Let $\phi:\Gamma\to\Lambda$ be an epimorphism of the fundamental groups of the
closed surfaces of $M_{g}$ and $M_{g^{\prime}}.$ Then

$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1$

if and only if $\phi$ factors through the free groups via epimorphisms
$\Gamma\to F_{n}\to\Lambda.$ Otherwise,
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=2.$

The paper is organized as follows. In Section 2, we recall the sectional
category, Ganea-Schwarz approach, Berstein-Schwarz classes, the Lusternik-
Schnirelman category, and the sequential topological complexity of space. In
Section 3, we introduce the sequential topological complexity of a map
borrowing the ideas from J.Scott’s paper [Sc]. We prove the basic inequalities
in the terms of LS-category maps, cuplength, the sequential topological
complexity of spaces, and composition of maps. In Section 4, we recall the
cohomological dimension of group homomorphisms and the product formula. We
prove Theorems 1.3 and Theorem 1.4, and give arbitrary product formula for
group homomorphisms. In Section 5, we give an analogue of the D-topological
complexity of space introduced by Farber and Oprea [FO]. We prove the the same
results hold for maps (Theorem 5.3) as well. In Section 6, we prove Theorem
1.1 and Theorem 1.2. We give sharp bounds for other discrete group
homomorphisms.

In the paper, we use the notation $H^{*}(\Gamma,A)$ for the cohomology of a
group $\Gamma$ with coefficient in $\Gamma$-module $A$. The cohomology groups
of a space $X$ with the fundamental group $\Gamma$ we denote as $H^{*}(X;A)$.
Thus, $H^{*}(\Gamma,A)=H^{*}(B\Gamma;A)$ where $B\Gamma=K(\Gamma,1)$.

##  2\. Preliminaries

###  2.1. Sectional Category

######  Definition 2.1.

Let $p:E\to B$ be a fibration. The sectional category of $p,$ denoted
$\operatorname{secat}(p),$ is the smallest integer $k$ such that $B$ can be
covered by $k+1$ open sets $U_{0},\cdots,U_{k}$ that each admit a partial
section $s_{i}:U_{i}\to E$ of $p$.

Note that sectional category of a fibration $p:E\to B.$ was first introduced
by Schwarz [Sch] in 1961, under the name “genus”. In 1978, James [Ja] replaced
the overworked term “genus” by “sectional category” and denoted the sectional
category of $p$ by $\operatorname{secat}(p).$

We state the following the well-known properties of the sectional category.
The proofs can be found in [Sch, Ja].

######  Properties 2.2.

Let $p^{\prime}:E^{\prime}\to B^{\prime}$ be a pull-back fibration of $p:E\to
B.$ a fibration. Then
$\operatorname{secat}(p^{\prime})\leq\operatorname{secat}(p).$

######  Properties 2.3.

Let $p^{\prime}:E^{\prime}\to B^{\prime}$ and $p:E\to B.$ be fibrations. Then
$\operatorname{secat}(p^{\prime}\times
p)\leq\operatorname{secat}(p^{\prime})+\operatorname{secat}(p).$

###  2.2. Ganea-Schwarz’s approach to $\operatorname{secat}$

Recall that an element of an iterated join $X_{0}*X_{1}*\cdots*X_{n}$ of
topological spaces is a formal linear combination
$t_{0}x_{0}+\cdots+t_{n}x_{n}$ of points $x_{i}\in X_{i}$ with $\sum t_{i}=1$,
$t_{i}\geq 0$, in which all terms of the form $0x_{i}$ are dropped. Given
fibrations $f_{i}:X_{i}\to Y$ for $i=0,...,n$, the fiberwise join of spaces
$X_{0},...,X_{n}$ is defined to be the space

$X_{0}*_{Y}X_{1}*_{Y}\cdots*_{Y}X_{n}=\\{\ t_{0}x_{0}+\cdots+t_{n}x_{n}\in
X_{0}*\cdots*X_{n}\ |\ f_{0}(x_{0})=\cdots=f_{n}(x_{n})\ \\}.$

The fiberwise join of fibrations $f_{0},...,f_{n}$ is the fibration

$f_{0}*_{Y}*\cdots*_{Y}f_{n}:X_{0}*_{Y}X_{1}*_{Y}\cdots*_{Y}X_{n}\longrightarrow
Y$

defined by taking a point $t_{0}x_{0}+\cdots+t_{n}x_{n}$ to $f_{i}(x_{i})$ for
any $i$ such that $t_{i}\neq 0$.

When $X_{i}=X$ and $f_{i}=f:X\to Y$ for all $i$ the fiberwise join of spaces
is denoted by $*^{n+1}_{Y}X$ and the fiberwise join of fibrations is denoted
by $*_{Y}^{n+1}f$.

######  Theorem 2.4 ([Sch, Ja]).

Let $p:E\to B$ be a fibrarition with B normal. Then
$\operatorname{secat}(p)\leq n$ if and only if the fibration
$*_{B}^{n+1}p:*_{B}^{n+1}E\to B$ admits a section.

###  2.3. Berstein–Schwarz-Schwarz cohomology class

The Berstein–Schwarz-Schwarz class of a discrete group $\Gamma$ is the first
obstruction $\beta_{\Gamma}$ to a lift of $B\Gamma=K(\Gamma,1)$ to the
universal covering $E\Gamma$. Note that $\beta_{\Gamma}\in
H^{1}(\Gamma,I(\Gamma))$ where $I(\Gamma)$ is the augmentation ideal of the
group ring ${\mathbb{Z}}\Gamma$ [Be],[Sch].

######  Theorem 2.5 (Universality [DR],[Sch]).

For any cohomology class $\alpha\in H^{k}(\Gamma,L)$, there is a homomorphism
of $\Gamma$-modules $I(\Gamma)^{k}\to L$ such that the induced homomorphism
for cohomology takes $(\beta_{\Gamma})^{k}\in H^{k}(\Gamma,I(\Gamma)^{\otimes
k})$ to $\alpha$, where $I(\Gamma)^{\otimes k}=I(\Gamma)\otimes\dots\otimes
I(\Gamma)$ and
$(\beta_{\Gamma})^{k}=\beta_{\Gamma}\smile\dots\smile\beta_{\Gamma}$.

###  2.4. Lusternik-Schnirelmann category

######  Definition 2.6.

The Lusternik-Schnirelmann category (for short, LS-category) of a topological
space $X$, denoted $\operatorname{cat}(X),$ is the minimal number $k$ such
that $X$ is covered by $k+1$ open sets $U_{0},...,U_{k}$ that are each
contractible in $X$.

######  Definition 2.7.

The LS-category of a map $f:X\to Y$, denoted $\operatorname{cat}(f),$ is the
minimal number $k$ such that $X$ is covered by $k+1$ open sets
$U_{0},...,U_{k}$ with nullhomotopic restrictions $f|_{U_{i}}:U_{i}\to Y$ for
all $i$.

For a path connected space $X$, we turn an inclusion of a point $*\to X$ into
a fibration $p^{X}_{0}:G_{0}(X)\to X$, whose fiber is known to be the loop
space $\Omega X$. The $n$-th Ganea space of $X$ is defined to be the space
$G_{n}(X)=*_{X}^{n+1}G_{0}(X)$, while the $n$-th Ganea fibration
$p^{X}_{n}:G_{n}(X)\to X$ is the fiberwise join $\ast^{n+1}_{X}p^{X}_{0}$.
Then the fiber of $p^{X}_{n}$ is $\ast^{n+1}\Omega X$.

###  2.5. The sequential topological complexity

######  Definition 2.8.

Let $X$ be a path-connected space.

  1. (1)

A sequential motion planner on a subset $Z\subset X^{r}$ is a map $s:Z\to PY$
such that $s(x_{0},x_{1},\cdots,x_{r-1})(\frac{j}{r-1})=x_{j}$ for all
$j=0,\cdots,r-1.$

  2. (2)

A sequential motion planning algorithm is a cover of $X^{r}$ by sets
$Z_{0},\cdots,Z_{k}$ such that on each $Z_{i}$ there is some sequential motion
planner $s_{i}:Z_{i}\to PY.$

  3. (3)

The sequential topological complexity of a space $X$, denoted $TC_{r}(X),$ is
the minimal number $k$ such that $X^{r}$ is covered by $k+1$ open sets
$U_{0},\cdots,U_{k}$ on which there are sequential motion planners. If no such
$k$ exists, we set $TC_{r}(X)=\infty.$

Note that above definition of the sequential topological complexity agrees
with Rudyak’s original definition [Rud] using exponential corresponds of path
space $X^{J_{r}},$ where $J_{r},r\in N$ is the wedge of $r$ closed intervals
$[0,1]_{i},i=1,..,r$ where the zero points $0_{i}\in[0,1]_{i}$ are identified.
The sequential topological complexity of a path-connected space $X,$ where
$r\geq 2,$ is defined as $TC_{r}(X)=\operatorname{secat}(\Updelta_{0}),$ where

$\Updelta_{0}:X^{J_{r}}\to
X^{r},\Updelta_{0}(\gamma)=(\gamma(1_{1}),\gamma(1_{2}),..,\gamma(1_{r-1}),\gamma(1_{r}))$

is a fibration. This correspondence is known as the exponential correspondence
of homeomorphic path spaces $PX$ and $X^{J_{r}}.$

${PX}$${X^{J_{r}}}$${X^{r}}$$\scriptstyle{p_{r}}$$\scriptstyle{\Updelta_{0}}$

where
$p_{r}(\gamma)=\big{(}\gamma(0),\gamma(\frac{1}{r-1}),\cdots,\gamma(\frac{r-2}{r-1}),\gamma(1)\big{)}$

Given $\gamma(t)\in PX,$ we get the component-wise paths
$\gamma_{i}(t)=\gamma\left(\dfrac{(i-1)t}{r-1}\right)\in
X^{J_{r}},i=1,\cdots,r.$ On the other hand, given the component-wise paths
$(\alpha_{1}(t),\cdots,\alpha_{r}(t))$ we can concatenate them the following
way
$\alpha_{1}(\bar{\alpha}_{2}\alpha_{3})\cdots(\bar{\alpha}_{r-1}\alpha_{r})$
to get the path in $PX$, where $\bar{\alpha_{i}}$ is the reverse path of
$\alpha_{i}$ for $i=1,\cdots,r-1.$ Thus, we use these two different evaluation
fibrations $p_{r}$ and $\Updelta_{0}$ for path space $PX$ interchangeably.

We lists some known properties of the sequential topological complexity.
Proofs can be found in [Rud, BGRT].

######  Properties 2.9.

Let X and Y be path-conected, normal spaces. Then

  1. (1)

$\operatorname{cat}(X^{r-1})\leq TC_{r}(X)\leq\operatorname{cat}(X^{r})$.

  2. (2)

$TC_{r}(X\times Y)\leq TC_{r}(X)+TC_{r}(Y)$.

  3. (3)

If X is an ANR, then $\operatorname{cup}(\ker\Updelta^{*})\leq TC_{r}(X)$.

  4. (4)

$TC_{r}(X)\leq n$ if and only if $p_{r,n}$ has a section where
$p_{r,n}=*_{X^{r}}^{n+1}p_{r}$ is the fiberwise join of a fibrartion $p_{r}.$

##  3\. The pullback of sequential topological complexity of map

Let $f:X\to Y$ be a map. Let $X^{r}$ and $Y^{r}$ be the Cartisian product of
$r$ copies of $X$ and $Y$ respectively, i.e $X^{r}:=X\times\cdots\times X$ and
$Y^{r}:=Y\times\cdots\times Y.$ Let us denote $f^{r}:=f\times\cdots\times
f:X^{r}\to Y^{r}$ and elements of $X^{r}$ and $Y^{r}$ are vectors
$\bar{x}=(x_{0},\cdots,x_{r-1})$ and $\bar{y}=(y_{0},\cdots,y_{r-1})$
respectively.

######  Definition 3.1.

Let $f:X\to Y$ be the above map.

  1. (1)

A sequential $f$-motion planner on a subset $Z\subset X^{r}$ is a map
$f_{Z}:{\mathbb{Z}}\to PY$ such that
$f_{Z}(\bar{x})(\frac{j}{r-1})=f_{Z}(x_{0},x_{1},\cdots,x_{r-1})(\frac{j}{r-1})=f(x_{j})$
for all $j=0,\cdots,r-1.$

  2. (2)

A sequential $f$-motion planning algorithm is a cover of $X^{r}$ by sets
$Z_{0},\cdots,Z_{k}$ such that on each $Z_{i}$ there is some sequential
$f$-motion planner $f_{i}:Z_{i}\to PY.$

  3. (3)

The (pullback) sequential topological complexity of map f, denoted
$TC_{r}(f),$ is the minimal number $k$ such that $X^{r}$ is covered by $k+1$
open sets $U_{0},\cdots,U_{k}$ on which there are sequential $f$-motion
planners. If no such $k$ exists, we set $TC_{r}(f)=\infty.$

Note that if $r=2,$ we recover Scott’s topological complexity for maps.
Further, if map $f$ is identity on space $X,$ we get Rudyak’s sequential
topological complexity of space $X.$

We get the following analogous theorem of Scott [Sc, Theorem 3.4] for the
sequential topological complexity for maps.

