# Intertwining category and complexity

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

We develop the theory of the intertwining distributional versions of the LS-
category and the sequential topological complexities of a space $X$, denoted
by $i\operatorname{\mathsf{cat}}(X)$ and $i\operatorname{\mathsf{TC}}_{m}(X)$,
respectively. We prove that they satisfy most of the nice properties as their
respective distributional counterparts $d\operatorname{\mathsf{cat}}(X)$ and
$d\operatorname{\mathsf{TC}}_{m}(X)$, and their classical counterparts
$\operatorname{\mathsf{cat}}(X)$ and $\operatorname{\mathsf{TC}}_{m}(X)$, such
as homotopy invariance and special behavior on topological groups. We show
that the notions of $i\operatorname{\mathsf{TC}}_{m}$ and
$d\operatorname{\mathsf{TC}}_{m}$ are different for each $m\geq 2$ by proving
that $i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1$ for all $m\geq 2$ for
Higman’s group ${\mathcal{H}}$. Using cohomological lower bounds, we also
provide various examples of locally finite CW complexes $X$ for which
$i\operatorname{\mathsf{cat}}(X)>1$, $i\operatorname{\mathsf{TC}}_{m}(X)>1$,
$i\operatorname{\mathsf{cat}}(X)=d\operatorname{\mathsf{cat}}(X)=\operatorname{\mathsf{cat}}(X)$,
and
$i\operatorname{\mathsf{TC}}(X)=d\operatorname{\mathsf{TC}}(X)=\operatorname{\mathsf{TC}}(X)$.

###### Key words and phrases:

Sequential intertwining topological complexity, intertwining Lusternik-
Schnirelmann category, resolvable paths, probability measures, sequential
distributional topological complexity, intertwined navigation algorithm

###### 2020 Mathematics Subject Classification:

Primary 55M30; Secondary 60B05, 54A10, 68T40.

##  1\. Introduction

In the field of robotics, the ultimate aim is to develop autonomously
functioning mechanical systems that can understand well-defined descriptions
of tasks and execute them without any further human intervention. One of the
simplest such tasks is navigating the system through a sequence of positions
inside a configuration space $X$. Constructing a robot that can accomplish
this task autonomously is sometimes called the motion planning problem.

We consider the following sequential motion planning problem. Given a system
with configuration space $X$, a number $m\geq 2$, and positions
$x_{1},x_{2},\ldots,x_{m}$ in $X$, find a continuous motion planning algorithm
that takes as input the ordered $m$-tuple of positions
$\overline{x}=(x_{1},x_{2},\ldots,x_{m})\in X^{m}$ and produces as output the
continuous motion of the system from $x_{1}$ to $x_{m}$ via the $m-2$
intermediate positions $x_{2},\ldots,x_{m-1}$ attained in that order.

M. Farber observed in [Far1] that even when $m=2$ and the system just needs to
move from an initial position to a final position in $X$, such a continuous
motion planning algorithm does not exist on whole $X\times X$ unless $X$ is
contractible. So, he developed in [Far1], [Far2] the notion of topological
complexity of $X$, denoted $\operatorname{\mathsf{TC}}(X)$, which is one less
than the minimum number of subspaces into which $X\times X$ needs to be
partitioned such that a continuous (partial) motion planning algorithm can
exist on each of the subspaces. In a sense, $\operatorname{\mathsf{TC}}(X)$
provides the minimum number of rules required to plan the above motion, and
thus, gives a measure of the complexity in planning the motion. For any
general $m\geq 2$, the idea of topological complexity was extended by Y. B.
Rudyak in [R] in a natural way to get the $m^{th}$ sequential topological
complexity of $X$, denoted $\operatorname{\mathsf{TC}}_{m}(X)$, which measures
the complexity of planning the sequential motion.

Ideally, one would want to minimize the complexity of motion planning. Given a
configuration space $X$, this can be achieved for some advanced robotic
systems with special features. In a recent work with A. Dranishnikov [DJ], we
considered systems that can break into finitely many weighted pieces at the
initial position $x$ so that all the pieces travel independently to the
desired final position $y$ where they can reassemble back into the system. The
continuous motion can now be planned on $X\times X$ using the unordered
collection of the system’s weighted pieces instead of using the system as a
whole. We let $n$ be the maximum number of pieces that the system can break
into while traveling between all possible pairs of positions $(x,y)$. To
measure the complexity of such a motion, the notion of the distributional
topological complexity of $X$, denoted $d\operatorname{\mathsf{TC}}(X)$, was
introduced in [DJ] as one less than the minimum value of $n$ for which the
system can achieve such a motion by traveling as smaller pieces. Whenever $X$
is non-contractible, the system must break into at least two pieces for a
continuous motion planning algorithm to exist.

If we consider repeated breaking and reassembling of such a system in the same
weighted pieces, then the sequential motions can be planned. This natural
extension of $d\operatorname{\mathsf{TC}}$ was obtained by us recently in [J],
where the generalized notion of the $m^{th}$ sequential distributional
topological complexity of $X$, denoted $d\operatorname{\mathsf{TC}}_{m}(X)$,
was studied. Around the same time, B. Knudsen and S. Weinberger independently
developed in [KW] their probabilistic version of
$\operatorname{\mathsf{TC}}_{m}$, called the $m^{th}$ sequential analog
topological complexity, and denoted $A\operatorname{\mathsf{TC}}_{m}(X)$ for a
space $X$.

In [Far2], [Far3], Farber had seen $\operatorname{\mathsf{TC}}$ as a measure
of the minimal level of randomness in motion planning algorithms. To measure
this randomness, he considered ordered probability distributions. This is
equivalent to the breaking of the system into an ordered collection of pieces
as opposed to the unordered collection of pieces considered in the definition
of $d\operatorname{\mathsf{TC}}_{m}$. Thus,
$d\operatorname{\mathsf{TC}}_{m}(X)\leq\operatorname{\mathsf{TC}}_{m}(X)$.
Similarly, since [KW] argued that their notions measure analog randomness by
considering unordered probability distributions, it follows that
$A\operatorname{\mathsf{TC}}_{m}(X)\leq\operatorname{\mathsf{TC}}_{m}(X)$.
Even though there is a formal difference in the definition of
$d\operatorname{\mathsf{TC}}_{m}$ and $A\operatorname{\mathsf{TC}}_{m}$ (see
[Dr, Section 1] and Section 1.C), it was conjectured in [KW, Section 1] that
the two notions coincide on metric spaces. In that direction, we showed in [J,
Section 8] that $d\operatorname{\mathsf{TC}}_{m}(X)\leq
A\operatorname{\mathsf{TC}}_{m}(X)$. So, given an advanced system with a
configuration space $X$, the notion of $d\operatorname{\mathsf{TC}}_{m}(X)$
seems the most promising so far for minimizing the complexity of the motion
planning algorithms on $X$.

###  1.A. Intertwining motion

A further improvement to $d\operatorname{\mathsf{TC}}$ was proposed in the
Epilogue of our earlier work [DJ]. The basic idea is again of breaking and
reassembling the system, but this time, while traveling between a pair of
positions $x$ and $y$ in $X$, we allow the pieces of the system to intertwine
with each other and not necessarily have them travel independently as in the
case of $d\operatorname{\mathsf{TC}}$. More precisely, during the motion, the
weighted pieces are allowed to join with each other and break further into
differently weighted pieces in a controlled way such that the unordered
probability distribution of the motion from $x$ to $y$ remains unchanged. When
two pieces join, their weights add up. When a piece splits into two or more
pieces, its weight splits accordingly to give weights to the new pieces.
Unlike the case of $d\operatorname{\mathsf{TC}}$ where a specific motion of
the system from $x$ to $y$ was obtained only from a unique unordered
collection of weighted pieces, this new approach allows multiple unordered
collections of weighted pieces of the system to produce the same motion (see
Section 3.B for various such examples). Therefore, we now measure the
complexity of the motion planning using only the image of the system’s motion
rather than using the unordered collections of its weighted pieces that create
that motion.

Of course, this approach offers more freedom of motion; therefore, one can
expect to have even lesser navigational complexity using this approach. To
measure it, the notion of the intertwining distributional topological
complexity of $X$, denoted $i\operatorname{\mathsf{TC}}(X)$, was proposed in
[DJ] as one less than the minimum value of $n$ for which the system can
achieve a continuous motion by traveling as smaller pieces that can possibly
intertwine with each other. With this approach, a sequential motion of the
system can also be planned in the same way as before, whose complexity can
then be measured by the generalized notion of the $m^{th}$ sequential
intertwining topological complexity of $X$ that we will denote by
$i\operatorname{\mathsf{TC}}_{m}(X)$ and study in this paper. It is
intuitively clear that $i\operatorname{\mathsf{TC}}_{m}(X)\leq
d\operatorname{\mathsf{TC}}_{m}(X)$.

###  1.B. Comparison

At this stage, one may ask: why another notion of complexity and a new
sequence of invariants? As mentioned above, our target is to improve robot
motion planning by minimizing the complexity of motion planning for systems
with a given configuration space. These new notions of
$d\operatorname{\mathsf{TC}}_{m}$ and $i\operatorname{\mathsf{TC}}_{m}$ make
it possible by offering lower bounds to the respective notion of
$\operatorname{\mathsf{TC}}_{m}$ and reducing complexity in some cases. For
the simple problem of planning the motion of a rotating line in
${\mathbb{R}}^{k+1}$ that is fixed along a revolving joint at a base point,
$d\operatorname{\mathsf{TC}}$ offers a significantly better solution than
$\operatorname{\mathsf{TC}}$. While in the classical setting of
$\operatorname{\mathsf{TC}}$, the minimum number of rules required to plan
such a motion equals one more than the immersion dimension of
${\mathbb{R}}P^{k}$ when $k\neq 1,3,7$ and $k+1$ otherwise [FTY], in the
distributional setting, only two rules are enough to plan this motion for all
$k\geq 1$, [DJ], [KW].

Of course, another important reason is that these new notions bring with them
their LS-category versions ($i\operatorname{\mathsf{cat}}$ and
$d\operatorname{\mathsf{cat}}$) and encourage us to get examples of spaces on
which they disagree with the old notions, hence creating different theories.
Several ways in which the theory of the distributional invariants differs from
that of the classical invariants are highlighted in [DJ], [J]. In the same
spirit, it was shown in [Dr, Section 6] that
$i\operatorname{\mathsf{TC}}({\mathcal{H}})=1$ for Higman’s group
${\mathcal{H}}$ (which was introduced in [Hi]). As a consequence,
$i\operatorname{\mathsf{cat}}({\mathcal{H}})=1$ is obtained, thereby breaking
the Eilenberg–Ganea theorem [EG] in the case of intertwining invariants. It is
noteworthy that a version of the Eilenberg–Ganea theorem was proven for
torsion-free groups in the distributional case, [KW], [Dr]. To see more
differences between the theories of the intertwining and the distributional
invariants, we show in Section 5 of this paper that
$d\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})$ is at least $2(m-1)$ whereas
$i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1$. So, there are cases where
these new notions offer strict improvements over the previous notions as far
as motion planning is concerned. Hence, it seems worthwhile to develop their
properties and understand them better.

###  1.C. Continuous motion planning algorithm

For a metric space $Z$, let $\mathcal{B}(Z)$ denote the set of probability
measures on $Z$ and for any $n\geq 1$, let

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

denote the space of probability measures on $Z$ supported by at most $n$
points, equipped with the Lévy-Prokhorov metric [P]. The support of any
$\mu\in\mathcal{B}_{n}(Z)$ is given by $\operatorname{{\rm supp}}(\mu)=\\{z\in
Z\mid\lambda_{z}>0\\}$. There is an embedding
$\omega_{n}:Z\hookrightarrow{\mathcal{B}}_{n}(Z)$ defined as
$\omega_{n}:x\to\delta_{x}$ that sends $x\in Z$ to the Dirac measure
$\delta_{x}$ supported at $x$.

When $Z$ is non-metrizable, an alternate description of the space
${\mathcal{B}}_{n}(Z)$ with another topology is given as follows. Let $*^{n}Z$
denote the $n^{th}$ iterated join of $Z$ in the sense of [Mi]. The group
$S_{n}$ acts on $*^{n}Z$ by permutation of coordinates. The orbit space of
this action is the $n^{th}$ iterated symmetric join $\text{Sym}(*^{n}Z)$. Then
as in [KK], ${\mathcal{B}}_{n}(Z)=\text{Sym}(*^{n}Z)/\sim$ where
$t_{1}x_{1}+t_{2}x_{1}+\cdots+t_{n}x_{n}\sim(t_{1}+t_{2})x_{1}+\cdots+t_{n}x_{n}$.
This is also called the barycenter space of $Z$. Let
$q:\text{Sym}(*^{n}Z)\to{\mathcal{B}}_{n}(Z)$ be the quotient map of the above
relation that sends unordered formal linear combinations of size $n$ to
probability measures of support at most $n$. Using $q$, the quotient topology
$\mathscr{T}_{1}$ is induced on ${\mathcal{B}}_{n}(Z)$. In [KW], the topology
$\mathscr{T}_{1}$ was considered while defining the analog invariants, which
could be different from the metric topology $\mathscr{T}_{2}$ considered in
[DJ], [J]. It was explained in [J, Section 8] that the identity map
$\mathcal{I}:({\mathcal{B}}_{n}(Z),\mathscr{T}_{1})\to({\mathcal{B}}_{n}(Z),\mathscr{T}_{2})$
is continuous when $Z$ is a metric space. Furthermore, if $Z$ is a locally
finite CW complex or a discrete space, then it is possible to show that
$\mathcal{I}$ is a homotopy equivalence, [Dr, Section 1]. So, the definitions
of the distributional invariants and the analog invariants agree on such
spaces. Therefore, we can study the intertwining invariants on these spaces as
lower bounds to their respective distributional invariants by working in the
space $({\mathcal{B}}_{n}(Z),\mathscr{T}_{2})$.

Let $Z=P(X)=\\{f\hskip 2.84526pt|\hskip 2.84526ptf:[0,1]\to X\\}$ be 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 $\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}$, where
$t_{i}=(i-1)/(m-1)$ for all $1\leq i\leq m$. Also, for Dirac measures
$\delta_{x_{i}}$, let

$P(\delta_{x_{1}},\ldots,\delta_{x_{m}})=\\{g\in P({\mathcal{B}}_{n}(X))\mid
g(t_{i})=\delta_{x_{i}}\text{ for all }i\\}.$

Consider the continuous map $\Phi_{n}:{\mathcal{B}}_{n}(P(X))\to
P({\mathcal{B}}_{n}(X))$ defined as

$\Phi_{n}\left(\sum\lambda_{\alpha}\alpha\right)(t)=\sum\lambda_{\alpha}\alpha(t)$

for all $t\in I=[0,1]$. Since we are concerned with the image of the system’s
motion as a consequence of repeated breaking and reassembling and possible
intertwining of pieces, the desired sequential motion planning algorithm is a
continuous assignment of each $\overline{x}\in X^{m}$ to a path in
$P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$. But since we also want the
distribution of the sequential motion from $x_{1}$ to $x_{m}$ to remain
unchanged, we additionally require our path in
$P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$ to have a preimage in
${\mathcal{B}}_{n}(P(X))$ under the map $\Phi_{n}$. Thus, the desired
$n$-intertwined $m$-navigation algorithm on $X$ is a continuous map

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

such that for each $\overline{x}\in X^{m}$, we have $s_{m}(\overline{x})\in
P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$ and
$\Phi_{n}^{-1}(s_{m}(\overline{x}))\neq\emptyset$. Such paths in
$P({\mathcal{B}}_{n}(X))$ that have a preimage in ${\mathcal{B}}_{n}(P(X))$
under $\Phi_{n}$ are called resolvable paths and their preimages are called
their resolver measures. Like the case of $d\operatorname{\mathsf{TC}}_{m}$
[J, Section 1.B], as we aim to minimize the maximum number of pieces $n$ into
which the system can break, we obtain the notion of
$i\operatorname{\mathsf{TC}}_{m}$.

