# On Sequential Versions of Distributional Topological Complexity

Ekansh Jauhari  Ekansh Jauhari, Department of Mathematics, University of
Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA.
<EMAIL_ADDRESS>
###### Abstract.

We define a (non-decreasing) sequence $\\{\operatorname{{\rm
dTC}}_{m}(X)\\}_{m\geq 2}$ of higher versions of distributional topological
complexity ($\operatorname{{\rm dTC}}$) of a space $X$ introduced by
Dranishnikov and Jauhari [DJ]. This sequence generalizes $\operatorname{{\rm
dTC}}(X)$ in the sense that $\operatorname{{\rm
dTC}}_{2}(X)=\operatorname{{\rm dTC}}(X)$, and is a direct analog to the
classical sequence $\\{\operatorname{{\rm TC}}_{m}(X)\\}_{m\geq 2}$. We show
that like $\operatorname{{\rm TC}}_{m}$ and $\operatorname{{\rm dTC}}$, the
sequential versions $\operatorname{{\rm dTC}}_{m}$ are also homotopy
invariants. Also, $\operatorname{{\rm dTC}}_{m}(X)$ relates with the
distributional LS-category ($d\operatorname{cat}$) of products of $X$ in the
same way as $\operatorname{{\rm TC}}_{m}(X)$ relates with the classical LS-
category ($\operatorname{cat}$) of products of $X$. We show that in general,
$\operatorname{{\rm dTC}}_{m}$ is a different concept than $\operatorname{{\rm
TC}}_{m}$ for each $m\geq 2$, but we also provide various examples of spaces
$X$ for which the sequences $\\{\operatorname{{\rm TC}}_{m}(X)\\}_{m\geq 2}$
and $\\{\operatorname{{\rm dTC}}_{m}(X)\\}_{m\geq 2}$ coincide.

###### Key words and phrases:

Sequential distributional topological complexity, distributed navigation
algorithm, distributional sectional category, distributional Lusternik-
Schnirelmann category, sequential topological complexity

###### 2010 Mathematics Subject Classification:

Primary 55M30; Secondary 68T40, 70B15.

##  1\. Introduction

In robotics, an autonomously functioning mechanical system, such as a robot,
is required to move from a specified initial position to a specified desired
position inside a configuration space in the most optimal way possible. By
optimal, we mean with the least possible number of instabilities or
discontinuities of motion. This is a motion planning problem whose input is
the positions and output is the motion of the system between them. In many
practical situations, to obtain a more precise and controlled motion, such a
system is additionally required to pass through a fixed number of intermediate
positions at some specified timestamps. This adds more constraints to the
motion planning problem.

This paper is motivated by the following sequential motion planning problem.
Given an autonomous mechanical system with configuration space $X$, a number
$m\geq 2$, and positions $x_{1},x_{2},\ldots,x_{m}$ in $X$, we want to
construct a “nice” algorithm to get from $x_{1}$ to $x_{m}$ via the $m-2$
intermediate positions $x_{2},\ldots,x_{m-1}$ attained in that order for each
ordered tuple $(x_{1},x_{2},\ldots,x_{m})\in X^{m}$. While studying this
problem for $m=2$, M. Farber [Far1] introduced the notion of topological
complexity, $\operatorname{{\rm TC}}(X)$, of a space $X$, as the minimal
degree of instability of motion planning algorithms for systems with
configuration space $X$. This notion was generalized for each $m\geq 2$ by Y.
B. Rudyak [R] to $m^{th}$ sequential topological complexity of the space,
denoted $\operatorname{{\rm TC}}_{m}(X)$.

In a recent joint work with A. Dranishnikov [DJ], we considered this problem
for $m=2$ for advanced autonomous systems, such as robots like Terminator $2$,
that can break into a finite number of pieces at the initial position so that
all the pieces travel independently to the desired final position where they
reassemble back into the system. For such systems, the notion of
distributional topological complexity, $\operatorname{{\rm dTC}}(X)$, of a
space $X$, was introduced in [DJ] as the minimal number of pieces into which
the system needs to break for a continuous motion between all possible pairs
of positions in the configuration space $X$. For such advanced systems,
$\operatorname{{\rm dTC}}$ seems to be an improvement of $\operatorname{{\rm
TC}}$. In this paper, we consider the generalized problem for $m\geq 2$ and
introduce a natural generalization of $\operatorname{{\rm dTC}}$, namely the
$m^{th}$ sequential distributional topological complexity, denoted
$\operatorname{{\rm dTC}}_{m}$, which, we think, offers a better solution to
the above sequential motion planning problem for some advanced autonomous
systems.

###  1.A. Planning a sequential motion

Consider an advanced autonomous system that is required to reach a position
$x_{i}\in X$ at a time $t_{i}\in[0,1]$ for each $1\leq i\leq m$, where
$t_{i}<t_{i+1}$. The idea for the sequential motion of the system is as
follows.

  1. (1)

At time $t_{1}$, the system breaks into finitely many weighted pieces at
position $x_{1}$. All the pieces travel independently for time $t_{2}-t_{1}$
to reach position $x_{2}$ at time $t_{2}$.

  2. (2)

At time $t_{2}$ and position $x_{2}$, all the pieces reassemble back into the
system. Then, the system breaks again at $x_{2}$ into the same weighted
pieces. These pieces travel to reach $x_{3}$ at $t_{3}$.

  3. (3)

The process continues like this, where the system breaks into the same
weighted pieces at time $t_{i}$ and position $x_{i}$, the pieces travel
independently to reach $x_{i+1}$ at $t_{i+1}$ where they reassemble into the
system, and then the system breaks again in the same manner.

  4. (4)

Finally, the pieces reach position $x_{m}$ at time $t_{m}$ where they
reassemble into the system at last.

We assume that our system above has mass $1$, i.e., the sum of the non-
negative weights of the pieces involved is $1$.

The “weights of the pieces” represent the percentage of the portion of the
original system in the corresponding pieces. For example, if the weights are
$0.23,0.16,$ and $0.61$, then the system breaks into pieces whose
capacity/mass are $23\%,16\%$, and $61\%$ of its original capacity/mass. In
another sense, the “weight of a piece” represents the probability that at a
given instance, the system is moving/traveling through that particular piece.

The input for our system is an ordered $m$-tuple of positions. Note that we
require our system to first analyze the $m$ positions and then decide the
weighted pieces it needs for a continuous motion between those $m$ positions
in the given order by the process of repetitive breaking and reassembling.
This is different, less chaotic, and better than letting the system decide its
weighted pieces at each individual $t_{i}$ to travel between $x_{i}$ and
$x_{i+1}$ for $1\leq i\leq m-1$.

Let $n\geq 1$ be the maximum number of pieces that the system can break into
while traveling between all possible ordered $m$-tuples of positions in $X$.
If no such $n$ exists, we conclude that the system is extremely complicated
for practical purposes and we say that its corresponding sequential
distributional topological complexity is infinite. Such a case is hardly found
in real-world situations and is therefore less interesting, both from a
physical as well as a theoretical point of view.

###  1.B. Continuous motion planning algorithm

Let $\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}$ be an ordered $m$-tuple. We
define an $n$-distributed $m$-sequence of $\overline{x}$ to be an unordered
collection of $n$ paths $\phi_{j}$ in $X$, with respective non-negative
weights $\lambda_{j}$, such that

  1. (1)

the sum of the weights is $1$, i.e., $\sum_{j=1}^{n}\lambda_{j}=1$, and

  2. (2)

for each $1\leq j\leq n$, we have $\phi_{j}(t_{i})=x_{i}$ for all $1\leq i\leq
m$.

Hence, the desired sequential motion planning algorithm is a continuous
assignment of each $\overline{x}\in X^{m}$ to an $n$-distributed $m$-sequence
of $\overline{x}$, which is an unordered probability distribution. This is in
contrast to Farber’s algorithm [Far2, Section 12] for $m=2$ whose output is an
ordered probability distribution.

In detail, the continuous algorithm we seek for the above problem is a natural
generalization of the one for $m=2$ from [DJ] in the following sense. For a
metric space $Z$, let $\mathcal{B}(Z)$ denote the set of probability measures
on $Z$ and

$\mathcal{B}_{n}(Z)=\left\\{\mu\in\mathcal{B}(Z)\mid|\operatorname{{\rm
supp}}(\mu)|\leq n\right\\}$

denote the space of probability measures on $Z$ supported by at most $n$
points, equipped with the Lévy-Prokhorov metric [P]. If
$\mu\in\mathcal{B}_{n}(Z)$, then

$\mu=\sum_{z\in F\subset Z,\ |F|\leq n}\lambda_{z}z,$

where $\lambda_{z}\geq 0$ and $\sum\lambda_{z}=1$, and $\operatorname{{\rm
supp}}(\mu)=\\{z\in Z\mid\lambda_{z}>0\\}$. Let $Z=P(X)=\\{f\hskip
2.84526pt|\hskip 2.84526ptf:[0,1]\to X\\}$, the path space of $X$ with the
compact-open topology, and

$P(\overline{x})=\\{f\in P(X)\hskip 2.84526pt|\hskip
2.84526ptf(t_{i})=x_{i}\text{ for all }1\leq i\leq m\\}$

for any $\overline{x}\in X^{m}$. Then $\mathcal{B}_{n}(P(\overline{x}))$ is
the space of all $n$-distributed $m$-sequences of $\overline{x}$. The desired
$m$-navigation algorithm for an advanced system with configuration space $X$
is then a continuous map

$s_{m}:X^{m}\to\mathcal{B}_{n}(P(X))$

such that $s_{m}(\overline{x})\in\mathcal{B}_{n}(P(\overline{x}))$ for all
$\overline{x}\in X^{m}$.

Observe that the breaking of the system into $n$ pieces while traveling is
like an $n^{th}$ degree discontinuity of its motion. Since we are looking for
the most optimal algorithm, we want to minimize this number $n$. This gives us
the notion of the $m^{th}$ sequential distributional topological complexity,
$\operatorname{{\rm dTC}}_{m}(X)$, of a space $X$.

###  1.C. About this paper

At this stage, we mention that typically, one takes

$t_{i}=\frac{i-1}{m-1}\hskip 14.22636pt\text{for all}\hskip 14.22636pt1\leq
i\leq m,$

as in [R], [LS]. So, in this paper, we will take $t_{i}=(i-1)/(m-1)$ to get
equally-timed motions and we will prove all our results for these $t_{i}$.
Thus, the sequential distributional topological complexity is discussed in
this paper only in this sense. However, we note that with minor modifications,
suitable analogs of most of our statements and results from Sections 3, 4, and
6 will also hold for any arbitrary $t_{i}\in[0,1]$, satisfying $t_{i}<t_{i+1}$
for all $1\leq i\leq m-1$, in case of a more general, parametrized motion
discussed in [CFW], [FP].

Here, we are taking the normalized version of $\operatorname{{\rm TC}}_{m}$ in
the sense that if $X$ is a contractible space, then $\operatorname{{\rm
TC}}_{m}(X)=0$ for all $m\geq 2$.

The paper is organized as follows. In Section 2, we set the ground for our
paper and recall the notion and characterization of the classical sequential
topological complexity, $\operatorname{{\rm TC}}_{m}$, for $m\geq 2$. In
Section 3, we formally define sequential versions, $\operatorname{{\rm
dTC}}_{m}$, of distributional topological complexity for $m\geq 2$, prove
their homotopy invariance, generalize results for dTC to $\operatorname{{\rm
dTC}}_{m}$, and compare these new numerical invariants $\operatorname{{\rm
dTC}}_{m}(X)$ for a space $X$ with $\operatorname{{\rm TC}}_{m}(X)$ and
$d\operatorname{cat}$ of $X^{m-1}$ and $X^{m}$. Section 4 is devoted to
finding sharp lower bounds to $\operatorname{{\rm dTC}}_{m}(X)$ in Alexander-
Spanier cohomology, using the cohomology of the symmetric products of $X^{m}$,
in a way slightly different from the one used to bound $\operatorname{{\rm
dTC}}(X)$. In Section 5, we formally define the notion of the distributional
sectional category of a fibration, prove its homotopy invariance, and
characterize $\operatorname{{\rm dTC}}_{m}$ and $d\operatorname{cat}$ from
that perspective. In Section 6, we rigorously prove that $\operatorname{{\rm
dTC}}_{m+1}(X)$ and $d\operatorname{cat}(X^{m})$ agree for CW $H$-spaces $X$
for all $m\geq 1$. Finally, Section 7 involves explicit computations and
estimates of $\operatorname{{\rm dTC}}_{m}$ for various classes of closed
manifolds, thereby extending some computations done for dTC.

When the author finished writing this paper, he learned about the draft [KW]
in private communication (this draft has very recently been made public on
arXiv as a preprint), which introduces the notions of analog LS-category
$(\operatorname{{\rm acat}})$ and analog sequential topological complexity
($\operatorname{{\rm ATC}}_{m}$) with a different motivation and presents some
interesting results, particularly for aspherical spaces. [KW] conjectured that
their invariants (defined for compactly generated Hausdorff spaces) coincide
with the distributional invariants on metrizable spaces. In this paper, we
take a step in that direction and prove in Section 8 that the distributional
invariants give a sharp lower bound to the respective analog invariants on
metrizable spaces. In particular,
$d\operatorname{cat}(X)\leq\operatorname{{\rm acat}}(X)$ and
$\operatorname{{\rm dTC}}_{m}(X)\leq\operatorname{{\rm ATC}}_{m}(X)$ for all
$m\geq 2$. This is especially relevant because, unlike the case of
$d\operatorname{cat}$ and $\operatorname{{\rm dTC}}_{m}$, no non-trivial lower
bounds to $\operatorname{{\rm acat}}$ and $\operatorname{{\rm ATC}}_{m}$ are
known, even for finite CW complexes. Using our computations from this paper,
we also complete some computations for $\operatorname{{\rm ATC}}_{m}$.

##  2\. Preliminaries

All the topological spaces considered in this paper are path-connected metric
spaces. First, we prove an easy result that will be used in the subsequent
sections. For any $X$, $m\geq 2$, and $a_{i}\in(1,\infty)$ such that
$a_{i}>a_{i+1}$ for all $1\leq i\leq m-2$, let

$T_{m}(X)=\left\\{\left(f_{1},\ldots,f_{m}\right)\in(P(X))^{m}\mid
f_{i}(1)=f_{i+1}(0)\text{ for all }1\leq i\leq m-1\right\\}$

and $\theta_{m}:T_{m}(X)\to P(X)$ be defined as
$\theta_{m}\left(f_{1},\ldots,f_{m}\right)=f_{1}\star\cdots\star f_{m}$, where

$\left(f_{1}\star\cdots\star f_{m}\right)(t)=\begin{cases}f_{1}(a_{1}t)&:0\leq
t\leq\frac{1}{a_{1}}\\\
f_{2}\left(\frac{a_{2}(a_{1}t-1)}{a_{1}-a_{2}}\right)&:\frac{1}{a_{1}}\leq
t\leq\frac{1}{a_{2}}\\\ \hskip 17.07164pt\vdots&\hskip 22.76219pt\vdots\\\
f_{m-1}\left(\frac{a_{m-1}(a_{m-2}\hskip
1.42262ptt-1)}{a_{m-2}-a_{m-1}}\right)&:\frac{1}{a_{m-2}}\leq
t\leq\frac{1}{a_{m-1}}\\\ f_{m}\left(\frac{1-a_{m-1}\hskip
1.42262ptt}{1-a_{m-1}}\right)&:\frac{1}{a_{m-1}}\leq t\leq 1\end{cases}$

######  2.1 Lemma.

The map $\theta_{m}$ is continuous for each $m\geq 2$.