######  Theorem 3.2.

Let $f:X\to Y$ be a map and let $Z\subset X^{r}$. The following are
equivalent:

  1. (1)

There is a partial section $s:Z\to(f^{r})^{*}PY$ of the pullback fibrations
$(f^{r})^{*}\Updelta_{0}^{Y}$.

  2. (2)

There is a sequential $f$-motion planner $f_{Z}:Z\to PY$.

  3. (3)

The projections from $f^{r}(Z)$ to the j+1 factor of $Y^{r}$ are homotopic,
where $j=0,..,r-1$.

  4. (4)

$f^{r}|_{Z}$ can be deformed into the diagonal $\Updelta Y$ of $Y^{r}$.

  5. (5)

There is a map $h:Z\to X$ such that $f^{r}\Updelta h$ is homotopic to
$f^{r}|_{Z}$.

###### Proof.

$(1\Rightarrow 2)$ Let $s:Z\to(f^{r})^{*}PY$ be a partial section of the
pullback fibration $(f^{r})^{*}\Updelta_{0}^{Y}$ and consider the pullback
diagram of the maps $f^{r}$ and $\Updelta_{0}^{Y}$

${(f^{r})^{*}PY}$${PY}$${X^{r}}$${Y^{r}}$$\scriptstyle{(f^{r})^{*}\Updelta_{0}^{Y}}$$\scriptstyle{\bar{f^{r}}}$$\scriptstyle{\Updelta_{0}^{Y}}$$\scriptstyle{f^{r}}$

where $(f^{r})^{*}PY=\\{(\bar{x},\gamma)\in X^{r}\times
PY|\gamma(\frac{i}{r-1})=f(x_{i})\,\text{for all}\,i=0,..,r-1.\\}$

We define the sequential $f$-motion planner $f_{Z}:Z\to PY$ by
$f_{Z}=\bar{f^{r}}s$. It is easy to see that

$\Updelta_{0}^{Y}f_{Z}=\Updelta_{0}^{Y}(\bar{f^{r}})s=(f^{r})((f^{r})^{*}\Updelta_{0}^{Y})s=(f^{r})Id_{Z}=f^{r}|_{Z}.$

Hence, $f_{Z}$ is a sequential $f$-motion planner on Z.

$(2\Rightarrow 1)$ It just follows from the definition of pullback of maps.
(See [Sc, Theorem 3.4]).

$(2\Rightarrow 3)$ Let $pr_{j}:f^{r}(Z)\subset Y^{r}\to Y$ be a projection
onto the $j^{th}$ factor of $Y^{r}$. It suffices to show that $pr_{j}$ is
homotopic to $pr_{j+1}$ for all $j=0,..,r-2.$ By (2), there is a sequential
$f$-motion planner on $Z$ and let $\bar{x}=(x_{0},\cdots,x_{r-1})\in Z.$

We define the homotopy

$H_{j}(\bar{x},t):=f_{Z}(\bar{x})\big{(}\frac{t+j}{r-1}\big{)}.$

Then
$H_{j}(\bar{x},0)=f_{Z}(\bar{x})(\frac{j}{r-1})=f(x_{j})=pr_{j}(f^{r}(\bar{x}))$
and
$H_{j}(\bar{x},1)=f_{Z}(\bar{x})(\frac{j+1}{r-1})=f(x_{j+1})=pr_{j+1}(f^{r}(\bar{x})).$
Thus, all projections from $f^{r}(Z)$ to the $j^{th}$ factor of $Y^{r}$ are
homotopic for $j=0,\cdots,r-1.$

$(3\Rightarrow 4,5)$ Since any two projections from $f^{r}(Z)$ to the $j^{th}$
factor of $Y^{r}$ are homotopic, we fix a homotopy $H_{j}:f^{r}(Z)\times I\to
Y^{r}$ from $pr_{j}$ to $pr_{j+1}$ for $j=0,\cdots,r-2.$ For given $\bar{x}\in
Z,$ this homotopy $H_{i}(f^{r}(\bar{x}),t)$ is a path from $f(x_{j})$ to
$f(x_{j+1})$ and denote this path as $\alpha_{\bar{x}}^{j}.$ We define a
concatenation of paths as
$\gamma_{\bar{x}}^{i}=*^{i}_{j=0}\alpha_{\bar{x}}^{j}.$ Note that
$\gamma_{\bar{x}}^{i}$ is a path from $f(x_{0})$ to $f(x_{i}).$ Now we define
a homotopy $H:f^{r}(Z)\times I\to Y^{r}$ as

$H(f^{r}(\bar{x}),t)=(H_{0}(f^{r}(\bar{x}),t(1-t)),\gamma_{\bar{x}}^{1}(1-t),\cdots,\gamma_{\bar{x}}^{r-1}(1-t)).$

Then
$H(f^{r}(\bar{x}),0)=(H_{0}(f^{r}(\bar{x}),0),\gamma_{\bar{x}}^{1}(1),\cdots,\gamma_{\bar{x}}^{r-1}(1))=(f(x_{0}),f(x_{1}),\cdots,f(x_{r-1}))=f^{r}(\bar{x})$
and
$H(f^{r}(\bar{x}),1)=(H_{0}(f^{r}(\bar{x}),0),\gamma_{\bar{x}}^{1}(0),\cdots,\gamma_{\bar{x}}^{r-1}(0))=(f(x_{0}),f(x_{0}),\cdots,f(x_{r-1}))\in\Updelta(Y).$
This gives a deformation of $f^{r}|_{Z}$ into $\Updelta(Y),$ showing (4). Also
observe that $h$ can be chosen to be $pr_{0}:Z\subset X^{r}\to X$ is the
projection map into the first coordinate.

$(5\Rightarrow 4)$ Note that the image of $f^{r}\Updelta h$ will always be
contained in $\Updelta(Y).$

$(4\Rightarrow 2)$ Let $H:Z\times I\to Y^{r}$ be a deformation of $f^{r}|_{Z}$
to $\Updelta(Y).$ We define a map $f_{Z}:Z\to Y^{r}$

$f_{Z}(\bar{x})(t)=\begin{cases}p_{j}H(\bar{x},2(r-1)t-2j)&\mbox{if}\,\,\frac{j}{r-1}\leq
t\leq\frac{2j+1}{2(r-1)}\\\
p_{j+1}H(\bar{x},2j+2-2(r-1)t)&\mbox{if}\,\,\frac{2j+1}{2(r-1)}\leq
t\leq\frac{j+1}{(r-1)}\end{cases}$

where $j=0,\cdots,r-2$ and $pr_{j}$ and $pr_{j+1}$ are projection of $Y^{r}$
to $Y$ with $j^{th}$ and $(j+1)^{th}$ coordinates respectively. Then $f_{Z}$
is well-defined and continous since $H(\bar{x},t)\in\Updelta(Y)$ for all
$\bar{x}\in Z.$ Moreover,

$f_{Z}(\bar{x})(\frac{j}{r-1})=pr_{j}H(\bar{x},2(r-1)\frac{j}{r-1}-2j)=pr_{j}H(\bar{x},0)=pr_{j}(f^{r}(\bar{x}))=f(x_{j}),$

and

$f_{Z}(\bar{x})(\frac{j+1}{r-1})=pr_{j+1}H(\bar{x},2j+2-2(r-1)\frac{j+1}{r-1})=pr_{j+1}H(\bar{x},0)=pr_{j+1}(f^{r}(\bar{x}))=f(x_{j+1})$

so that $f_{Z}$ is an sequential $f$-motion planner on Z.

∎

We can also define the sequential topological complexity of a map $f:X\to Y,$
$TC_{r}(f):=\operatorname{secat}((f^{r})^{*}\Updelta_{0}^{Y}).$ It was well-
defined by Theorem 3.2 part 2. Since sectional category is homotopy invariant
[Sch], the sequential topological complexity of map is also homotopy
invariant.

We will use the following proposition on the proof of main results in this
chapter. This result is a sequential analogue of [Sc, Proposition 3.8] which
relates the LS-category of map and the sequential topological complexity of
map.

######  Proposition 3.3.

Let $f:X\to Y$ be a map. Then

1\. $TC_{r}(f)\leq\min\\{TC_{r}(X),TC_{r}(Y)\\}$
2\. $\operatorname{cat}(f^{r-1})\leq TC_{r}(f)\leq\operatorname{cat}(f^{r})$

###### Proof.

1\. It is clear that $TC_{r}(f)\leq TC_{r}(Y)$ by the sectional category. Now
suppose that $TC_{r}(X)=k$ and let $U_{0},U_{1},...,U_{k}$ be the open cover
of $X^{r}$ with motion planners $s_{i}:U_{i}\to X^{r}.$ We define
$f_{i}:U_{i}\to PY$ by $f_{i}=fs_{i}.$ Then it is easy to see that

$f_{i}(x_{0},x_{1},..,x_{r-1})(\frac{j}{r-1})=f(s_{i}(x_{0},...,x_{r-1})(\frac{j}{r-1}))=f(x_{j})$

for all $j=0,..,r-1.$ Therefore, $f_{i}$ are the sequential $f$-motion
planners and $TC_{r}(f)\leq k.$

2\. First we prove that $\operatorname{cat}(f^{r-1})\leq TC_{r}(f).$ Suppose
$TC_{r}(f)=k$ and let $U_{0},U_{1},..,U_{k}$ be an open cover of $X^{r}$ with
sequential $f$-motion planners $f_{i}:U_{i}\to PY.$ Let $a$ and $b$ be a
choice of base points for $X$ and $Y$ respectively such that $f(a)=b.$ Let
$\gamma_{j}$ be paths from $b$ to $f(x_{j})$ for all $j=1,..,r-1$ as we
constructed in the proof of Theorem 3.2 $(3\Rightarrow 4,5)$. We define
$V_{i}=\\{\bar{x}=(x_{1},..,x_{r-1})|(a,x_{1},..,x_{r-1})\in U_{i}\\}.$ Then
$V_{0},..,V_{k}$ is clearly an open cover for $X^{r-1}.$ Now we define a
nullhomotopy $H_{i}:V_{i}\times I\to Y^{r-1}$ of $f^{r-1}$ on $V_{i}$ by

$H_{i}(\bar{x},t)=(f_{i}(a,\bar{x})(t(1-t)),\gamma^{1}(1-t),...,\gamma^{r-2}(1-t)).$

Then

$H_{i}(\bar{x},0)=(f(a),\gamma^{1}(1),..,\gamma^{r-1}(1))=(b,f(x_{1}),...,f(x_{r-1}))$

and

$H_{i}(\bar{x},1)=(f(a),\gamma^{1}(0),...,\gamma^{r-1}(0))=(b,b,...,b)$

Therefore, $\operatorname{cat}(f^{r-1})\leq k.$

Next, we show that $TC_{r}(f)\leq\operatorname{cat}(f^{r}).$ Suppose that
$\operatorname{cat}(f^{r})=k$ and let $U_{0},U_{1},..,U_{k}$ be an open cover
of $X^{r}$ such that each $f^{r}|_{U_{i}}$ is nullhomotopic. Let
$H_{i}:U_{i}\times I\to Y^{r}$ be a nullhomotopy of $f^{r}|_{U_{i}}$ to some
$\bar{y}^{i}=(y_{0}^{i},..,y_{r-1}^{i})\in Y^{r}.$ Since $Y$ is a path-
connected, let $\beta_{j}^{i}$ be paths from $y_{j}^{i}$ to $y_{j+1}^{i}$ for
all $j=0,..,r-2.$ Now we define $f_{i}:U_{i}\to PY$ by

$f_{i}(\bar{x})(t)=\begin{cases}pr_{j}(H_{i}(\bar{x},3(r-1)t-3j)),&\mbox{if}\,\,\frac{j}{r-1}\leq
t\leq\frac{3j+1}{3(r-1)}\\\
\beta_{j}^{i}(3(r-1)t-3j-1)&\mbox{if}\,\,\frac{j+1}{3(r-1)}\leq
t\leq\frac{3j+2}{3(r-1)}\\\
pr_{j+1}(H_{i}(\bar{x},3(r-1)t-3j-2))&\mbox{if}\,\,\frac{3j+2}{3(r-1)}\leq
t\leq\frac{j+1}{r-1}\\\
pr_{r-2}(H_{i}(\bar{x},3(r-1)t-3(r-2)))&\mbox{if}\,\,\frac{r-2}{r-1}\leq
t\leq\frac{3(r-2)+1}{3(r-1)}\\\
\beta_{r-2}^{i}(3(r-1)t-3(r-2)-1)&\mbox{if}\,\,\frac{3(r-2)+1}{3(r-1)}\leq
t\leq\frac{3(r-2)+2}{3(r-1)}\\\
pr_{r-1}(H_{i}(\bar{x},3(r-1)-3(r-1)t))&\mbox{if}\,\,\frac{3(r-2)+2}{3(r-1)}\leq
t\leq 1\end{cases}$

where $j=0,..,r-3$, and $pr_{j}:Y^{r}\to Y,$ $pr_{r-2}:Y^{r}\to Y,$ and
$p_{r-1}:Y^{r}\to Y,$ are projections into $j+1$, $r-1$ and $r$ coordinates,
respectively.