###  1.D. About this paper

This paper is organized as follows. In Section 2, we revisit the theory of the
classical and the distributional notions of LS-category and (sequential)
topological complexity. In Section 3, we study resolvers and the space of
resolvable paths. The intertwining invariants are formally defined and studied
in Section 4 where we prove several properties for them, such as their
homotopy invariance, behavior on products of spaces, and their relationship
with each other and with their respective distributional counterparts. In
Section 5, we establish that the notions of $d\operatorname{\mathsf{TC}}_{m}$
and $i\operatorname{\mathsf{TC}}_{m}$ are different for each $m\geq 2$ by
showing that $i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1$ for Higman’s
group ${\mathcal{H}}$. Section 6 contains some simple characterizations of
$i\operatorname{\mathsf{TC}}_{m}$ and $i\operatorname{\mathsf{cat}}$ in terms
of pullback. In Section 7, we obtain lower bounds to
$i\operatorname{\mathsf{cat}}(X)$ and $i\operatorname{\mathsf{TC}}_{m}(X)$,
first using the cohomology of the barycenter spaces [KK] of $X$ and $X^{m}$,
respectively, and then using the cohomology of their respective symmetric
squares. We also discuss the difficulty in obtaining better lower bounds.
Finally, we end this paper in Section 8 by performing some computations using
our lower bounds and providing examples of spaces $X$ for which
$i\operatorname{\mathsf{cat}}(X)>1$, $i\operatorname{\mathsf{TC}}_{m}(X)>1$,
$i\operatorname{\mathsf{cat}}(X)=d\operatorname{\mathsf{cat}}(X)$, and
$i\operatorname{\mathsf{TC}}(X)=d\operatorname{\mathsf{TC}}(X)$.

We use the following notations and conventions in this paper. All the
topological spaces considered are path-connected ANR spaces. The composition
of functions $f:X\to Y$ and $g:Y\to Z$ is denoted by $gf$. The symbol “$\hskip
1.99168pt="$ is used to denote homeomorphisms and isomorphisms, and the symbol
“$\hskip 1.70717pt\simeq"$ is used to denote homotopy equivalences of spaces
and maps. For any fixed $m\geq 2$, we agree to set timestamps
$t_{i}=(i-1)/(m-1)$ for all $1\leq i\leq m$.

##  2\. The classical and distributional invariants

We recall the classical definitions of the Lusternik-Schnirelmann category
[CLOT] and the sequential topological complexity [Far1], [R], [BGRT] of a
space.

The _Lusternik-Schnirelmann category_ (LS-category),
$\operatorname{\mathsf{cat}}(X)$, of $X$ is the minimal integer $n$ 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_ ,
$\operatorname{\mathsf{TC}}_{m}(X)$, of $X$ is the minimal integer $n$ such
that there is a covering $\\{V_{i}\\}$ of $X^{m}$ by $n+1$ open sets over each
of which there is a continuous map $s_{i}:V_{i}\to P(X)$ such that for each
$\overline{x}\in V_{i}\subset X^{m}$, we have that
$s_{i}(\overline{x})(t_{j})=x_{j}$ for all $1\leq j\leq m$.

We also recall the definitions of the distributional LS-category [DJ] and the
sequential distributional topological complexity [DJ], [J] of a space.

######  2.1 Definition.

The distributional LS-category, $d\operatorname{\mathsf{cat}}(X)$, of a space
$X$, is the minimal integer $n$ for which there exists a map
$H:X\to{\mathcal{B}}_{n+1}(P_{0}(X))$ such that
$H(x)\in{\mathcal{B}}_{n+1}(P(x,x_{0}))$ for all $x\in X$.

Here, the pointed space $(X,x_{0})$ is considered, and $P_{0}(X)$ denotes the
based path space containing paths in $X$ with endpoints $x_{0}$.

######  2.2 Definition.

For a given $m\geq 2$, the $m^{th}$ sequential distributional topological
complexity, $d\operatorname{\mathsf{TC}}_{m}(X)$, of a space $X$, is the
minimal integer $n$ for which there exists a map
$s_{m}:X^{m}\to{\mathcal{B}}_{n+1}(P(X))$ such that
$s_{m}(\overline{x})\in{\mathcal{B}}_{n+1}(P(\overline{x}))$ for all
$\overline{x}\in X^{m}$.

The distributional invariants $d\operatorname{\mathsf{cat}}$ and
$d\operatorname{\mathsf{TC}}_{m}$ satisfy [DJ], [J] most of the nice
properties and relations as the classical invariants
$\operatorname{\mathsf{cat}}$ and $\operatorname{\mathsf{TC}}_{m}$, [Far1],
[R], [BGRT]. However, in general, $d\operatorname{\mathsf{cat}}$ and
$\operatorname{\mathsf{cat}}$ are different notions, and similarly, so are
$d\operatorname{\mathsf{TC}}_{m}$ and $\operatorname{\mathsf{TC}}_{m}$ for
each $m\geq 2$. In particular, for all $n\geq 1$,
$d\operatorname{\mathsf{cat}}({\mathbb{R}}P^{n})=1$ due to [DJ, Example 3.13],
and $d\operatorname{\mathsf{TC}}_{m}({\mathbb{R}}P^{n})\leq 2m+1$ follows from
[J, Section 8.A] (see also [KW, Corollary 6.6]). As explained in Section 1.B,
these examples lead to various differences between the theories of the
classical and the distributional invariants (see also Remark 4.9). Another
such instance of that is Remark 4.13.

###  2.A. Lower bounds for $\operatorname{\mathsf{cat}}$ and
$\operatorname{\mathsf{TC}}_{m}$

Sharp lower bounds to these invariants come from cohomology. We first recall
some basic notions.

The cup-length of a space $X$ with coefficients in a ring $R$ (or
alternatively, the cup-length of the cohomology ring $H^{*}(X;R)$), denoted
$c\ell_{R}(X)$, 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.

For a fixed $m\geq 2$, let $\Delta:X\to X^{m}$ be the diagonal map that
induces the homomorphism $\Delta^{*}:H^{*}(X^{m};R)\to H^{*}(X;R)$. The cup-
length of the ideal $\operatorname{{\rm Ker}}(\Delta^{*})$ is called the
$m^{th}$ zero-divisors cup-length of $X$, denoted $zc\ell^{m}_{R}(X)$. When
$m=2$, we simply denote it by $zc\ell_{R}(X)$.

It is well-known that for any ring $R$,
$c\ell_{R}(X)\leq\operatorname{\mathsf{cat}}(X)$, see [CLOT], and that
$zc\ell^{m}_{R}(X)\leq\operatorname{\mathsf{TC}}_{m}(X)$, see [Far1], [R],
[BGRT].

###  2.B. Lower bounds for $d\operatorname{\mathsf{cat}}$ and
$d\operatorname{\mathsf{TC}}_{m}$

For a space $Y$ and $k\geq 1$, its $k^{th}$ symmetric power $SP^{k}(Y)$ is
defined as the orbit space of the action of the symmetric group $S_{k}$ on the
product space $Y^{k}$ by permutation of coordinates. In this section, we
regard each $[y_{1},\ldots,y_{m}]\in SP^{k}(Y)$ as a formal sum $\sum
n_{i}y_{i}$, where $n_{i}\geq 1$ and $\sum n_{i}=k$, subject to the
equivalence $n_{1}y+n_{2}y=(n_{1}+n_{2})y$. We define the diagonal embedding
$\partial^{Y}_{k}:Y\to SP^{k}(Y)$ as $\partial^{Y}_{k}(y)=[y,\ldots,y]=ky$.
The following result in singular cohomology will be very useful later.

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

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

We note that Proposition 2.3 was proven in [DJ] for rational coefficients but
the same proof holds for real coefficients as well!

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

######  2.4 Theorem ([DJ]).

Suppose that $\alpha_{i}^{*}\in H^{k_{i}}\left(SP^{n!}(X);R\right)$, $1\leq
i\leq n$, for some ring $R$ and $k_{i}\geq 1$. Let $\alpha_{i}\in
H^{k_{i}}(X;R)$ be the image of $\alpha_{i}^{*}$ under the induced
homomorphism $(\partial^{X}_{n!})^{*}$ such that
$\alpha_{1}\smile\cdots\smile\alpha_{n}\neq 0$. Then
$d\operatorname{\mathsf{cat}}(X)\geq n.$

Let the map $\Delta_{n}:SP^{n!}(X)\to SP^{n!}(X^{m})$ be induced from $\Delta$
by functoriality. Then the following lower bound is obtained in Alexander-
Spanier cohomology.

######  2.5 Theorem ([J]).

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

###  2.C. A characterization of $d\operatorname{\mathsf{cat}}$ and
$d\operatorname{\mathsf{TC}}_{m}$

For a fibration $p:E\to B$, let $E_{n}(p)=\\{\mu\in
B_{n}(E)\mid\operatorname{{\rm supp}}(\mu)\subset p^{-1}(x)\text{ for some
}x\in B\\}$ and $\mathcal{B}_{n}(p):E_{n}(p)\to B$ be the fibration defined as
$\mathcal{B}_{n}(p)(\mu)=x$ when $\mu\in\mathcal{B}_{n}(p^{-1}(x)).$

Let $p_{0}:P_{0}(X)\to X$ and $\pi_{m}:P(X)\to X^{m}$ be the fibrations
defined as

$p_{0}:\phi\mapsto\phi(0)\hskip 14.22636pt\text{and}\hskip
14.22636pt\pi_{m}:\psi\mapsto\left(\psi\left(t_{1}\right),\psi\left(t_{2}\right),\ldots,\psi\left(t_{m}\right)\right).$

The following characterizations were obtained in [DJ] and [J], respectively.

  * •

$d\operatorname{\mathsf{cat}}(X)<n$ if and only if there is a section to
${\mathcal{B}}_{n}(p_{0}):P_{0}(X)_{n}(p_{0})\to X$.

  * •

Similarly, $d\operatorname{\mathsf{TC}}_{m}(X)<n$ if and only if there exists
a section to the fibration ${\mathcal{B}}_{n}(\pi_{m}):P(X)_{n}(\pi_{m})\to
X^{m}$.

###  2.D. Some preliminaries

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

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

The proof of the following statement can be found in [J, Lemma 2.1].

######  2.6 Lemma.

The map $\theta_{m}:T_{m}(X)\to P(X)$ is continuous for each $m\geq 2$.

This construction was used in [J, Sections 3 and 5.B] to prove various
properties for $d\operatorname{\mathsf{TC}}_{m}$. Here, we will use it in
Sections 4 and 6 for similar purposes.

##  3\. Resolvable paths

###  3.A. Topological aspects

For a fixed $n\geq 1$, recall the continuous mapping
$\Phi_{n}:{\mathcal{B}}_{n}(P(X))\to P({\mathcal{B}}_{n}(X))$ defined in the
introduction. The notion of resolvable paths was first introduced in [DJ] as
follows.

######  3.1 Definition.

A path $f:I\to{\mathcal{B}}_{n}(X)$ is called resolvable if there exists a
measure $\mu\in{\mathcal{B}}_{n}(P(X))$ such that $\Phi_{n}(\mu)=f$. In this
case, $\mu$ is called a resolver of $f$.

Let ${\mathcal{P}}({\mathcal{B}}_{n}(X))\subset P({\mathcal{B}}_{n}(X))$
denote the subspace of resolvable paths.

######  3.2 Remark.

Let $f\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$ be resolved by
$\mu=\sum\lambda_{\phi}\phi\in{\mathcal{B}}_{n}(P(X))$. Let there exist some
$t_{0}\in I$ and $x\in X$ such that $f(t_{0})=\delta_{x}$. Then

$\sum\lambda_{\phi}\phi(t_{0})=f(t_{0})=\delta_{x}$

implies that $\phi(t_{0})=x$ for each $\phi\in\operatorname{{\rm supp}}(\mu)$.

In [Dr], the definition of resolvable paths was reformulated as follows.

######  3.3 Definition.

The subspace $RP({\mathcal{B}}_{n}(X))\subset P({\mathcal{B}}_{n}(X))$ is
called the space of resolvable paths of $P({\mathcal{B}}_{n}(X))$ if for each
$f\in RP({\mathcal{B}}_{n}(X))$, there exists a continuous map
$F:I\times\\{1,\ldots,n\\}\to X$ and a measure
$\nu\in{\mathcal{B}}_{n}(\\{1,\ldots,n\\})$ such that for all $t\in I$,
$f(t)={\mathcal{B}}_{n}(F)(t,\nu)$, where the map
${\mathcal{B}}_{n}(F):I\times{\mathcal{B}}_{n}(\\{1,\ldots,n\\})\to{\mathcal{B}}_{n}(X)$
is defined using the functoriality of ${\mathcal{B}}_{n}$ in the obvious way.

It is easy to see that both these definitions coincide.

######  3.4 Lemma.

For any $n$, the subspaces $RP({\mathcal{B}}_{n}(X))$ and
${\mathcal{P}}({\mathcal{B}}_{n}(X))$ are the same.

###### Proof.

Let us fix some $x_{0}\in X$. For brevity, let $J^{\prime}$ denote the
unordered set $\\{1,\ldots,n\\}$. Let
$f\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$ so that there exists $\mu=\sum_{i\in
J}\lambda_{i}\phi_{i}\in{\mathcal{B}}_{n}(P(X))$ for some $J\subset
J^{\prime}$ such that $f(t)=\sum_{i\in J}\lambda_{i}\phi_{i}(t)$ for each
$t\in I$. Let us define the map $F:I\times J\to X$ as $F(t,i)=\phi_{i}(t)$.
When $i\in J^{\prime}\setminus J$, let $F(t,i)=x_{0}$ for all $t$. This gives
a continuous map $F:I\times J^{\prime}\to X$. Also, define $\nu=\sum_{i\in
J}\lambda_{i}\hskip
1.13809pti\in{\mathcal{B}}_{n}(J)\subset{\mathcal{B}}_{n}(J^{\prime})$. We see
that

${\mathcal{B}}_{n}(F):\left(t,\nu\right)\mapsto\sum_{i\in J}\lambda_{i}\hskip
1.13809ptF(t,i)=\sum_{i\in J}\lambda_{i}\hskip 1.13809pt\phi_{i}(t)=f(t).$

Thus, $f\in RP({\mathcal{B}}_{n}(X))$ and therefore,
${\mathcal{P}}({\mathcal{B}}_{n}(X))\subset RP({\mathcal{B}}_{n}(X))$.

Conversely, if $f\in RP({\mathcal{B}}_{n}(X))$, then there exists a map
$F:I\times J^{\prime}\to X$ and $\nu=\sum_{i\in
J}\lambda_{i}\alpha_{i}\in{\mathcal{B}}_{n}(J^{\prime})$ for some $J\subset
J^{\prime}$ such that $f(t)=\sum_{i\in J}\lambda_{i}F(t,\alpha_{i})$. For each
$i\in J$, define the path $\phi_{i}:I\to X$ as $\phi_{i}(t)=F(t,\alpha_{i})$.
Finally, define $\mu=\sum_{i\in
J}\lambda_{i}\phi_{i}\in{\mathcal{B}}_{n}(P(X))$. Clearly, $\Phi_{n}(\mu)=f$.
Thus, $f\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$ and so,
$RP({\mathcal{B}}_{n}(X))\subset{\mathcal{P}}({\mathcal{B}}_{n}(X))$. ∎

Definition 3.1 turns out to be more convenient to use in most cases and is
used throughout the paper. However, we will also make use of the equivalent
Definition 3.3 in one of our computations in Section 5.