###### Proof.

Let us take some $f=(f_{1},\ldots,f_{m})\in T_{m}(X)$ and any sub-basis
neighborhood $U\subset P(X)$ of $\theta_{m}(f)$. By definition of the compact-
open topology, $U=(K,W)=\left\\{\gamma\in P(X)\mid\gamma(K)\subset W\right\\}$
for some compact $K\subset I$ and open $W\subset X$. Define sets
$K_{1}=\left\\{a_{1}t\mid t\in K\cap[0,1/a_{1}]\right\\}$,

$K_{m}=\left\\{\frac{1-a_{m-1}\hskip 1.42262ptt}{1-a_{m-1}}\hskip
4.2679pt\middle|\hskip 4.2679ptt\in
K\cap\left[\frac{1}{a_{m-1}},1\right]\right\\},$

and for all $2\leq i\leq m-1$ the sets

$K_{i}=\left\\{\frac{a_{i}\left(a_{i-1}\hskip
1.42262ptt-1\right)}{a_{i-1}-a_{i}}\hskip 4.2679pt\middle|\hskip 4.2679ptt\in
K\cap\left[\frac{1}{a_{i-1}},\frac{1}{a_{i}}\right]\right\\}.$

Then for each $1\leq i\leq m$, the sets $K_{i}$ are compact. Therefore,

$V=T_{m}(X)\cap\prod_{i=1}^{m}(K_{i},W)$

is open in $T_{m}(X)$. For any $(F_{1},\ldots,F_{m})\in T_{m}(X)$, by
definition, $F_{i}\in(K_{i},W)$ for all $1\leq i\leq m$ if and only if
$F_{1}\star\cdots\star F_{m}\in(K,W)$. Hence, $V$ is a neighborhood of $f$ and
$\theta_{m}(V)\subset U$. Thus, $\theta_{m}$ is continuous at $f\in T_{m}(X)$.
∎

Let us now recall the classical definitions of the Lusternik-Schnirelmann
category [Ja], [CLOT] and the sequential topological complexity [Far1], [R] of
a space.

The _Lusternik-Schnirelmann category_ (LS-category), $\operatorname{cat}(X)$,
of $X$ is the minimal number $n$ such that there is a covering $\\{U_{i}\\}$
of $X$ by $n+1$ open sets each of which is contractible in $X$.

For given $m\geq 2$, the $m^{th}$ _sequential topological complexity_ ,
$\operatorname{{\rm TC}}_{m}(X)$, of $X$ is the minimal number $n$ such that
there is a covering $\\{U_{i}\\}$ of $X^{m}$ by $n+1$ open sets over each of
which there is a continuous map $s_{i}:U_{i}\to P(X)$ such that for each
$\overline{x}=(x_{1},\ldots,x_{m})\in U_{i}\subset X^{m}$,
$s_{i}(\overline{x})(t_{j})=x_{j}$ for all $1\leq j\leq m$.

We also recall the definition of distributional LS-category [DJ] of a space.
Recall that a space $X$ is called $k$-contractible to a point $x_{0}\in X$ if
there is a continuous map $H:X\to\mathcal{B}_{k}(P(X))$ satisfying
$H(x)\in\mathcal{B}_{k}(P(x,x_{0}))$ for all $x\in X$.

The distributional LS-category, $d\operatorname{cat}(X)$, of a space $X$, is
the minimal number $n$ such that $X$ is $(n+1)$-contractible to some fixed
$x_{0}\in X$.

###  2.A. Schwarz-Ganea’s approach to $\operatorname{{\rm TC}}_{m}$

Let $p:E\to B$ be a fibration. For any $n\geq 1$, the iterated fiberwise join
of $n$-copies of $E$ along $p$, denoted $\ast^{n}_{B}\hskip 1.42262ptE$, is
defined as the space

$\ast^{n}_{B}\hskip 1.42262ptE=\left\\{\sum_{i=1}^{n}\lambda_{i}e_{i}\hskip
2.84526pt\middle|\hskip 2.84526pte_{i}\in E,\hskip
1.42262pt\sum_{i=1}^{n}\lambda_{i}=1,\hskip 1.42262pt\lambda_{i}\geq 0,\hskip
1.42262ptp\left(e_{i}\right)=p\left(e_{j}\right)\right\\},$

where each element is a formal ordered linear combination of elements such
that all terms where $\lambda_{i}=0$ are dropped. Similarly, the iterated
fiberwise join of $n$-copies of $p$, denoted $\ast^{n}_{B}\hskip
1.42262ptp:\ast^{n}_{B}\hskip 1.42262ptE\to B$, is defined as the fibration

$\ast^{n}_{B}\hskip
1.42262ptp\left(\sum_{i=1}^{n}\lambda_{i}e_{i}\right)=p\left(e_{i}\right)$

for any $i$ with $\lambda_{i}>0$. Now, we recall the notion of the sectional
category, also known as the Schwarz genus, of a fibration.

The sectional category of a fibration $p:E\to B$, denoted $\text{secat}(p)$,
is the minimal number $n$ such that the fibration $\ast^{n+1}_{B}\hskip
1.42262ptp:\ast^{n+1}_{B}\hskip 1.42262ptE\to B$ admits a section.

Let us fix some $m\geq 2$. Given a space $X$, let $\pi_{m}^{X}:P(X)\to X^{m}$
be the fibration
$\pi_{m}^{X}:\phi\mapsto(\phi(t_{1}),\phi(t_{2}),\ldots,\phi(t_{m}))$. Let us
denote the space $\ast^{n+1}_{X^{m}}\hskip 2.84526ptP(X)$ by
$\mathcal{G}_{m,n}(X)$ and the fibration $\ast^{n+1}_{X^{m}}\hskip
2.84526pt\pi_{m}^{X}$ by $\pi_{m,n}^{X}$. Then the following theorem gives the
Schwarz-Ganea characterization of sequential topological complexity.

######  2.2 Theorem ([Sch]).

For any $X$, $\operatorname{{\rm TC}}_{m}(X)\leq n$ if and only if the
fibration

$\pi_{m,n}^{X}:\mathcal{G}_{m,n}(X)\to X^{m}$

admits a section.

Hence, $\operatorname{{\rm TC}}_{m}(X)=\text{secat}(\pi_{m}^{X})$ for any
space $X$.

###  2.B. Cohomological lower bounds of $\operatorname{{\rm TC}}_{m}$

The cup-length of a space $X$ with coefficients in a ring $R$ (or
alternatively, the cup-length of the cohomology ring $H^{*}(X;R)$) is the
maximal length $k$ of a non-zero cup product
$\alpha_{1}\smile\cdots\smile\alpha_{k}\neq 0$ of cohomology classes
$\alpha_{i}$ of positive dimensions. Let $\Delta:X\to X^{m}$ be the diagonal
map that induces $\Delta^{*}:H^{*}(X^{m};R)\to H^{*}(X;R)$. The elements of
$\operatorname{{\rm Ker}}(\Delta^{*})$ are called the $m^{th}$ zero-divisors
of $X$. Then the cup-length of the ideal of the $m^{th}$ zero-divisors of $X$
is a sharp lower bound of $\operatorname{{\rm TC}}_{m}(X)$ [R], [BGRT].

We note that this lower bound can be obtained from a more general result of A.
S. Schwarz [Sch].

##  3\. Sequential Distributional Topological Complexity

Given a metric space $Z$, we equip $\mathcal{B}_{n}(Z)$ with the Lévy-
Prokhorov metric [P], [DJ] as mentioned in the introduction. We focus on the
case $Z=P(X)$ when $X$ is a path-connected metric space.

We recall that for any $\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}$, a
$k$-distributed $m$-sequence of $\overline{x}$ is an unordered collection of
$k$ weighted paths $\phi_{j}$ in $X$ such that for each $1\leq j\leq k$,
$\phi_{j}(t_{i})=x_{i}$ for all $1\leq i\leq m$, where the weights are non-
negative and the sum of the weights is $1$.

######  3.1 Definition.

A $k$-distributed $m$-navigation algorithm on a space $X$ is a continuous map

$s_{m}:X^{m}\to\mathcal{B}_{k}(P(X))$

that satisfies $s_{m}(\overline{x})\in\mathcal{B}_{k}(P(\overline{x}))$ for
all $\overline{x}\in X^{m}$.

######  3.2 Definition.

For a given $m\geq 2$, the $m^{th}$ sequential distributional topological
complexity, or alternatively, the $m^{th}$ higher distributional topological
complexity, $\operatorname{{\rm dTC}}_{m}(X)$, of a space $X$, is the minimal
number $n$ such that $X$ admits an $(n+1)$-distributed $m$-navigation
algorithm.

Each of the following results generalizes the respective statements for
$\operatorname{{\rm dTC}}$ proved in [DJ, Section 3].

######  3.3 Proposition.

For each $m\geq 2$, $\operatorname{{\rm dTC}}_{m}$ is a homotopy invariant.

###### Proof.

Let us fix some $m\geq 2$. Let $f:X\to Y$ be a homotopy domination with a
continuous right homotopy inverse $g:Y\to X$. We only need to prove that
$\operatorname{{\rm dTC}}_{m}(Y)\leq\operatorname{{\rm dTC}}_{m}(X)$. Since
$fg\simeq\mathbbm{1}_{Y}$, there exists a homotopy $h:Y\to P(Y)$ such that
$h:y\mapsto h_{y}$ for each $y\in Y$, where $h_{y}(0)=y$ and $h_{y}(1)=fg(y)$.
Let $\operatorname{{\rm dTC}}_{m}(X)=n$ and

$s_{m}:X^{m}\to\mathcal{B}_{n+1}(P(X))$

be an $(n+1)$-distributed $m$-navigation algorithm on $X$. Let $f$ induce a
continuous map $f_{*}:\mathcal{B}_{n+1}(P(X))\to\mathcal{B}_{n+1}(P(Y))$ due
to the functoriality of $\mathcal{B}_{n+1}$. Then the composition

$f_{*}s_{m}\hskip 2.84526ptg^{m}:Y^{m}\to\mathcal{B}_{n+1}(P(Y))$

maps each $\overline{y}\in Y^{m}$ to an $(n+1)$-distributed $m$-sequence of
the ordered $m$-tuple
$f^{m}g^{m}(\overline{y})=\\{fg(y_{1}),\ldots,fg(y_{m})\\}$. Let
$s_{m}g^{m}(\overline{y})=\sum\lambda_{\phi}\phi$. For each such element
$\phi\in\operatorname{{\rm supp}}(s_{m}g^{m}(\overline{y}))$, we write
$\widetilde{\phi}=\\{\phi_{1},\ldots,\phi_{m-1}\\}$, where

$\phi_{i}(s)=\phi\left(\frac{s+i-1}{m-1}\right)$

for all $1\leq i\leq m-1$. So, $\phi_{i}$ is a path in $X$ from $g(y_{i})$ to
$g(y_{i+1})$. Using these paths and the homotopy $h$, let us define

$\widehat{\phi}=\left(h_{y_{1}}\cdot
f\phi_{1}\cdot\overline{h}_{y_{2}}\right)\star\left(h_{y_{2}}\cdot
f\phi_{2}\cdot\overline{h}_{y_{3}}\right)\star\cdots\star\left(h_{y_{m-1}}\cdot
f\phi_{m-1}\cdot\overline{h}_{y_{m}}\right),$

where $\cdot$ denotes the usual concatenation and $\star$ denotes
concatenation done using the map $\theta_{m-1}$ from Lemma 2.1 with
$a_{i}=1/t_{i+1}=(m-1)/i$ for all $1\leq i\leq m-2$. Then the continuous map
$\sigma_{m}:Y^{m}\to\mathcal{B}_{n+1}(P(Y))$ defined by

$\sigma_{m}(\overline{y})=\sum_{\phi\hskip 2.84526pt\in\hskip
2.84526pt\operatorname{{\rm supp}}(s_{m}g^{m}(\overline{y}))}\hskip
2.84526pt\lambda_{\phi}\hskip 1.42262pt\widehat{\phi}$

is an $(n+1)$-distributed $m$-navigation algorithm on $Y$. Therefore, we
obtain $\operatorname{{\rm dTC}}_{m}(Y)\leq n=\operatorname{{\rm
dTC}}_{m}(X)$. ∎

The following proposition is straightforward.

######  3.4 Proposition.

For each $m\geq 2$, $\operatorname{{\rm dTC}}_{m}(X)\leq\operatorname{{\rm
TC}}_{m}(X)$.

######  3.5 Proposition.

For each $m\geq 2$, $d\operatorname{cat}(X^{m-1})\leq\operatorname{{\rm
dTC}}_{m}(X)$.

###### Proof.

Let $\operatorname{{\rm dTC}}_{m}(X)=n$. Then there exists an
$(n+1)$-distributed $m$-navigation algorithm on $X$, say

$s_{m}:X^{m}\to\mathcal{B}_{n+1}(P(X)).$

For a fixed basepoint $x_{0}\in X$, let $J_{m}:X^{m-1}\hookrightarrow X^{m}$
be the map

$J(x_{1},\ldots,x_{m-1})=(x_{1},\ldots,x_{m-1},x_{0}).$

Let $\widetilde{x}=(x_{1},\ldots,x_{m-1})$ and
$s_{m}J(\widetilde{x})=\sum\lambda_{\phi}\phi$. For each such $\phi$, we can
write $\phi^{\prime}=(\phi_{1},\ldots,\phi_{m-1})$, where

$\phi_{i}(s)=\phi\left(\frac{s(m-i)+i-1}{m-1}\right)$

for each $1\leq i\leq m-1$. Then $\phi^{\prime}(0)=\widetilde{x}$ and
$\phi^{\prime}(1)=\widetilde{x_{0}}=(x_{0},\ldots,x_{0})\in X^{m-1}$. So, the
continuous map $\sigma_{m}:X^{m-1}\to\mathcal{B}_{n+1}(P_{0}(X^{m-1}))$
defined by

$\sigma_{m}(\widetilde{x})=\sum_{\phi\hskip 2.84526pt\in\hskip
2.84526pt\operatorname{{\rm supp}}(s_{m}J(\widetilde{x}))}\hskip
2.84526pt\lambda_{\phi}\hskip 1.42262pt\phi^{\prime}$

is an $(n+1)$-contraction of $X^{m-1}$ to $\widetilde{x_{0}}$. Hence,
$d\operatorname{cat}(X^{m-1})\leq n=\operatorname{{\rm dTC}}_{m}(X)$. ∎

######  3.6 Proposition.

For each $m\geq 2$, $\operatorname{{\rm dTC}}_{m}(X)\leq
d\operatorname{cat}(X^{m})$.