Since $H_{i}(\bar{x},1)=\bar{y}^{i}=(y_{0}^{i},..,y_{r-1}^{i}),$ $f_{i}$ is
continuous. Furthermore,

$f_{i}(\bar{x})(\frac{j}{r-1})=pr_{j}(H_{i}(\bar{x},3(r-1)\frac{j}{r-1}-3j))=pr_{j}(f^{r})(\bar{x})=f(x_{j})$
for all $j=0,...,r-3$,

$f_{i}(\bar{x})(\frac{r-2}{r-1})=pr_{r-2}(H_{i}(\bar{x},3(r-1)\frac{r-2}{r-1}-3(r-2)))=p_{r-2}(f^{r})(\bar{x})=f(x_{r-2})$,
and

$f_{i}(\bar{x})(1)=p_{r-1}(H_{i}(\bar{x},3(r-1)-3(r-1)1))=p_{r-1}(f^{r})(\bar{x})=f(x_{r-1})$

so that the $f_{i}$ are sequential $f$-motion planners. Therefore,
$TC_{r}(f)\leq k.$ ∎

######  Remark 3.4.

This proof recovers Jamie Scott’s proof ([Sc], Proposition 3.8) when $r=2.$

######  Theorem 3.5.

Let $f:G\to H$ be any map such that $G$ is an topological group. Then
$TC_{r}(f)=\operatorname{cat}(f^{r-1}).$

###### Proof.

By Proposition 3.3, it suffices to show that
$TC_{r}(f)\leq\operatorname{cat}(f^{r-1}).$ Let $e\in G$ be the unit. Let us
denote $\bar{e}=(e,\cdots,e)\in G^{r-1}$ and $(e,\bar{e})\in G\times G^{r-1}.$
Suppose that $\operatorname{cat}(f^{r-1})=k$ and let $U_{0},\dots,U_{k}$ be
open covers of $G^{r-1}$ together nullhomotopies $H_{i}:U_{i}\times I\to
H^{r-1}$ of $f^{r-1}|_{U_{i}}$ to $f^{r-1}(\bar{e}).$ Indeed, this gives a
path from $f^{r-1}(\bar{b})$ to $(f(a),\cdots,f(a))=f(a)f^{r-1}(\bar{e}),$
which is continous and depends on only $\bar{b}\in U_{i}$ since
$\gamma_{\bar{b}}(t)=f(a)H_{i}(a^{-1}\bar{b},t)$

Now we define the open cover of $G^{r}=G\times G^{r-1}$ as follows

$W_{i}=\\{(a,\bar{g})\in(G\times G^{r-1})|a^{-1}\bar{b}\in U_{i}\\}$ for all
$i=0,\cdots,k.$ Clearly, $\\{W_{i}\\}$ are open subsets and covers the
topological group $G^{r}.$

Note that path between two vectors can be thought as collection of component-
wise paths, i.e,
$\gamma_{\bar{b}}(t)=(\gamma_{b_{1}}(t),\cdots,\gamma_{b_{r-1}}(t)).$ Abusing
notation, we let
$\gamma_{\bar{b}}(t)=(1_{f(a)},\gamma_{b_{1}}(t),\cdots,\gamma_{b_{r-1}}(t))$
where $1_{f(a)}$ is constant path at $f(a)$ for all $t\in[0,1].$

Thus, we obtain that the map $f^{r}$ restricts on each open set $W_{i}$,
$f^{r}|_{W_{i}}$ deforms into the diagonal $\Updelta(Y)$ of $Y^{r}$, namely
$(f(a),f(a),..,f(a)).$ By Theorem 3.2, we get an sequential $f$-motion planner
on $W_{i}$ for each $i=0,\cdots,k$ (See, Theorem 3.2, proof of $(4\Rightarrow
2)$).

Therefore, $TC_{r}(f)\leq k.$

∎

######  Remark 3.6.

Note that the statement is true for $H$-space. Because of notational
difficulties, we prove here only for topological groups, which suffices us to
prove our main results.

The cuplength lower bound for topological complexity of a map was proven in
[Sc] for map with an ANR X with coefficients in any
$\pi_{1}(Y)\times\pi_{1}(Y)$ module. The exact same proof works for the
sequential topological complexity of a map. To completeness of our work, we
state it here and give the proof:

######  Theorem 3.7.

Let $f:X\to Y$ be any map with X an ANR. Suppose that
$u_{i}\in\tilde{H}^{*}(X^{r};A_{i})$ are such that
$u_{i}\in\ker(\Updelta_{X}^{*})\cap\operatorname{im}(f^{r})^{*}$ and
$u_{0}\cup\cdots\cup u_{k}\neq 0,$ where each $A_{i}$ are $\pi_{1}(Y)^{r}$
modules. Then $TC_{r}(f)\geq k+1.$ Indeed,

$\operatorname{cup}(\ker(\Updelta_{X}^{*})\cap\operatorname{im}(f^{r})^{*})\leq
TC_{r}(f).$

###### Proof.

Suppose $TC_{r}(f)\leq k$ so that there are open sets $U_{0},\cdots,U_{k}$
covering $X^{r}$ such that on each $U_{i}$ there is a map $h:U_{i}\to X$
satisfying $(f^{r})\Updelta h\cong(f^{r})|_{U_{i}}$ (see Thoerem 3.2 part
(5)). Thus, for each $i$ we have a commutative diagram

${\cdots}$${H^{n}(X^{r},U_{i};A_{i})}$${H^{n}(X^{r};A_{i})}$${H^{n}(U_{i};A_{i})}$${\cdots}$${H^{n}(X;A_{i})}$${H^{n}(Y^{r};A_{i})}$${H^{n}(X^{r};A_{i})}$$\scriptstyle{q_{i}^{*}}$$\scriptstyle{j_{i}^{*}}$$\scriptstyle{h_{i}^{*}}$$\scriptstyle{(f^{r})^{*}}$$\scriptstyle{(f^{r})^{*}}$$\scriptstyle{\Updelta_{X}^{*}}$

where the top row is the long exact sequence of the pair $(X^{r},U_{i}),$ and
the maps $q_{i}$ and $j_{i}$ are inclusion maps.

Now let
$u_{0},..,u_{k}\in\ker(\Updelta_{X}^{*})\cap\operatorname{im}(f^{r})^{*}$
where each $u_{i}$ is taken in coefficients $A_{i}$ and has degree at least 1.
Then there is some $v_{i}\in H^{n}(Y^{r};A_{i})$ such that
$(f^{r})^{*}v_{i}=u_{i}.$ Since each $u_{i}\in\ker\Updelta_{X}^{*},$ it
follows that

$j_{i}^{*}u_{i}=j_{i}^{*}(f^{r})^{*}v_{i}=h_{i}^{*}\Updelta_{X}^{*}(f^{r})^{*}v_{i}=h_{i}^{*}\Updelta_{X}^{*}u_{i}=h_{i}^{*}(0)=0$

so that $u_{i}\in\ker j_{i}^{*}.$ By exactness, there is some $w_{i}\in
H^{*}(X^{r},U_{i};A_{i})$ such that $q_{i}^{*}w_{i}=u_{i}.$ Now consider
$w_{0}\cup\cdots\cup w_{k},$ which lies in

$H^{*}(X^{r},U_{0}\cup\cdots\cup U_{k};A_{0}\otimes\cdots\otimes
A_{k})=H^{*}(X^{r},X^{r};A_{0})\otimes\cdots\otimes A_{k})=0$

so that $w_{0}\cup\cdots\cup w_{k}=0;$ hence,

$u_{0}\cup\cdots\cup u_{k}=q_{0}^{*}w_{0}\cup\cdots\cup
q_{k}^{*}w_{k}=q^{*}(w_{0}\cup\cdots\cup w_{k})=0$

where $q:X^{r}\to(X^{r},X^{r})$ is the inclusion map of pairs. Thus, the
theorem follows. ∎

Similarly, the proof of topological complexity of composite maps can be
identically applied for the sequential topological complexity of composite
maps.

######  Proposition 3.8.

$TC_{r}(gf)\leq\min\\{TC_{r}(g),TC_{r}(f)\\}$ for any maps $f:X\to Y$ and
$g:Y\to Z.$

###### Proof.

Suppose that $TC_{r}(f)=k$ so that there is an open cover $U_{0},\cdots,U_{k}$
of $X^{r}$ such that each restriction $(f^{r})|_{U_{i}}$ can be deformed into
$\Updelta(Y)$ via some homotopy $F_{i}:U_{i}\times I\to Y^{r}.$ Now define a
homotopy $H_{i}:U_{i}\times I\to Z^{r}$ by $H_{i}=(g^{r})F_{i}$, which defines
a deformation of $(gf)^{r}|_{U_{i}}$ into $\Updelta(Z);$ hence,
$TC_{r}(gf)\leq k=TC_{r}(f).$

Now suppose that $TC_{r}(g)=k$ so that there is an open cover
$V_{0},\cdots,V_{k}$ of $Y^{r}$ such that each restriction $(g^{r})|_{V_{i}}$
can be deformed into $\Updelta(Z)$ via some homotopy $G_{i}:V_{i}\times I\to
Z^{r.}$ Now let $U_{i}=(f^{r})^{-1}(V_{i})$ and define a homotopy
$H_{i}:U_{i}\times I\to Z^{r}$ by $H_{i}(\bar{x},t)=G_{i}(f^{r}(\bar{x}),t),$
which defines a deformation of $(gf)^{r}|_{U_{i}}$ into $\Updelta(Z).$
Therefore, $TC_{r}(gf)\leq k=TC_{r}(g).$

∎

######  Corollary 3.9.

Let $f:X\to Y$ be a map, and let $r:X\to A$ be a retraction. Then
$TC_{r}(f)=TC_{r}(\bar{f})$ where $\bar{f}:=f|_{A}$.

###### Proof.

By Proposition 3.8, we obtain that $TC_{r}(f)\leq TC_{r}(\bar{f}).$

Since $\bar{f}=f|_{A}$, we have $TC_{r}(\bar{f})\leq TC_{r}(f).$ ∎

##  4\. Cohomological dimension of group homomorphisms

Let $\Gamma$ be a discrete group. A projective resolution $P_{*}(\Gamma)$ of
${\mathbb{Z}}$ for the group $\Gamma$ is an exact sequence of projective
$\Gamma-$ modules

$\cdots\to P_{n}(\Gamma)\to P_{n-1}(\Gamma)\to\cdots P_{2}(\Gamma)\to
P_{1}\to{\mathbb{Z}}\Gamma\to{\mathbb{Z}}.$

The cohomology of $\Gamma$ with coefficients in a $\Gamma-$ module M can be
defined as homology of the cochain complexe $Hom_{\Gamma}(P_{*}(\Gamma),M)$.

We recall the cohomological dimension $\operatorname{cd}(\phi)$ of a group
homomorphism $\phi:\Gamma\to\Lambda$ was introduced by Mark Grant [Gr] as the
maximum of $k$ such that there is a ${\mathbb{Z}}\Lambda$-module $M$ with the
nonzero induced homomorphism $\phi^{*}:H^{k}(\Lambda,M)\to H^{k}(\Gamma,M)$.
When $\phi$ is the identity homomorphism, we recover the classical
cohomological dimension of a discrete group $\Gamma$, the shortest length of
projective resolution $P_{*}(\Gamma)$ [Br]. The following Theorem of
Dranishnikov and De Saha gives the chacterisation of group homomorphism in
term of the projective resolutions.