We now study the homotopy type of the space
${\mathcal{P}}({\mathcal{B}}_{n}(X))$ of resolvable paths.

######  3.5 Lemma.

For any $n\geq 1$, the space ${\mathcal{P}}({\mathcal{B}}_{n}(X))$ deforms to
${\mathcal{B}}_{n}(X)$.

###### Proof.

Let us define a map $f:{\mathcal{B}}_{n}(X)\to P({\mathcal{B}}_{n}(X))$ as
$f:\mu\mapsto\phi_{\mu}$, where $\phi_{\mu}$ is the trivial loop at
$\mu\in{\mathcal{B}}_{n}(X)$, i.e., if $\mu=\sum\lambda_{\alpha}\alpha$, then
$\phi_{\mu}(t)=\sum\lambda_{\alpha}\alpha$. For any $x\in X$, let $c_{x}\in
P(X)$ denote the trivial loop at $x$. If
$\nu=\sum\lambda_{\alpha}c_{\alpha}\in{\mathcal{B}}_{n}(P(X))$, then $\nu$ is
a resolver of $\phi_{\mu}$. So, the image of $f$ is in
${\mathcal{P}}({\mathcal{B}}_{n}(X))$. Define
$g:{\mathcal{P}}({\mathcal{B}}_{n}(X))\to{\mathcal{B}}_{n}(X)$ to be the
evaluation map $g:\psi\mapsto\psi(0)$. Then clearly,
$gf=\mathbbm{1}_{{\mathcal{B}}_{n}(X)}$. Finally, define a homotopy
$H:{\mathcal{P}}({\mathcal{B}}_{n}(X))\times I\to P({\mathcal{B}}_{n}(X))$ as

$H:(\psi,t)(s)\mapsto\psi(s(1-t))$

for all $s,t\in I$. Of course, we get $H(\psi,0)(s)=\psi(s)$ and
$H(\psi,1)(s)=\psi(0)$ for all $s\in I$. Now, see that if
$\mu=\sum\lambda_{\alpha}\alpha\in{\mathcal{B}}_{n}(P(X))$ is a resolver of
$\psi\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$ and for each $t\in I$, if
$\gamma_{t}:I\to I$ is given by $\gamma_{t}(s)=s(1-t)$, then the measure
$\nu=\sum\lambda_{\alpha}\hskip
0.85358pt\alpha\gamma_{t}\in{\mathcal{B}}_{n}(P(X))$ is a resolver of
$H(\psi,t)\in P({\mathcal{B}}_{n}(X))$. Hence, the image of $H$ is in
${\mathcal{P}}({\mathcal{B}}_{n}(X))$ and so, $H$ is a homotopy between
$\mathbbm{1}_{{\mathcal{P}}({\mathcal{B}}_{n}(X))}$ and $fg$. ∎

######  3.6 Corollary.

The space ${\mathcal{P}}({\mathcal{B}}_{n}(X))$ is homotopy equivalent to
$P({\mathcal{B}}_{n}(X))$.

###### Proof.

We know that for any space $Y$, the path space $P(Y)$ deforms to $Y$. Upon
taking $Y={\mathcal{B}}_{n}(X)$ and using Lemma 3.5, the statement follows. ∎

For a metric space $(X,d)$ and $n\geq 1$, we define
$\operatorname{exp}_{n}(X)=\\{A\subset X\mid|A|\leq n\\}$ to be the set of all
subsets of $X$ having cardinality at most $n$. We equip
$\operatorname{exp}_{n}(X)$ with the topology induced by the Hausdorff metric
$d_{H}$. So, for any $A,B\in\operatorname{exp}_{n}(X)$,

$d_{H}(A,B)=\max\left\\{\sup_{a\in A}d(a,B),\hskip 1.42262pt\sup_{b\in
B}d(A,b)\right\\}.$

It is easy to see that $\operatorname{exp}_{1}(X)=X$,
$\operatorname{exp}_{2}(X)=SP^{2}(X)$, and
$\operatorname{exp}_{n}(X)\subset\operatorname{exp}_{n+1}(X)$ for all $n\geq
1$.

Consider a support function $\operatorname{{\rm
supp}}:{\mathcal{B}}_{n}(X)\to\operatorname{exp}_{n}(X)$ that maps each
measure $\mu\in{\mathcal{B}}_{n}(X)$ to its support $\operatorname{{\rm
supp}}(\mu)\in\operatorname{exp}_{n}(X)$. For $n\geq 2$, this function need
not be continuous. For any $f\in P({\mathcal{B}}_{n}(X))$, the composition
$\operatorname{{\rm supp}}\hskip 0.85358ptf:I\to\operatorname{exp}_{n}(X)$ can
be viewed as the variation of the support of $f$ with respect to the time $t$.

######  3.7 Proposition.

If $f\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$, then the composition
$\operatorname{{\rm supp}}f$ is continuous.

###### Proof.

The continuity of $\operatorname{{\rm supp}}f$ can be checked by the
sequential criterion. Let $\\{t_{n}\\}$ be a sequence in $I$ converging to
$t$. Let $A=(\operatorname{{\rm supp}}f)(t)=\operatorname{{\rm supp}}(f(t))$
and for each $n$, let $A_{n}=(\operatorname{{\rm
supp}}f)(t_{n})=\operatorname{{\rm supp}}(f(t_{n}))$. Since
$f\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$, there exists
$\mu=\sum\lambda_{\phi}\phi\in{\mathcal{B}}_{n}(X)$ such that
$f(s)=\sum\lambda_{\phi}\phi(s)$ for all $s\in I$. Let us fix some
$\phi\in\operatorname{{\rm supp}}(\mu)$. Then $\phi(t_{n})\in A_{n}$ and
$\phi(t)\in A$. Thus, for all $n$,

$d(A_{n},\phi(t)),\hskip 2.84526ptd(\phi(t_{n}),A)\leq
d(\phi(t_{n}),\phi(t)).$

Now, choose some $\epsilon>0$. Since $\phi$ is continuous, there exists some
$m_{\phi}\geq 1$ such that $d(\phi(t_{n}),\phi(t))<\frac{\epsilon}{2}$ for all
$n\geq m_{\phi}$. This happens for each fixed $\phi\in\operatorname{{\rm
supp}}(\mu)$. Let $m=\max\\{m_{\phi}\mid\phi\in\operatorname{{\rm
supp}}(\mu)\\}$. Then, it follows that for all $n\geq m$,

$\sup_{\phi\in\operatorname{{\rm
supp}}(\mu)}d(A_{n},\phi(t)),\sup_{\phi\in\operatorname{{\rm
supp}}(\mu)}d(\phi(t_{n}),A)\leq\frac{\epsilon}{2}.$

Note that $A$ is a subset of the multiset
$\\{\phi(t)\mid\phi\in\operatorname{{\rm supp}}(\mu)\\}$ and $A_{n}$ is a
subset of the multiset $\\{\phi(t_{n})\mid\phi\in\operatorname{{\rm
supp}}(\mu)\\}$ for each $n$. Thus, for each $n$,

$\sup_{a\in A}d(A_{n},a)\leq\sup_{\phi\in\operatorname{{\rm
supp}}(\mu)}d(A_{n},\phi(t))\hskip 11.38109pt\text{and}\hskip
11.38109pt\sup_{b\in A_{n}}d(b,A)\leq\sup_{\phi\in\operatorname{{\rm
supp}}(\mu)}d(\phi(t_{n}),A).$

From this, we conclude that $d_{H}(A_{n},A)\leq\frac{\epsilon}{2}<\epsilon$
for all $n\geq m$. So, the sequence $\\{A_{n}\\}$ converges to $A$ in
$\operatorname{exp}_{n}(X)$ and hence, the map $\operatorname{{\rm supp}}f$ is
continuous. ∎

In other words, the support of a resolvable path varies continuously with
time.

###  3.B. Some examples

Let us now look at various explicit examples of the resolvers of some
resolvable paths in ${\mathcal{P}}({\mathcal{B}}_{n}(X))$ for $n=2,3,$ and
$4$. For any two paths $\phi_{1},\phi_{2}\in P(X)$, if there exists some
$t_{1}\in(0,1)$ such that $\phi_{1}(t_{1})=\phi_{2}(t_{1})$, then we define a
path $(\phi_{1},\phi_{2})\in P(X)$ by pasting such that
$(\phi_{1},\phi_{2})(t)=\phi_{1}(t)$ when $t\in[0,t_{1}]$ and
$(\phi_{1},\phi_{2})(t)=\phi_{2}(t)$ when $t\in[t_{1},1]$.

######  3.8 Example.

Consider a resolvable path $f:I\to{\mathcal{B}}_{2}(X)$ described as follows.
At $t=0$, the path breaks into two strings of weights $\frac{1}{2}$ and
$\frac{1}{2}$. At $t=t_{1}$, the two strings intertwine and again break into
two strings of weights $\frac{1}{2}$ and $\frac{1}{2}$. Finally, the two
strings reassemble at $t=1$. Then any resolver of $f$ in
${\mathcal{B}}_{2}(P(X))$ must have both weights equal to $\frac{1}{2}$. Let
$\mu=\frac{1}{2}\alpha+\frac{1}{2}\beta$ be a resolver with
$\alpha(t_{1})=\beta(t_{1})$ such that we have
$f(t)=\frac{1}{2}\alpha(t)+\frac{1}{2}\beta(t)$. It is easy to see that
$\nu=\frac{1}{2}(\alpha,\beta)+\frac{1}{2}(\beta,\alpha)$ is another resolver
of $f$.

######  3.9 Example.

For a slightly more complicated situation, consider a resolvable path
$g:I\to{\mathcal{B}}_{3}(X)$ described as follows. At $t=0$, the path breaks
into two strings of weights $\frac{1}{2}$ and $\frac{1}{2}$. At $t=t_{1}$, the
two strings intertwine and further break into two strings of weights
$\frac{1}{4}$ and $\frac{3}{4}$. Finally, the two strings reassemble at $t=1$.
This path cannot have a resolver in ${\mathcal{B}}_{2}(P(X))$. In fact, any
resolver of $g$ in ${\mathcal{B}}_{3}(P(X))$ must have weights
$\frac{1}{4},\frac{1}{4}$, and $\frac{1}{2}$. Let
$\mu=\frac{1}{4}\alpha+\frac{1}{4}\beta+\frac{1}{2}\gamma$ be a resolver with
$\alpha(t_{1})=\beta(t_{1})=\gamma(t_{1})$. By definition of $g$, it follows
that $\beta=(\alpha,\gamma)$. Clearly,

$g(t)=\begin{cases}\frac{1}{2}\alpha(t)+\frac{1}{2}\gamma(t)&:t\in[0,t_{1}]\\\
\frac{1}{4}\alpha(t)+\frac{3}{4}\gamma(t)&:t\in[t_{1},1]\end{cases}.$

Thus, another resolver of $g$ is
$\nu=\frac{1}{2}(\alpha,\gamma)+\frac{1}{4}(\gamma,\alpha)+\frac{1}{4}\gamma$.

For three paths $\phi_{1},\phi_{2},\phi_{3}\in P(X)$ such that
$\phi_{1}(t_{1})=\phi_{2}(t_{1})$ and $\phi_{2}(t_{2})=\phi_{3}(t_{2})$ where
$0<t_{1}<t_{2}<1$, we define a path $(\phi_{1},\phi_{2},\phi_{3})\in P(X)$ by
pasting in a similar way as above such that
$(\phi_{1},\phi_{2},\phi_{3})(t)=\phi_{i}(t)$ if $t\in[t_{i-1},t_{i}]$ for
$i=1,2,3$, where we have $t_{0}=0$ and $t_{3}=1$.

######  3.10 Example.

Let $\psi:I\to{\mathcal{B}}_{4}(X)$ be a resolvable path described as follows.
At $t=0$, the path breaks into three strings of weights $\frac{1}{3}$ each. At
$t=t_{1}$, the strings intertwine and break further into three strings of
weights $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{6}$. At last, the strings
reassemble at $t=1$. This path cannot have a resolver in
${\mathcal{B}}_{3}(P(X))$. Let
$\mu_{1}=\frac{1}{3}\alpha+\frac{1}{6}\beta+\frac{1}{6}\gamma+\frac{1}{3}\delta\in{\mathcal{B}}_{4}(P(X))$
be a resolver of $h$. By definition, $\beta=(\gamma,\alpha)$. We see that

$\psi(t)=\begin{cases}\frac{1}{3}\alpha(t)+\frac{1}{3}\gamma(t)+\frac{1}{3}\delta(t)&:t\in[0,t_{1}]\\\
\frac{1}{2}\alpha(t)+\frac{1}{6}\gamma(t)+\frac{1}{3}\delta(t)&:t\in[t_{1},1].\end{cases}$

Three more resolvers of $\psi$ in ${\mathcal{B}}_{4}(P(X))$ are:
$\frac{1}{3}(\gamma,\alpha)+\frac{1}{6}\alpha+\frac{1}{6}(\alpha,\gamma)+\frac{1}{3}\delta$,
$\frac{1}{3}(\delta,\alpha)+\frac{1}{6}\gamma+\frac{1}{6}(\gamma,\alpha)+\frac{1}{3}(\alpha,\delta)$,
$\frac{1}{3}(\delta,\alpha)+\frac{1}{6}(\alpha,\gamma)+\frac{1}{6}\alpha+\frac{1}{3}(\gamma,\delta)$.

Given a resolvable path $\psi$, let us call a point $t_{0}\in(0,1)$ a
branching point of $\psi$ if there exists an $\epsilon>0$ such that two or
more strings in $(t_{0}-\epsilon,t_{0})$ intertwine at $t_{0}$ or a string
breaks at $t_{0}$ into two or more strings in $(t_{0},t_{0}+\epsilon)$.

######  3.11 Example.

We modify the path $\psi$ above by adding another branching point. Let
$h:I\to{\mathcal{B}}_{4}(X)$ be a resolvable path described as follows. At
$t=0$, the path breaks into three strings of weights $\frac{1}{3}$ each. At
$t=t_{1}>0$, the strings intertwine and break further into three strings of
weights $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{6}$. Then at
$t=t_{2}>t_{1}$, the strings again intertwine and break into two strings of
weights $\frac{2}{3}$ and $\frac{1}{3}$. Finally, the two strings reassemble
at $t=1$. Again, this path cannot have a resolver in
${\mathcal{B}}_{3}(P(X))$. Let
$\mu_{1}=\frac{1}{3}\alpha+\frac{1}{6}\beta+\frac{1}{6}\gamma+\frac{1}{3}\delta\in{\mathcal{B}}_{4}(P(X))$
be a resolver of $h$. By definition, we must have
$\beta=(\gamma,\alpha,\alpha)$ and $\gamma=(\gamma,\gamma,\alpha)$. Clearly,

$h(t)=\begin{cases}\frac{1}{3}\alpha(t)+\frac{1}{3}\gamma(t)+\frac{1}{3}\delta(t)&:t\in[0,t_{1}]\\\
\frac{1}{2}\alpha(t)+\frac{1}{6}\gamma(t)+\frac{1}{3}\delta(t)&:t\in[t_{1},t_{2}]\\\
\frac{2}{3}\alpha(t)+\frac{1}{3}\delta(t)&:t\in[t_{2},1]\end{cases}$

The path $h$ has many more resolvers in ${\mathcal{B}}_{4}(P(X))$ than the
resolvers of $\psi$. In fact, it is easy to find at least list $11$ other
resolvers of $h$.