###### Proof.

Let $d\operatorname{cat}(X^{m})=n$ and let us have an $(n+1)$-contraction

$H:X^{m}\to\mathcal{B}_{n+1}(P_{0}(X^{m}))$

of $X^{m}$ to its fixed basepoint $(x_{1}^{0},\ldots,x_{m}^{0})$. For some
$\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}$, let
$H(\overline{x})=\sum\lambda_{\phi}\phi$. Since $\phi\in\operatorname{{\rm
supp}}(H(\overline{x}))\subset P_{0}(X^{m})$, we can write

$\alpha_{i}=\text{proj}_{i}\hskip 1.42262pt\phi$

so that $\alpha_{i}(0)=x_{i}$ and $\alpha_{i}(1)=x_{i}^{0}$ for all $1\leq
i\leq m$. Let us fix a path $\gamma_{i}\in P(X)$ from $x_{i}^{0}$ and
$x_{i+1}^{0}$ for each $1\leq i\leq m-1$. Define

$\widehat{\phi}=\left(\alpha_{1}\cdot\gamma_{1}\cdot\overline{\alpha}_{2}\right)\star\left(\alpha_{2}\cdot\gamma_{2}\cdot\overline{\alpha}_{3}\right)\star\cdots\star\left(\alpha_{m-1}\cdot\gamma_{m-1}\cdot\overline{\alpha}_{m}\right),$

where $\star$ is the concatenation via the map $\theta_{m-1}$ with
$a_{i}=1/t_{i+1}=(m-1)/i$ for all $1\leq i\leq m-2$. Then the continuous map
$\sigma_{m}:X^{m}\to\mathcal{B}_{n+1}(P(X))$ defined by

$\sigma_{m}(\overline{x})=\sum_{\phi\hskip 2.84526pt\in\hskip
2.84526pt\operatorname{{\rm supp}}(H(\overline{x}))}\hskip
2.84526pt\lambda_{\phi}\hskip 1.42262pt\widehat{\phi}$

is an $(n+1)$-distributed $m$-navigation algorithm on $X$. Therefore, we
obtain $\operatorname{{\rm dTC}}_{m}(X)\leq n=d\operatorname{cat}(X^{m})$. ∎

The following statement justifies that $\\{\operatorname{{\rm
dTC}}_{m}(X)\\}_{m\geq 2}$ is a non-decreasing sequence for any fixed space
$X$.

######  3.7 Proposition.

For each $m\geq 2$, $\operatorname{{\rm dTC}}_{m}(X)\leq\operatorname{{\rm
dTC}}_{m+1}(X)$.

###### Proof.

The inequality $\operatorname{{\rm dTC}}_{m}(X)\leq
d\operatorname{cat}(X^{m})\leq\operatorname{{\rm dTC}}_{m+1}(X)$ follows
directly from Propositions 3.5 and 3.6. ∎

######  3.8 Proposition.

For any $m\geq 2$, $\max\\{\operatorname{{\rm dTC}}_{m}(X),\operatorname{{\rm
dTC}}_{m}(Y)\\}\leq\operatorname{{\rm dTC}}_{m}(X\times Y)$ for any given
spaces $X$ and $Y$.

###### Proof.

Let us fix some $b\in Y$ and define $J^{m}:X^{m}\to(X\times Y)^{m}$ by

$J^{m}\left(x_{1},x_{2},\ldots,x_{m}\right)=\left(\left(x_{1},b\right),\left(x_{2},b\right),\ldots,\left(x_{m},b\right)\right).$

Let us denote $\overline{x}=(x_{1},\ldots,x_{m})$. Let $\operatorname{{\rm
dTC}}_{m}(X\times Y)=n$ with an $(n+1)$-distributed $m$-navigation algorithm

$s_{m}:(X\times Y)^{m}\to\mathcal{B}_{n+1}(P(X\times Y))$

on $X\times Y$. Let $s_{m}J^{m}(\overline{x})=\sum\lambda_{\phi}\phi$. If
$\gamma:P(X\times Y)\to P(X)$ is defined as

$\gamma:f\mapsto\text{proj}_{1}\hskip 1.42262ptf,$

then it induces a map $\gamma^{*}:\mathcal{B}_{n+1}(P(X\times
Y))\to\mathcal{B}_{n+1}(P(X))$ by functoriality of $\mathcal{B}_{n+1}$. Hence,
$\sigma_{m}=\gamma^{*}s_{m}J^{m}:X^{m}\to\mathcal{B}_{n+1}(P(X))$ defined as

$\sigma_{m}(\overline{x})=\sum_{\phi\hskip 2.84526pt\in\hskip
2.84526pt\operatorname{{\rm supp}}(s_{m}J^{m}(\overline{x}))}\hskip
2.84526pt\lambda_{\phi}\hskip 2.84526pt\text{proj}_{1}\hskip 1.42262pt\phi$

is an $(n+1)$-distributed $m$-navigation algorithm on $X$. This gives us the
inequality $\operatorname{{\rm dTC}}_{m}(X)\leq n=\operatorname{{\rm
dTC}}_{m}(X\times Y)$. By similar arguments, the other inequality
$\operatorname{{\rm dTC}}_{m}(Y)\leq n=\operatorname{{\rm dTC}}_{m}(X\times
Y)$ follows. ∎

######  3.9 Proposition.

For any $m\geq 2$ and $n\geq 1$, $\operatorname{{\rm
dTC}}_{m}({\mathbb{R}}P^{n})\leq 2^{m-1}-1$.

###### Proof.

Fix some $n\geq 1$ and let $X={\mathbb{R}}P^{n}$. Recall that
$\operatorname{{\rm dTC}}(X)=\operatorname{{\rm dTC}}_{2}(X)=1$ due to [DJ,
Example 3.13] with a $2$-distributed $2$-navigation algorithm

$H:X\times X\to\mathcal{B}_{2}(P(X)).$

For any $m\geq 3$, let $\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}$. For each
$1\leq i\leq m-1$, let

$H\left(x_{i},x_{i+1}\right)=b_{i}^{1}\hskip
1.42262pt\phi_{i}^{1}+b_{i}^{2}\hskip 1.42262pt\phi_{i}^{2},$

where $b_{i}^{1}+b_{i}^{2}=1$, $\phi_{i}^{1}(0)=x_{i}=\phi_{i}^{2}(0)$, and
$\phi_{i}^{1}(1)=x_{i+1}=\phi_{i}^{2}(1)$. Consider

$\phi_{1}^{k_{1}}\star\phi_{2}^{k_{2}}\star\cdots\star\phi_{m-1}^{k_{m-1}},$

where $k_{i}\in L=\\{1,2\\}$ for each $1\leq i\leq m-1$, and $\star$ denotes
concatenation done via the map $\theta_{m-1}$ with $a_{j}=1/t_{j+1}=(m-1)/j$
for all $1\leq j\leq m-2$. Then the continuous map
$s_{m}:X^{m}\to\mathcal{B}_{2^{m-1}}(P(X))$ defined by

$s_{m}(\overline{x})=\sum_{k_{1}\in L}\hskip 2.84526pt\sum_{k_{2}\in
L}\cdots\sum_{k_{m-1}\in L}\hskip 2.84526ptb_{1}^{k_{1}}b_{2}^{k_{2}}\cdots
b_{m-1}^{k_{m-1}}\hskip
1.42262pt\left(\phi_{1}^{k_{1}}\star\phi_{2}^{k_{2}}\star\cdots\star\phi_{m-1}^{k_{m-1}}\right)$

is an $(2^{m-1}-1)$-distributed $m$-navigation algorithm on $X$. Hence, we
obtain $\operatorname{{\rm dTC}}_{m}({\mathbb{R}}P^{n})\leq 2^{m-1}-1$. ∎

The same technique gives the following crude upper bound for any space $X$.

######  3.10 Proposition.

For any $m\geq 2$, $\operatorname{{\rm dTC}}_{m}(X)\leq(\operatorname{{\rm
dTC}}(X)+1)^{m-1}-1$.

For $m=2$ and $X={\mathbb{R}}P^{n}$, this upper bound is sharp [DJ, Example
3.13] for each $n\geq 1$. Using Proposition 3.4, we obtain

$\operatorname{{\rm dTC}}_{m}({\mathbb{R}}P^{n})\leq\operatorname{{\rm
TC}}_{m}({\mathbb{R}}P^{n})\leq\operatorname{cat}(({\mathbb{R}}P^{n})^{m})=mn.$

For a fixed $m\geq 2$, this upper bound is linear in $n$. So, our bound from
Proposition 3.9, which is independent of $n$, is significantly better when $n$
is large. However, using the analog invariants, we will improve this upper
bound in Section 8.A.

######  3.11 Remark.

Let us fix some $m\geq 3$. Due to Proposition 3.9, we obtain for all
$n\geq(2^{m-1}-1)/(m-2)$ that

$\operatorname{{\rm TC}}_{m}({\mathbb{R}}P^{n})-\operatorname{{\rm
dTC}}_{m}({\mathbb{R}}P^{n})\geq\operatorname{cat}(({\mathbb{R}}P^{n})^{m-1})-\operatorname{{\rm
dTC}}_{m}({\mathbb{R}}P^{n})$ $\geq n(m-1)-(2^{m-1}-1)\geq n.$

For $m=2$ and any $n\geq 1$, we get

$\operatorname{{\rm TC}}({\mathbb{R}}P^{n+1})-\operatorname{{\rm
dTC}}({\mathbb{R}}P^{n+1})\geq\operatorname{cat}({\mathbb{R}}P^{n+1})-\operatorname{{\rm
dTC}}({\mathbb{R}}P^{n+1})\geq n.$

Hence, the gap between $\operatorname{{\rm TC}}_{m}(X)$ and
$\operatorname{{\rm dTC}}_{m}(X)$ can be arbitrarily large for each $m\geq 2$.
So, in particular, $\operatorname{{\rm dTC}}_{m}$ is a different notion than
$\operatorname{{\rm TC}}_{m}$ for each $m\geq 2$ and the inequality in
Proposition 3.4 can be strict for all $m\geq 2$.

Since $\operatorname{{\rm dTC}}_{m}$ is a homotopy invariant, for a discrete
group $\Gamma$, we can define $\operatorname{{\rm
dTC}}_{m}(\Gamma):=\operatorname{{\rm dTC}}_{m}(B\Gamma)$, where
$B\Gamma=K(\Gamma,1)$ is a classifying space of the universal cover of spaces
having $\Gamma$ as their fundamental group. From Proposition 3.9, we get
$\operatorname{{\rm dTC}}_{m}({\mathbb{Z}}_{2})=\operatorname{{\rm
dTC}}_{m}({\mathbb{R}}P^{\infty})\leq 2^{m-1}-1$.

######  3.12 Remark.

We note that an analog of [FO, Theorem 2.1] does not hold in the case of
$\operatorname{{\rm dTC}}_{m}$ for any $m\geq 2$, at least for groups with
torsion. This is because for $\Gamma={\mathbb{Z}}_{2}$, since the subgroup
$K\subset{\mathbb{Z}}_{2}^{m}$ (as defined in [FO, Theorem 2.1]) is finite,
$\operatorname{{\rm cd}}(K)$ is infinite, where $\operatorname{{\rm cd}}$
denotes the cohomological dimension [Br]. But on the other hand,
$\operatorname{{\rm dTC}}_{m}({\mathbb{Z}}_{2})$ cannot exceed $2^{m-1}-1$.

##  4\. Cohomological Lower Bounds

###  4.A. Symmetric products

For a space $X$ and $k\geq 1$, its $k^{th}$ symmetric product $SP^{k}(X)$ is
defined as the orbit space of the action of the symmetric group $S_{k}$ on the
product space $X^{k}$ by permutation of coordinates. In this section, we
regard each $[x_{1},\ldots,x_{m}]\in SP^{k}(X)$ as a formal sum $\sum
n_{i}x_{i}$, where $n_{i}\geq 1$ and $\sum n_{i}=k$, subject to the
equivalence $n_{1}x+n_{2}x=(n_{1}+n_{2})x$. So, $n_{i}$ denotes the number of
times $x_{i}$ appears in $[x_{1},\ldots,x_{m}]$. Define $\delta_{k}:X\to
SP^{k}(X)$ as the diagonal embedding $\delta_{k}(x)=[x,x,\ldots,x]=kx$.

The following result in singular cohomology will be very useful in Section 7.

######  4.1 Proposition ([DJ, Proposition 4.3]).

For a finite simplicial complex $X$ and any $k\geq 1$, the induced
homomorphism $\delta_{k}^{*}:H^{*}(SP^{k}(X);{\mathbb{Q}})\to
H^{*}(X;{\mathbb{Q}})$ is surjective.

In this section, we regard $X$ as the subspace of $SP^{k}(X)$ under the
diagonal embedding $\delta_{k}$ and use the term inclusion to refer to
$\delta_{k}$.

###  4.B. Lower bound for $\operatorname{{\rm dTC}}_{m}$

######  4.2 Lemma.

For any $k\geq 1$, $SP^{k}(P(X))$ deforms to $SP^{k}(X)$.

###### Proof.

Let $\text{ev}:P(X)\to X$ be defined as the evaluation fibration
$\text{ev}:\phi\mapsto\phi(0)$. Let $f=SP^{k}(\text{ev}):SP^{k}(P(X))\to
SP^{k}(X)$ be induced by ev due to functoriality of the $k^{th}$ symmetric
product $SP^{k}$. Define $g:SP^{k}(X)\to SP^{k}(P(X))$ as

$g[a_{1},\ldots,a_{k}]=\left[c_{a_{1}},\ldots,c_{a_{k}}\right],$

where $c_{a_{i}}$ denotes the constant loop at $a_{i}\in X$. Clearly,
$fg=\mathbbm{1}_{SP^{k}(X)}$. Finally, define a map $h:SP^{k}(P(X))\times I\to
SP^{k}(X)$ such that

$h\left(\left[\phi_{1},\ldots,\phi_{k}\right](s),t\right)=\left[\phi_{1}(s(1-t)),\ldots,\phi_{k}(s(1-t))\right]$

for all $s\in[0,1]$. Then $h$ is a homotopy between
$\mathbbm{1}_{SP^{k}(P(X))}$ and $gf$. ∎

For fixed $m\geq 2$ and space $X$, the fibration $\pi_{m}:P(X)\to X^{m}$
defined in Section 2.A induces by functoriality
$\zeta_{n}=SP^{n!}(\pi_{m}):SP^{n!}(P(X))\to SP^{n!}(X^{m})$ for each $n\geq
1$. Let $\partial_{n}:X^{m}\to SP^{n!}(X^{m})$ be the inclusion (diagonal
embedding) in the above sense. Consider the following pullback diagram.

(4.1)
${\mathcal{D}_{n,m}}$${SP^{n!}(P(X))}$${X^{m}}$${SP^{n!}(X^{m})}$$\scriptstyle{a}$$\scriptstyle{\sigma_{n}}$$\scriptstyle{\zeta_{n}}$$\scriptstyle{\partial_{n}}$

Here, for $t_{j}=(j-1)/(m-1)$, we have

$\mathcal{D}_{n,m}=\left\\{\left(\left(x_{1},\ldots,x_{m}\right),\left[\phi_{1},\ldots,\phi_{n!}\right]\right)\hskip
2.84526pt\middle|\hskip 2.84526pt\phi_{i}(t_{j})=x_{j}\text{ for all }1\leq
j\leq m\right\\},$

and $\sigma_{n}$ is the canonical projection, the pullback of $\zeta_{n}$
along $\partial_{n}$.