######  Theorem 4.1 ([DD]).

Let $\phi:\Gamma\to\Lambda$ be a group homomorphism with
$\operatorname{cd}(\phi)=n$ and let
$\phi_{*}:(P_{*}(\Gamma),\partial_{*})\to(P_{*}(\Gamma),\partial^{\prime}_{*})$
be the chain map between projective resolutions of ${\mathbb{Z}}$ for $\Gamma$
and $\Lambda$ induced by $\phi$. Then $\phi_{*}$ is chain homotopic to
$\psi_{*}:P_{*}(\Gamma)\to P_{*}(\Lambda)$ with $\psi_{k}=0$ for $k>n.$

By Theorem 3.1 in [DK], we assume that group homomorphism
$\phi:\Gamma\to\Lambda$ is an epimorphism. We get the following
characterisation of group epimorphism.

######  Proposition 4.2.

Let $\phi:\Gamma\to\Lambda$ be an nonzero epimorphism of finitely generated
groups $\Gamma,\Lambda$. Then $\operatorname{cd}(\phi)=0$ if and only if
$\Gamma$ is the trivial group.

###### Proof.

Suppose $\Gamma$ is the trivial group. Then
$\phi^{*}:\tilde{H^{*}}(B\Lambda;M)\to\tilde{H^{*}}(B\Gamma;M)$ is the trivial
homomorphism for all $\Lambda-$ modules M. Thus, we obtain that
$\operatorname{cd}(\phi)=0.$

We prove this direction in two steps:

Step 1. Suppose $\Gamma$ is a finite group. Since $\phi(\Gamma)\neq 0,$ by
Theorem [Ku1] we obtain that $\operatorname{cd}(\phi)=\infty.$ Thus, it is a
contradiction to have $\operatorname{cd}(\phi)=0.$ Hence, $\Gamma$ is the
trivial group.

Step 2. Suppose $\Gamma$ is a torsion-free group. We obtain the following
commutative diagram

${\cdots}$${P_{3}(\Gamma)}$${P_{2}(\Gamma)}$${P_{1}(\Gamma)}$${{\mathbb{Z}}\Gamma}$${{\mathbb{Z}}}$${0}$${\cdots}$${P_{3}(\Lambda)}$${P_{2}(\Lambda)}$${P_{1}(\Lambda)}$${{\mathbb{Z}}\Lambda}$${{\mathbb{Z}}}$${0}$$\scriptstyle{\partial_{4}}$$\scriptstyle{\partial_{3}}$$\scriptstyle{\phi_{3}}$$\scriptstyle{\partial_{3}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\phi_{2}}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\epsilon}$$\scriptstyle{\phi_{0}}$$\scriptstyle{1}$$\scriptstyle{\partial^{\prime}_{4}}$$\scriptstyle{\partial^{\prime}_{3}}$$\scriptstyle{\partial^{\prime}_{2}}$$\scriptstyle{\partial^{\prime}_{1}}$$\scriptstyle{\epsilon^{\prime}}$

where the augmentation homomorphisms $\epsilon(g)=1$ and
$\epsilon^{\prime}(g^{\prime})$ for all $g\in\Gamma$ and all
$g^{\prime}\in\Lambda.$

Note that $P_{*}(\Gamma)$, ${\mathbb{Z}}\Gamma,$ and $I(\Gamma)$ are $\Gamma-$
modules, but also $\Lambda-$ modules via $\phi.$

The Berstein–Schwarz class of $\Lambda$, $\beta_{\Lambda}\in
H^{1}(B\Lambda;I(\Lambda))$, is given by the cocycle
$f_{\Lambda}:P_{1}(\Lambda)\to
I(\Lambda),f_{\Lambda}(c^{\prime})=\partial^{\prime}_{1}(c^{\prime})$ where
$I(\Lambda)$ is the augmentation ideal of ${\mathbb{Z}}\Lambda$.

Since $\phi_{*}$ is nonzero surjective, for each $c^{\prime}\in
P_{1}(\Lambda)$, there exists $c\in P_{1}(\Gamma)$ with $c^{\prime}=\phi(c).$
Since the above diagram is commutative, we get the nonzero cocycle
$f_{\Lambda}\phi_{1}\in Hom_{\Lambda}(P_{1}(\Gamma),I(\Lambda))$. Hence, this
implies $\phi^{*}:H^{1}(B\Lambda;I(\Lambda))\to H^{1}(B\Gamma;I(\Lambda))$ is
not trivial, i.e $\phi^{*}(\beta_{\Lambda})\neq 0.$ In other words, the
cohomological class $\phi^{*}(\beta{\Lambda})$ is given by a nonzero cocycle
$f_{\Lambda}\phi_{1}.$ This contradicts $\operatorname{cd}(\phi)=0.$ Hence,
$\Gamma$ is the trivial group. ∎

######  Remark 4.3.

Step 2 covers Step 1 since the proof uses only nonzero epimorphism. However,
for finite groups one does not need to introduce the Berstein-Schwarz class.

######  Corollary 4.4.

Let $\phi:F_{n}\to F_{m}$ be an epimorphism between free groups with $n$ and
$m$ generators respectively. Then
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1$. Moreover, the
Berstein–Schwarz classes of $F_{n}$ and $F_{m}$ are related.

###### Proof.

By Proposition 4.2, it is clear that $\operatorname{cd}(\phi)\geq 1$ since
$F_{n}$ is not trivial. Since
$\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)\leq\operatorname{cat}(F_{n})=1,$
we obtain that $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1.$

We recall that the Berstein–Schwarz class of discrete group $\Gamma$ is also
defined as the image of the connecting homomorphism
$\delta:H^{0}(B\Gamma;{\mathbb{Z}})\to H^{1}(B\Gamma;I(\Gamma))$, i.e.,
$\delta(1)=\beta_{\Gamma},$ where $I(\Gamma)$ is the augmentation ideal of the
group ring ${\mathbb{Z}}\Gamma$ in [DR].

Since $\phi:F_{n}\to F_{m}$ is an epimorphism, there exists a section
$s:F_{m}\to F_{n}$ with $\phi s=1_{F_{m}}.$

We have the following commutative diagram of the short exact sequences

${0}$${I(F_{n})}$${{\mathbb{Z}}F_{n}}$${{\mathbb{Z}}}$${0}$${0}$${I(F_{m})}$${{\mathbb{Z}}F_{m}}$${{\mathbb{Z}}}$${0}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\epsilon}$$\scriptstyle{1}$$\scriptstyle{\bar{s}}$$\scriptstyle{\epsilon}$$\scriptstyle{\bar{s}}$

where $\bar{\phi}$ and $\bar{s}$ are induced maps of $\phi$ and $s$
respectively.

The above short exact sequences give the following commutative diagrams

${H^{0}(BF_{n};{\mathbb{Z}})}$${H^{1}(BF_{n};I(F_{n}))}$${H^{1}(BF_{n};I(F_{m}))}$${H^{0}(BF_{m};{\mathbb{Z}})}$${H^{1}(BF_{m};I(F_{m}))}$$\scriptstyle{\cong}$$\scriptstyle{\delta}$$\scriptstyle{I(\phi)_{*}}$$\scriptstyle{s^{*}}$$\scriptstyle{I(s)_{*}}$$\scriptstyle{\delta}$$\scriptstyle{\phi^{*}}$

Note that we can treat $I(F_{n})$ is $F_{m}-$ module via $\phi$ while
$I(F_{m})$ is $F_{n}-$ module via $s$. Hence, all homomorphisms are well-
defined. The commutative diagram give us the conclusion of the corollary:
$\delta(1)=\beta_{F_{n}}=I(s)_{*}\circ\phi^{*}(\beta_{F_{m}})$ and
$\delta(1)=\beta_{F_{m}}=s^{*}\circ I(\phi)_{*}(\beta_{F_{n}})$ ∎

Let $\Gamma=\pi_{1}(M_{g})$ and $\Lambda_{1}(M_{g^{\prime}})$ be the
fundamental groups of orientable, closed surfaces $M_{g}$ and $M_{g^{\prime}}$
where $g,g^{\prime}$ are positive genus numbers.

######  Theorem 4.5.

Let $\phi:\Gamma\to\Lambda$ be an epimorphism of the fundamental groups of the
closed surfaces of $M_{g}$ and $M_{g^{\prime}}.$ Then

$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1$

if and only if $\phi$ factors through the free groups via epimorphisms
$\Gamma\to F_{n}\to\Lambda.$ Otherwise,
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=2.$

###### Proof.

By Proposition 4.2, we always obtain $\operatorname{cd}(\phi)\geq 1$ since
$\Gamma$ is not the trivial group.

We consider two steps:

Step 1. Now we suppose the epimorphism $\phi:\Gamma\to\Lambda$ factors through
the free groups via epimorphisms $\Gamma\to F_{n}\to\Lambda.$ Since
$\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)$ (see [DK]), it suffices
to show $\operatorname{cat}(\phi)\leq 1.$

This easily follows from the classical LS-category. Indeed,

$\operatorname{cat}(\phi)\leq\min\\{\operatorname{cat}(h),\operatorname{cat}(g)\\}\leq\operatorname{cat}(F_{n})=1,$

where $h:B\Gamma\to BF_{n}$ and $g:BF_{n}\to B\Lambda$ are maps that induced
epimorphisms $h_{*}:\Gamma\to F_{n}$ and $g_{*}:F_{n}\to\Lambda.$

Now conversely $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1,$ then
using the Ganea-Schwarz approach to LS-category of induced map $B\phi,$ we
obtain that the induced map $B\phi:B\Gamma\to B\Lambda$ can be lifted to the
Ganea’s space $G_{1}(B\Lambda).$ Note that $G_{1}(B\Lambda)$ is homotopy
equivalent to 1-dimensional complex $\Sigma\Lambda$, the reduced suspension of
$\Lambda$ [CLOT]. Then the epimorphism $\phi$ factors through a free group
$F_{n}$ via epimorphisms $\Gamma\to F_{n}\to\Lambda.$

Step 2. By the classical LS-category,
$\operatorname{cd}(\phi)\leq\min\\{\operatorname{cat}(\Gamma),\operatorname{cat}(\Lambda)\\}=2$
since $\operatorname{cat}(\Gamma)=2$ and $\operatorname{cat}(\Lambda)=2.$ By
Step 1, we have that $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)\neq 1$
since the epimorphism $\phi$ does not factor through the free groups via
epimorphisms $\Gamma\to F_{n}\to\Lambda$. Since $\operatorname{cd}(\phi)\geq
1$, we get the conclusion of Theorem
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=2.$ ∎

######  Remark 4.6.

We only use the surface groups in Step 2. Note that the proof of Step 1 only
requires a nonzero epimorphism and
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1.$ Otherwise, consider any
degree one map $f:T^{2}\to S^{2}$, then $\operatorname{cat}(f)=1$ and
$\operatorname{cd}(f_{*})=0.$ Moreover, the epimorphism $f_{*}$ is not
nonzero. In the case of aspherical spaces, there is also the open problem that
states that there is a group epimorphism $\phi$ of finite aspherical spaces
with $\operatorname{cd}(\phi)=1$ and $\operatorname{cat}(\phi)=2.$ (More
details, see [DK, DD]). That is why we need the condition
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1$ for finite aspherical
spaces instead of just $\operatorname{cd}(\phi)=1.$

We need the product formula for cohomological dimension of group homomorphisms
to prove the main results.

######  Theorem 4.7 ([DD]).

For every homomorphism $\phi:\Gamma\to\Lambda$ of geometrically finite groups
$\operatorname{cd}(\phi\times\phi)=2\operatorname{cd}(\phi).$

######  Corollary 4.8.

For every homomorphism $\phi:\Gamma\to\Lambda$ of geometrically finite groups
$\operatorname{cd}(\phi^{n})=n\operatorname{cd}(\phi),$ where
$\phi^{n}=\phi\times\cdots\times\phi:\Gamma^{n}\to\Lambda^{n}.$

###### Proof.

The first inequality $\operatorname{cd}(\phi^{n})\leq
n\operatorname{cd}(\phi)$ follows by [DD, Lemma 6.1] and the induction on $n$.

To show the inequality $\operatorname{cd}(\phi^{n})\geq
n\operatorname{cd}(\phi).$ By induction on $k$ we obtain the equality
$\operatorname{cd}(\phi^{2^{k}})=2^{k}\operatorname{cd}(\phi)$ from Theorem
4.7.