From Examples 3.10 and 3.11, we see that even if we increase the number of
branching points of a resolvable path by one, the number of resolvers can
increase significantly.

######  3.12 Example.

Consider the resolvable path $f:I\to B_{4}(X)$ described in Example 3.8.
Certainly, we have two of its resolvers $\mu,\nu\in{\mathcal{B}}_{4}(P(X))$
defined above. Another resolver $\delta\in{\mathcal{B}}_{4}(P(X))$ is defined
as follows:
$\delta=\frac{1}{4}\alpha+\frac{1}{4}\beta+\frac{1}{4}(\alpha,\beta)+\frac{1}{4}(\beta,\alpha)$.
We note that in ${\mathcal{B}}_{3}(P(X))$, however, $f$ has exactly two
resolvers.

So, even if we keep the number of branching points of a resolvable path fixed,
its number of resolvers can increase as we increase the value of the maximum
possible size $n$ of the support of its resolver measures.

We end this section by giving two more examples of resolvable paths whose
resolvers have supports of different sizes.

######  3.13 Example.

Let $k:I\to{\mathcal{B}}_{3}(X)$ be described as follows. At $t=0$, the path
breaks into two strings of weights $\frac{2}{3}$ and $\frac{1}{3}$. At
$t=t_{1}$, the two strings intertwine and again break into two strings of
weights $\frac{2}{3}$ and $\frac{1}{3}$. Finally, the two strings reassemble
at $t=1$. This path has a resolver
$\mu=\frac{2}{3}\alpha+\frac{1}{3}\beta\in{\mathcal{B}}_{3}(P(X))$, where
$\alpha(t)$ is taken to correspond with the string of weight $\frac{2}{3}$ and
$\beta(t)$ is taken for the string of weight $\frac{1}{3}$, and
$\alpha(t_{1})=\beta(t_{1})$, so that
$k(t)=\frac{2}{3}\alpha(t)+\frac{1}{3}\beta(t)$. But another resolver of $k$
is $\nu=\frac{1}{3}\alpha+\frac{1}{3}(\alpha,\beta)+\frac{1}{3}(\beta,\alpha)$
and its support size is $3$.

######  3.14 Example.

Certainly, the weights do not have to be rational numbers. In the above
example for $k:I\to B_{4}(X)$, if the weights of the two strings are taken to
be $1-\frac{1}{\sqrt{3}}$ and $\frac{1}{\sqrt{3}}$ instead, then besides the
usual resolver $\mu=\frac{1}{\sqrt{3}}\alpha+(1-\frac{1}{\sqrt{3}})\beta$, a
resolver having support size $4$ is
$\nu=\frac{1}{2\sqrt{3}}\alpha+\frac{1}{2\sqrt{3}}(\alpha,\beta)+\frac{1}{2\sqrt{3}}(\beta,\alpha)+(1-\frac{\sqrt{3}}{2})\beta$.

##  4\. The intertwining invariants

###  4.A. Sequential intertwining complexities

For $\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}$, recall that
$P(\delta_{x_{1}},\ldots,\delta_{x_{m}})=\\{f\in P({\mathcal{B}}_{n}(X))\mid
f(t_{i})=\delta_{x_{i}}\text{ for all }i\\}$.

######  4.1 Definition.

An $n$-intertwined $m$-navigation algorithm on a space $X$ is a map

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

satisfying $s_{m}(\overline{x})\in P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$
for each $\overline{x}\in X^{m}$.

In view of Remark 3.2, it is easy to see that the path
$s_{m}(\overline{x})\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$ is resolved by
measures in ${\mathcal{B}}_{n}(P(\overline{x}))$.

######  4.2 Definition.

The $m^{th}$ sequential intertwining topological complexity,
$i\operatorname{\mathsf{TC}}_{m}(X)$, of a space $X$ is the minimal integer
$n$ such that $X$ admits an $(n+1)$-intertwined $m$-navigation algorithm.

It is clear that $i\operatorname{\mathsf{TC}}_{m}(X)=0$ for some $m\geq 2$ if
and only if $X$ is contractible.

When $m=2$, $i\operatorname{\mathsf{TC}}_{2}(X)$ is denoted by
$i\operatorname{\mathsf{TC}}$ and simply called the intertwining topological
complexity of $X$, [DJ].

We now explicitly prove some basic properties for
$i\operatorname{\mathsf{TC}}_{m}$ that are very similar to the properties of
$d\operatorname{\mathsf{TC}}_{m}$ (see [J, Section 3]) and
$\operatorname{\mathsf{TC}}_{m}$ (see [R, Section 3] and [BGRT, Section 3]).
The proofs are partly inspired by the corresponding proofs for
$d\operatorname{\mathsf{TC}}_{m}$ obtained in [J, Section 3].

######  4.3 Proposition.

If $f:X\to Y$ is a homotopy domination, then for each $m\geq 2$,
$i\operatorname{\mathsf{TC}}_{m}(Y)\leq i\operatorname{\mathsf{TC}}_{m}(X)$.

###### Proof.

Since $f:X\to Y$ is a homotopy domination, there exists a continuous map
$g:Y\to X$ such that $fg\simeq\mathbbm{1}_{Y}$. Let
$i\operatorname{\mathsf{TC}}_{m}(X)=n-1$. Then there exists a $n$-intertwined
$m$-navigation algorithm $s_{m}:X^{m}\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$.
Let $g^{m}:Y^{m}\to X^{m}$ be the product map and let
$f_{*}:{\mathcal{B}}_{n}(X)\to{\mathcal{B}}_{n}(Y)$ and
$\widetilde{f}:P({\mathcal{B}}_{n}(X))\to P({\mathcal{B}}_{n}(Y))$ be induced
by $f$ due to functoriality. Consider the map

$\widetilde{f}s_{m}\hskip 0.85358ptg^{m}:Y^{m}\to P({\mathcal{B}}_{n}(Y)).$

Let $h:Y\to P(Y)$ be a homotopy such that $h:y\to h_{y}$ for each $y\in Y$,
where $h_{y}(0)=y$ and $h_{y}(1)=fg(y)$. We write
$H_{y}=\omega_{n}h_{y}:I\to{\mathcal{B}}_{n}(Y)$ as a path in
${\mathcal{B}}_{n}(Y)$ from $\delta_{y}$ to $\delta_{fg(y)}$, where we recall
that $\omega_{n}:Y\hookrightarrow{\mathcal{B}}_{n}(Y)$ is the Dirac embedding.
For each $1\leq i\leq m-1$, let $k_{i}:I\to I$ be the map

$k_{i}:s\mapsto\frac{s+i-1}{m-1}.$

For a fixed $\overline{y}\in Y^{m}$, let $s_{m}\hskip
0.85358ptg^{m}(\overline{y})=\psi$ and $\psi_{i}=\psi k_{i}$. In the notations
of Section 2.D, using the map $\theta_{m-1}:T_{m-1}({\mathcal{B}}_{n}(Y))\to
P({\mathcal{B}}_{n}(Y))$ with $a_{i}=1/t_{i+1}=(m-1)/i$ for all $1\leq i\leq
m-2$, we define a map $\sigma:Y^{m}\to P({\mathcal{B}}_{n}(Y))$ as

$\sigma(\overline{y})=\left(H_{y_{1}}\cdot
f_{*}\psi_{1}\cdot\overline{H}_{y_{2}}\right)\star\left(H_{y_{2}}\cdot
f_{*}\psi_{2}\cdot\overline{H}_{y_{3}}\right)\star\cdots\star\left(H_{y_{m-1}}\cdot
f_{*}\psi_{m-1}\cdot\overline{H}_{y_{m}}\right).$

Here, $\cdot$ denotes the operation of the usual concatenation of paths.
Continuity of $\sigma$ follows from the continuity of $\theta_{m-1}$ in Lemma
2.6. Clearly, $\sigma(\overline{y})\in
P(\delta_{y_{1}},\ldots,\delta_{y_{m}})$. We now show that
$\sigma(\overline{y})$ is resolvable. Let
$\psi\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$ be resolved by
$\mu=\sum\lambda_{\alpha}\alpha$. Thus, $\alpha(t_{i})=g(y_{i})$ for all $i$.
As before, define

$\widetilde{\alpha}=\left(h_{y_{1}}\cdot f\alpha
k_{1}\cdot\overline{h}_{y_{2}}\right)\star\left(h_{y_{2}}\cdot f\alpha
k_{2}\cdot\overline{h}_{y_{3}}\right)\star\cdots\star\left(h_{y_{m-1}}\cdot
f\alpha k_{m-1}\cdot\overline{h}_{y_{m}}\right).$

Then the measure
$\nu=\sum\lambda_{\alpha}\widetilde{\alpha}\in{\mathcal{B}}_{n}(P(Y))$ is a
resolver of $\sigma(\overline{y})$. So, $\sigma$ is an $n$-intertwined
$m$-navigation algorithm on $Y$. Thus, $i\operatorname{\mathsf{TC}}_{m}(Y)\leq
n-1$. ∎

######  4.4 Corollary.

$i\operatorname{\mathsf{TC}}_{m}$ is a homotopy invariant of spaces for each
$m\geq 2$.

###### Proof.

Let $X\simeq Y$. Then there exist continuous maps $f:X\to Y$ and $g:Y\to X$
such that $fg\simeq\mathbbm{1}_{Y}$ and $gf\simeq\mathbbm{1}_{X}$. Since $f$
is a homotopy domination of $g$, we get
$i\operatorname{\mathsf{TC}}_{m}(Y)\leq i\operatorname{\mathsf{TC}}_{m}(X)$
from Proposition 4.3. Similarly, since $g$ is a homotopy domination of $f$, we
get $i\operatorname{\mathsf{TC}}_{m}(X)\leq
i\operatorname{\mathsf{TC}}_{m}(Y)$ from Proposition 4.3. Therefore,
$i\operatorname{\mathsf{TC}}_{m}(X)=i\operatorname{\mathsf{TC}}_{m}(Y)$ and
thus, $i\operatorname{\mathsf{TC}}_{m}$ is a homotopy invariant. ∎

######  4.5 Corollary.

$\max\\{i\operatorname{\mathsf{TC}}_{m}(X),i\operatorname{\mathsf{TC}}_{m}(Y)\\}\leq\min\\{i\operatorname{\mathsf{TC}}_{m}(X\vee
Y),i\operatorname{\mathsf{TC}}_{m}(X\times Y)\\}$ for all $m\geq 2$ and spaces
$X$ and $Y$.

###### Proof.

Let us fix $z_{0}$ as the wedge basepoint of $X\vee Y$. Let $r_{X}:X\vee Y\to
X$ and $r_{Y}:X\vee Y\to Y$ be the collapsing maps such that
$r_{X}(Y)=\\{z_{0}\\}$ and $r_{Y}(X)=\\{z_{0}\\}$. Let
$\iota_{X}:X\hookrightarrow X\vee Y$ and $\iota_{Y}:Y\hookrightarrow X\vee Y$
be the inclusions. Then, $r_{X}$ and $r_{Y}$ are homotopy dominations of
$\iota_{X}$ and $\iota_{Y}$, respectively. Hence, we get from Proposition 4.3
that $i\operatorname{\mathsf{TC}}_{m}(X)\leq
i\operatorname{\mathsf{TC}}_{m}(X\vee Y)$ and
$i\operatorname{\mathsf{TC}}_{m}(Y)\leq i\operatorname{\mathsf{TC}}_{m}(X\vee
Y)$. Next, let $\operatorname{proj}_{X}:X\times Y\to X$ and
$\operatorname{proj}_{Y}:X\times Y\to Y$ be the projection maps. Fix some
$a\in X$ and $b\in Y$ and let $\iota^{\prime}_{X}:X\hookrightarrow X\times Y$
and $\iota^{\prime}_{Y}:Y\hookrightarrow X\times Y$ be the inclusions
$\iota^{\prime}_{X}:x\mapsto(x,b)$ and $\iota^{\prime}_{Y}:y\mapsto(a,y)$.
Then, $\operatorname{proj}_{X}$ and $\operatorname{proj}_{Y}$ are homotopy
dominations of $\iota^{\prime}_{X}$ and $\iota^{\prime}_{Y}$, respectively.
The inequalities $i\operatorname{\mathsf{TC}}_{m}(X)\leq
i\operatorname{\mathsf{TC}}_{m}(X\times Y)$ and
$i\operatorname{\mathsf{TC}}_{m}(Y)\leq
i\operatorname{\mathsf{TC}}_{m}(X\times Y)$ then follow from Proposition 4.3.
∎

Corollary 4.5 can also be proven independently, without using Proposition 4.3,
by using ideas from [J, Proposition 3.8].

######  4.6 Proposition.

For any $m\geq 2$, $i\operatorname{\mathsf{TC}}_{m}(X)\leq
d\operatorname{\mathsf{TC}}_{m}(X)\leq\operatorname{\mathsf{TC}}_{m}(X)$.

###### Proof.

Let $d\operatorname{\mathsf{TC}}_{m}(X)=n-1$ so that there exists a map
$k_{m}:X^{m}\to{\mathcal{B}}_{n}(P(X))$ such that
$k_{m}(\overline{x})\in{\mathcal{B}}_{n}(P(\overline{x}))$. Define
$s_{m}=\Phi_{n}\hskip 0.56905ptk_{m}:X^{m}\to P({\mathcal{B}}_{n}(X))$. By
definition, $s_{m}(\overline{x})\in{\mathcal{P}}({\mathcal{B}}_{n}(X))$. Since
$\Phi_{n}$ maps ${\mathcal{B}}_{n}(P(\overline{x}))$ to
$P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$, the map $s_{m}$ is an
$n$-intertwined $m$-navigation algorithm. Hence,
$i\operatorname{\mathsf{TC}}_{m}(X)\leq
n-1=d\operatorname{\mathsf{TC}}_{m}(X)$. The inequality
$d\operatorname{\mathsf{TC}}_{m}(X)\leq\operatorname{\mathsf{TC}}_{m}(X)$ is
well-known, see [DJ], [J]. ∎

######  4.7 Corollary.

If $d\operatorname{\mathsf{TC}}_{m}(X)=1$ for any $m\geq 2$, then
$i\operatorname{\mathsf{TC}}_{m}(X)=1$.

######  4.8 Example.

$i\operatorname{\mathsf{TC}}({\mathbb{R}}P^{n})=1$ and
$i\operatorname{\mathsf{TC}}_{m}({\mathbb{R}}P^{n})\leq 2m+1$ for all $n\geq
1$ and $m\geq 2$.

######  4.9 Remark.

In the classical theory, it is known due to [GLO1, Corollary 3.5] that
$\operatorname{\mathsf{TC}}(X)=1\iff X\simeq S^{2k-1}$ for some $k$. As noted
in [DJ, Section 3.3], this does not remain valid for
$d\operatorname{\mathsf{TC}}$ due to the example of ${\mathbb{R}}P^{n}$ for
$n\geq 2$. For the same reason, this also does not hold in the case of
$i\operatorname{\mathsf{TC}}$. Furthermore, as noted in [J, Remark 3.12] for
$d\operatorname{\mathsf{TC}}_{m}$, the conclusion of [FO, Theorem 2.1] from
the classical theory of $\operatorname{\mathsf{TC}}_{m}$ does not hold in the
theory of $i\operatorname{\mathsf{TC}}_{m}$ also since
$i\operatorname{\mathsf{TC}}({\mathbb{R}}P^{n})\leq 2m+1$ whereas the
cohomological dimension of any finite group is infinite.