######  4.3 Lemma.

If $\operatorname{{\rm dTC}}_{m}(X)<n$, then there exist sets
$A_{1},A_{2},\ldots,A_{n}$ that cover $X^{m}$ and over each of which
$\sigma_{n}$ has a section.

###### Proof.

Since $\operatorname{{\rm dTC}}_{m}(X)<n$, there exists an $n$-distributed
$m$-navigation algorithm

$s_{m}:X^{m}\to\mathcal{B}_{n}(P(X))$

on $X^{m}$. For each $1\leq i\leq n$, define

$A_{i}=\left\\{\overline{x}=(x_{1},\ldots,x_{m})\in
X^{m}\mid|\operatorname{{\rm supp}}(s_{m}(\overline{x}))|=i\right\\}.$

We define a continuous map $H_{i}:A_{i}\to SP^{n!}(P(X))$ by

$H_{i}(\overline{x})=\sum_{\phi\hskip 2.84526pt\in\hskip
2.84526pt\operatorname{{\rm supp}}(s_{m}(\overline{x}))}\hskip
2.84526pt\frac{n!}{i}\phi.$

For each $\overline{x}\in X^{m}$, see that
$\zeta_{n}H_{i}(\overline{x})=(n!\hskip 1.42262ptx_{1},\ldots,n!\hskip
1.42262ptx_{m})=n!\hskip 1.42262pt\overline{x}$. Hence,
$\zeta_{n}H_{i}=\partial_{n}$. So, the following diagram commutes.

(4.2)
${A_{i}}$${\smash[b]{\mathcal{D}_{n,m}}}$${SP^{n!}(P(X))}$${A_{i}}$${SP^{n!}(X^{m})}$$\scriptstyle{\mathbbm{1}_{A_{i}}}$$\scriptstyle{H_{i}}$$\scriptstyle{\tau_{i}}$$\scriptstyle{a}$$\scriptstyle{\sigma_{n}}$$\scriptstyle{\zeta_{n}}$$\scriptstyle{\partial_{n}}$

Here, $\tau_{i}:A_{i}\to\mathcal{D}_{n,m}$ exists due to the universal
property of pullback and we have $\sigma_{n}\tau_{i}=\mathbbm{1}_{A_{i}}$. So,
we have found a section of $\sigma_{n}$ over each $A_{i}$. ∎

For a fixed $m\geq 2$, let $\Delta:X\to X^{m}$ be the diagonal map and let the
map $\Delta_{n}:SP^{n!}(X)\to SP^{n!}(X^{m})$ be induced from $\Delta$ by
functoriality. We use an idea from [Sh, Section 2] to prove the following in
Alexander-Spanier cohomology [Sp].

######  4.4 Theorem.

Suppose that $\alpha_{i}^{*}\in H^{k_{i}}(SP^{n!}(X^{m});R)$, $1\leq i\leq n$,
for some ring $R$ and $k_{i}\geq 1$, are cohomology classes such that
$\Delta_{n}^{*}(\alpha_{i}^{*})=0$. Let $\alpha_{i}$ be their images under the
induced homomorphism $\partial_{n}^{*}$ such that
$\alpha_{1}\smile\alpha_{2}\smile\cdots\smile\alpha_{n}\neq 0$. Then,

$\operatorname{{\rm dTC}}_{m}(X)\geq n.$

###### Proof.

We have the following commutative diagram.

(4.3)
${H^{*}(SP^{n!}(P(X));R)}$${H^{*}(SP^{n!}(X);R)}$${H^{*}(SP^{n!}(X^{m});R)}$$\scriptstyle{g^{*}}$$\scriptstyle{\Delta_{n}^{*}}$$\scriptstyle{\zeta_{n}^{*}}$

Due to Lemma 4.2, $g^{*}$ is an isomorphism. So, in particular,
$\operatorname{{\rm Ker}}(\Delta_{n}^{*})\subset\operatorname{{\rm
Ker}}(\zeta_{n}^{*})$. Suppose that $\operatorname{{\rm dTC}}_{m}(X)<n$. Then
from Lemma 4.3, there exists a cover $\\{A_{i}\\}_{i=1}^{n}$ of $X^{m}$ such
that $\sigma_{n}$ has a section $\tau_{i}$ over each $A_{i}$, i.e.,
$\sigma_{n}\tau_{i}=\mathbbm{1}_{A_{i}}$. Due to this and Diagram 4.1, the
following diagram commutes for each $1\leq i\leq n$.

(4.4)
${H^{k_{i}}(A_{i};R)}$${H^{k_{i}}(\mathcal{D}_{n,m};R)}$${H^{k_{i}}(SP^{n!}(X^{m});R)}$${H^{k_{i}}(SP^{n!}(P(X));R)}$$\scriptstyle{\sigma_{n}^{*}}$$\scriptstyle{{\tau_{i}^{*}\vphantom{1}}}$$\scriptstyle{\zeta_{n}^{*}}$$\scriptstyle{\partial_{n}^{*}}$$\scriptstyle{a^{*}}$

Clearly, $\tau_{i}^{*}\sigma_{n}^{*}=\mathbbm{1}_{H^{k_{i}}(A_{i};R)}$.
Therefore, $\sigma_{n}^{*}$ is injective. Since
$\Delta_{n}^{*}(\alpha_{i}^{*})=0$, we get $\zeta_{n}^{*}(\alpha_{i}^{*})=0$,
and thus

$a^{*}(\zeta_{n}^{*}(\alpha_{i}^{*}))=0=\sigma_{n}^{*}(\partial_{n}^{*}(\alpha_{i}^{*})).$

Now, because $\sigma_{n}^{*}$ is injective,
$\partial_{n}^{*}(\alpha_{i}^{*})=0$. From the long exact sequence of the pair
$(SP^{n!}(X^{m}),A_{i})$ in Alexander-Spanier cohomology,

$\cdots\to
H^{k_{i}}(SP^{n!}(X^{m}),A_{i};R)\xrightarrow{j_{i}^{*}}H^{k_{i}}(SP^{n!}(X^{m});R)\xrightarrow{\partial_{n}^{*}}H^{k_{i}}(A_{i};R)\to\cdots,$

there exists $\overline{\alpha_{i}^{*}}\in H^{k_{i}}(SP^{n!}(X^{m}),A_{i};R)$
such that $j_{i}^{*}(\overline{\alpha_{i}^{*}})=\alpha_{i}^{*}$. Further, let
$\partial_{n}^{*}(\overline{\alpha_{i}^{*}})=\overline{\alpha}_{i}\in
H^{k_{i}}(X^{m},A_{i};R)$. For $j$ and $j^{\prime}$ denoting the sums of maps
$j_{i}$ and $j_{i}^{\prime}$, respectively, and $k=\sum k_{i}$, we get the
following commutative diagram.

(4.5)
${H^{k}(X^{m};R)}$${H^{k}\left(X^{m},\bigcup_{i=1}^{n}A_{i};R\right)}$${H^{k}\left(SP^{n!}(X^{m});R\right)}$${H^{k}\left(SP^{n!}(X^{m}),\bigcup_{i=1}^{n}A_{i};R\right)}$$\scriptstyle{(j^{\prime})^{*}}$$\scriptstyle{\partial_{n}^{*}}$$\scriptstyle{\partial_{n}^{*}}$$\scriptstyle{j^{*}}$

The cup product
$\overline{\alpha_{1}^{*}}\smile\cdots\smile\overline{\alpha_{n}^{*}}$ in the
bottom-right goes to the non-zero cup product
$\alpha_{1}\smile\cdots\smile\alpha_{n}\neq 0$ in the top-left. But in the
process, it factors through
$\overline{\alpha}_{1}\smile\cdots\smile\overline{\alpha}_{n}\in
H^{k}(X^{m},X^{m};R)=0$. This is a contradiction. Hence, we must have
$\operatorname{{\rm dTC}}_{m}(X)\geq n$. ∎

######  4.5 Remark.

If $X$ is a locally finite CW complex, then Theorem 4.4 for $m=2$ is recovered
from [DJ, Theorem 4.12] due to the fact that the Alexander-Spanier cohomology
groups coincide with the singular cohomology groups for locally finite CW
complexes [Sp], and the following commutative diagram.

(4.6)
${H^{*}(SP^{n!}(X^{2});R)}$${H^{*}\left(SP^{n!}(X);R\right)}$${H^{*}\left(X^{2};R\right)}$${H^{*}\left(X;R\right)}$$\scriptstyle{\Delta_{n}^{*}}$$\scriptstyle{\partial_{2}^{*}}$$\scriptstyle{\Delta^{*}}$

Hence, $\Delta_{n}^{*}(\alpha_{i}^{*})=0$ implies that
$\Delta^{*}(\alpha_{i})=0$, i.e., $\alpha_{i}$ is a $2^{nd}$ zero-divisor.

##  5\. Distributional Sectional Category

Let $p:E\to B$ be a Hurewicz fibration. For each $n\geq 1$, define a space

$E_{n}(p)=\bigcup_{x\in B}\mathcal{B}_{n}(p^{-1}(x))=\left\\{\mu\in
B_{n}(E)\hskip 2.84526pt\middle|\hskip 2.84526pt\operatorname{{\rm
supp}}(\mu)\subset p^{-1}(x),\hskip 2.84526ptx\in B\right\\}$

and a continuous map $\mathcal{B}_{n}(p):E_{n}(p)\to B$ such that

$\mathcal{B}_{n}(p)(\mu)=x\hskip 11.38109pt\text{whenever}\hskip
11.38109pt\mu\in\mathcal{B}_{n}(p^{-1}(x)).$

######  5.1 Proposition ([DJ, Proposition 5.1]).

If $p:E\to B$ is a Hurewicz fibration, then so is
$\mathcal{B}_{n}(p):E_{n}(p)\to B$.

######  5.2 Definition.

Given a fibration $p:E\to B$, its distributional Schwarz genus, or
alternatively, its distributional sectional category, denoted $d\hskip
1.42262pt\text{secat}(p)$, is the minimal number $n$ such that
$\mathcal{B}_{n+1}(p):E_{n+1}(p)\to B$ admits a section.

######  5.3 Proposition.

$d\hskip 1.42262pt\textup{secat}$ is a homotopy invariant.

###### Proof.

Given fibrations $p:E\to B$ and $q:Z\to C$ and the following commutative
diagram, where the horizontal maps are homotopy equivalences,

(5.1)
${E}$${Z}$${B}$${C}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{{f^{\prime}\vphantom{1}}}$$\scriptstyle{q}$$\scriptstyle{g}$$\scriptstyle{{g^{\prime}\vphantom{1}}}$

we need to show that $d\hskip 1.42262pt\text{secat}(p)=d\hskip
1.42262pt\text{secat}(q)$. For any $k\geq 1$, we first obtain fibrations
$\mathcal{B}_{k}(p):E_{k}(p)\to B$ and $\mathcal{B}_{k}(q):Z_{k}(p)\to C$ by
Proposition 5.1. The functoriality of $\mathcal{B}_{k}$ gives a map
$f_{*}:\mathcal{B}_{k}(E)\to\mathcal{B}_{k}(Z)$ so that

$f_{*}\left(\sum\lambda_{r}r\right)=\sum\lambda_{r}\hskip 1.42262ptf(r).$

Let $\mu=\sum\lambda_{r}r\in\mathcal{B}_{k}(p^{-1}(x))\subset E_{k}(p)$ for
some $x\in B$. So, $\text{supp}(\mu)\subset p^{-1}(x)$, which means $p(r)=x$
whenever $\lambda_{r}>0$. Note that $q(f(r))=g(p(r))=g(x)$ whenever
$\lambda_{r}>0$. So, $\text{supp}(f_{*}(\mu))\subset q^{-1}(g(x))$. Let $F$
denote the restriction of $f_{*}$ to $E_{k}(p)$. Thus, $F:E_{k}(p)\to
Z_{k}(p)$ is defined. Similarly, $F^{\prime}:Z_{k}(p)\to E_{k}(p)$ is
obtained. If $\mu\in\mathcal{B}_{k}(p^{-1}(x))\subset E_{k}(p)$ for some $x\in
B$, then

$\mathcal{B}_{k}(q)\hskip 1.42262ptF(\mu)=g(x)=g\hskip
1.42262pt\mathcal{B}_{k}(p)(\mu).$

Similarly, if $\vartheta\in B_{k}(q^{-1}(y))\subset Z_{k}(q)$ for some $y\in
B$, we get

$\mathcal{B}_{k}(p)\hskip
1.42262ptF^{\prime}(\vartheta)=g^{\prime}(y)=g^{\prime}\hskip
1.42262pt\mathcal{B}_{k}(q)(\vartheta).$

So, we get the following commutative diagram.

(5.2)
${E_{k}(p)}$${Z_{k}(q)}$${B}$${C}$$\scriptstyle{F}$$\scriptstyle{\mathcal{B}_{k}(p)}$$\scriptstyle{{F^{\prime}\vphantom{1}}}$$\scriptstyle{\mathcal{B}_{k}(q)}$$\scriptstyle{g}$$\scriptstyle{{g^{\prime}\vphantom{1}}}$

Let $d\hskip 1.42262pt\text{secat}(p)=n-1$. Then by definition, there exists a
section $s:B\to E_{n}(p)$ of $\mathcal{B}_{n}(p)$. Consider $Fsg^{\prime}:C\to
Z_{n}(q)$. See that

$\mathcal{B}_{n}(q)Fsg^{\prime}=g\hskip
1.42262pt\mathcal{B}_{n}(p)sg^{\prime}=gg^{\prime}\simeq\mathbbm{1}_{C}.$

So, $Fsg^{\prime}$ is a homotopy section of $\mathcal{B}_{n}(q)$. Since
$\mathcal{B}_{n}(q)$ is a fibration, the homotopy section $Fsg^{\prime}$ gives
a section of $\mathcal{B}_{n}(q)$. So, $d\hskip 1.42262pt\text{secat}(q)\leq
n-1=d\hskip 1.42262pt\text{secat}(p)$. Similarly, if $d\hskip
1.42262pt\text{secat}(q)=m-1$ and $s^{\prime}:C\to Z_{m}(q)$ is a section of
$\mathcal{B}_{m}(q)$, then $F^{\prime}s^{\prime}g$ helps obtain a section of
$\mathcal{B}_{m}(p)$, thereby implying that $d\hskip
1.42262pt\text{secat}(p)\leq m-1=d\hskip 1.42262pt\text{secat}(q)$. ∎

Let $h:A\to B$ be any continuous map. It has a fibrational substitute, say
$p_{h}:E\to B$. Since any two fibrational substitutes of a given continuous
map are fiberwise homotopy equivalent, using Proposition 5.3, we can define

$d\hskip 1.42262pt\text{secat}(h):=d\hskip
1.42262pt\text{secat}\left(p_{h}\right).$

######  5.4 Remark.

We can also define the distributional sectional category of group
homomorphisms between discrete groups. Let $\phi:\Gamma\to\Lambda$ be such a
homomorphism. This gives a continuous map $B\phi:B\Gamma\to B\Lambda$ that
induces $\phi$ at the level of fundamental groups. We define $d\hskip
1.42262pt\text{secat}(\phi):=d\hskip 1.42262pt\text{secat}(B\phi).$

###  5.A. For $\operatorname{{\rm dTC}}_{m}$

For $m\geq 2$, upon taking $p=\pi_{m}^{X},E=P(X),$ and $B=X^{m}$, and
observing that an $(n+1)$-distributed $m$-navigation algorithm on $X$ produces
a section of the fibration $\mathcal{B}_{n+1}(\pi_{m}^{X})$ and vice-versa, we
get the following Schwarz-Ganea-type characterization of $\operatorname{{\rm
dTC}}_{m}$.