For $n<2^{k}$ the equality
$2^{k}\operatorname{cd}(\phi)=\operatorname{cd}(\phi^{2^{k}})\leq\operatorname{cd}(\phi^{n})+\operatorname{cd}(\phi^{2^{k}-n})\leq\operatorname{cd}(\phi^{n})+(2^{k}-n)\operatorname{cd}(\phi)$
implies $n\operatorname{cd}(\phi)\leq\operatorname{cd}(\phi^{n}).$

∎

##  5\. The $D-$topological complexity introduced by Farber and Oprea

Let $\Updelta$ be the diagonal subgroup of $\Lambda^{r}.$ Let $D_{\Lambda}$ be
the minimal family of subgroups of $\Lambda^{r}$ containing the diagonal
subgroup $\Updelta$ and the trivial group, which is closed under conjugations
and taking finite intersections.

In [FO], Farber and Oprea introduced $D-$topological complexity of a path-
connected topological space. Here we modify their definition to give
$D_{\Lambda}-$topological complexity of a map $f:X\to Y$ between path-
connected topological spaces $X,Y$. Let $\pi_{1}(X,x_{0})=\Gamma$,
$\pi_{1}(Y,y_{0})=\Lambda$ and $y_{0}=f(x_{0})$.

######  Definition 5.1.

The $D_{\Lambda}-$topological complexity, $TC_{r}^{D_{\Lambda}}(f),$ is
defined as the minimal number $k$ such that $X^{r}$ is covered by k+1 open
subsets $\\{U_{0},\cdots,U_{k}\\}$ with property that for each $U_{i}$ and for
any choice of the base point $\bar{u}\in U_{i}$, the induced the homomorphism
$f_{*}^{r}|_{U_{i}}:\pi_{1}(U_{i},\bar{u})\to\pi_{1}(Y^{r},f^{r}(\bar{u}))$
takes values in a subgroup conjugate to the diagonal $\Updelta$ of
$\Lambda^{r}$.

######  Remark 5.2.

If the map is the identity map, then we recover Farber’ and Oprea’ invariant.
This definition is also equivalent to saying that the projections from
$f^{r}(Z)$ to the $j+1$ factor of $Y^{r}$ are homotopic, where $j=0,..,r-1.$

Let $X$ and $Y$ be $CW-$complexes and let $X^{(n)}$ (respectively $Y^{(n)}$)
denote the $n$-skeleton of $X$ (respectively $Y$). We recall that a continuous
map $f:X\to Y$ is said to be cellular if it takes n-skeletons of $X$ to
$n$-skeletons of $Y$, $f(X^{(n)})\subset Y^{(n)}$, for all non-negative
integers $n$. Note that we always assume that CW-complex has only one zero
cell.

We obtain the following the generalised version of Lemma 4.2 in [FO] and
Theorem 6.2 in [EFMO].

######  Theorem 5.3.

Let $f:X\to Y$ be a cellular map between finite aspherical spaces with
$\pi_{1}(X)=\Gamma,\pi_{1}(Y)=\Lambda$. Then
$TC_{r}^{D_{\Lambda}}(f)=TC_{r}(f)=\operatorname{secat}((f^{r})^{*}q)$ where
$q:\hat{Y^{r}}\to Y$ be the connected covering space corresponding to diagonal
subgroup $\Updelta_{\Lambda}$ of $\Lambda^{r}$.

###### Proof.

Since $f:X\to Y$ is a cellular map, the proof of the first equality follows
from Lemma 4.2 in [FO] and Remark 5.2.

The second equality follows from the following commutative diagram

${(f^{r})^{*}PY}$${PY}$${\hat{Y^{r}}}$${X^{r}}$${Y^{r}}$$\scriptstyle{\bar{f^{r}}}$$\scriptstyle{(f^{r})^{*}p_{r}}$$\scriptstyle{p_{r}}$$\scriptstyle{q}$$\scriptstyle{f^{r}}$

We recall that the Deck transformation $\Lambda$ acts on the universal cover
of Y, $\tilde{Y}$ [Ha]. The product of Deck transformations, $\Lambda^{r}$
acts component-wise on the product of universal covers of $Y^{r}$,
$\tilde{Y^{r}}.$ The orbit space
$\hat{Y^{r}}:=\tilde{Y^{r}}/\Updelta_{\Lambda}$ where the action of
$\Lambda^{r}$ on $\tilde{Y^{r}}$ restrict on the diagonal subgroup
$\Updelta_{\Lambda}.$

Note that the fibrations $p_{r}$ and $q$ are a fiber homotopy equivalence
since $\hat{Y^{r}}$ and $PY$ are aspherical spaces. Thus, the pull-back
fibrations $(f^{r})^{*}p_{r}$ and $(f^{r})^{*}q$ are a fiber homotopy
equivalent. Since $TC_{r}(f)=\operatorname{secat}((f^{r})^{*}p_{r})$ by
Theorem 3.2, we get the second equality.

∎

######  Remark 5.4.

By Theorem 5.3, we always have the following inequalities
$TC_{r}^{D_{\Lambda}}(f)\leq TC_{r}(f)$ for any map of cell complexes. This
inequality can be strict in general. For example, take Hopf bundle $S^{1}\to
S^{3}\overset{h}{\to}S^{2}.$ Since spaces are simply-connected,
$TC_{r}^{D_{\Lambda}}(h)=0$. However, $TC_{r}(h)\geq 1$ since $h$ is not null-
homotopic, $\operatorname{cat}(h)=1$ and
$\operatorname{cat}(h)\leq\operatorname{cat}(h^{r-1})\leq TC_{r}(h)$ by
Proposition 3.3.

##  6\. Group homomorphisms

Due to the homotopy invariance of classifying spaces, there is a one-to-one
correspondence between group homomorphism $\phi:\Gamma\to\Lambda$ and homotopy
classes of maps $B\phi:B\Gamma\to B\Lambda$ that induce $\phi$ on the
fundamental group. Hence, the following definition is well-defined.

######  Definition 6.1.

Let $\phi:\Gamma\to\Lambda$ be a homomorphism.

  1. (1)

The LS-category of $\phi$, $\operatorname{cat}(\phi),$ is defined to be
$\operatorname{cat}(B\phi).$

  2. (2)

The sequential topological complexity of $\phi$, $TC_{r}(\phi),$ is defined to
be $TC_{r}(B\phi).$

###  6.1. Reduction to epimorphisms

Let $\phi:\Gamma\to\Lambda$ be a nonzero homomorphism. Let us denote
$\pi:=\phi(\Gamma)$, $\psi:=\phi:\Gamma\to\pi$, and $i:\pi\to\Lambda$ is the
inclusion homomorphism.

With Dranishnikov [DK], we proved that
$\operatorname{cat}(\phi)=\operatorname{cat}(\psi)$,
$\operatorname{cat}(i)=\operatorname{cat}(\pi),$
$\operatorname{cd}(\phi)=\operatorname{cd}(\psi)$, and
$\operatorname{cd}(i)=\operatorname{cd}(\pi).$ However, these phenomena do not
hold for the sequential topological complexity of group homomorphisms in the
general. Thus, we prove them under some assumption on groups.

######  Theorem 6.2.

Let $\phi:\Gamma^{\prime}\to\Gamma$ be an injective homomorphism and $\Gamma$
is an abelian group. Then $TC_{r}(\phi)=TC_{r}(\Gamma^{\prime}).$

###### Proof.

By Proposition 3.3, we get $TC_{r}(\phi)\leq TC_{r}(\Gamma^{\prime}).$ To
prove other direction, we use the Ganea-Schwarz characterisation of the
sequential topological complexity. Note that
$TC_{r}(\phi)=\operatorname{secat}((\phi^{r})^{*}u_{\Updelta})$ and
$TC_{r}(\Gamma^{\prime})=\operatorname{secat}(u^{\prime}_{\Updelta}).$
Existing lift $\tilde{\phi^{r}}$ of map $\phi^{r}$ with respect to
$u_{\Updelta}$ gives a section $s$ of $u_{\Updelta},$ i.e.
$\tilde{\phi^{r}}=s\phi^{r}.$ Because of pull-back diagram, this section $s$
is also the section of $(\phi^{r})^{*}u_{\Updelta}.$ Hence, we conclude that
the composition of this section $s$ and the dash arrow is a section of
$u^{\prime}_{\Updelta}.$

${B\Updelta_{\Gamma^{\prime}}}$${(\phi^{r})^{*}B\Updelta_{\Gamma}}$${B\Updelta_{\Gamma}}$${B(\Gamma^{\prime})^{r}}$${B\Gamma^{r}}$$\scriptstyle{u^{\prime}_{\Updelta}}$$\scriptstyle{\bar{\phi^{r}}}$$\scriptstyle{(\phi^{r})^{*}u_{\Updelta}}$$\scriptstyle{u_{\Updelta}}$$\scriptstyle{\phi^{r}}$

Now it is left to prove the dash arrow exists. This is equivalent to the
following lifting problem
$(\phi^{r})^{*}u_{\Updelta}:(\phi^{r})^{*}B\Updelta_{\Gamma}\to
B(\Gamma^{\prime})^{r}$ for the given diagonal cover
$u^{\prime}_{\Updelta}:B(\Updelta_{\Gamma^{\prime}})\to
B(\Gamma^{\prime})^{r}.$ Note that $B(\Updelta(\Gamma)^{\prime})$ is homotopy
equivalent to $B\Gamma^{\prime}$. By lifting criterion in [Ha], the lift
exists if and only if $(\phi^{r})^{*}(\Gamma)$ is subgroup of
$\Updelta_{\Gamma^{\prime}}$. Thus, the dashed arrow exists since the
condition trivially satisfies when $\Gamma$ is the abelian group.

∎

###  6.2. A group epimorphism

We use the notation $T(A)$ for the torsion subgroup of an abelian group $A$.
For finitely generated abelian groups we define the rank
$rank(A)=rank(A/T(A))$.

We recall the Invariant Factor Decomposition for finite abelian group $T(A)$.

######  Theorem 6.3 (Invariant Factor Decomposition (IDF) for Finite Abelian
Groups).

Every finite abelian group $T(A)$ can be written uniquely as
$T(A)=Z_{n_{1}}\times...\times Z_{n_{k}}$ where the integers $n_{i}\geq 2$ are
the invariant factors of $T(A)$ that satisfy $n_{1}|n_{2}|...|n_{k}$ and
${\mathbb{Z}}_{n_{i}}$ are cyclic group of order $n_{i},i=1,\cdots,k.$

######  Definition 6.4.

Given a finite abelian group $T(A)$, the Smith Normal number $k(T(A))$ of
$T(A)$ is the number $k$ from Theorem 6.3.

######  Theorem 6.5.

Let $\phi:\Gamma\to\Lambda$ be an epimorphism of finitely generated abelian
groups. Then

$TC_{r}(\phi)=(r-1)\operatorname{cd}(\phi).$

In particular, if $\Gamma$ is free abelain groups, then
$TC_{r}(\phi)=(r-1)\operatorname{cd}(\phi)=(r-1)(\operatorname{rank}(\Lambda)+k(T(\Lambda)))$
where $k(T(\lambda))$ is the Smith Normal number for given finite abelian
group $T(\Lambda).$

###### Proof.

Since the groups $\Gamma$ and $\Lambda$ are finitely generated abelian groups,
we may assume $\Gamma={\mathbb{Z}}^{n}\oplus T(\Gamma)$ and
$\Lambda={\mathbb{Z}}^{m}\oplus T(\Lambda)$ for some $m$ and $n$.

By Theorem 2 in [Ku1], if both groups $\Gamma,\Lambda$ have torsion and
$\phi(T(\Gamma))\neq 0$, then $\operatorname{cd}(\phi)=\infty.$

Since $\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)$ and
$\operatorname{cat}(\phi)\leq TC_{r}(\phi),$ we get $TC_{r}(\phi)=\infty$ and
equality holds trivially.

Thus, we consider $\phi$ with $\phi(T(\Lambda))=0$. Such $\phi$ factors
through the epimorphism $\bar{\phi}:{\mathbb{Z}}^{n}\to\Lambda$. In view of
the retraction $B\Gamma\to B{\mathbb{Z}}^{n}$, we obtain that
$\operatorname{cd}(\bar{\phi})=\operatorname{cd}(\phi)$ and
$TC_{r}(\bar{\phi})=TC_{r}(\phi)$ by Corollary 3.9. Therefore, we may assume
that $\Gamma={\mathbb{Z}}^{n}$.