###  4.B. Intertwining category

For a fixed basepoint $x_{0}\in X$, let us define a subspace
${\mathcal{P}}_{0}({\mathcal{B}}_{k}(X))=\\{f\in{\mathcal{P}}({\mathcal{B}}_{k}(X))\mid
f(1)=\delta_{x_{0}}\\}$. Then in the same spirit as
$i\operatorname{\mathsf{TC}}_{m}$, an analogous intertwining version of the
LS-category can be defined.

######  4.10 Definition ([DJ, Dr]).

The intertwining Lusternik-Schnirelmann category,
$i\operatorname{\mathsf{cat}}(X)$, of a space $X$ is the minimal integer $n$
such that there exists a continuous map
$H:X\to{\mathcal{P}}_{0}({\mathcal{B}}_{n+1}(X))$ satisfying
$H(x)(0)=\delta_{x}$ for all $x\in X$.

Of course, $H(x)(1)=\delta_{x_{0}}$ and $H(x)$ is resolved by measures in
${\mathcal{B}}_{n}(P(x,x_{0}))$.

Each of the above results for $i\operatorname{\mathsf{TC}}_{m}$ holds for
$i\operatorname{\mathsf{cat}}$ as well and the proofs are just minor
modifications of the respective proofs for $i\operatorname{\mathsf{TC}}_{m}$.

######  4.11 Proposition.

  1. (1)

$i\operatorname{\mathsf{cat}}(X)\leq
d\operatorname{\mathsf{cat}}(X)\leq\operatorname{\mathsf{cat}}(X)$.

  2. (2)

If $f:X\to Y$ is a homotopy domination, then
$i\operatorname{\mathsf{cat}}(Y)\leq i\operatorname{\mathsf{cat}}(X)$.

  3. (3)

$i\operatorname{\mathsf{cat}}$ is a homotopy invariant of spaces.

  4. (4)

$\max\\{i\operatorname{\mathsf{cat}}(X),i\operatorname{\mathsf{cat}}(Y)\\}\leq\min\\{i\operatorname{\mathsf{cat}}(X\vee
Y),i\operatorname{\mathsf{cat}}(X\times Y)\\}$.

By definition, $i\operatorname{\mathsf{cat}}(X)=0$ if and only if $X$ is
contractible. Therefore, by Proposition 4.11 (1),
$d\operatorname{\mathsf{cat}}(X)=1\implies i\operatorname{\mathsf{cat}}(X)=1$.

######  4.12 Example.

For any $n\geq 1$,
$i\operatorname{\mathsf{cat}}(S^{n})=1=i\operatorname{\mathsf{cat}}({\mathbb{R}}P^{n})$.

######  4.13 Remark.

It follows from Proposition 4.11 (1), [J, Proposition 3.5], and [J,
Proposition 3.9] that

$i\operatorname{\mathsf{cat}}({\mathbb{R}}P^{k}\times{\mathbb{R}}P^{k})\leq
d\operatorname{\mathsf{cat}}({\mathbb{R}}P^{k}\times{\mathbb{R}}P^{k})\leq
d\operatorname{\mathsf{TC}}_{3}({\mathbb{R}}P^{k})\leq 3$

for any $k\geq 1$. For a path-connected CW complex $X$, [GLO1, Proposition
2.2] says that if $\pi_{1}(X)$ is not torsion-free, then
$\operatorname{\mathsf{cat}}(X^{n})\geq 2n$. Since
$\pi_{1}({\mathbb{R}}P^{k})={\mathbb{Z}}_{2}$,
$i\operatorname{\mathsf{cat}}({\mathbb{R}}P^{k})$ and
$d\operatorname{\mathsf{cat}}({\mathbb{R}}P^{k})$ are not at least $2$, and
$i\operatorname{\mathsf{cat}}(({\mathbb{R}}P^{k})^{2})$ and
$d\operatorname{\mathsf{cat}}(({\mathbb{R}}P^{k})^{2})$ are not at least $4$,
we conclude that an analog of [GLO1, Proposition 2.2] does not hold in the
case of $i\operatorname{\mathsf{cat}}$ or $d\operatorname{\mathsf{cat}}$, at
least for $n\leq 2$.

######  4.14 Proposition.

If $(X,x_{0})$ and $(Y,y_{0})$ are pointed spaces with non-degerate
basepoints, then $i\operatorname{\mathsf{cat}}(X\vee
Y)=\max\\{i\operatorname{\mathsf{cat}}(X),i\operatorname{\mathsf{cat}}(Y)\\}$.

###### Proof.

Since the inclusions $x_{0}\hookrightarrow X$ and $y_{0}\hookrightarrow Y$ are
cofibrations, using the proof of [CLOT, Lemma 1.25], it can be shown that the
loops over $x_{0}$ and $y_{0}$ are homotopic to the respective trivial loops.
Let $i\operatorname{\mathsf{cat}}(X)=n-1$ and
$i\operatorname{\mathsf{cat}}(Y)=m-1$, so that there exist maps
$H:X\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X))$ and
$K:Y\to{\mathcal{P}}_{0}({\mathcal{B}}_{m}(Y))$ satisfying
$H(x)(0)=\delta_{x}$, $H(x)(1)=\delta_{x_{0}}$, $K(y)(0)=\delta_{y}$, and
$K(y)(1)=\delta_{y_{0}}$. Using the fact that loops over $x_{0}$ (resp.
$y_{0}$) are contractible, we can get maps
$H^{\prime}:X\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X))$ and
$K^{\prime}:Y\to{\mathcal{P}}_{0}({\mathcal{B}}_{m}(Y))$ from $H$ and $K$,
respectively, such that $H^{\prime}(x_{0})(t)=\delta_{x_{0}}$ and
$K^{\prime}(y_{0})(t)=\delta_{y_{0}}$ for all $t\in I$. We form the wedge
$(X\vee Y,z_{0})$ by identifying $x_{0}$ and $y_{0}$. Note that $z_{0}$ is a
non-degenerate basepoint and thus, $z_{0}\hookrightarrow X\vee Y$ is a
cofibration. Without loss of generality, let $n\geq m$. Then $H^{\prime}$ and
$K^{\prime}$ can be extended to get maps
$H^{\prime\prime}:X\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X\vee Y))$ and
$K^{\prime\prime}:Y\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X\vee Y))$,
respectively. Finally, we define $\sigma:X\vee Y\to
P_{0}({\mathcal{B}}_{n}(X\vee Y))$ as

$\sigma(z)=\begin{cases}H^{\prime\prime}(z)&:z\in X\\\
K^{\prime\prime}(z)&:z\in Y\end{cases}$

This map is well-defined because
$H^{\prime\prime}(z_{0})(t)=\delta_{z_{0}}=K^{\prime\prime}(z_{0})(t)$ for all
$t$, and it is continuous by pasting because $X$ and $Y$ are closed in $X\vee
Y$. Clearly, the image of $\sigma$ is in
${\mathcal{P}}_{0}({\mathcal{B}}_{n}(X\vee Y))$. Thus,
$i\operatorname{\mathsf{cat}}(X\vee Y)\leq
n-1=\max\\{i\operatorname{\mathsf{cat}}(X),i\operatorname{\mathsf{cat}}(Y)\\}$.
The reverse inequality follows from Proposition 4.11 (4). ∎

######  4.15 Lemma ([KW, Proposition 3.3]).

If $p:E\to X$ is a degree $k$ covering map, then the map
$p^{*}:X\to{\mathcal{B}}_{k}(E)$ defined by

$p^{*}:x\mapsto\frac{1}{k}\hskip 1.70717pt\sum_{\widetilde{x}\in
p^{-1}(x)}\delta_{\widetilde{x}}$

is continuous and $p_{*}p^{*}=\omega_{k}:X\to{\mathcal{B}}_{k}(X)$, where
$p_{*}:{\mathcal{B}}_{k}(E)\to{\mathcal{B}}_{k}(X)$.

The following result is inspired by [KW, Proposition 6.5].

######  4.16 Proposition.

If $p:E\to X$ is a degree $k$ covering map, then for all $m\geq 2$,
$i\operatorname{\mathsf{TC}}_{m}(X)\leq
km(i\operatorname{\mathsf{TC}}_{m}(E)+1)-1$ and
$i\operatorname{\mathsf{cat}}(X)\leq k(i\operatorname{\mathsf{cat}}(E)+1)-1$.

###### Proof.

First, we prove this statement for $i\operatorname{\mathsf{cat}}$. Let
$i\operatorname{\mathsf{cat}}(E)=n-1$ so that there exists a map
$H:E\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(E))$. This induces
$H_{*}:{\mathcal{B}}_{k}(E)\to{\mathcal{B}}_{k}({\mathcal{P}}_{0}({\mathcal{B}}_{n}(E)))$.
Furthermore, we have our usual map
$\Phi_{k}:{\mathcal{B}}_{k}({\mathcal{P}}_{0}({\mathcal{B}}_{n}(E)))\to
P_{0}({\mathcal{B}}_{k}({\mathcal{B}}_{n}(E)))$. But
${\mathcal{B}}_{k}({\mathcal{B}}_{n}(E))={\mathcal{B}}_{nk}(E)$. So,
$\Phi_{k}:{\mathcal{B}}_{k}({\mathcal{P}}_{0}({\mathcal{B}}_{n}(E)))\to
P_{0}({\mathcal{B}}_{nk}(E))$. Also, $p$ induces
$\widetilde{p}:P({\mathcal{B}}_{nk}(E))\to P({\mathcal{B}}_{nk}(X))$. Finally,
define

$\sigma=\widetilde{p}\hskip 1.13809pt\Phi_{k}\hskip 1.13809ptH_{*}\hskip
0.85358ptp^{*}:X\to P({\mathcal{B}}_{nk}(X)).$

This composition is continuous because $p^{*}$ is continuous due to Lemma
4.15. By definition, $\sigma(x)(0)=\delta_{x}\in{\mathcal{B}}_{nk}(X)$ and
$\sigma(x)(1)=\delta_{x_{0}}\in{\mathcal{B}}_{nk}(X)$. Since $H(x)$ is
resolvable, $H_{*}$ is induced by $H$, and $\sigma$ factors through
$\Phi_{k}$, it follows that $\sigma(x)$ is resolvable. Therefore,
$i\operatorname{\mathsf{cat}}(X)\leq kn-1$.

To prove this for $i\operatorname{\mathsf{TC}}_{m}$ for a fixed $m\geq 2$, we
first observe that the product map $p_{m}:E^{m}\to X^{m}$ defined as
$p_{m}(\overline{x})=(p(e_{1}),\ldots,p(e_{m}))$ is a covering map of degree
$km$. Thus, the induced map $p_{m}^{*}:X^{m}\to{\mathcal{B}}_{km}(E^{m})$ is
continuous by Lemma 4.15. Now, letting
$i\operatorname{\mathsf{TC}}_{m}(E)=q-1$, there exists a $q$-intertwined
$m$-navigation algorithm $s:E^{m}\to{\mathcal{P}}({\mathcal{B}}_{q}(E))$ that
induces by functoriality
$s_{*}:{\mathcal{B}}_{km}(E^{m})\to{\mathcal{B}}_{km}({\mathcal{P}}({\mathcal{B}}_{q}(E)))$.
As before, we form the composition

$\sigma_{m}=\widetilde{p}\hskip 1.13809pt\Phi_{k}\hskip 1.13809pts_{*}\hskip
0.85358ptp_{m}^{*}:X^{m}\to P({\mathcal{B}}_{qkm}(X)).$

It follows by construction that $\sigma_{m}(\overline{x})\in
P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$ for each $\overline{x}\in X^{m}$ and
the image of $\sigma_{m}$ is indeed in ${\mathcal{P}}({\mathcal{B}}_{qk}(X))$.
Therefore, $i\operatorname{\mathsf{TC}}_{m}(X)\leq kmq-1$. ∎

###  4.C. Relations between $i\operatorname{\mathsf{cat}}$ and
$i\operatorname{\mathsf{TC}}_{m}$

The following statements show that $i\operatorname{\mathsf{TC}}_{m}$ relates
with the $i\operatorname{\mathsf{cat}}$ of products of $X$ in mostly the same
way as $d\operatorname{\mathsf{TC}}_{m}$ (resp.
$\operatorname{\mathsf{TC}}_{m}$) relates with the
$d\operatorname{\mathsf{cat}}$ (resp. $\operatorname{\mathsf{cat}}$) of
products of $X$, [J], [R], [BGRT].

######  4.17 Proposition.

For any space $X$, $i\operatorname{\mathsf{cat}}(X)\leq
i\operatorname{\mathsf{TC}}(X)$.

###### Proof.

Let $i\operatorname{\mathsf{TC}}(X)=n-1$. Then there exists a $n$-intertwined
navigation algorithm $s:X\times X\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$. For
a fixed basepoint $x_{0}\in X$, let $J:X\to X\times X$ be the inclusion
$J:x\mapsto(x,x_{0})$. Taking the composition
$sJ:X\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$, we see that
$sJ(x)(0)=\delta_{x}$ and $sJ(x)(1)=\delta_{x_{0}}$ for all $x\in X$. Hence,
$i\operatorname{\mathsf{cat}}(X)\leq n-1$. ∎

######  4.18 Proposition.

For any $m\geq 2$, $i\operatorname{\mathsf{TC}}_{m}(X)\leq
i\operatorname{\mathsf{cat}}(X^{m})$.

###### Proof.

Let $i\operatorname{\mathsf{cat}}(X^{m})=n-1$. There exists a map
$H:X^{m}\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X^{m}))$ with
$H(\overline{x})(0)=\delta_{\overline{x}}$ and
$H(\overline{x})(1)=\delta_{\overline{x_{0}}}$ for all $\overline{x}\in X^{m}$
and a fixed basepoint $\overline{x_{0}}=(x_{1}^{0},\ldots,x_{m}^{0})\in
X^{m}$. Let $\operatorname{proj}_{i}:X^{m}\to X$ be the projection onto the
$i^{th}$ coordinate that induces the map
$(\operatorname{proj}_{i})_{*}:{\mathcal{B}}_{n}(X^{m})\to{\mathcal{B}}_{n}(X)$.
Let $H(\overline{x})=\phi$, and define
$\phi_{i}=(\operatorname{proj}_{i})_{*}\phi$ and
$\overline{\phi_{i}}=(\operatorname{proj}_{i})_{*}\overline{\phi}$. For each
$i$, let $\gamma_{i}\in P(X)$ be a path such that $\gamma_{i}(0)=x_{i}^{0}$
and $\gamma_{i}(1)=x_{i+1}^{0}$. This gives
$\gamma_{i}^{*}=\omega_{n}\gamma_{i}\in P({\mathcal{B}}_{n}(X))$. Then, using
Lemma 2.6, we define a continuous map $\sigma:X^{m}\to
P({\mathcal{B}}_{n}(X))$ as

$\sigma(\overline{x})=\left(\phi_{1}\cdot\gamma_{1}^{*}\cdot\overline{\phi}_{2}\right)\star\left(\phi_{2}\cdot\gamma_{2}^{*}\cdot\overline{\phi}_{3}\right)\star\cdots\star\left(\phi_{m-1}\cdot\gamma_{m-1}^{*}\cdot\overline{\phi}_{m}\right).$

Clearly, $\sigma(\overline{x})\in P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$. We
still need to verify the resolvability of $\sigma(\overline{x})$. Let
$\mu=\sum\lambda_{\alpha}\alpha\in{\mathcal{B}}_{n}(P(X^{m}))$ be a resolver
of $\phi\in{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X^{m}))$. Let
$\alpha_{i}=\operatorname{proj}_{i}\alpha$ and
$\overline{\alpha_{i}}=\operatorname{proj}_{i}\overline{\alpha}$. As before,
we define a joined path

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

Then the measure
$\nu=\sum\lambda_{\alpha}\widetilde{\alpha}\in{\mathcal{B}}_{n}(P(X))$ is a
resolver of $\sigma(\overline{x})$. Hence, $\sigma$ defines an $n$-intertwined
$m$-navigation algorithm on $X$ and so,
$i\operatorname{\mathsf{TC}}_{m}(X)\leq n-1$. ∎

######  4.19 Proposition.

For each $m\geq 2$, $i\operatorname{\mathsf{TC}}_{m}(X)\leq
i\operatorname{\mathsf{TC}}_{m+1}(X)$.