######  5.5 Proposition.

For $m\geq 2$, $\operatorname{{\rm dTC}}_{m}(X)\leq n$ if and only if the
fibration

$\mathcal{B}_{n+1}(\pi_{m}^{X}):P(X)_{n+1}(\pi_{m}^{X})\to X^{m}$

admits a section.

Therefore, for any space $X$ and $m\geq 2$, we conclude that

$\operatorname{{\rm dTC}}_{m}(X)=d\hskip
1.42262pt\text{secat}\left(\pi_{m}^{X}\right).$

######  5.6 Remark.

This realization provides an alternate proof of Proposition 3.3 in light of
Proposition 5.3.

If $\Delta_{m}^{X}:X\to X^{m}$ denotes the diagonal map, then since
$\pi_{m}^{X}$ is the fibrational substitute of $\Delta_{m}^{X}$, we have
$\operatorname{{\rm dTC}}_{m}(X)=d\hskip
1.42262pt\text{secat}(\Delta_{m}^{X})$.

###  5.B. For $d\operatorname{cat}$

For any given $m\geq 1$, we now obtain a different characterization of
$d\operatorname{cat}(X^{m})$ as follows. Let $\xi_{m}^{X}:P(X)\to X^{m}$ be
the fibration

$\xi_{m}^{X}:\phi\mapsto\left(\phi\left(\frac{1}{m}\right),\phi\left(\frac{2}{m}\right),\ldots,\phi\left(\frac{m-1}{m}\right),\phi\left(1\right)\right).$

From this, we can form the fibration
$\mathcal{B}_{n+1}(\xi_{m}^{X}):P(X)_{n+1}(\xi_{m}^{X})\to X^{m}$. For some
fixed basepoint $x_{0}\in X$, let

$\mathcal{P}_{n+1}(X)=\left\\{\sum\lambda_{\psi}\hskip
1.42262pt\psi\in\mathcal{B}_{n+1}(P(X))\hskip 2.84526pt\middle|\hskip
2.84526pt\psi(0)=x_{0}\right\\}.$

If $\mathcal{A}_{n+1}(X)=P(X)_{n+1}(\xi_{m}^{X})\cap\mathcal{P}_{n+1}(X)$,
then we let the map $\mathcal{C}_{n+1}(\xi_{m}^{X})$ denote the restriction of
the fibration $\mathcal{B}_{n+1}(\xi_{m}^{X})$ to the subspace
$\mathcal{A}_{n+1}(X)$. The following statement generalizes [DJ, Proposition
5.3].

######  5.7 Lemma.

For $m\geq 1$, $d\operatorname{cat}(X^{m})\leq n$ if and only if the map

$\mathcal{C}_{n+1}\left(\xi_{m}^{X}\right):\mathcal{A}_{n+1}(X)\to X^{m}$

admits a section.

###### Proof.

Let $d\operatorname{cat}(X^{m})\leq n$. Then for a fixed basepoint
$\overline{x}_{0}=(x_{1}^{0},\ldots,x_{m}^{0})\in X^{m}$, there exists an
$(n+1)$-contraction

$H:X^{m}\to\mathcal{B}_{n+1}(P_{0}(X^{m}))$

of $X^{m}$ to $\overline{x}_{0}$. For $\overline{x}=(x_{1},\ldots,x_{m})\in
X^{m}$, let $H(\overline{x})=\sum\lambda_{\phi}\phi$. Let

$\beta_{i}=\text{proj}_{i}\hskip 0.85358pt\overline{\phi},$

where $\overline{\phi}(t)=\phi(1-t)$. Let $\gamma_{i}\in P(X)$ be a path from
$x_{i}^{0}$ to $x_{i+1}^{0}$ for each $i$. Define

$\widehat{\phi}=\beta_{1}\star\left(\overline{\beta}_{1}\cdot\gamma_{1}\cdot\beta_{2}\right)\star\left(\overline{\beta}_{2}\cdot\gamma_{2}\cdot\beta_{3}\right)\star\cdots\star\left(\overline{\beta}_{m-1}\cdot\gamma_{m-1}\cdot\beta_{m}\right),$

where $\star$ is the concatenation via the map $\theta_{m}$ in Lemma 2.1 with
$a_{i}=i/m$ for all $1\leq i\leq m-1$. Note that
$\widehat{\phi}(0)=x_{1}^{0}$, which we shall regard as the basepoint of $X$.
Finally, define $K:X^{m}\to\mathcal{B}_{n+1}(P(X))$ as

$K(\overline{x})=\sum_{\phi\hskip 2.84526pt\in\hskip
2.84526pt\operatorname{{\rm supp}}(H(\overline{x}))}\hskip
2.84526pt\lambda_{\phi}\hskip 1.42262pt\widehat{\phi}.$

By definition, the range of $K$ is contained in $\mathcal{A}_{n+1}(X)$.
Clearly, we have that
$\mathcal{C}_{n+1}\left(\xi_{m}^{X}\right)K=\mathbbm{1}_{X^{m}}$. Therefore,
$K$ is a section of $\mathcal{C}_{n+1}(\xi_{m}^{X})$. For the converse, let

$\psi:X^{m}\to\mathcal{A}_{n+1}(X)$

be a section of $\mathcal{C}_{n+1}(\xi_{m}^{X})$, with a chosen basepoint
$x_{0}\in X$. Let $\psi(\overline{x})=\sum\lambda_{\phi}\phi$. For each such
$\phi\in\operatorname{{\rm supp}}(\psi(\overline{x}))$, we can write
$\phi^{\prime}=(\phi_{1},\phi_{2},\ldots,\phi_{m})$, where

$\phi_{i}(s)=\phi\left(\frac{i(1-s)}{m}\right)$

for each $1\leq i\leq m$. Then $\phi^{\prime}(0)=\overline{x}$ and
$\phi^{\prime}(1)=(x_{0},\ldots,x_{0})\in X^{m}$. So, the continuous map
$\sigma_{m}:X^{m}\to\mathcal{B}_{n+1}(P_{0}(X^{m}))$ defined by

$\sigma_{m}(\overline{x})=\sum_{\phi\hskip 2.84526pt\in\hskip
2.84526pt\operatorname{{\rm supp}}(\psi(\overline{x}))}\hskip
2.84526pt\lambda_{\phi}\hskip 1.42262pt\phi^{\prime}$

is an $(n+1)$-contraction of $X^{m}$ to $(x_{0},\ldots,x_{0})$. Hence,
$d\operatorname{cat}(X^{m})\leq n$. ∎

Hence, for any space $X$ and $m\geq 1$, it follows that

$d\operatorname{cat}(X^{m})=d\hskip
1.42262pt\text{secat}\left(\xi_{m}^{X}\right).$

For a fixed $x_{0}\in X$, let $P_{0}(X)$ denote the space of paths issued to
$x_{0}$ and let $p_{0}^{X}:P_{0}(X)\to X$ be the evaluation fibration at
$t=0$. Then, in particular for $m=1$, we have $d\operatorname{cat}(X)=d\hskip
1.42262pt\text{secat}(\xi_{1}^{X})=d\hskip 1.42262pt\text{secat}(p_{0}^{X}).$

######  5.8 Remark.

This gives an alternate proof of the homotopy invariance of
$d\operatorname{cat}$ [DJ, Proposition 3.2].

Also, for the inclusion map $\iota_{X}:x_{0}\hookrightarrow X$, we have
$d\operatorname{cat}(X)=d\hskip 1.42262pt\text{secat}(\iota_{X}).$

###  5.C. A lower bound for $d\hskip 1.42262pt\textup{secat}$

Let $p:E\to B$ be a continuous map and for a fixed $n\geq 1$, let
$p_{n}:SP^{n!}(E)\to SP^{n!}(B)$ be induced by the functor $SP^{n!}$. As in
Section 4.A, we let $\delta_{n}:B\to SP^{n!}(B)$ be the diagonal embedding. In
light of Schwarz’s cohomological lower bound [Sch] for $\text{secat}(p)$, and
motivated by the analogy between the definitions of $d\hskip
1.42262pt\text{secat}(p)$ and $\text{secat}(p)$, we propose the following
generalization of Theorem 4.4 and [DJ, Theorem 4.7].

######  5.9 Conjecture.

Suppose $\alpha_{i}^{*}\in H^{k_{i}}(SP^{n!}(B);R)$, $1\leq i\leq n$, for some
ring $R$ and $k_{i}\geq 1$, are cohomology classes such that
$p_{n}^{*}(\alpha_{i}^{*})=0$. Let $\alpha_{i}$ be their images under the
induced homomorphism $\delta_{n}^{*}$ such that
$\alpha_{1}\smile\cdots\smile\alpha_{n}\neq 0$, then $d\hskip
1.42262pt\textup{secat}(p)\geq n$.

##  6\. For $H$-spaces

######  6.1 Definition.

A CW complex $X$ is called a $H$-space if there exists a basepoint $e\in X$,
called an identity of $X$, and a continuous map $\nu:X\times X\to X$, called a
product on $X$, such that $\nu(x,e)=x=\nu(e,x)$ for all $x\in X$.

For classical sequential topological complexity and LS-category, when $X$ is a
$H$-space, then $\operatorname{{\rm TC}}_{m+1}(X)=\operatorname{cat}(X^{m})$
holds for all $m\geq 1$ [LS]. For dTC and $d\operatorname{cat}$, when $G$ is a
topological group, then $\operatorname{{\rm dTC}}(G)=d\operatorname{cat}(G)$
[DJ]. The values dTC and $d\operatorname{cat}$ agree for $H$-spaces as well.

Here, using ideas from [LS, Theorem 1], we obtain the following
generalization.

######  6.2 Theorem.

If $X$ is a $H$-space, then $\operatorname{{\rm
dTC}}_{m+1}(X)=d\operatorname{cat}(X^{m})$ for all $m\geq 1$.

###### Proof.

Since $X$ is a $H$-space, we have an identity element $e\in X$ and a product
$\nu:X\times X\to X$ as above. Following the proof of [LS, Theorem 1], we
obtain a continuous map $D:X\times X\to X$ such that $\text{proj}_{1}\bullet
D=\text{proj}_{2}$, where $\bullet$ is the product in $[X\times X,X]$ induced
by $\nu$. Define $f_{m}:X^{m+1}\to X^{m}$ by

$f_{m}\left(x,x_{1},x_{2},\ldots,x_{m}\right)=\left(D\left(x,x_{1}\right),D\left(x,x_{2}\right),\ldots,D\left(x,x_{m}\right)\right).$

Let $d\operatorname{cat}(X^{m})=n$. Consider the following pullback diagram.

(6.1)
${Q_{m}}$${\mathcal{A}_{n+1}(X)}$${X^{m+1}}$${X^{m}}$$\scriptstyle{q_{m}}$$\scriptstyle{\mathcal{C}_{n+1}(\xi_{m}^{X})}$$\scriptstyle{f_{m}}$

Here, we regard $e\in X$ as the basepoint of $X$ to form
$\mathcal{P}_{n+1}(X)$. By definition, the subset $Q_{m}\subset
X^{m+1}\times\mathcal{A}_{n+1}(X)$ is

$Q_{m}=\left\\{\left(\left(x,x_{1},\ldots,x_{m}\right),\sum\lambda_{\phi}\phi\right)\hskip
2.84526pt\middle|\hskip
2.84526pt\phi(0)=e,\phi\left(\frac{i}{m}\right)=D\left(x,x_{i}\right)\text{
for all }1\leq i\leq m\right\\}$

and $q_{m}$ is the projection onto $X^{m+1}$. Define a map
$\Theta_{m}:Q_{m}\to\mathcal{B}_{n+1}(P(X))$ by

$\Theta_{m}:\left(\left(x,x_{1},\ldots,x_{m}\right),\sum\lambda_{\phi}\phi\right)\mapsto\sum\lambda_{\phi}\hskip
1.42262pt\nu(x,\phi),$

where for each $x\in X$, the path $\nu(x,\phi)$ is defined as
$\nu(x,\phi)(t)=\nu(x,\phi(t))$. So,

$\mathcal{B}_{n+1}(\pi_{m+1})\hskip
2.84526pt\Theta_{m}\left(\left(x,x_{1},\ldots,x_{m}\right),\sum\lambda_{\phi}\phi\right)=\left(x,\nu\left(x,D\left(x,x_{1}\right)\right),\ldots,\nu\left(x,D\left(x,x_{m}\right)\right)\right).$

Like in the proof of [LS, Theorem 1], let us define maps $p_{i}:X^{m+1}\to X$
as $p_{i}(a_{1},\ldots,a_{m+1})=a_{i}$, and $p_{1,i}:X^{m+1}\to X^{2}$ as
$p_{1,i}(a_{1},\ldots,a_{m+1})=(a_{1},a_{i+1})$. Then

$\nu(x,D(x,x_{i}))=(\text{proj}_{1}\bullet D)\hskip
2.84526ptp_{1,i}(x,x_{1},\ldots,x_{m})$

and $\text{proj}_{2}\hskip 2.84526ptp_{1,i}=p_{i+1}$. Since
$\text{proj}_{1}\bullet D\simeq\text{proj}_{2}$, we get

$\mathcal{B}_{n+1}(\pi_{m+1})\hskip
2.84526pt\delta_{m}=\left(p_{1},(\text{proj}_{1}\bullet D)\hskip
2.84526ptp_{1,1},\ldots,(\text{proj}_{1}\bullet D)\hskip
2.84526ptp_{1,m}\right)q_{m}$ $\simeq\left(p_{1},\text{proj}_{2}\hskip
2.84526ptp_{1,1},\ldots,\text{proj}_{2}\hskip 2.84526ptp_{1,m}\right)\hskip
1.13809ptq_{m}=(p_{1},p_{2},\ldots,p_{m+1})\hskip 2.84526ptq_{m}=q_{m}.$

So, $\mathcal{B}_{n+1}(\pi_{m+1})\Theta_{m}\simeq q_{m}$. Since
$d\operatorname{cat}(X^{m})=n$, there exists a section of
$\mathcal{C}_{n+1}(\xi_{m}^{X})$, say $H:X^{m}\to\mathcal{A}_{n+1}(X)$, due to
Proposition 5.7. Then we get the following commutative diagram.