Since $B{\mathbb{Z}}^{n}$ is an $n$ dimensional torus (Topological group), we
apply Theorem 3.5 . Hence, we get that
$TC_{r}(\phi)=\operatorname{cat}(\phi^{r-1}).$ By Theorem 1 in [Ku1], the
analogue of Eilenberg and Ganea Theorem holds for finitely generated abelian
groups, i.e. $\operatorname{cat}(\phi^{r-1})=\operatorname{cd}(\phi^{r-1}).$

By Corollary 4.8, $\operatorname{cd}(\phi^{r-1})=(r-1)\operatorname{cd}(\phi)$
and by Theorem 2 in [Ku1],
$\operatorname{cd}(\phi)=\operatorname{rank}(\Lambda)+k(T(\Lambda))$

Combining these all equalities, we get the theorem:

$TC_{r}(\phi)=\operatorname{cat}(\phi^{r-1})=\operatorname{cd}(\phi^{r-1})=(r-1)\operatorname{cd}(\phi)=(r-1)(rank(\Lambda)+k(T(\Lambda))).$
∎

######  Remark 6.6.

Note that even $r=2,$ this theorem is the generalized result of [Sc, Theorem
6.12 and Colollary 6.13].

######  Theorem 6.7.

Let $\phi:\Gamma\to\Lambda$ be a nonzero epimorphism of free groups. Then

$TC_{r}(\phi)=\begin{cases}r-1&\mbox{if}\,\,\,\Lambda\cong{\mathbb{Z}}\\\
r&\mbox{if}\,\,\,\Lambda\cong F_{m},m>1\end{cases}$

###### Proof.

We prove in two steps:

Step 1. Suppose $\Lambda\cong{\mathbb{Z}}.$

The upper bound: By Proposition 3.3, $TC_{r}(\phi)\leq TC_{r}(S^{1})=r-1$
since $S^{1}=K({\mathbb{Z}},1)$ and $TC_{r}(S^{1})=r-1$ by [Rud].

The lower bound: By Proposition 3.3 $\operatorname{cat}(\phi^{r-1})\leq
TC_{r}(\phi)$. Since
$\operatorname{cd}(\phi^{r-1})=(r-1)\operatorname{cd}(\phi)$ by Corollary 4.8
and $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1$ by Corollary 4.4, we
get that $r-1\leq\operatorname{cat}(\phi^{r-1}).$ Hence, we obtain that
$TC_{r}(\phi)=r-1.$

Step 2. $\Lambda\cong F_{m}$ where $F_{m}$ is the free group with $m>1$
generators. Let $\Gamma\cong F_{n}$ where $n\geq m$ since $\phi$ is
surjective.

The upper bound: Similar to Step 1, $TC_{r}(\phi)\leq r$ since
$TC_{r}(F_{n})=r$ by [FO].

The lower bound: We apply the Theorem 3.7 to show that $TC_{r}(\phi)\geq r.$

We first recall that the canonical cohomological class

$\alpha_{(F_{m})}\in H^{1}(BF_{m}^{r};I(F_{m}^{r-1}))$

is given by a crossed homomorphism $f_{r,m}:F_{m}^{r}\to I(F_{m}^{r-1})$ such
that
$f_{r,m}(g_{1},g_{2},\cdots,g_{r})=(g_{1}g_{2}^{-1}-1,g_{2}g_{3}^{-1}-1,,\cdots,g_{r-1}g_{r}^{-1}-1)$
in [EFMO]. Note that the crossed homomorphism $f_{r,m}$ restricted the
diagonal $\Updelta_{F_{m}}$ of $F_{m}^{r}$ is trivial, so
$\alpha_{(F_{m})}\in\ker(\Updelta_{F_{m}}^{*}).$

Similarly to Corollary 4.4, the canonical cohomological classes of
$\alpha_{(F_{m})}\in H^{1}(BF_{m}^{r};I(F_{m}^{r-1}))$ and
$\alpha_{(F_{n})}\in H^{1}(BF_{n}^{r};I(F_{n}^{r-1}))$ are related. Indeed, we
have the following commutative diagrams:

${F_{n}^{r}}$${I(F_{n}^{r-1})}$${F_{m}^{r}}$${I(F_{n}^{r-1})}$$\scriptstyle{f_{r,n}}$$\scriptstyle{\phi^{r}}$$\scriptstyle{I(\phi^{r-1})}$$\scriptstyle{f_{r,m}}$$\scriptstyle{s^{r}}$$\scriptstyle{I(s^{r-1})}$

where $s:F_{m}\to F_{n}$ is a section for the epimorphism $\phi:F_{n}\to
F_{m},$ $I(s^{r-1}),I(\phi^{r-1}),$ $s^{r}$, and $\phi^{r}$ are the obvious
morphisms.

Since $\alpha_{(F_{m})}\neq 0$, we obtain that
$(\phi^{r})^{*}:H^{1}(BF_{m}^{r};I(F_{m}^{r-1}))\to
H^{1}(BF_{n}^{r};I(F_{m}^{r-1}))$ is not trivial homomorphism,i.e.
$(\phi^{r})^{*}\alpha_{(F_{m})}\neq 0.$

Note that $(\alpha_{(F_{m})})^{r}\neq 0$ in
$H^{r}(BF_{m}^{r};I(F_{m}^{r-1})^{\otimes r})$ where

$I(F_{m}^{r-1})^{\otimes r}=I(F_{m}^{r-1})\otimes
I(F_{m}^{r-1})\otimes\cdots\otimes I(F_{m}^{r-1}).$

Thus,

$(\phi^{r})^{*}(\alpha_{(F_{m})})^{r}\neq 0.$

By Theorem 3.7, we obtain that $r\leq TC_{r}(\phi).$ Hence, we conclude that
$TC_{r}(\phi)=r$ when $\Lambda\cong F_{m}$ and $m>1.$

∎

We extend the results for other discrete groups using author’s previous work
on LS-category of group homomorphisms [DK, Ku1, Ku2] and Corollary 4.8, so we
give the sharp estimates. This phenomena is expected since the exact value of
the sequential topological complexity of discrete group $\Gamma,$
$TC_{r}(\Gamma)$ is known for a few classes of groups (See [FO, EFMO, BGRT,
Rud]).

######  Theorem 6.8.

Let $\phi:\Gamma\to\Lambda$ be an epimorphism of surface groups. Then

$r-1\leq TC_{r}(\phi)\leq
r\,\,\,\,\textit{if}\,\,\,\,\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1.$

Otherwise, $2(r-1)\leq TC_{r}(\phi)\leq 2r$ if
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=2.$

###### Proof.

By Proposition 3.3, we obtain that $\operatorname{cat}(\phi^{r-1})\leq
TC_{r}(\phi)$ Since
$\operatorname{cd}(\phi^{r-1})\leq\operatorname{cat}(\phi^{r-1})$ and by
Corollary 4.8, we obtain that
$\operatorname{cd}(\phi^{r-1})=(r-1)\operatorname{cd}(\phi)=r-1$ where
$\operatorname{cd}(\phi)=1.$

Since $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=1,$ by Theorem 4.5 the
epimorphism $\phi$ factors through the free groups $F_{n}$ via epimorphisms
$\Gamma\to F_{n}\to\Lambda.$ By Proposition 3.8, we obtain that

$TC_{r}(\phi)\leq\min\\{TC_{r}(h),TC_{r}(g)\\}\leq TC_{r}(F_{n})=r$

where the last equality is due [FO], $h:\Gamma\to F_{n}$ and
$g:F_{n}\to\Lambda.$

With similar above argument, we obtain that $2(r-1)\leq TC_{r}(\phi)\leq 2r$
if $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=2.$ ∎

We recall the upper central series of a group $\Gamma$ is a chain of subgroups

${e}=Z_{0}\leqslant Z_{1}\leqslant....\leqslant Z_{n}\leqslant....$

where $Z_{1}=Z(\Gamma)$ is the center of the group, and $Z_{i+1}$ is the
preimage under the canonical epimorphism $\Gamma\to\Gamma/Z_{i}$ of the center
of $\Gamma/Z_{i}$. A group $\Gamma$ is nilpotent if $Z_{n}=\Gamma$ for some
$n$.

A group $\Gamma$ is called a virtually nilpotent if it has a nilpotent
subgroup $\Gamma^{\prime}$ of finite index.

######  Theorem 6.9.

Let $\phi:\Gamma\to\Lambda$ be an epimorphisms between finitely generated,
torsion-free nilpotent (virtually nilpotent) groups. Then

$(r-1)\operatorname{cd}(\Lambda)\leq TC_{r}(\phi)\leq
r\operatorname{cd}(\Lambda).$

###### Proof.

By Proposition 3.3, we obtain that $\operatorname{cat}(\phi^{r-1})\leq
TC_{r}(\phi)\leq\operatorname{cat}(\phi^{r}).$ Since the Cartesian product of
nilpotent (virtually nilpotent) groups is nilpotent (virtually nilpotent),
Theorem 4.1 in [DK] gives the following equality,
$\operatorname{cat}(\phi^{k})=\operatorname{cd}(\phi^{k})$ for each finite k.
By Corollary 4.8, $\operatorname{cd}(\phi^{k})=k\operatorname{cd}(\phi)$ since
the groups are geometrically finite. Since
$\operatorname{cd}(\phi)=\operatorname{cd}(\Lambda)$ in Theorem 4.2 [Ku2], we
obtain the result by the combining all above equalities.

∎

We recall an almost nilpotent group is the extension group of the infinite
cyclic group by the torsion-free nilpotent groups. More explicitly, one can
think that it is the semidirect product of the torsion-free nilpotent group
with infinite cyclic group ${\mathbb{Z}}\ltimes_{\phi}\Gamma,$ where
$\phi:{\mathbb{Z}}\to Aut(\Gamma)$ is a one-parameter group.

######  Theorem 6.10.

Let $f:{\mathbb{Z}}\ltimes_{\phi}\Gamma\to{\mathbb{Z}}\ltimes_{\nu}\Lambda$ be
an epimorphism between torsion-free almost nilpotent groups such that
$f|_{{\mathbb{Z}}}=Id.$ Then

$(r-1)(\operatorname{cd}(\Lambda)+1)\leq TC_{r}(\phi)\leq
r(\operatorname{cd}(\Lambda)+1).$

###### Proof.

Similarly to Theorem 6.9, we have that $\operatorname{cat}(\phi^{r-1})\leq
TC_{r}(\phi)\leq\operatorname{cat}(\phi^{r}).$ Since
$\operatorname{cat}(f)=\operatorname{cd}(f)=\operatorname{cd}(\Lambda)+1$ in
[Ku2] and Corollary 4.8, it suffices to show
$\operatorname{cat}(f^{k})=\operatorname{cd}(f^{k}).$

Note that we always have the following inequality
$\operatorname{cd}(f^{k})\leq\operatorname{cat}(f^{k})$ (see [DK]).

The reverse inequality comes from the product formula for LS-category of maps
and Corollary 4.8. Indeed, $\operatorname{cat}(f^{k})\leq
k\operatorname{cat}(f)=k\operatorname{cd}(f)=\operatorname{cd}(f^{k})$.

∎

We would like to end the paper by asking the following questions:

######  Question 6.11.

Is the following equality $\operatorname{cd}(\phi^{r-1})\leq
TC_{r}(\phi)\leq\operatorname{cd}(\phi^{r})$ true for the epimorphisms of
geometrically finite groups?

This question is motivated by the following fact
$\operatorname{cd}(\Gamma^{r-1})\leq
TC_{r}(\Gamma)\leq\operatorname{cd}(\Gamma^{r}).$

######  Question 6.12.

What is the gap of LS-category and cohomological dimension of group
homomorphisms of geometrically finite groups?

All known examples regarding the Question 6.12 have a gap of 1 (See [DK, DD,
Gr]). Possible negative answer for Question 6.11 comes the constructing the
arbitrary large gap for Question 6.12.

## Acknowledgments

I would like to thank my advisor, Alexander Dranishnikov, for all of his help
and encouragement throughout this project.