###### Proof.

Let $i\operatorname{\mathsf{TC}}_{m+1}(X)=n-1$ so that there exists an
$n$-intertwined $(m+1)$-navigation algorithm
$s:X^{m+1}\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$. Let us fix some $x_{0}\in
X$ and define a map $J:X^{m}\to X^{m+1}$ such that

$J:(x_{1},\ldots,x_{m})\mapsto(x_{0},x_{1},\ldots,x_{m})$

for all $\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}$. Let
$sJ(\overline{x})=\phi$. Then $\phi((i-1)/m)=\delta_{x_{i-1}}$ for all $2\leq
i\leq m+1$ and $\phi(0)=\delta_{x_{0}}$. In other words,
$\phi(i/m)=\delta_{x_{i}}$ for all $1\leq i\leq m$. Let $H:I\to I$ be the map

$H:s\mapsto\frac{(m-1)s+1}{m}.$

Then for $\phi^{\prime}=\phi H:I\to{\mathcal{B}}_{n}(X)$, we get
$\phi^{\prime}((i-1)/(m-1))=\delta_{x_{i}}$ for all $1\leq i\leq m$. Hence, we
define a continuous map $K:X^{m}\to P({\mathcal{B}}_{n}(X))$ such that
$K(\overline{x})=\phi^{\prime}$. If
$\mu=\sum\lambda_{\alpha}\alpha\in{\mathcal{B}}_{n}(P(X))$ is a resolver of
$\phi$, then for $\alpha^{\prime}=\alpha H$, the measure
$\nu=\sum\lambda_{\alpha}\alpha^{\prime}\in{\mathcal{B}}_{n}(P(X))$ is a
resolver of $\phi^{\prime}$. Hence, $K$ defines an $n$-intertwined
$m$-navigation algorithm on $X$. Therefore,
$i\operatorname{\mathsf{TC}}_{m}(X)\leq n-1$. ∎

Just like in the classical setting where the equality
$\operatorname{\mathsf{cat}}(X)=\operatorname{\mathsf{TC}}(X)$ holds [Far2,
Lemma 8.2] for all topological groups $X$, in the distributional setting also,
the equality $d\operatorname{\mathsf{cat}}(X)=d\operatorname{\mathsf{TC}}(X)$
holds, see [DJ, Theorem 3.15]. We now prove that such an equality holds for
the intertwining invariants as well.

######  4.20 Proposition.

If $X$ is a topological group, then $i\operatorname{\mathsf{TC}}_{m+1}(X)\leq
i\operatorname{\mathsf{cat}}(X^{m})$ for all $m\geq 1$.

###### Proof.

Let $i\operatorname{\mathsf{cat}}(X^{m})=n-1$ and
$\overline{e}=(e,\ldots,e)\in X^{m}$ be the basepoint, where $e$ is the
identity. This can be chosen as the basepoint since $X$ is path-connected.
There exists a map $H:X^{m}\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X^{m}))$
satisfying $H(\overline{x})(0)=\delta_{\overline{x}}$ and
$H(\overline{x})(1)=\delta_{\overline{e}}$ for each $\overline{x}\in X^{m}$.
Since $X$ is a topological group, the map $J_{m}:X^{m+1}\to X^{m}$ defined as

$J_{m}:\left(x_{1},x_{2},\ldots,x_{m+1}\right)\mapsto\left(x_{2}^{-1}x_{1},x_{3}^{-1}x_{2},\ldots,x_{m+1}^{-1}x_{m}\right),$

is continuous. Here, $x_{i}^{-1}\in X$ denotes the group-theoretic inverse of
$x_{i}\in X$ and $x_{i+1}^{-1}x_{i}\in X$ is the group-theoretic product due
to the product map $X\times X\to X$. Let
$HJ_{m}(x_{1},\ldots,x_{m+1})=\phi\in{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X^{m}))$.
Using projections $\operatorname{proj}_{i}:X^{m}\to X$, we get induced maps
$(\operatorname{proj}_{i})_{*}:{\mathcal{B}}_{n}(X^{m})\to{\mathcal{B}}_{n}(X)$
and paths $\phi_{i}=(\operatorname{proj}_{i})_{*}\phi$ in
$P_{0}({\mathcal{B}}_{n}(X))$ that start at $\delta_{x_{i+1}^{-1}x_{i}}$ and
end at $\delta_{e}$ for each $1\leq i\leq m$. The product map $X\times X\to X$
induces by functoriality of ${\mathcal{B}}_{n}$ a map
$\mathcal{M}:X\times{\mathcal{B}}_{n}(X)\to{\mathcal{B}}_{n}(X)$ defined as

$\mathcal{M}:\left(x,\sum b_{y}y\right)\mapsto\sum b_{y}\hskip 1.70717ptxy.$

For each $1\leq i\leq m$, we define a path $x_{i+1}\phi_{i}\in
P({\mathcal{B}}_{n}(X))$ such that for all $t\in I$,

$\left(x_{i+1}\phi_{i}\right)(t)=\mathcal{M}\left(x_{i+1},\phi_{i}(t)\right).$

Clearly, $(x_{i+1}\phi_{i})(0)=\delta_{x_{i}}$ and
$(x_{i+1}\phi_{i})(1)=\delta_{x_{i+1}}$. Finally, we define a mapping
$s:X^{m+1}\to P({\mathcal{B}}_{n}(X))$ where, as usual,

$s(x_{1},\ldots,x_{m+1})=x_{2}\phi_{1}\star x_{3}\phi_{2}\star\cdots\star
x_{m+1}\phi_{m}.$

Here, we are using the map $\theta_{m}:T_{m}({\mathcal{B}}_{n}(X))\to
P({\mathcal{B}}_{n}(X))$ from Section 2.D with $a_{i}=1/t_{i+1}=m/i$ for all
$1\leq i\leq m-1$. By definition of $s$, we have that
$s(x_{1},\ldots,x_{m+1})\in P(\delta_{x_{1}},\ldots,\delta_{x_{m}+1})$. We
still need to check if $s(x_{1},\ldots,x_{m+1})$ is resolvable. Let
$\mu=\sum\lambda_{\alpha}\alpha\in{\mathcal{B}}_{n}(P_{0}(X^{m}))$ be a
resolver of $\phi\in{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X^{m}))$ and
$\alpha_{i}=\operatorname{proj}_{i}\alpha$. Then,
$\mu_{i}=\sum\lambda_{\alpha}\alpha_{i}\in{\mathcal{B}}_{n}(P_{0}(X))$ becomes
a resolver of $\phi_{i}$ for each $i$. Let $x_{i+1}\alpha_{i}\in P(X)$ be
given by $(x_{i+1}\alpha_{i})(t)=x_{i+1}\alpha_{i}(t)$ for all $t\in I$. As
before, we define a path

$\widehat{\alpha}=x_{2}\alpha_{1}\star x_{3}\alpha_{2}\star\cdots\star
x_{m+1}\alpha_{m}.$

Then $\nu=\sum\lambda_{\alpha}\widehat{\alpha}\in{\mathcal{B}}_{n}(P(X))$ is a
resolver of $s(x_{1},\ldots,x_{m+1})$. Hence, $s$ defines an $n$-intertwined
$(m+1)$-navigation algorithm. So, $i\operatorname{\mathsf{TC}}_{m+1}(X)\leq
n-1$. ∎

######  4.21 Corollary.

If $X$ is a topological group, then
$i\operatorname{\mathsf{TC}}(X)=i\operatorname{\mathsf{cat}}(X)$.

###### Proof.

From Proposition 4.20, we get $i\operatorname{\mathsf{TC}}(X)\leq
i\operatorname{\mathsf{cat}}(X)$ by taking $m=1$. The reverse inequality
follows from Proposition 4.17. ∎

##  5\. For Higman’s group ${\mathcal{H}}$

For each $m\geq 2$, we are interested in finding spaces $X$ (possibly
depending on the choice of $m$) for which
$i\operatorname{\mathsf{TC}}_{m}(X)<d\operatorname{\mathsf{TC}}_{m}(X)$. For
$m=2$, such an example has been found in [Dr, Proposition 6.4]. To obtain
examples for each $m\geq 2$, we proceed as follows.

Let $\Gamma$ be a discrete group and let $B\Gamma$ denote the homotopy class
of its classifying spaces. Since $i\operatorname{\mathsf{TC}}_{m}$ and
$i\operatorname{\mathsf{cat}}$ are homotopy invariants of spaces by Corollary
4.4 and Proposition 4.11 (3), respectively, we can define
$i\operatorname{\mathsf{TC}}_{m}(\Gamma):=i\operatorname{\mathsf{TC}}_{m}(B\Gamma)$
and
$i\operatorname{\mathsf{cat}}(\Gamma):=i\operatorname{\mathsf{cat}}(B\Gamma)$.
Let ${\mathcal{H}}$ denote Higman’s group [Hi] which is presented as follows.

${\mathcal{H}}=\langle x,y,z,w\mid
xyx^{-1}y^{-2},yzy^{-1}z^{-2},zwz^{-1}w^{-2},wxw^{-1}x^{-2}\rangle.$

It is well-known that ${\mathcal{H}}$ is a torsion-free, discrete group of
cohomological dimension $2$ and it is acyclic, i.e., the (co)homology groups
of $B{\mathcal{H}}$ vanish in positive degrees. Furthermore, $B{\mathcal{H}}$
is a CW complex of finite type. The $m^{th}$ sequential topological complexity
of ${\mathcal{H}}$ was computed in [FO, Theorem 2.2] as $2m$ for each $m\geq
2$ (also see [GLO2, Theorem 4.1] for the special case $m=2$).

In [Dr], Dranishnikov showed that
$i\operatorname{\mathsf{TC}}({\mathcal{H}})=1$. We use their technique to
obtain the generalized result that
$i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1$ for all $m\geq 2$. We first
prepare as follows.

Let us fix some $m\geq 2$. For each $1\leq j\leq m$, consider the polynomial

$z_{j}(t)=\prod_{i=1,i\neq j}^{m}(t-t_{i})=\prod_{i=1,i\neq
j}^{m}\left(t-\frac{i-1}{m-1}\right)$

in the variable $t\in I$. For any fixed $j$, the only roots of $z_{j}$ are
$t_{i}$ for all $i\neq j$. So, clearly, $z_{j}(t_{j})\neq 0$ and
$z_{j}(t_{i})=0$ for all $i\neq j$. Let

$y_{j}(t)=\frac{z_{j}(t)}{z_{j}(t_{j})}.$

Then $y_{j}(t_{j})=1$ and $y_{j}(t_{i})=0$ for all $i\neq j$. Next, for each
$1\leq j\leq m$, define

$w_{j}(t)=\frac{\left(y_{j}(t)\right)^{2}}{\sum_{i=1}^{m}\left(y_{i}(t)\right)^{2}}.$

By definition, $\sum_{i=1}^{m}\left(y_{j}(t)\right)^{2}\neq 0$ for all $t$, so
$w_{j}(t)$ is well-defined and continuous. We observe the following.

  * •

For each $j$, $w_{j}(t)\in[0,1]$ for all $t\in I$.

  * •

For each $j$, $w_{j}(t_{j})=1$ and $w_{j}(t_{i})=0$ for all $i\neq j$.

  * •

$\sum_{i=1}^{m}w_{i}(t)=1$ for all $t\in I$.

Let $\ast^{m}X$ denote the iterated join of $X$ in the sense of [Mi]. We
define a map $q_{m}:X^{m}\times I\to\ast^{m}X$ as

$q_{m}:\left(\left(x_{1},x_{2},\ldots,x_{m}\right),t\right)\mapsto\sum_{i=1}^{m}w_{i}(t)\hskip
1.42262ptx_{i}.$

Since each $w_{i}$ is continuous, the map $q_{m}$ is continuous as well, [Mi,
Section 2]. This produces a continuous map $Q_{m}:X^{m}\to P(\ast^{m}X)$ such
that for all $\overline{x}\in X^{m}$, we have
$Q_{m}(\overline{x})(t_{i})=x_{i}$ for each $1\leq i\leq m$.

We now extend the proof of the equality
$i\operatorname{\mathsf{TC}}({\mathcal{H}})=1$ from [Dr] as follows.

######  5.1 Proposition.

For Higman’s group ${\mathcal{H}}$,
$i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1$ for all $m\geq 2$.

###### Proof.

Let us fix some $m$. Since ${\mathcal{H}}$ is not contractible, we only need
to show that $i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})\leq 1$. As noted
above, $H_{i}(B{\mathcal{H}})=0$ for all $i>0$. Let $\\{\text{pt}\\}$ denote a
one-point space. Since, $H_{i}(B{\mathcal{H}})=0=H_{i}(\\{\text{pt}\\})$ for
each $i>0$, Dold’s Theorem [D] implies that
$\widetilde{H}_{i}(SP^{2}(B{\mathcal{H}}))=\widetilde{H}_{i}(SP^{2}(\\{\text{pt}\\}))=0$
for each $i>0$. Hence, $SP^{2}(B{\mathcal{H}})$ is acyclic. Furthermore,
$\pi_{1}(SP^{2}(B{\mathcal{H}}))=\widetilde{H}_{1}(B{\mathcal{H}})=0$. Then by
Whitehead’s theorem and Hurewicz’s theorem [Ha], $SP^{2}(B{\mathcal{H}})$ is a
contractible CW complex and hence an absolute extensor. We note that the
disjoint union $\sqcup_{i=1}^{m}B{\mathcal{H}}$ is closed in the iterated join
$\ast^{m}B{\mathcal{H}}$. So, if $\partial:B{\mathcal{H}}\to
SP^{2}(B{\mathcal{H}})$ denotes the diagonal embedding given by
$\partial(x)=2x$, then there exists an extension

(5.a)
${\bigsqcup_{i=1}^{m}B{\mathcal{H}}}$${SP^{2}(B{\mathcal{H}})}$${\ast^{m}B{\mathcal{H}}}$$\scriptstyle{\bigsqcup_{i=1}^{m}\partial}$

of the map $\sqcup_{i=1}^{m}\partial$. This produces a map
$f:P(\ast^{m}B{\mathcal{H}})\to P(SP^{2}(B{\mathcal{H}}))$ of paths. The map
$\phi:SP^{2}(B{\mathcal{H}})\to{\mathcal{B}}_{2}(B{\mathcal{H}})$ defined as

$\phi:x+y\mapsto\frac{1}{2}x+\frac{1}{2}y$

is an embedding that produces a map
$\phi^{\prime}:P(SP^{2}(B{\mathcal{H}}))\to
P({\mathcal{B}}_{2}(B{\mathcal{H}}))$. Finally, consider the composition
$\psi=\phi^{\prime}fQ_{m}:(B{\mathcal{H}})^{m}\to
P({\mathcal{B}}_{2}(B{\mathcal{H}}))$, where the map
$Q_{m}:(B{\mathcal{H}})^{m}\to P(\ast^{m}B{\mathcal{H}})$ is defined above. By
our construction of maps, for any $\overline{x}\in(B{\mathcal{H}})^{m}$, we
have that $\psi(\overline{x})\in P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$. We
still need to check if $\psi(\overline{x})$ is resolvable. See that each path
$g\in P(SP^{2}(B{\mathcal{H}}))$ is resolvable in the sense of Definition 3.3
upon replacing the role of the functor ${\mathcal{B}}_{2}$ by the functor
$SP^{2}$ (see also the proof of [Dr, Proposition 6.4]). Hence, its image
$\phi^{\prime}(g)\in P({\mathcal{B}}_{2}(B{\mathcal{H}}))$ is resolvable in
the sense of of Definition 3.3. Then by Lemma 3.4, it follows that the image
of $\psi$ is in ${\mathcal{P}}({\mathcal{B}}_{2}(B{\mathcal{H}}))$. Thus,
$\psi$ is the required $2$-intertwined $m$-navigation algorithm. So,
$i\operatorname{\mathsf{TC}}_{m}(B{\mathcal{H}})=i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1$.
∎

We note that in contrast, the exact values
$d\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})$ are not known.