(6.2)
${X^{m+1}}$${\smash[b]{Q_{m}}}$${\mathcal{A}_{n+1}(X)}$${X^{m+1}}$${X^{m}}$$\scriptstyle{\mathbbm{1}_{X^{m+1}}}$$\scriptstyle{Hf_{m}}$$\scriptstyle{s}$$\scriptstyle{q_{m}}$$\scriptstyle{\mathcal{C}_{n+1}(\xi_{m}^{X})}$$\scriptstyle{f_{m}}$

Here, $s:X^{m+1}\to Q_{m}$ exists due to universal property of pullback and we
have $\mathbbm{1}_{X^{m+1}}=q_{m}s$. Thus,

$\mathbbm{1}_{X^{m+1}}=q_{m}s\simeq\left(\mathcal{B}_{n+1}(\pi_{m+1})\hskip
1.42262pt\Theta_{m}\right)s=\mathcal{B}_{n+1}(\pi_{m+1})\left(\Theta_{m}s\right).$

So, $\Theta_{m}s$ is a homotopy section of $\mathcal{B}_{n+1}(\pi_{m+1})$. But
since $\mathcal{B}_{n+1}(\pi_{m+1})$ is a fibration because of Proposition
5.1, $\Theta_{m}s$ gives a section of $\mathcal{B}_{n+1}(\pi_{m+1})$. So, by
Proposition 5.5, we get the inequality $\operatorname{{\rm dTC}}_{m+1}(X)\leq
n=d\operatorname{cat}(X^{m})$. The other inequality
$d\operatorname{cat}(X^{m})\leq\operatorname{{\rm dTC}}_{m+1}(X)$ follows from
Proposition 3.5. ∎

######  6.3 Remark.

In view of Proposition 6.2, the lower bound of $\operatorname{{\rm dTC}}_{m}$
in Proposition 3.5 is sharp for all $m\geq 2$.

######  6.4 Corollary.

If $G$ is a topological (Lie) group, then $\operatorname{{\rm
dTC}}_{m+1}(G)=d\operatorname{cat}(G^{m})$ for all $m\geq 1$.

######  6.5 Example.

For $k=1,3,$ and $7$, we obtain $\operatorname{{\rm
dTC}}_{m}(S^{k})=d\operatorname{cat}((S^{k})^{m-1})=m-1$ from Theorem 6.2 and
[DJ, Proposition 6.7].

######  6.6 Example.

For the $n$-torus $T^{n}$, due to Theorem 6.2 and [DJ, Proposition 6.7],

$\operatorname{{\rm
dTC}}_{m}(T^{n})=d\operatorname{cat}((S^{1})^{n(m-1)})=n(m-1).$

Like in the case of [LS, Corollary 3.5] for classical invariants, it follows
from Theorem 6.2 that

$\operatorname{{\rm
dTC}}_{m}(X^{k})=d\operatorname{cat}(X^{k(m-1)})=\operatorname{{\rm
dTC}}_{k+1}(X^{m-1})$

if $X$ is a $H$-space. Thus, $\operatorname{{\rm
dTC}}(T^{2})=2=\operatorname{{\rm dTC}}_{3}(S^{1})$ due to Example 6.5, and
$\operatorname{{\rm dTC}}_{4}(T^{2})=6=\operatorname{{\rm dTC}}_{3}(T^{3})$.

##  7\. Estimates and Computations

Recall that Alexander-Spanier cohomology coincides with singular cohomology on
locally finite CW complexes [Sp]. Hence, for all the spaces discussed in this
section, Theorem 4.4 will hold in singular cohomology as well. So, here we
will work only with the singular cohomology groups.

###  7.A. $\operatorname{{\rm dTC}}_{m}$ of surfaces

######  7.1 Proposition.

If $X$ is a closed orientable manifold, or if $X$ is a closed non-orientable
surface of genus $>1$, then $\operatorname{{\rm dTC}}_{m}(X)\geq m-1$ for all
$m\geq 2$.

###### Proof.

Let $X\in\\{M^{n},N_{g}\hskip 2.84526pt|\hskip 2.84526ptn\geq 1,g\geq 2\\}$,
where $M^{n}$ is a closed orientable manifold of dimension $n$ and $N_{g}$ is
a closed non-orientable surface of genus $g$. For $m=2$, the statement is
obvious because $X$ is not contractible. Let us fix some $m\geq 3$. Let

$d=\begin{cases}n&:X=M^{n}\text{ for some }n\geq 1\\\ 1&:X=N_{g}\text{ for any
}g\geq 2\end{cases}$

Clearly, $H^{d}(X;{\mathbb{Q}})\neq 0$. So, there exists some $v\in
H^{d}(X;{\mathbb{Q}})$ such that $v\neq 0$. Due to Proposition 4.1, there
exists $w\in H^{d}(SP^{(m-1)!}(X);{\mathbb{Q}})$ such that
$\partial_{m-1}^{*}(w)=v$, where $\partial_{m-1}:X\to SP^{(m-1)!}(X)$ is the
diagonal embedding. For $1\leq i\leq m$, let $\text{proj}_{i}:X^{m}\to X$ be
the projection onto the $i^{th}$ coordinate. By functoriality, we get
$r_{i}=SP^{(m-1)!}(\text{proj}_{i}):SP^{(m-1)!}(X^{m})\to SP^{(m-1)!}(X)$. For
brevity, let $Y=SP^{(m-1)!}(X)$ and $Z=SP^{(m-1)!}(X^{m})$. Then for each $i$,
the following diagram commutes.

(7.1)
${Y}$${Z}$${Y}$${X}$${X^{m}}$${X}$$\scriptstyle{r_{i}}$$\scriptstyle{\Delta_{m-1}}$$\scriptstyle{\partial_{m-1}}$$\scriptstyle{\text{proj}_{i}}$$\scriptstyle{\partial_{m-1}^{m}}$$\scriptstyle{\Delta}$$\scriptstyle{\partial_{m-1}}$

Here, $\partial_{m-1}^{m}:X^{m}\to Z$ is the diagonal embedding and $\Delta$
and $\Delta_{m-1}$ are the maps defined in Section 4.B. For each fixed $1\leq
i\leq m-1$, $\text{proj}_{i}$ and $\text{proj}_{m}$ induce the map
$\phi_{i}:H^{d}(X;{\mathbb{Q}})\oplus H^{d}(X;{\mathbb{Q}})\to
H^{d}(X^{m};{\mathbb{Q}})$, and similarly, $r_{i}$ and $r_{m}$ induce
$\psi_{i}:H^{d}(Y;{\mathbb{Q}})\oplus H^{d}(Y;{\mathbb{Q}})\to
H^{d}(Z;{\mathbb{Q}})$ defined, respectively, as

$\phi_{i}(x\oplus y)=\text{proj}_{i}^{*}(x)-\text{proj}_{m}^{*}(y)\hskip
14.22636pt\text{and}\hskip 14.22636pt\psi_{i}(a\oplus
b)=r_{i}^{*}(a)-r_{m}^{*}(b).$

Let us denote $\phi_{i}(v\oplus v)$ by $\alpha_{i}$ and $\psi_{i}(w\oplus w)$
by $\alpha_{i}^{*}$. Then due to Diagram 7.1, we have the following
commutative diagram for each $1\leq i\leq m-1$.

(7.2) ${w\oplus w}$${\alpha_{i}^{*}}$${w-w=0}$${H^{d}(Y;{\mathbb{Q}})\oplus
H^{d}(Y;{\mathbb{Q}})}$${H^{d}(Z;{\mathbb{Q}})}$${H^{d}(Y;{\mathbb{Q}})}$${H^{d}(X;{\mathbb{Q}})\oplus
H^{d}(X;{\mathbb{Q}})}$${H^{d}(X^{m};{\mathbb{Q}})}$${H^{d}(X;{\mathbb{Q}})}$${v\oplus
v}$${\alpha_{i}}$${v-v=0}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\psi_{i}}$$\scriptstyle{\partial_{m-1}^{*}\oplus\hskip
2.84526pt\partial_{m-1}^{*}\hskip
2.84526pt}$$\scriptstyle{\Delta_{m-1}^{*}}$$\scriptstyle{(\partial_{m-1}^{m})^{*}}$$\scriptstyle{\partial_{m-1}^{*}}$$\scriptstyle{\phi_{i}}$$\scriptstyle{\Delta^{*}}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\in}$

On the top-right, $w-w=0$ because the top (resp. bottom) row when restricted
to either of the components $H^{d}(Y;{\mathbb{Q}})$ (resp.
$H^{d}(X;{\mathbb{Q}})$) is an isomorphism. Due to [R, Proposition 3.5], we
have

$\alpha_{1}\smile\cdots\smile\alpha_{m-1}\neq 0.$

Hence, taking $k_{i}=d$ for all $1\leq i\leq m-1$, we have $\alpha_{i}^{*}\in
H^{k_{i}}(SP^{(m-1)!}(X^{m});{\mathbb{Q}})$ as in the statement of Theorem
4.4. Therefore, $\operatorname{{\rm dTC}}_{m}(X)\geq m-1$. ∎

######  7.2 Remark.

More generally, for any space $X$ for which $H^{d}(X;{\mathbb{Q}})\neq 0$ for
some $d\geq 1$, the above technique can be used to conclude
$\operatorname{{\rm dTC}}_{m}(X)\geq m-1$.

The following simple consequence generalizes Example 6.5.

######  7.3 Corollary.

For any $k\geq 1$ and $m\geq 2$, $\operatorname{{\rm dTC}}_{m}(S^{2k-1})=m-1$.

###### Proof.

From [R, Section 4], $\operatorname{{\rm TC}}_{m}(S^{2k-1})=m-1$. So, we
obtain

$m-1\leq\operatorname{{\rm dTC}}_{m}(S^{2k-1})\leq\operatorname{{\rm
TC}}_{m}(S^{2k-1})=m-1$

due to Propositions 7.1 and 3.4. ∎

######  7.4 Remark.

In view of Corollary 7.3, the lower bounds obtained in Theorem 4.4 and
Proposition 7.1 are sharp for all $m\geq 2$. Also, the inequality in
Proposition 3.6 can be strict for all $m\geq 2$: this comes from Corollary 7.3
and [DJ, Proposition 6.7]. Furthermore, the inequality in Proposition 3.7 can
be strict, and $\\{\operatorname{{\rm dTC}}_{m}(X)\\}_{m\geq 2}$ can be a
strictly increasing linear sequence.

The following statement for closed orientable surfaces $\Sigma_{g}$ of genus
$g\geq 2$ generalizes [DJ, Proposition 6.9] for all $m,g\geq 2$.

######  7.5 Proposition.

For any $m,g\geq 2$, $\operatorname{{\rm dTC}}_{m}(\Sigma_{g})=2m$.

###### Proof.

Let us fix some $m,g\geq 2$. Let $a_{i},b_{i}\in
H^{1}(\Sigma_{g};{\mathbb{Q}})$ be the generators for $1\leq i\leq g$ such
that $a_{i}b_{j}=a_{i}a_{j}=b_{i}b_{j}=0$ for $i\neq j$ and
$a_{i}^{2}=b_{i}^{2}=0$. Due to Proposition 4.1, there exist
$a_{i}^{*},b_{i}^{*}\in H^{1}(SP^{(2m)!}(\Sigma_{g});{\mathbb{Q}})$ such that
$\partial_{2m}^{*}(a_{i}^{*})=a_{i}$ and $\partial_{2m}(b_{i}^{*})=b_{i}$. We
take the maps $\text{proj}_{i}^{*}$ and $r_{i}^{*}$ as defined in Proposition
7.1 for $X=\Sigma_{g}$. For brevity, take $Y=SP^{(2m)!}(\Sigma_{g})$ and
$Z=SP^{(2m)!}(\Sigma_{g}^{m})$. For $2\leq i\leq m$, consider the maps
$\psi,\phi_{i}:H^{1}(X;{\mathbb{Q}})\oplus H^{1}(X;{\mathbb{Q}})\to
H^{1}(X^{m};{\mathbb{Q}})$ defined as

$\psi(c\oplus d)=\text{proj}_{1}^{*}(c)\oplus\text{proj}_{2}^{*}(d)\hskip
14.22636pt\text{and}\hskip 14.22636pt\phi_{i}(c\oplus
d)=\text{proj}_{1}^{*}(c)-\text{proj}_{i}^{*}(d).$

Similarly, for all $2\leq i\leq m$, define
$\Psi,\Phi_{i}:H^{1}(Y;{\mathbb{Q}})\oplus H^{1}(Y;{\mathbb{Q}})\to
H^{1}(Z;{\mathbb{Q}})$ as

$\Psi(u\oplus v)=r_{1}^{*}(u)\oplus r_{2}^{*}(v)\hskip
14.22636pt\text{and}\hskip 14.22636pt\Phi_{i}(u\oplus
v)=r_{1}^{*}(u)-r_{i}^{*}(v).$

For each $2\leq i\leq m$, let $\alpha_{i}=\phi_{i}(a_{1}\oplus a_{1})$ and
$\beta_{i}=\phi_{i}(b_{1}\oplus b_{1})$, and $\gamma_{1}=\psi(a_{2}\oplus
a_{2})$ and $\gamma_{2}=\psi(b_{2}\oplus b_{2})$. Similarly, let
$\alpha_{i}^{*}=\Phi_{i}(a_{1}^{*}\oplus a_{1}^{*})$ and
$\beta_{i}^{*}=\Phi_{i}(b_{1}^{*}\oplus b_{1}^{*})$, and
$\gamma_{1}^{*}=\Psi(a_{2}^{*}\oplus a_{2}^{*})$ and
$\gamma_{2}^{*}=\Psi(b_{2}^{*}\oplus b_{2}^{*})$. For each $i$, we have the
following commutative diagram.

(7.3) ${H^{1}(Y;{\mathbb{Q}})\oplus
H^{1}(Y;{\mathbb{Q}})}$${H^{1}(Z;{\mathbb{Q}})}$${H^{1}(Y;{\mathbb{Q}})}$${H^{1}(X;{\mathbb{Q}})\oplus
H^{1}(X;{\mathbb{Q}})}$${H^{1}(X^{m};{\mathbb{Q}})}$${H^{1}(X;{\mathbb{Q}})}$$\scriptstyle{\Phi_{i}}$$\scriptstyle{\partial_{2m}^{*}\oplus\hskip
2.84526pt\partial_{2m}^{*}\hskip
2.84526pt}$$\scriptstyle{\Delta_{2m}^{*}}$$\scriptstyle{(\partial_{2m}^{m})^{*}}$$\scriptstyle{\partial_{2m}^{*}}$$\scriptstyle{\phi_{i}}$$\scriptstyle{\Delta^{*}}$

The same diagram exists upon replacing $\phi_{i}$ and $\Phi_{i}$ with $\psi$
and $\Psi$, respectively. We note that
$(\partial_{2m}^{m})^{*}(\alpha_{i}^{*})=\alpha_{i}$ and
$(\partial_{2m}^{m})^{*}(\beta_{i}^{*})=\beta_{i}$ for all $2\leq i\leq m$,
and $(\partial_{2m}^{m})^{*}(\gamma_{j}^{*})=\gamma_{j}$ for $j=1,2$. Also,

$\Delta^{*}(\alpha_{i})=\alpha_{i}-\alpha_{i}=0=\beta_{i}-\beta_{i}=\Delta^{*}(\beta_{i}),\hskip
5.69054pt\Delta^{*}(\gamma_{j})=\gamma_{j}-\gamma_{j}=0.$

due to [etal, Proposition 3.2]. Hence, by arguments used in Proposition 7.1,
we conclude that
$\alpha_{i}^{*},\beta_{i}^{*},\gamma_{j}^{*}\in\operatorname{{\rm
Ker}}(\Delta_{2m}^{*})$ for $2\leq i\leq m$ and $j=1,2$. Hence, the cohomology
classes are as in Theorem 4.4. Therefore, we have $\operatorname{{\rm
dTC}}_{m}(\Sigma_{g})\geq 2m$. But we have $\operatorname{{\rm
TC}}_{m}(\Sigma_{g})=2m$ from [etal, Proposition 3.2]. Hence,
$\operatorname{{\rm dTC}}_{m}(\Sigma_{g})=2m$ follows from Proposition 3.4. ∎

###  7.B. $\operatorname{{\rm dTC}}_{m}$ of spheres and their products

######  7.6 Proposition.

For any $k\geq 1$ and $m\geq 2$, $\operatorname{{\rm dTC}}_{m}(S^{2k})=m$.