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30’000, 60’000, 90’000, 120’000 | 66.0 | 76.0 | 22.4 | 0.4 | 0.4 | 0.2 | 0.2 | 0.0 | 0.4 | 0.0 | 0.0 | 3.10 | 3.19
100, 800, 6’400, 24’000, 123’000 | 66.0 | 68.0 | 14.8 | 6.6 | 6.0 | 1.2 | 1.4 | 0.6 | 0.0 | 0.2 | 1.2 | 35.32 | 36.18
0.75 | Gaussian | 100, 110’000 | 30.6 | 0.0 | 51.2 | 30.8 | 18.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.86 | 1.88
1’000, 110’000 | 30.4 | 0.0 | 61.0 | 30.6 | 8.4 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 2.27 | 2.34
10’000, 110’000 | 30.4 | 24.6 | 65.4 | 6.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4.74 | 4.89
50’000, 110’000 | 30.4 | 30.2 | 66.8 | 0.4 | 2.6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.87 | 1.93
100, 1’000, 116’000 | 30.4 | 0.0 | 53.8 | 0.0 | 7.4 | 30.6 | 8.2 | 0.0 | 0.0 | 0.0 | 0.0 | 2.19 | 2.25
100, 10’000, 116’000 | 30.8 | 0.0 | 51.6 | 25.0 | 13.4 | 6.0 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 6.02 | 6.02
1’000, 10’000, 116’000 | 30.4 | 0.0 | 60.8 | 25.6 | 3.8 | 5.0 | 4.8 | 0.0 | 0.0 | 0.0 | 0.0 | 6.11 | 6.14
30’000, 60’000, 116’000 | 30.6 | 30.2 | 66.2 | 0.4 | 0.8 | 0.2 | 2.2 | 0.0 | 0.0 | 0.0 | 0.0 | 2.86 | 2.87
100, 1’000, 10’000, 120’000 | 30.2 | 0.0 | 48.8 | 0.0 | 11.6 | 25.6 | 3.8 | 4.8 | 5.4 | 0.0 | 0.0 | 5.92 | 6.10
30’000, 60’000, 90’000, 120’000 | 30.4 | 29.2 | 66.0 | 1.0 | 0.8 | 0.2 | 0.6 | 0.2 | 2.0 | 0.0 | 0.0 | 2.61 | 2.69
100, 800, 6’400, 24’000, 123’000 | 30.2 | 0.0 | 48.8 | 0.0 | 10.8 | 21.2 | 3.6 | 7.2 | 2.8 | 2.0 | 3.6 | 7.24 | 7.49
Consistency | 100, 110’000 | 46.0 | 0.0 | 34.6 | 48.8 | 16.6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.39 | 1.42
1’000, 110’000 | 46.2 | 0.0 | 41.2 | 49.0 | 9.8 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.52 | 1.55
10’000, 110’000 | 46.2 | 43.8 | 45.8 | 5.2 | 5.2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 4.65 | 4.70
50’000, 110’000 | 46.4 | 48.6 | 48.2 | 0.6 | 2.6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.87 | 1.89
100, 1’000, 116’000 | 46.4 | 0.0 | 33.8 | 0.0 | 8.8 | 49.2 | 8.2 | 0.0 | 0.0 | 0.0 | 0.0 | 1.48 | 1.50
100, 10’000, 116’000 | 46.2 | 0.0 | 34.4 | 43.6 | 12.2 | 5.4 | 4.4 | 0.0 | 0.0 | 0.0 | 0.0 | 5.52 | 5.61
1’000, 10’000, 116’000 | 47.0 | 0.0 | 41.0 | 44.2 | 5.2 | 5.6 | 4.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5.52 | 5.60
30’000, 60’000, 116’000 | 46.4 | 47.6 | 47.8 | 1.2 | 0.4 | 0.4 | 2.6 | 0.0 | 0.0 | 0.0 | 0.0 | 2.75 | 2.79
100, 1’000, 10’000, 120’000 | 46.4 | 0.0 | 32.4 | 0.0 | 8.2 | 44.2 | 5.2 | 5.0 | 5.0 | 0.0 | 0.0 | 5.32 | 5.40
30’000, 60’000, 90’000, 120’000 | 46.8 | 47.6 | 47.0 | 1.0 | 1.4 | 1.0 | 0.0 | 0.0 | 2.0 | 0.0 | 0.0 | 2.60 | 2.64
100, 800, 6’400, 24’000, 123’000 | 46.4 | 0.0 | 33.6 | 0.0 | 7.2 | 41.0 | 4.4 | 6.0 | 1.4 | 2.2 | 4.2 | 6.37 | 6.47

##### Ablation of $\bm{\alpha}$ and $\bm{\beta}$

(a)

(b)

(c)

Figure 16: Expected probability of certifying (a), abstaining (b) or
continuing to the next phase (c) of CertifyAdp after the first phase,
conditioned on the underlying probability of the majority class. We consider
various $\alpha$ and $\beta$. We are in the setting with
$r=\sigma_{\epsilon}=0.25$ with stage sizes
$n_{j}=\\{100,1^{\prime}000,10^{\prime}000,120^{\prime}000\\}$, i.e. the first
phase whose simulated outcomes we observe here has $100$ samples.

In Fig. 16, we illustrate the effect of different $\alpha$ and $\beta$ on the
first phase of certification with CertifyAdp where we consider just 100
perturbations. Increasing $\alpha$ reduces the confidence $1-\alpha$ required
for certification and hence makes early certification more likely for the same
underlying true success probabilities (see Fig. 16(a)). Increasing $\beta$
reduces the confidence $1-\beta$ that certification will not be possible
required for early termination and hence makes early termination more likely
for the same underlying true success probabilities (see Fig. 16(b)). Overall,
we observe that evaluating even only $100$ perturbations allows us to, with
high probability, abstain from or certify samples with a wide range of true
success probabilities. We only have to continue with the costly certification
process for samples in a narrow band of true success probabilities (see Fig.
16(c)), greatly reducing the expected certification cost.

####  H.4.2 Additional K-Consensus Experiments

In Table 25, we present a more detailed version of Table 5. We emphasize that
for $K=5$ and $K=10$ for ResNet110 and ResNet20, respectively, we achieve the
same ACR as with the full ensemble while reducing the certification time by a
factor of $1.59$ and $2.01$, respectively.

Table 25: Effect of $K$-Consensus aggregation on Consistency trained ensembles
of 10 ResNet110 and 50 ResNet20 on CIFAR10 at $\sigma_{\epsilon}=0.25$.

Architecture | $K$ | ACR | Radius r | TimeRF | KCR [%]
---|---|---|---|---|---
0.0 | 0.25 | 0.50 | 0.75
ResNet110 | 2 | 0.576 | 77.2 | 70.2 | 60.0 | 50.4 | 3.25 | 85.8
3 | 0.581 | 77.0 | 70.0 | 60.6 | 51.6 | 2.29 | 79.7
5 | 0.583 | 76.8 | 70.4 | 60.4 | 51.6 | 1.59 | 74.2
10 | 0.583 | 76.8 | 70.4 | 60.4 | 51.6 | 1.00 | 0.0
ResNet20 | 2 | 0.544 | 72.2 | 65.2 | 57.0 | 48.4 | 6.50 | 87.7
3 | 0.549 | 72.6 | 65.0 | 57.4 | 50.0 | 4.41 | 82.4
5 | 0.550 | 72.8 | 65.0 | 57.0 | 50.2 | 2.99 | 76.4
10 | 0.551 | 73.0 | 64.8 | 57.0 | 50.2 | 2.01 | 69.8
50 | 0.551 | 73.0 | 64.8 | 57.0 | 50.2 | 1.00 | 0.0


# IR Models and the COVID-19 Pandemic: A Comparative Study of Performance and
Challenges

Moksh Shukla
Indian Institute of Technology Kanpur
Kanpur, India
<EMAIL_ADDRESS>
&Nitik Jain
Indian Institute of Technology Kanpur
Kanpur, India
<EMAIL_ADDRESS>
&Shubham Gupta
Indian Institute of Technology Kanpur
Kanpur, India
<EMAIL_ADDRESS>

###### Abstract

This research study investigates the efficiency of different information
retrieval (IR) systems in accessing relevant information from the scientific
literature during the COVID-19 pandemic. The study applies the TREC framework
to the COVID-19 Open Research Dataset (CORD-19) and evaluates BM25,
Contriever, and Bag of Embeddings IR frameworks. The objective is to build a
test collection for search engines that tackle the complex information
landscape during a pandemic. The study uses the CORD-19 dataset to train and
evaluate the IR models and compares the results to those manually labeled in
the TREC-COVID IR Challenge. The results indicate that advanced IR models like
BERT and Contriever better retrieve relevant information during a pandemic.
However, the study also highlights the challenges in processing large datasets
and the need for strategies to focus on abstracts or summaries. Overall, the
research highlights the importance of effectively tailored IR systems in
dealing with information overload during crises like COVID-19 and can guide
future research and development in this field.

_K_ eywords Information Retrieval Systems $\cdot$ COVID-19 $\cdot$ TREC
framework

##  1 Introduction

Accessing the scientific literature through the most effective information
retrieval (IR or search) technologies is crucial for locating evidence. Hence,
rapid implementation of IR systems optimised for such a context and a
comparison of their efficacy were required. In consequence, the COVID-19
global health crisis has exacerbated the issue. The immediate need for
information on a worldwide scale has resulted in an exponential increase in
scientific literature publication. Relevant and dependable information has
become an urgent necessity.

Concerning the application of IR in such a pandemic, there are numerous
fundamental research concerns that must be answered, including the
identification of essential IR modalities, the development of domain-specific
search engines, and the quantitative evaluation of search engine performance.
In an effort to combat the pandemic, massive COVID-19-related corpora are
being compiled, often containing erroneous data that is no longer amenable to
human analysis.

The challenge evaluation is a standard method for evaluating IR systems on a
wide scale. The Text Retrieval Conference (TREC), organised by the US National
Institute of Standards and Technology, is the largest and most well-known
approach (NIST). The TREC framework was applied to the COVID-19 Open Research
Dataset (CORD-19), a dynamic repository of scientific papers on COVID-19 and
historical coronavirus research connected to COVID-19. The dataset is
described in full under Section corpus. The major objective of the TREC-COVID
Competition was to develop a collection of test data for evaluating search
engines’ ability to navigate the complicated information landscape during
events such as a pandemic. In this research, we evaluate various Information
Retrieval frameworks, including BM25, Contriever, Bag of Embeddings, etc.,
based on their ability to rank documents according to their relevance to input
queries.

In this study, we begin by utilising document data from the CORD-19 dataset
and pre-processing it with its metadata. We opted to examine several IR models
because the hand classified data categorising documents as relevant, slightly
relevant, or irrelevant was unavailable. Our baseline model is BM25, which
compares the relevance of searches to the document by utilising the Abstract
and Title. Additionally, we retrieve the list of relevant documents from other
faster models and compare them to the standard documents. Finally, we compare
our results to those manually labelled according to the TREC-COVID IR
Challenge.

##  2 Related Work

The COVID-19 pandemic has led to a global surge in scientific literature, and
the need for efficient information retrieval (IR) systems has become more
critical than ever. The CORD-19 dataset, a large-scale open access resource of
scientific papers on COVID-19 and related historical coronavirus research, has
emerged as a valuable resource for COVID-19 research. As a result, several
studies have been conducted to explore the use of IR systems and natural
language processing (NLP) and machine learning (ML) techniques to improve IR
performance on the CORD-19 dataset.

One notable study conducted by [1] developed the CORD-19 dataset, which has
been widely used in COVID-19 research. Other studies have used NLP and ML
techniques to improve IR performance on the CORD-19 dataset. For example, [2]
developed a deep learning-based text mining framework for COVID-19 literature,
while [3] used unsupervised topic modeling and machine learning techniques to
analyze the COVID-19 "infodemic" using the CORD-19 dataset.

In addition to improving IR performance, the CORD-19 dataset has been used for
drug repurposing and knowledge graph construction. [1] developed a COVID-19
literature knowledge graph construction and drug repurposing report generation
system using the CORD-19 dataset. [4] used the CORD-19 dataset to mine the
literature for COVID-19 research and identify potential drug candidates for
further investigation.

Studies have also evaluated the CORD-19 dataset and its use in IR systems. [5]
provided an overview of the CORD-19 dataset and its potential uses, while [3]
applied domain adaptation techniques to improve IR performance on the CORD-19
dataset. Overall, the CORD-19 dataset has been widely used and evaluated in
the context of COVID-19 research, particularly in the development and
evaluation of IR systems using NLP and ML techniques, as well as for drug
repurposing and knowledge graph construction.

The COVID-19 pandemic has highlighted the need for rapid implementation and
evaluation of IR systems. The CORD-19 dataset has emerged as a valuable
resource for COVID-19 research and has been extensively studied to improve IR
performance, analyze the "infodemic," and identify potential drug candidates.
As such, the CORD-19 dataset has become a crucial component of COVID-19
research, with many studies evaluating its effectiveness and exploring its
potential uses.

In conclusion, the CORD-19 dataset has been widely used and evaluated in the
context of COVID-19 research, particularly in the development and evaluation
of IR systems using NLP and ML techniques, as well as for drug repurposing and
knowledge graph construction. The dataset has provided valuable insights and
has played a significant role in advancing COVID-19 research.