######  5.2 Remark.

Since ${\mathcal{H}}$ is torsion-free, it follows from [Dr, Theorem 5.3] (see
also [KW, Theorem 7.4] for a version of the Eilenberg–Ganea theorem [EG] for
the analog invariants) that $d\operatorname{\mathsf{cat}}({\mathcal{H}})=2$.
Thus, from Proposition 5.1 and [J, Proposition 3.5], we get for each $m\geq 2$
that

$i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1<2(m-1)=d\operatorname{\mathsf{cat}}({\mathcal{H}}^{m-1})\leq
d\operatorname{\mathsf{TC}}_{m}({\mathcal{H}}).$

So, we have shown that for each $m\geq 2$, the first inequality in Proposition
4.6 can be strict. Hence, in general, the notions of
$i\operatorname{\mathsf{TC}}_{m}$ and $d\operatorname{\mathsf{TC}}_{m}$ are
different! Furthermore, as noted in [Dr, Corollary 6.5], we get
$i\operatorname{\mathsf{cat}}({\mathcal{H}})=1$ due to Propositions 5.1 and
4.17 (1). So, while an analog of the Eilenberg–Ganea theorem holds (only) for
torsion-free groups in the distributional setting, such an analog is not
possible in the intertwining setting, even for torsion-free groups.

For a fixed $X$, the non-decreasing sequence
$\\{i\operatorname{\mathsf{TC}}_{m}(X)\\}_{m\geq 2}$ can, in fact, be constant
due to Proposition 5.1. Besides $X=B{\mathcal{H}}$, another example for which
the sequence is again constant at $1$ is $X={\mathbb{R}}P^{\infty}$. This
follows from Corollary 4.7 and [J, Section 8.A] due to [KW, Corollary 7.3].

######  5.3 Example.

Since
$i\operatorname{\mathsf{cat}}({\mathcal{H}})=1=i\operatorname{\mathsf{cat}}({\mathbb{Z}}_{2})$,
we get for their free product the equality
$i\operatorname{\mathsf{cat}}({\mathcal{H}}\ast{\mathbb{Z}}_{2})=1$ due to
Proposition 4.14.

######  5.4 Remark.

In the case of classical $\operatorname{\mathsf{TC}}_{m}$, if $X$ is a non-
contractible, finite CW complex, then $\operatorname{\mathsf{TC}}_{m}(X)\geq
m-1$, see [R, Proposition 3.5]. However, due to Proposition 5.1, such a
statement does not hold in the case of $i\operatorname{\mathsf{TC}}_{m}$ for
any $m\geq 3$.

The classifying space $B{\mathcal{H}}$ of Higman’s group is an example of a
finite-dimensional aspherical CW complex for which
$i\operatorname{\mathsf{cat}}$ and $i\operatorname{\mathsf{TC}}_{m}$ values
agree for all $m$, even though it is neither a topological group nor a
(product of) co-$H$-space(s). Such an example is not known in the
distributional setting or the classical setting.

##  6\. Some simple characterizations

Let $\pi_{m}:{\mathcal{P}}({\mathcal{B}}_{n}(X))\to({\mathcal{B}}_{n}(X))^{m}$
be the evaluation map

$\pi_{m}:\phi\mapsto\left(\phi(t_{1}),\phi(t_{2}),\ldots,\phi(t_{m})\right),$

where $t_{i}=(i-1)/(m-1)$ for all $1\leq i\leq m$. Recall the inclusion
$\omega_{n}:X\to{\mathcal{B}}_{n}(X)$ defined as
$\omega_{n}:x\mapsto\delta_{x}$. Consider the following pullback diagram,

(6.a)
${{\mathscr{U}}_{n,m}}$${{\mathcal{P}}({\mathcal{B}}_{n}(X))}$${X^{m}}$${({\mathcal{B}}_{n}(X))^{m}}$$\scriptstyle{\eta_{m}}$$\scriptstyle{q_{m}}$$\scriptstyle{\pi_{m}}$$\scriptstyle{\omega_{n}^{m}}$

where
$\omega_{n}^{m}(\overline{x})=(\omega_{n}(x_{1}),\omega_{n}(x_{2}),\ldots,\omega_{n}(x_{m}))=(\delta_{x_{1}},\ldots,\delta_{x_{m}})$.

######  6.1 Proposition.

$i\operatorname{\mathsf{TC}}_{m}(X)<n$ if and only if there exists a section
to the map $q_{m}:{\mathscr{U}}_{n,m}\to X^{m}$.

###### Proof.

First, let $i\operatorname{\mathsf{TC}}_{m}(X)<n$. Then there exists an
$n$-intertwined $m$-navigation algorithm
$s_{m}:X^{m}\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$. Note that $\pi_{m}\hskip
1.13809pts_{m}(\overline{x})=(\delta_{x_{1}},\ldots,\delta_{x_{m}})=\omega_{n}^{m}(\overline{x})$.
So, by the universal property of the above pullback, there exists
$K:X^{m}\to{\mathscr{U}}_{n,m}$ such that $q_{m}K=\mathbbm{1}_{X^{m}}$. So,
$q_{m}$ admits a section. Conversely, let $K:X^{m}\to{\mathscr{U}}_{n,m}$ be a
section of $q_{m}$, i.e., $q_{m}K=\mathbbm{1}_{X^{m}}$. Note that

$\pi_{m}(\eta_{m}K)=(\pi_{m}\eta_{m})K=(\omega_{n}^{m}q_{m})K=\omega_{n}^{m}(q_{m}K)=\omega_{n}^{m}.$

Hence, $(\eta_{m}K)(\overline{x})\in P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$.
So, $\eta_{m}K$ defines an $n$-intertwined $m$-navigation algorithm on $X$.
Therefore, $i\operatorname{\mathsf{TC}}_{m}(X)\leq n-1$. ∎

Similarly, if we consider the evaluation map
$p_{0}:{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X))\to{\mathcal{B}}_{n}(X)$ defined
as $p_{0}:\phi\mapsto\phi(0)$ and let $q:{\mathscr{V}}_{n}\to X$ be the
pullback of $p_{0}$ along $\omega_{n}$ for the pullback space
${\mathscr{V}}_{n}$, then the following statement follows immediately.

######  6.2 Proposition.

$i\operatorname{\mathsf{cat}}(X)<n$ if and only if there exists a section to
the map $q:{\mathscr{V}}_{n}\to X$.

######  6.3 Remark.

We note that the evaluation maps
$\widehat{\pi_{m}}:P({\mathcal{B}}_{n}(X))\to({\mathcal{B}}_{n}(X))^{m}$ and
$\widehat{p_{0}}:P_{0}({\mathcal{B}}_{n}(X))\to{\mathcal{B}}_{n}(X)$ are
fibrations. So, due to Corollary 3.6, they are the fibrational substitutes of
the maps $\pi_{m}$ and $p_{0}$, respectively. In the above setting, the maps
$\pi_{m}$ and $p_{0}$ cannot be replaced by their fibrational substitutes
$\widehat{\pi_{m}}$ and $\widehat{p_{0}}$ because then, the reverse
implications will not hold in the above propositions.

Let
${\mathcal{P}}_{1}({\mathcal{B}}_{n}(X))=\\{f\in{\mathcal{P}}({\mathcal{B}}_{n}(X))\mid
f(0)=\delta_{x_{0}}\\}$ for some fixed $x_{0}\in X$. We define an evaluation
map
$\xi_{m}:{\mathcal{P}}_{1}({\mathcal{B}}_{n}(X))\to({\mathcal{B}}_{n}(X))^{m}$
as

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

As before, we consider the following pullback diagram.

(6.b)
${{\mathscr{W}}_{n,m}}$${{\mathcal{P}}_{1}({\mathcal{B}}_{n}(X))}$${X^{m}}$${({\mathcal{B}}_{n}(X))^{m}}$$\scriptstyle{\rho_{m}}$$\scriptstyle{r_{m}}$$\scriptstyle{\xi_{m}}$$\scriptstyle{\omega_{n}^{m}}$

The following statement may be compared with [J, Lemma 5.7].

######  6.4 Proposition.

For any $m\geq 1$, if $i\operatorname{\mathsf{cat}}(X^{m})<n$, then there
exists a section to $r_{m}:{\mathscr{W}}_{n,m}\to X^{m}$.

###### Proof.

If $i\operatorname{\mathsf{cat}}(X^{m})<n$, then there exists a continuous map
$H:X^{m}\to{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X^{m}))$ such that for some
fixed basepoint $\overline{x_{0}}=(x_{1}^{0},\ldots,x_{m}^{0})\in X^{m}$,
$H(\overline{x})(0)=\delta_{\overline{x}}$ and
$H(\overline{x})(1)=\delta_{\overline{x_{0}}}$ for all $\overline{x}\in
X^{m}$. Let $H(\overline{x})=\phi$. For projections
$\operatorname{proj}_{i}:X^{m}\to X$, let
$\phi_{i}=(\operatorname{proj}_{i})_{*}\overline{\phi}\in
P({\mathcal{B}}_{n}(X))$ be paths from $\delta_{x_{i}^{0}}$ to
$\delta_{x_{i}}$. For each $i$, let $\gamma_{i}\in P(X)$ be a path from
$x_{i}^{0}$ to $x_{i+1}^{0}$ and $\gamma_{i}^{*}=\omega_{n}\gamma_{i}\in
P({\mathcal{B}}_{n}(X))$. Finally, define a map $K:X^{m}\to
P({\mathcal{B}}_{n}(X))$ such that

$K(\overline{x})=\phi_{1}\star\left(\overline{\phi}_{1}\cdot\gamma_{1}^{*}\cdot\phi_{2}\right)\star\left(\overline{\phi}_{2}\cdot\gamma_{2}^{*}\cdot\phi_{3}\right)\star\cdots\star\left(\overline{\phi}_{m-1}\cdot\gamma_{m-1}^{*}\cdot\phi_{m}\right),$

where we use the map $\theta_{m}:T_{m}({\mathcal{B}}_{n}(X))\to
P({\mathcal{B}}_{n}(X))$ in Section 2.D with $a_{i}=i/m$ for all $1\leq i\leq
m-1$. The continuity of $K$ follows from Lemma 2.6. For all $\overline{x}\in
X^{m}$, we have $K(\overline{x})(0)=\delta_{x_{1}^{0}}$ and
$K(\overline{x})(i/m)=\delta_{x_{i}}$ for all $1\leq i\leq m$. Now, we check
the resolvability of $K(\overline{x})$. Let
$\mu=\sum\lambda_{\alpha}\alpha\in{\mathcal{B}}_{n}(P(X))$ be a resolver of
$\phi\in{\mathcal{P}}_{0}({\mathcal{B}}_{n}(X))$. Letting
$\alpha_{i}=\operatorname{proj}_{i}\overline{\alpha}$, we can define a path

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

Then the measure
$\nu=\sum\lambda_{\alpha}\widetilde{\alpha}\in{\mathcal{B}}_{n}(P(X))$ is a
resolver of $K(\overline{x})$. Hence, the image of $K$ is in
${\mathcal{P}}_{1}({\mathcal{B}}_{n}(X))$. Clearly, $\xi_{m}K=\omega_{n}^{m}$.
So, by the universal property of the pullback in Diagram 6.b, we obtain a
section $s:X^{m}\to{\mathscr{W}}_{n,m}$ to the map $r_{m}$. ∎

##  7\. Cohomological lower bounds

In the case of intertwining motion, there is a lot of liberty with the system
while navigating between given positions. Indeed, the intertwining of strings
of a resolvable path can happen in various ways. Due to this liberty, for most
configuration spaces, very few rules may be needed to plan the motion. For any
non-contractible configuration space, we know that at least two rules are
needed. So, to enrich the theory of the intertwining invariants, it becomes
important to give conditions and examples of spaces for which at least three
rules are needed to plan the motion.

The aim of this section is to obtain lower bounds for
$i\operatorname{\mathsf{cat}}(X)$ and $i\operatorname{\mathsf{TC}}_{m}(X)$. As
in the case of $d\operatorname{\mathsf{cat}}(X)$ and
$d\operatorname{\mathsf{TC}}_{m}(X)$, we aim to find these in terms of the
cohomology of some functor of the space $X$ and its products $X^{m}$. As a
starting point, we check if the map
$\omega_{n}:X\hookrightarrow{\mathcal{B}}_{n}(X)$ can be of help. For a
coefficient ring $R$, this induces a homomorphism
$\omega_{n}^{*}:H^{*}({\mathcal{B}}_{n}(X);R)\to H^{*}(X;R)$.

######  7.1 Proposition.

If $\alpha\in H^{k}({\mathcal{B}}_{n}(X);R)$ for some $k\geq 1$ and ring $R$
such that $\omega_{n}^{*}(\alpha)\neq 0$, then
$i\operatorname{\mathsf{cat}}(X)\geq n$.

###### Proof.

If possible, let us assume $i\operatorname{\mathsf{cat}}(X)<n$. Then there
exists a continuous map $H:X\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$ such that
$H(x)(0)=\delta_{x}$ and $H(x)(1)=\delta_{x_{0}}$ for each $x\in X$. Let
$c:X\to{\mathcal{B}}_{n}(X)$ be the constant map $c:x\mapsto\delta_{x_{0}}$.
We define a homotopy $K:X\times I\to{\mathcal{B}}_{n}(X)$ by

$K(x,t)=H(x)(t).$

From this, we get that $\omega_{n}\simeq c$. In particular, this means
$\omega_{n}^{*}=0$. This contradicts the hypothesis that
$\omega_{n}^{*}(\alpha)\neq 0$. Hence, $i\operatorname{\mathsf{cat}}(X)\geq
n$. ∎

Let $\Delta:X\to X\times X$ be the diagonal map. Then using the same technique
as above, the following can be shown easily.

######  7.2 Proposition.

If $\alpha\in H^{k}(({\mathcal{B}}_{n}(X))^{2};R)$ for some $k\geq 1$ and ring
$R$ such that $(\omega_{n}\times\omega_{n})^{*}(\alpha)\neq 0$ and
$\Delta^{*}((\omega_{n}\times\omega_{n})^{*}(\alpha))=0$, then
$i\operatorname{\mathsf{TC}}(X)\geq n$.

Recall Diagram 6.a. Let
$\Delta_{n}^{\prime}:{\mathcal{B}}_{n}(X)\to({\mathcal{B}}_{n}(X))^{m}$ be the
diagonal map.