###### Proof.

Let $u\in H^{2k}(S^{2k};{\mathbb{Q}})$ such that $u\neq 0$. By Proposition
4.1, there exists $w\in H^{2k}(SP^{m!}(S^{2k});{\mathbb{Q}})$ such that
$\partial_{m}^{*}(w)=u$. We consider the maps $\text{proj}_{i}^{*}$ and
$r_{i}^{*}$, for all $1\leq i\leq m$, from Proposition 7.1 here for
$X=S^{2k}$. Define

$\phi:\bigoplus_{i=1}^{m}H^{2k}(X;{\mathbb{Q}})\to H^{2k}(X^{m};{\mathbb{Q}})$

using the maps $\text{proj}_{i}^{*}$ as

$\phi\left(a_{1}\oplus\cdots\oplus
a_{m}\right)=\bigoplus_{i=1}^{m-1}\text{proj}_{i}^{*}\left(a_{i}\right)-\text{proj}_{m}^{*}\left(a_{m}\right).$

Again, we let $Y=SP^{m!}(X)$ and $Z=SP^{m!}(X^{m})$. Similarly, define

$\psi:\bigoplus_{i=1}^{m}H^{2k}(Y;{\mathbb{Q}})\to H^{2k}(Z;{\mathbb{Q}})$

using the maps $r_{i}^{*}$ as

$\psi\left(b_{1}\oplus\cdots\oplus
b_{m}\right)=\bigoplus_{i=1}^{m-1}r_{i}^{*}\left(b_{i}\right)-r_{m}^{*}\left(b_{m}\right).$

Let $\alpha=\phi(u\oplus\cdots\oplus u\oplus(m-1)u)$ and
$\alpha^{*}=\psi(w\oplus\cdots\oplus w\oplus(m-1)w)$. Then we get the
following commutative diagram.

(7.4)
${\bigoplus_{i=1}^{m-1}w\oplus(m-1)w}$${\alpha^{*}}$${(m-1)w-(m-1)w=0}$${\bigoplus_{i=1}^{m}H^{2k}(Y;{\mathbb{Q}})}$${H^{2k}(Z;{\mathbb{Q}})}$${H^{2k}(Y;{\mathbb{Q}})}$${\bigoplus_{i=1}^{m}H^{2k}(X;{\mathbb{Q}})}$${H^{2k}(X^{m};{\mathbb{Q}})}$${H^{2k}(X;{\mathbb{Q}})}$${\bigoplus_{i=1}^{m-1}u\oplus(m-1)u}$${\alpha}$${(m-1)u-(m-1)u=0}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\psi}$$\scriptstyle{\bigoplus_{i=1}^{m}\partial_{m}^{*}\hskip
2.84526pt}$$\scriptstyle{\Delta_{m}^{*}}$$\scriptstyle{(\partial_{m}^{m})^{*}}$$\scriptstyle{\partial_{m}^{*}}$$\scriptstyle{\phi}$$\scriptstyle{\Delta^{*}}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\in}$

Hence, $(\partial_{m}^{m})^{*}(\alpha^{*})=\alpha$. Due to the computations
done in [R, Section 4], $\alpha^{m}\neq 0$. Therefore, taking $k_{i}=2k$ and
$\alpha_{i}^{*}=\alpha^{*}$ for all $1\leq i\leq m$ in the statement of
Theorem 4.4, we get $\operatorname{{\rm dTC}}_{m}(S^{2k})\geq m$. But
$\operatorname{{\rm TC}}_{m}(S^{2k})=m$ due to [R, Section 4]. Hence,
$\operatorname{{\rm dTC}}_{m}(S^{2k})=m$ follows from Proposition 3.4. ∎

For cases $g=0,1$, we have from this and Example 6.6, respectively, that
$\operatorname{{\rm dTC}}_{m}(\Sigma_{0})=m$ and $\operatorname{{\rm
dTC}}_{m}(\Sigma_{1})=2(m-1)$.

######  7.7 Remark.

In view of Proposition 7.6 and [DJ, Proposition 6.7], the upper bound of
$\operatorname{{\rm dTC}}_{m}$ in Proposition 3.6 is sharp for all $m\geq 2$.
Also, the inequalities in Propositions 3.5 and 7.1 can be strict for all
$m\geq 2$.

Example 6.6 is recovered from the following more general result which, in
particular, extends [DJ, Proposition 6.8] to products of spheres all $m\geq
2$.

######  7.8 Proposition.

For any $m\geq 2$ and $n\geq 1$,

$\operatorname{{\rm
dTC}}_{m}\left(\prod_{j=1}^{n}S^{k_{j}}\right)=n(m-1)+l_{n},$

where $l_{n}$ is the number of $k_{j}$ that are even.

###### Proof.

For $n=1$, this holds due to Corollary 7.3 and Proposition 7.6. Here, we prove
this result for $n=2$. Let us fix some $m\geq 2$ and take $X=S^{k_{1}}\times
S^{k_{2}}$. For $j=1,2$, define a map $J_{j}:S^{k_{j}}\hookrightarrow X$ as
the inclusion into the $j^{th}$ coordinate, $\pi_{j}:X\to S^{k_{j}}$ as the
projection onto the $j^{th}$ coordinate,
$\Delta_{j}:S^{k_{j}}\hookrightarrow(S^{k_{j}})^{m}$ as the diagonal
embedding, and for each $1\leq t\leq m$ a map $p_{t}^{j}:(S^{k_{j}})^{m}\to
S^{k_{j}}$ as the projection onto the $t^{th}$ copy of $S^{k_{j}}$ and
$r_{t}^{j}=J_{j}\hskip 1.42262ptp_{t}^{j}\pi_{j}^{m}:X^{m}\to X$. Let

$d_{j}=\begin{cases}m&:k_{j}\text{ is even}\\\ m-1&:k_{j}\text{ is
odd}\end{cases}\hskip 14.22636pt\text{ and }\hskip
14.22636pte_{j}=\begin{cases}m&:k_{j}\text{ is even}\\\ 2&:k_{j}\text{ is
odd}\end{cases}$

Let $d=d_{1}+d_{2}$. Define $\partial:X\to Y$ and $\partial_{m}:X^{m}\to Z$ as
the diagonal embeddings, where $Y=SP^{d!}(X)$ and $Z=SP^{d!}(X^{m})$. Define
$R_{t}^{j}=SP^{d!}(r_{t}^{j}):Z\to Y$. For all $1\leq t\leq m$ and $j=1,2$,
let $(\text{proj}_{t}^{j})^{*}$ be the maps as defined in Proposition 7.1 for
$S^{k_{j}}$. We define the homomorphisms

$\phi_{i}^{j}:\bigoplus_{t=1}^{e_{j}}H^{k_{j}}(S^{k_{j}};{\mathbb{Q}})\to
H^{k_{j}}((S^{k_{j}})^{m};{\mathbb{Q}})$

for each $1\leq i\leq d_{j}$ and $j=1,2$ as

$\phi_{i}^{j}\left(\bigoplus_{t=1}^{e_{j}}a_{t}\right)=\begin{cases}\bigoplus_{t=1}^{e_{j}-1}(\text{proj}_{t}^{j})^{*}\left(a_{t}\right)-(\text{proj}_{m}^{j})^{*}\left(a_{m}\right)&:k_{j}\text{
is even}\\\
(\text{proj}_{i}^{j})^{*}\left(a_{1}\right)-(\text{proj}_{m}^{j})^{*}\left(a_{2}\right)&:k_{j}\text{
is odd}\end{cases}$

Let $u_{j}\in H^{k_{j}}(S^{k_{j}};{\mathbb{Q}})$ such that $u_{j}\neq 0$.
Then, it follows from Diagrams 7.2 and 7.4 that

(7.5)
$\left(\Delta_{j}^{*}\phi_{i}^{j}\right)\left(\bigoplus_{t=1}^{e_{j}}u_{j}\right)=\left(e_{j}-1\right)u_{j}-\left(e_{j}-1\right)u_{j}=0$

for each $i$ and $j$. In the same spirit, we define homomorphisms

$\psi_{i}^{j}:\bigoplus_{t=1}^{e_{j}}H^{k_{j}}(X;{\mathbb{Q}})\to
H^{k_{j}}(X^{m};{\mathbb{Q}})$

for each $1\leq i\leq d_{j}$ and $j=1,2$ as

$\psi_{i}^{j}\left(\bigoplus_{t=1}^{e_{j}}b_{t}\right)=\begin{cases}\bigoplus_{t=1}^{e_{j}-1}(r_{t}^{j})^{*}\left(b_{t}\right)-(r_{m}^{j})^{*}\left(b_{m}\right)&:k_{j}\text{
is even}\\\
(r_{i}^{j})^{*}\left(b_{1}\right)-(r_{m}^{j})^{*}\left(b_{2}\right)&:k_{j}\text{
is odd}\end{cases}$

Finally, we define

$\Psi_{i}^{j}:\bigoplus_{t=1}^{e_{j}}H^{k_{j}}(Y;{\mathbb{Q}})\to
H^{k_{j}}(Z;{\mathbb{Q}})$

for each $1\leq i\leq d_{j}$ and $j=1,2$ as

$\Psi_{i}^{j}\left(\bigoplus_{t=1}^{e_{j}}z_{t}\right)=\begin{cases}\bigoplus_{t=1}^{e_{j}-1}(R_{t}^{j})^{*}\left(z_{t}\right)-(R_{m}^{j})^{*}\left(z_{m}\right)&:k_{j}\text{
is even}\\\
(R_{i}^{j})^{*}\left(z_{1}\right)-(R_{m}^{j})^{*}\left(z_{2}\right)&:k_{j}\text{
is odd}\end{cases}$

Let $\eta=SP^{d!}(\Delta_{1}\times\Delta_{2}):Y\to Z$. Then for $j=1,2$, the
following diagram commutes for each $1\leq i\leq d_{j}$.

(7.6)
${\bigoplus_{t=1}^{e_{j}}H^{k_{j}}(Y;{\mathbb{Q}})}$${H^{k_{j}}(Z;{\mathbb{Q}})}$${H^{k_{j}}(Y;{\mathbb{Q}})}$${\bigoplus_{t=1}^{e_{j}}H^{k_{j}}(X;{\mathbb{Q}})}$${H^{k_{j}}(X^{m};{\mathbb{Q}})}$${H^{k_{j}}(X;{\mathbb{Q}})}$${\bigoplus_{t=1}^{e_{j}}H^{k_{j}}(S^{k_{j}};{\mathbb{Q}})}$${H^{k_{j}}((S^{k_{j}})^{m};{\mathbb{Q}})}$${H^{k_{j}}(S^{k_{j}};{\mathbb{Q}})}$$\scriptstyle{\Psi_{i}^{j}}$$\scriptstyle{\bigoplus_{t=1}^{e_{j}}\partial^{*}\hskip
2.84526pt}$$\scriptstyle{\eta^{*}}$$\scriptstyle{\partial_{m}^{*}}$$\scriptstyle{\partial^{*}}$$\scriptstyle{\psi_{i}^{j}}$$\scriptstyle{(\Delta_{1}\times\Delta_{2})^{*}}$$\scriptstyle{\phi_{i}^{j}}$$\scriptstyle{\bigoplus_{t=1}^{e_{j}}\pi_{j}^{*}\hskip
2.84526pt}$$\scriptstyle{\Delta_{j}^{*}}$$\scriptstyle{(\pi_{j}^{m})^{*}}$$\scriptstyle{\pi_{j}^{*}}$

Let $y_{j}=\pi_{j}^{*}(u_{j})\in H^{k_{j}}(X;{\mathbb{Q}})$. For each $1\leq
p\leq d_{1}$ and $1\leq q\leq d_{2}$, let us write

$\alpha_{p}=\psi_{p}^{1}\left(\bigoplus_{t=1}^{e_{1}-1}y_{1}\oplus(e_{1}-1)y_{1}\right)\hskip
14.22636pt\text{and}\hskip
14.22636pt\beta_{q}=\psi_{q}^{2}\left(\bigoplus_{t=1}^{e_{2}-1}y_{2}\oplus(e_{2}-1)y_{2}\right).$

Then, it follows from the proof of [BGRT, Theorem 3.10] that

$\alpha_{1}\smile\cdots\smile\alpha_{d_{1}}\smile\beta_{1}\smile\cdots\smile\beta_{d_{2}}\neq
0.$

Also, from Equation 7.5 and Diagram 7.6,

$(\Delta_{1}\times\Delta_{2})^{*}(\alpha_{p})=0=(\Delta_{1}\times\Delta_{2})^{*}(\beta_{q}).$

Due to Proposition 4.1, there exist some $y_{j}^{*}\in
H^{k_{j}}(Y;{\mathbb{Q}})$ such that $\partial^{*}(y_{j}^{*})=y_{j}$. For each
$1\leq p\leq d_{1}$ and $1\leq q\leq d_{2}$, let

$\alpha_{p}^{*}=\Psi_{p}^{1}\left(\bigoplus_{t=1}^{e_{1}-1}y_{1}^{*}\oplus(e_{1}-1)y_{1}^{*}\right)\hskip
14.22636pt\text{and}\hskip
14.22636pt\beta_{q}^{*}=\Psi_{q}^{2}\left(\bigoplus_{t=1}^{e_{2}-1}y_{2}^{*}\oplus(e_{2}-1)y_{2}^{*}\right).$

Then, $\partial_{m}^{*}(\alpha_{p}^{*})=\alpha_{p}$ and
$\partial_{m}^{*}(\beta_{q}^{*})=\beta_{q}$ by Diagram 7.6. The top (resp.
middle and bottom) row when restricted on the left to either of the components
$H^{k_{j}}(Y;{\mathbb{Q}})$ (resp. $H^{k_{j}}(X;{\mathbb{Q}})$ and
$H^{k_{j}}(S^{k_{j}};{\mathbb{Q}})$) is an isomorphism. Therefore, we get

$\eta^{*}(\alpha_{p}^{*})=\alpha_{p}^{*}-\alpha_{p}^{*}=0=\beta_{q}^{*}-\beta_{q}^{*}=\eta^{*}(\beta_{q}^{*})$

for all $1\leq p\leq d_{1}$ and $1\leq q\leq d_{2}$. We note that
$\Delta_{1}\times\Delta_{2}:X\to X^{m}$ is the diagonal embedding. Hence, the
inequality $\operatorname{{\rm dTC}}_{m}(S^{k_{1}}\times S^{k_{2}})\geq
d=2(m-1)+l_{2}$ follows from Theorem 4.4, where $l_{2}\in\\{0,1,2\\}$
depending on the parity of $k_{1}$ and $k_{2}$.