##  3 Document Set Description

TREC-COVID makes use of the CORD-19 document collection. CORD-19 contains
fresh articles and preprints on COVID-19, as well as previous studies on
coronaviruses such as SARS and MERS. The CORD-19 release on April 10, 2020,
which will be utilised for the first round of TREC-COVID, has 51K papers, with
full text available for 39K. We train and evaluate our models using whole
51000 documents.

###  3.1 File Structure:

  1. 1.

CORD-19 \- folder holding the CORD-19 papers and metadata as of May 19, 2020.

  2. 2.

topics-rnd3.csv \- file containing each topic’s topic-id, query, question, and
narrative.

  3. 3.

docids-rnd3.txt \- the set of documents that can be projected to be relevant;
these papers do not include those that have been rated in prior rounds for a
topic.

  4. 4.

qrels.csv \- human annotated relevant and irrelevant document categorization
for the 40 queries.

###  3.2 Topics:

TREC-COVID topics were written by its organisers with biomedical training, and
were inspired by consumer questions submitted to the National Library of
Medicine, discussions by medical influencers on social media, and suggestions
solicited on Twitter in late March 2020 via the #COVIDSearch tag. They are
representative of the pandemic’s high-level concerns. An initial set of 30
subjects was established, with 5 new topics added for each subsequent round.
As a result, we ran our retrieval on 40 topics, which can be accessed in the
topics-rnd3.csv file.

The subject file is an xml file that contains all of the round’s topics. A
topic is formatted as follows (this is an example topic, not part of the
official topic set):

Each topic is composed of three fields:

  1. 1.

query: a short keyword query

  2. 2.

question: a more precise natural language question

  3. 3.

narrative: a more detailed description that expands on the question,
frequently specifying specific types of facts that would fall under the topic
score

  4. 4.

average: in addition, we generate another set of questions that are the
average of the aforementioned three embeddings. This is a frequently used and
acknowledged technique in NLP.

Query | Question | Narrative
---|---|---
| Coronavirus
---
response to
weather changes
| How does the coronavirus
---
respond to changes in the
weather?
| Seeking a variety of information about virus survival in
---
various weather/climate settings, as well as
information about virus transmission in
various climatic circumstances.
| Coronavirus
---
social distancing
impact
| Has social distancing had
---
an impact on slowing the
spread of COVID-19?
| Seeking particular information on studies that have
---
investigated COVID-19 transmission in one or more
social (or non-social)
techniques.
| Coronavirus
---
outside body
| How long can the
---
coronavirus live outside the
body?
| Seeking a variety of information about the virus’s survivability
---
outside the human body (surfaces, liquids, etc.)
while remaining alive for
transmission to another human.
| coronavirus
---
asymptomatic
| What is known about those
---
infected with Covid-19 but
are asymptomatic?
| Studies on patients who are known to be Covid-19 infected
---
but show no symptoms?
| Coronavirus
---
hydroxy-
chloroquine
| What evidence is there for
---
the value of
hydroxychloroquine in
treating Covid-19?
| Basic scientific or clinical research evaluating the
---
benefits and risks of treating Covid-19 with
hydroxychloroquine.
Table 1: Few illustrative examples of topics for TREC-COVID task.

##  4 Methodology

###  4.1 Pre-processing

In the data-set, for each document, we are provided with two types of JSON
files: pdf_json and pmc_json, which differ only in their format. For
simplicity, we used only the pdf_json form. We first cleared the data set from
NaN values. Then we extracted the abstract of each document and stored it
separately along with the document ID. Later, we used this data for various
tasks detailed in his report.

For the BM25 model as well as the Bag of Embeddings model the pre-processing
done was very similar to the one we did in our class assignments. The corpus
consisted of .json parsed files for each research article as mentioned above
and we leveraged that to read the abstract, title and the body text as well
(only for BM25), created a single string for the whole article and moved ahead
with the pre-processing and cleaning part. The pre-processing and the cleaning
involved the following things:-

  * •

Lower Casing

  * •

Email and url removal

  * •

Non ASCII removal

  * •

Special Characters removal

  * •

Stop words removal

  * •

Stripping extra white space and stemming

  * •

Tokenizing

For the queries also same pre-processing was done before running the query

###  4.2 Models

We implemented and compared the following IR models:

####  4.2.1 BM25 - The Baseline Model [6]

A type of ranking function that ranks a set of documents based on the query
terms that exist in each document, independent of how the query terms inside a
document are related. The BM25 term weighting algorithms have been widely and
successfully employed in a variety of collections and search activities.

The ranking function: BM25 is a bag-of-words retrieval algorithm that ranks a
set of documents based on the query phrases that exist in each document,
regardless of how close they are to each other. It refers to a group of
scoring functions that have somewhat varied components and parameters. The
following is one of the more notable instantiations of the function.

Given a query $Q$, containing keywords $q_{1},\ldots,q_{n}$, the BM25 score of
a document $D$ is:

$\operatorname{score}(D,Q)=\sum_{i=1}^{n}\operatorname{DF}\left(q_{i}\right)\cdot\frac{f\left(q_{i},D\right)\cdot\left(k_{1}+1\right)}{f\left(q_{i},D\right)+k_{1}\cdot\left(1-b+b\cdot\frac{|D|}{\mathrm{svgdl}}\right)}$

where $f\left(q_{i},D\right)$ is $q_{i}$ ’s term frequency in the document
$D,|D|$ is the length of the document $D$ in words, and avgdl is the average
document length in the text collection from which documents are drawn. $k_{1}$
and $b$ are free parameters, usually chosen, in absence of an advanced
optimization, as $k_{1}\in[1.2,2.0]$ and
$b=0.75\mathrm{IDF}\left(q_{i}\right)$ is the IDF (inverse document frequency)
weight of the query term $q_{i}$. It is usually computed as:

$\operatorname{IDF}\left(q_{i}\right)=\ln\left(\frac{N-n\left(q_{i}\right)+0.5}{n\left(q_{i}\right)+0.5}+1\right)$

where $N$ is the total number of documents in the collection, and
$n\left(q_{i}\right)$ is the number of documents containing $q_{i}$.

####  4.2.2 Contriever [7]

Neural network-based information retrieval has achieved state-of-the-art
performance on datasets and problems where extensive training sets are
available. Unfortunately, they do not translate well to new domains or
applications without training data and are frequently surpassed by
unsupervised term-frequency approaches like BM25. Contriever is a basic self-
supervised IR model based on contrastive learning that is competitive with
BM25.

Contrastive Learning: Contrastive learning is a method that takes advantage of
the fact that every document is distinctive in some way. This signal is the
only information provided in the absence of manual monitoring. Using a
contrastive loss (citecontrastive), the final algorithm learns by
differentiating between documents. This loss compares pairs of document
representations that are either positive (from the same document) or negative
(from distinct documents). Formally, the contrastive textttInfoNCE loss is
defined as follows:

$\mathcal{L}\left(q,k_{+}\right)=\frac{\exp\left(s\left(q,k_{+}\right)/\tau\right)}{\sum_{i=0}^{K}\exp\left(s\left(q,k_{i}\right)/\tau\right)},$

This loss increases the relevance value of comparable examples and decreases
the relevance score of dissimilar ones. This loss function can also be
interpreted as follows: given the query representation $q$, the objective is
to recover or retrieve the representation $k_{+}$ corresponding to the
positive document from among all the negatives $k_{i}$. The left-hand side
representations in the score $s$ are referred to as questions, whereas the
right-hand side representations are referred to as keys.

Building positive pairings from a single document: A fundamental feature of
contrastive learning is how to construct positive pairs from a single input.
This stage in computer vision is applying two independent data augmentations
to the same image, yielding two "views" that constitute a positive pair. While
contriever primarily considers similar independent text modifications, they
also investigate dependent transformations that aim to lessen the correlation
across perspectives. They are:

  1. 1.

Inverse Cloze Task

  2. 2.

Independent cropping

  3. 3.

Further data augmentation.

Building large sets of negative pairs: The maintenance of a high number of
negative pairs is a crucial feature of contrastive learning. The majority of
common frameworks handle negatives differently, and the two utilised here are:

  1. 1.

Negative pairs within a batch.

  2. 2.

Negative pairs across a batch.

Figure 1: BERT input representation. The input embeddings are the sum of the
token embeddings, the segmentation embeddings and the position embeddings

####  4.2.3 Bag-of-Embeddings [8]

Instead of a single point in the semantic space, a word is represented as a
probability distribution that encompasses the complete semantic area. It
corresponds specifically to the multivariate normal distribution with
spherical covariance. Given a second embedded word, the "overlap" between
these two distributions can be used to determine the degree of similarity
between the two words.

####  4.2.4 BERT Embeddings [9]

Google created the state-of-the-art Bidirectional Encoder Representation from
Transformers (BERT) technique for natural language processing pre-training.
Our corpus will be embedded with a BERT embedding that has been pre-trained.

##  5 Results

In this section of our study paper, we compare the retrieval performance of
three distinct models, namely BERT, Contriever, and BM25. To evaluate the
models, the top 50 most relevant documents were extracted for each of the 40
questions in the dataset using each of the three models. This procedure
generated three distinct lists of document IDs for each query, one list for
each model.

To establish a fair comparison across the models, we determined the average
score for the extracted documents, as described in the section on methodology.
We next did a pairwise intersection of the document ID lists from the three
models for each query. This analysis generated three unique pairs: BERT and
BM25, BERT and Contriever, and BM25 and Contriever.

The amount of shared documents obtained for each query by each pair of models
provides insight into the degree of overlap between the pages deemed relevant
by each pair. Finally, we computed the mean and standard deviation of the
number of common documents across all queries for each pair in order to
standardise the results and gain a deeper understanding of the consistency and
agreement across the models in terms of document retrieval performance.

The following table summarizes our findings:

Model Comparison | BERT-Contriever | BERT-BM25 | BM25-Contriever
---|---|---|---
Mean | 36.725 | 18.1 | 11.775
Standard Deviation | 17.927 | 13.647 | 11.014
Table 2: Summary of the results comparing the retrieval performance of BERT,
Contriever, and BM25

The mean and standard deviation are greatest for the BERT-Contriever pair,
followed by the BERT-BM25 pair, and then the BM25-Contriever pair. This
indicates that BERT and Contriever have the greatest degree of agreement about
document retrieval, but BM25 has the least overlap with both BERT and
Contriever. By analysing the mean and standard deviation of the number of
common documents obtained by each pair of models, we can draw inferences
regarding the similarity and dissimilarity of their performance in obtaining
relevant documents in our research topic.

##  6 Conclusion

In conclusion, our study emphasises the significance of information retrieval
systems for conveniently gaining access to essential scientific publications
during a pandemic. The application of the TREC framework to the COVID-19 Open
Research Dataset (CORD-19) allowed for the evaluation and comparison of
various information retrieval (IR) frameworks such as BM25, Contriever, and
Bag of Embeddings, with the primary objective of constructing a test
collection for search engines dealing with the complex information landscape
during events such as a pandemic.

The CORD-19 dataset, consisting of publications and preprints on COVID-19 and
relevant historical coronavirus research, was utilised to train and assess the
IR models. Our methodology entailed preprocessing document data, utilising
metadata and abstracts to compare the performance of several IR models, with
BM25 serving as the baseline. In addition, the outcomes were compared to those
manually labelled in the TREC-COVID IR Challenge.

According to the results of our investigation, the performance of the IR
models varied, with BERT and Contriever exhibiting the most document overlap,
followed by BERT-BM25 and BM25-Contriever. These findings demonstrate the
capacity of advanced IR models such as BERT and Contriever to retrieve
pertinent information during a pandemic. However, the study also revealed the
difficulties associated with processing huge datasets due to limited computer
resources, necessitating the employment of tactics such as concentrating on
abstracts and summaries. This study highlights the significance of well-
tailored IR systems for managing the information deluge during crises such as
the COVID-19 pandemic. This study’s findings can influence future research and
development of IR systems in order to enhance their performance in quickly
developing situations, hence facilitating decision-making and response
efforts.

##  7 Challenges and Future Work

Our main challenge was the availability of adequate computational resources to
process the vast dataset. Despite our best efforts, including taking only the
abstract and summary, using a summarised version, or limiting the number of
documents processed, we were constrained by the limited computational
resources. Moreover, we encountered another challenge when utilizing Sentence-
BERT on an entire paragraph. To address this issue, we initially employed
Pegasus, followed by S-BERT.

In future work, we aim to explore more advanced algorithms and hybrid models
to enhance retrieval performance. Our current approach relies on vanilla
models for all three tasks. Additionally, fine-tuning BERT specifically for
the COVID-19 dataset represents another promising avenue for future research.

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