In general, for any $n\geq 3$, the same technique can be used to get the
inequality

$\operatorname{{\rm dTC}}_{m}\left(\prod_{j=1}^{n}S^{k_{j}}\right)\geq
n(m-1)+l_{n}.$

But due to [BGRT, Corollary 3.12], $\operatorname{{\rm
TC}}_{m}(\prod_{j=1}^{n}S^{k_{j}})=n(m-1)+l_{n}$. Hence, $\operatorname{{\rm
dTC}}_{m}(\prod_{j=1}^{n}S^{k_{j}})=n(m-1)+l_{n}$ follows from Proposition
3.4. ∎

For cases $g=0,1$, this gives $\operatorname{{\rm
dTC}}_{m}(\Sigma_{0}\times\prod_{j=1}^{n}S^{k_{j}})=m(n+1)-n+l_{n}$ and
$\operatorname{{\rm
dTC}}_{m}(\Sigma_{1}\times\prod_{j=1}^{n}S^{k_{j}})=(m-1)(n+2)+l_{n}$.

######  7.9 Remark.

Due to Proposition 7.8, the inequality in Proposition 3.8 can be strict for
all $m\geq 2$ for non-contractible spaces $X$ and $Y$.

The techniques used in the proof of Propositions 7.8 and 7.5 can be used more
generally to obtain the following.

######  7.10 Proposition.

For any $m,g\geq 2$ and $n\geq 1$,

$\operatorname{{\rm
dTC}}_{m}\left(\Sigma_{g}\times\prod_{j=1}^{n}S^{k_{j}}\right)=m(n+2)-n+l_{n},$

where $l_{n}$ is the number of $k_{j}$ that are even.

###### Proof.

Take $A=\Sigma_{g}$ and $B=\prod_{j=1}^{n}S^{k_{j}}$. From Proposition 7.5,
there are $2m$ number of cohomology classes, say $\alpha_{i}\in
H^{1}(A^{m};{\mathbb{Q}})$, that vanish under the mapping
$\Delta_{1}^{*}:H^{1}(A^{m};{\mathbb{Q}})\to H^{1}(A;{\mathbb{Q}})$. From
Proposition 7.8, there are $d=n(m-1)+l_{n}$ classes, say $\beta_{k}\in
H^{e}(B^{m};{\mathbb{Q}})$, that vanish under
$\Delta_{2}^{*}:H^{e}(B^{m};{\mathbb{Q}})\to H^{e}(B;{\mathbb{Q}})$, where
$e=\sum_{j=1}^{n}k_{j}$. In the proof of Proposition 7.8 for $n=2$, replace
the roles of spheres $S^{k_{1}}$ and $S^{k_{2}}$ with $A$ and $B$,
respectively, and let $X=A\times B$. Then from a direct analog of Diagram 7.6
for $A$ and $B$, we get images $\alpha_{i}^{*}\in H^{1}(X^{m};{\mathbb{Q}})$
and $\beta_{k}^{*}\in H^{e}(X^{m};{\mathbb{Q}})$ such that

$(\Delta_{1}\times\Delta_{2})^{*}(\alpha_{i}^{*})=0=(\Delta_{1}\times\Delta_{2})^{*}(\beta_{k}^{*}),$

and elements $\alpha_{i}^{**}\in H^{1}(Z;{\mathbb{Q}})$ and $\beta_{k}^{**}\in
H^{e}(Z;{\mathbb{Q}})$ satisfying

$\eta^{*}(\alpha_{i}^{**})=0=\eta^{*}(\beta_{k}^{**})$

such that $\partial_{m}^{*}(\alpha_{i}^{**})=\alpha_{i}^{*}$ and
$\partial_{m}^{*}(\beta_{k}^{**})=\beta_{k}^{*}$ for all $1\leq i\leq 2m$ and
$1\leq k\leq d$. From the proof of [BGRT, Theorem 3.10], we deduce that

$\alpha_{1}^{*}\smile\cdots\smile\alpha_{2m}^{*}\smile\beta_{1}^{*}\smile\cdots\smile\beta_{d}^{*}\neq
0.$

Hence, $\alpha_{i}^{**},\beta_{i}^{**}\in H^{*}(Z;{\mathbb{Q}})$ are $2m+d$
number of cohomology classes satisfying the conditions of Theorem 4.4.
Therefore, $\operatorname{{\rm dTC}}_{m}(X)\geq 2m+d=m(n+2)-n+l_{n}$. Note
that $X^{m}$ is a normal space, $\operatorname{{\rm TC}}_{m}(A)=2m$ by [etal,
Proposition 3.2], and $\operatorname{{\rm TC}}_{m}(B)=n(m-1)+l_{n}$ by [BGRT,
Corollary 3.12]. So,

$m(n+2)-n+l_{n}\leq\operatorname{{\rm dTC}}_{m}(X)\leq\operatorname{{\rm
TC}}_{m}(X)\leq\operatorname{{\rm TC}}_{m}(A)+\operatorname{{\rm
TC}}_{m}(B)=m(n+2)-n+l_{n},$

where the second-last inequality is due to [BGRT, Proposition 3.11]. ∎

For classical $\operatorname{{\rm TC}}_{m}$, the above set of inequalities
immediately implies that

$\operatorname{{\rm
TC}}_{m}\left(\Sigma_{g}\times\prod_{j=1}^{n}S^{k_{j}}\right)=m(n+2)-n+l_{n}.$

######  7.11 Remark.

The sequences $\\{\operatorname{{\rm dTC}}_{m}(X)\\}_{m\geq 2}$ and
$\\{\operatorname{{\rm TC}}_{m}(X)\\}_{m\geq 2}$ coincide for various classes
of spaces $X$: spheres, products of spheres, $\Sigma_{g}$, and product of
$\Sigma_{g}$ with products of spheres. In particular, the upper bound of
$\operatorname{{\rm dTC}}_{m}$ in Proposition 3.4 is sharp for all these
classes of spaces.

##  8\. The Analog Invariants

For a metric space $(X,d)$ and $n\geq 1$, let us consider the following
quotient map from [KW, Definition 2.1]:

$q:\bigsqcup_{i=1}^{n}\left(X^{i}\times\Delta^{i-1}\right)\to\bigsqcup_{i=1}^{n}\left(X^{i}\times\Delta^{i-1}\right)/\sim\hskip
5.69054pt=:\mathscr{P}_{n}(X).$

Let $\mathscr{T}_{1}$ be the quotient topology on $\mathscr{P}_{n}(X)$. For
any fibration $p:E\to B$, define
$\mathscr{P}_{n}(p):=\\{\mu\in\mathscr{P}_{n}(X)\mid|\hskip
0.85358ptp(\operatorname{{\rm supp}}(\mu))|=1\\}$. See that as sets,
$\mathscr{P}_{n}(X)=\mathcal{B}_{n}(X)$ and $\mathscr{P}_{n}(p)=E_{n}(p)$. Let
us choose some $\mu=\lambda_{x}\hskip 1.42262ptx\in\mathcal{B}_{n}(X)$ and fix
it. Let $\delta=\text{min}\\{d(x,y)\mid x,y\in\text{supp}(\mu),x\neq y\\}$.
Then for $\epsilon<\delta/2$, the collection of the sets

$\mathscr{U}(\mu,\epsilon)=\left\\{\sum s_{y}\hskip 1.42262pty\hskip
2.84526pt\middle|\hskip 2.84526pt-\frac{\epsilon}{n}<\sum_{y\in
B(x,\epsilon)}s_{y}-\lambda_{x}<\frac{\epsilon}{n}\right\\}$

defines a local basis at $\mu$ in the topology induced by the Lévy-Prokhorov
metric on $\mathcal{B}_{n}(X)$ [DJ]. Let us denote by $\mathscr{T}_{2}$ the
Lévy-Prokhorov topology. These sets are also considered in Construction 8.3 of
[KW], and it follows from the proof of the first part of their Lemma 8.4 that
the sets $\mathscr{U}(\mu,\epsilon)$ are open in the quotient topology of
$\mathscr{P}_{n}(X)$. Hence, $\mathscr{T}_{2}\subset\mathscr{T}_{1}$. So, the
identity map
$\mathcal{I}^{\prime}:(\mathscr{P}_{n}(X),\mathscr{T}_{1})\to(\mathcal{B}_{n}(X),\mathscr{T}_{2})$
is continuous and thus, so is its restriction
$\mathcal{I}:(\mathscr{P}_{n}(p),\mathscr{T}_{1})\to(E_{n}(p),\mathscr{T}_{2})$.

###  8.A. Comparison between the invariants

For a fibration $p:E\to B$, its analog sectional category, denoted
$\text{asecat}(p)$, is the minimal number $n$ such that the continuous map
$\varphi_{n+1}(p):(\mathscr{P}_{n+1}(p),\mathscr{T}_{1})\to B$, defined as
$\varphi_{n+1}(p):\mu\mapsto p(\operatorname{{\rm supp}}(\mu))$, admits a
section [KW, Definition 5.1].

######  8.1 Lemma.

For any fibration $p:E\to B$, $d\hskip
1.42262pt\textup{secat}(p)\leq\textup{asecat}(p)$.

###### Proof.

Let $\text{asecat}(p)=n-1$. So, there exists a section of $\varphi_{n}(p)$,
say

$\kappa:B\to(\mathscr{P}_{n}(p),\mathscr{T}_{1}).$

Consider $\kappa^{\prime}=\mathcal{I}\hskip
1.42262pt\kappa:B\to(E_{n}(p),\mathscr{T}_{2})$. Since $\mathcal{I}$ is
continuous, so is $\kappa^{\prime}$. It is clear that
$\varphi_{n}(p)=\mathcal{B}_{n}(p)\hskip 1.42262pt\mathcal{I}$. Next, see that

$\mathbbm{1}_{B}=\varphi_{n}(p)\hskip 1.42262pt\kappa=\mathcal{B}_{n}(p)\hskip
1.42262pt\mathcal{I}\hskip 1.42262pt\kappa=\mathcal{B}_{n}(p)\hskip
1.42262pt\kappa^{\prime}.$

So, $\kappa^{\prime}$ is a section of
$\mathcal{B}_{n}(p):(E_{n}(p),\mathscr{T}_{1})\to B$. Thus, $d\hskip
1.42262pt\text{secat}(p)\leq n-1$. ∎

Since $d\hskip 1.42262pt\text{secat}$, asecat, and secat are all homotopy
invariants, we have for any continuous map $f:X\to Y$ the inequalities

(8.1) $d\hskip
1.42262pt\textup{secat}(f)\leq\textup{asecat}(f)\leq\textup{secat}(f),$

where the last inequality is because of [KW, Corollary 5.5].

######  8.2 Corollary.

$d\operatorname{cat}(X)\leq\operatorname{{\rm acat}}(X)$ and
$\operatorname{{\rm dTC}}_{m}(X)\leq\operatorname{{\rm ATC}}_{m}(X)$ for all
$m\geq 2$.

###### Proof.

This follows directly from Lemma 8.1 since $d\operatorname{cat}(X)=d\hskip
1.42262pt\text{secat}(p_{0}^{X})$ and $\operatorname{{\rm dTC}}_{m}(X)=d\hskip
1.42262pt\text{secat}(\pi_{m}^{X})$, and $\operatorname{{\rm
acat}}(X)=\text{asecat}(p_{0}^{X})$ and $\operatorname{{\rm
ATC}}_{m}(X)=\text{asecat}(\pi_{m}^{X})$ for the evaluation fibrations
$p_{0}^{X}:P_{0}(X)\to X$ and $\pi_{m}^{X}:P(X)\to X^{m}$. ∎

This and [KW, Corollary 6.6] together imply

$\operatorname{{\rm dTC}}_{m}({\mathbb{R}}P^{n})\leq\operatorname{{\rm
ATC}}_{m}({\mathbb{R}}P^{n})\leq 2m+1$

for all $n\geq 1$, thereby significantly improving our bound from Proposition
3.9. Similarly, for a finite group $\Gamma$, we get

(8.2) $\operatorname{{\rm dTC}}_{m}(\Gamma)=\operatorname{{\rm
dTC}}_{m}(B\Gamma)\leq\operatorname{{\rm ATC}}_{m}(B\Gamma)=|\Gamma|-1$

due to Corollary 8.2 and [KW, Corollary 7.3]. Thus, taking
$\Gamma=\mathbb{Z}_{2}$, we conclude $\operatorname{{\rm
dTC}}_{m}({\mathbb{R}}P^{\infty})=1$, which is another major improvement to
our calculation.

We note that Equation 8.2 can provide us with many more examples of spaces $X$
where the inequality in Proposition 3.4 is strict.

###  8.B. Completing some computations for $\operatorname{{\rm acat}}$ and
$\operatorname{{\rm ATC}}_{m}$

With the help of our computations from Sections 6 and 7, and [DJ, Section 6],
we complete some computations of [KW] using Corollary 8.2.

######  8.3 Remark.

For a torsion-free, discrete group $\Gamma$, $\operatorname{{\rm
ATC}}_{m}(B\Gamma)\leq\operatorname{{\rm cd}}(\Gamma^{m})$ holds for all
$m\geq 2$ [KW, Corollary 7.7]. For closed orientable surfaces $\Sigma_{g}$
with $g\geq 1$, this gives $\operatorname{{\rm ATC}}_{m}(\Sigma_{g})\leq 2m$.
However, this is obvious in light of Equation 8.1 and doesn’t help much in
finding the exact value $\operatorname{{\rm ATC}}_{m}$, unless one gets into
the technicalities of [KW, Theorem 7.9] for cases $g\geq 2$. Also, for the
$n$-torus $T^{n}$, the above inequality yields $\operatorname{{\rm
ATC}}_{m}(T^{n})\leq nm$. This upper bound is poorer than the bound $n(m-1)$
implied by Equation 8.1. Hence, it’s also not helpful.

For non-aspherical spaces $X$, such as spheres $S^{n}$ for $n\geq 2$ and their
products, there is no general recipe as such for computing
$a\operatorname{cat}(X)$ or $\operatorname{{\rm ATC}}_{m}(X)$.

We now show in the following two examples that the exact values of
$\operatorname{{\rm ATC}}_{m}$ can be determined easily for various classes of
nice closed manifolds, including some of the non-aspherical spaces.

######  8.4 Example.

Due to Remark 7.11, Equation 8.1, and Corollary 8.2, if $X$ is a sphere, a
product of spheres, $\Sigma_{g}$, or a product of $\Sigma_{g}$ with a product
of spheres, we have

$\operatorname{{\rm TC}}_{m}(X)=\operatorname{{\rm
dTC}}_{m}(X)\leq\operatorname{{\rm ATC}}_{m}(X)\leq\operatorname{{\rm
TC}}_{m}(X).$

######  8.5 Example.

Similarly, due to [DJ, Propositions 6.1, 6.4, 6.6, and 6.7], if $X$ is a
closed surface, a sphere, a product of spheres, or ${\mathbb{C}}P^{n}$, we
have

$\operatorname{cat}(X)=d\operatorname{cat}(X)\leq\operatorname{{\rm
acat}}(X)\leq\operatorname{cat}(X).$

In light of the above examples, all the lower bounds in Corollary 8.2 are
sharp.

## Acknowledgement

The author would like to thank Alexander Dranishnikov for his kind guidance
and various insightful and helpful discussions.

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