######  7.3 Proposition.

If $\alpha\in H^{k}(({\mathcal{B}}_{n}(X))^{m};R)$ for some $k\geq 1$, $m\geq
2$ and ring $R$ such that $(\Delta_{n}^{\prime})^{*}(\alpha)=0$ and
$(\omega_{n}^{m})^{*}(\alpha)\neq 0$, then
$i\operatorname{\mathsf{TC}}_{m}(X)\geq n$.

###### Proof.

In view of Lemma 3.5, there exists a map
$f:{\mathcal{B}}_{n}(X)\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$ such that
$f^{*}$ is an isomorphism in cohomology and
$f^{*}\pi_{m}^{*}=(\Delta^{\prime}_{n})^{*}$. So, since
$(\Delta_{n}^{\prime})^{*}(\alpha)=0$, we have that $\pi_{m}^{*}(\alpha)=0$.
If possible, let $i\operatorname{\mathsf{TC}}_{m}(X)<n$. Then there exists a
continuous map $s_{m}:X^{m}\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$ such that
$s_{m}(\overline{x})\in P(\delta_{x_{1}},\ldots,\delta_{x_{m}})$ for each
$\overline{x}\in X^{m}$. This means $\pi_{m}s_{m}=\omega_{n}^{m}$. Thus, by
the universal property of the pullback in Diagram 6.a, we get a section
$K:X^{m}\to{\mathscr{U}}_{n,m}$ of the map
$q_{m}:X^{m}\to{\mathscr{U}}_{n,m}$. Hence, we have that
$K^{*}\eta_{n}^{*}\pi_{m}^{*}=(\omega_{n}^{m})^{*}$. But then
$\pi_{m}^{*}(\alpha)=0\implies(\omega_{n}^{m})^{*}(\alpha)=0$. This is a
contradiction. Therefore, $i\operatorname{\mathsf{TC}}_{m}(X)\geq n$. ∎

We now investigate if the above lower bounds are useful. The homology of
${\mathcal{B}}_{n}(X)$ has been studied in [KK]. We refer the reader to [KK,
Sections 10 and 11] for some examples and computations.

######  7.4 Proposition.

If $X$ is a locally finite CW complex or a discrete space, then the induced
map $\omega_{n}^{*}$ is trivial for all $n\geq 2$ when coefficients are in
${\mathbb{Q}}$ or ${\mathbb{R}}$. Furthermore, with any coefficients,
$\omega_{2}^{*}$ is trivial.

###### Proof.

Recall that $\text{Sym}(\ast^{n}X)$ denotes the $n^{th}$ iterated symmetric
join. Due to [KK, Theorem 1.1], the quotient map
$q:\text{Sym}(\ast^{n}X)\to({\mathcal{B}}_{n}(X),\mathscr{T}_{1})$ defined in
Section 1.C is a homotopy equivalence. If $\Sigma$ denotes the reduced
suspension functor, then we have

$\Sigma({\mathcal{B}}_{n}(X),\mathscr{T}_{1})\simeq\Sigma\text{Sym}(\ast^{n}X)\simeq
SP^{n}(\Sigma X)/SP^{n-1}(\Sigma X)$

for all $n\geq 2$, where the last equivalence is due to [KK, Theorem 1.3].
Since the identity map
$\mathcal{I}:({\mathcal{B}}_{n}(X),\mathscr{T}_{1})\to({\mathcal{B}}_{n}(X),\mathscr{T}_{2})$
is also a homotopy equivalence (see Section 1.C and [Dr, Section 1]), we get
in homology the induced homomorphism

$(\Sigma\omega_{n})_{*}:H_{*}(\Sigma X;R)\to H_{*}(SP^{n}(\Sigma
X),SP^{n-1}(\Sigma X);R)$

because $(SP^{n}(\Sigma X),SP^{n-1}(\Sigma X))$ forms a good pair, [Ha]. When
$R\in\\{{\mathbb{Q}},{\mathbb{R}}\\}$, the map $H_{*}(\Sigma X;R)\to
H_{*}(SP^{n-1}(\Sigma X);R)$ induced by the diagonal embedding is injective
for each $n\geq 2$ by taking $Y=\Sigma X$ and $k=n-1$ in Proposition 2.3.
Furthermore, when $n=2$, the above map induced by the diagonal is just the
identity map. Hence, it is injective with respect to all the coefficients. So,
in all these cases, it follows that $(\Sigma\omega_{n})_{*}$ is trivial. From
this, we deduce that $\omega_{n}^{*}$ is trivial in all the above-mentioned
cases. ∎

Let $\xi_{n}:SP^{n}(X)\to\text{Sym}(*^{n}X)$ be the canonical embedding

$\xi_{n}:\sum_{i=1}^{n}x_{i}\mapsto\left[\frac{1}{n}x_{1},\ldots,\frac{1}{n}x_{n}\right].$

For any ring $R$, this induces $\xi_{n}^{*}:H^{*}(\text{Sym}(*^{n}X);R)\to
H^{*}(SP^{n}(X);R)$.

######  7.5 Corollary.

The map $\xi_{n}^{*}$ is not surjective for any $n\geq 2$ when
$R\in\\{{\mathbb{Q}},{\mathbb{R}}\\}$.

###### Proof.

We note that $\omega_{n}=(\mathcal{I}q)\xi_{n}\partial_{n}$, where
$\partial_{n}:X\to SP^{n}(X)$ is the diagonal embedding. When
$R\in\\{{\mathbb{Q}},{\mathbb{R}}\\}$, the map $\partial_{n}^{*}$ is
surjective by Proposition 2.3. Also, the induced map $(\mathcal{I}q)^{*}$ is
surjective. So, if $\xi_{n}^{*}$ were surjective, then $\omega_{n}^{*}$ will
also be surjective. Of course, this is not possible because of Proposition
7.4. ∎

So, in the above cases, the lower bounds provided by Propositions 7.1, 7.2,
and 7.3 are not useful. In particular, our methods do not give a lower bound
$2$ to $i\operatorname{\mathsf{cat}}(X)$ or
$i\operatorname{\mathsf{TC}}_{m}(X)$.

The next idea is to proceed using the symmetric products as in [DJ], [J]. But,
we can’t directly adopt the approach of [DJ, Lemma 4.5] because the resolvers
of a resolvable path can have wildly different characteristics — see Section
3.B. So, in general, it is difficult to characterize resolvable paths.

###  7.A. Preparation

In this subsection, we deal with the case $n=2$ to provide various conditions
in which $2$ could be the lower bound to the intertwining invariants of
spaces.

Let $\mu=af+bg\in{\mathcal{B}}_{2}(P(X))$ with $a,b>0$ and $f\neq g$ as paths
in $P(X)$. Then we can see $\mu$ as an element of $\text{Sym}(*^{2}P(X)))$ and
write it as $[af,bg]$. Consider the natural map
$\Psi:\text{Sym}(*^{2}P(X)))\to P(\text{Sym}(*^{2}X)))$ defined as
$\Psi:\mu\mapsto\widetilde{\mu}$, where we have
$\widetilde{\mu}(t)=[af(t),bg(t)]$ for all $t\in I$. Let
$Y(\mu)=\widetilde{\mu}(I)\subset\text{Sym}(*^{2}X)$. Let
$q(\mu):Y(\mu)\to({\mathcal{B}}_{2}(X),\mathscr{T}_{1})$ be the restriction of
the canonical quotient map
$q:\text{Sym}(*^{2}X)\to({\mathcal{B}}_{2}(X),\mathscr{T}_{1})$ to $Y(\mu)$.
Let us recall from Section 1.C the continuous map
$\mathcal{I}:({\mathcal{B}}_{2}(X),\mathscr{T}_{1})\to({\mathcal{B}}_{2}(X),\mathscr{T}_{2})$.

######  7.6 Lemma.

The map
$\widetilde{q}(\mu)=\mathcal{I}q(\mu):Y(\mu)\to({\mathcal{B}}_{2}(X),\mathscr{T}_{2})$
is an embedding.

###### Proof.

Note that the $Y(\mu)$ is compact and $({\mathcal{B}}_{2}(X),\mathscr{T}_{2})$
is Hausdorff. Hence, $\widetilde{q}(\mu)$ is a closed map. Now, we show that
$q(\mu)$ is injective. Let $s,t\in I$ be such that

$q(\mu)([af(t),bg(t)])=q(\mu)([af(s),bg(s)])=\nu.$

If $|\operatorname{{\rm supp}}(\nu)|=1$, then $f(t)=g(t)=f(s)=g(s)$, which
obviously implies that $[af(t),bg(t)]=[af(s),bg(s)]$. On the other hand, if
$|\operatorname{{\rm supp}}(\nu)|=2$, then $f(t)\neq g(t)$ and $f(s)\neq
g(s)$. If $a>b$, then we must have $f(t)=f(s)$ and $g(t)=g(s)$, and hence,
$[af(t),bg(t)]=[af(s),bg(s)]$. If $a=b$, then we get
$[f(t),g(t)]=[f(s),g(s)]$, which implies $[af(t),bg(t)]=[af(s),bg(s)]$. This
proves that $q(\mu)$ is injective. Since $\mathcal{I}$ is injective, the map
$\widetilde{q}(\mu)$ is injective. Since $\widetilde{q}(\mu)$ is closed, it is
an embedding. ∎

Let us denote $\widetilde{q}(\mu)(Y(\mu))\subset{\mathcal{B}}_{2}(X)$ by
$Z(\mu)$. Lemma 7.6 gives us a continuous map
$\widetilde{q}(\mu)^{-1}:Z(\mu)\to\text{Sym}(*^{2}X)$. We know that the map
$r:*^{2}X\to X^{2}$ defined by $r:(cx,dy)\mapsto(x,y)$ that drops weights is
continuous and ${\mathbb{Z}}_{2}$-equivariant, [Mi]. Quotient by
${\mathbb{Z}}_{2}$ gives a continuous weight-dropping map
$\widetilde{r}:\text{Sym}(*^{2}X)\to SP^{2}(X)$. Therefore, we have a
continuous mapping $\widetilde{r}\hskip
1.13809pt\widetilde{q}(\mu)^{-1}:Z(\mu)\to SP^{2}(X)$ such that for all $t\in
I$,

$\widetilde{r}\hskip 1.13809pt\widetilde{q}(\mu)^{-1}:af(t)+bg(t)\mapsto
f(t)+g(t).$

It is easy to check that if $\mu=af+bg$ and
$\nu=a^{\prime}f^{\prime}+b^{\prime}g^{\prime}$ are two such resolvers of
$\phi\in{\mathcal{P}}({\mathcal{B}}_{2}(X))$, then
$\widetilde{\mu}(t)=\widetilde{\nu}(t)$ for each $t\in I$. Hence,
$Y(\mu)=Y(\nu)$, and as maps, $\widetilde{q}(\mu)=\widetilde{q}(\nu)$ and
$\widetilde{r}\hskip 1.13809pt\widetilde{q}(\mu)^{-1}=\widetilde{r}\hskip
1.13809pt\widetilde{q}(\nu)^{-1}$.

######  7.7 Remark.

We note that these facts do not hold for measures in ${\mathcal{B}}_{n}(P(X))$
when $n\geq 3$. Let
$\mu=\frac{1}{2}\alpha+\frac{1}{4}(\gamma,\beta)+\frac{1}{4}\gamma\in{\mathcal{B}}_{3}(P(X))$,
in the notations of Section 3.B. Clearly,
$\widetilde{q}(\mu)(t)=\frac{1}{2}\alpha(t)+\frac{1}{2}\gamma(t)$ for all
$t\in(0,t_{1})$. Let $t,s\in(0,t_{1})$ such that
$\alpha(t)=\gamma(s)\neq\gamma(t)=\alpha(s)$. Then

$\widetilde{q}(\mu)\left[\frac{1}{2}\alpha(t),\frac{1}{4}\gamma(t),\frac{1}{4}\gamma(t)\right]=\frac{1}{2}\alpha(t)+\frac{1}{2}\gamma(t)=\widetilde{q}(\mu)\left[\frac{1}{2}\alpha(s),\frac{1}{4}\gamma(s),\frac{1}{4}\gamma(s)\right]$

Hence, $\widetilde{q}(\mu)$ is not injective and thus, it is not an embedding.

######  7.8 Remark.

Consider a resolvable path $\phi:I\to{\mathcal{B}}_{3}(X)$ described as
follows. At $t=0$, the path breaks into two strings of weights $\frac{1}{2}$
each. Then at $t=t_{1}$, the strings intertwine and break into three strings
of weights $\frac{1}{2},\frac{1}{4}$, and $\frac{1}{4}$. Finally, they
recombine at $t=1$. While the measure $\mu\in{\mathcal{B}}_{3}(P(X))$
described above is one resolver of $\phi$, another is
$\nu=\frac{1}{2}(\gamma,\alpha)+\frac{1}{4}(\alpha,\beta)+\frac{1}{4}(\alpha,\gamma)\in{\mathcal{B}}_{3}(P(X))$.
By the description of $\phi$, the paths $\alpha$ and $\gamma$ are not
identical on $(0,t_{1})$. Thus, there exists some $t\in(0,t_{1})$ such that
$\alpha(t)\neq\gamma(t)$. Then for this $t$, we have that

$\widetilde{\mu}(t)=\left[\frac{1}{2}\alpha(t),\frac{1}{4}\gamma(t),\frac{1}{4}\gamma(t)\right]\neq\left[\frac{1}{2}\gamma(t),\frac{1}{4}\alpha(t),\frac{1}{4}\alpha(t)\right]=\widetilde{\nu}(t).$

Therefore, the outputs are not resolver invariant anymore. Hence, this
particular method does not extend to the general case $n\geq 3$ for several
reasons.

###  7.B. For $i\operatorname{\mathsf{cat}}$

Recall the diagonal embedding $\partial_{2}:X\to SP^{2}(X)$ defined in Section
2.B as $\partial_{2}:x\mapsto[x,x]=2x$, which we called the inclusion of $X$
into $SP^{2}(X)$.

######  7.9 Lemma.

If $i\operatorname{\mathsf{cat}}(X)<2$, then there exist sets $A_{1}$ and
$A_{2}$ that cover $X$ such that the inclusion $A_{i}\to SP^{2}(X)$ is null-
homotopic for $i=1,2$.

###### Proof.

If $i\operatorname{\mathsf{cat}}(X)<2$, then for a basepoint $x_{0}\in X$,
there exists a continuous map $H:X\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$ such
that $H(x)(0)=\delta_{x}$ and $H(x)(1)=\delta_{x_{0}}$ for all $x\in X$.
Define

$A_{1}=\left\\{x\in X\hskip 2.84526pt\middle|\hskip 2.84526ptH(x)\text{ has a
resolver of support }1\right\\}.$

Clearly, if $x\in A_{1}$, then $H(x)$ has a resolver in $P(X)$ and thus,
$H(x)(t)\in X$ for all $t\in I$. So, we define $K_{1}:A_{1}\times I\to
SP^{2}(X)$ as the composition

$K_{1}:(x,t)\mapsto H(x)(t)\xmapsto{\partial_{2}}2H(x)(t).$

Let $A_{2}=X\setminus A_{1}$. For each $x\in A_{2}$, we choose and fix a
resolver of $H(x)$, say
$\mu_{x}=a_{x}\phi_{1}^{x}+b_{x}\phi_{2}^{x}\in{\mathcal{B}}_{2}(P(X))$. Note
that $H(x)(t)\in Z(\mu_{x})$ for all $t\in I$. Define a map $K_{2}:A_{2}\times
I\to SP^{2}(X)$ as the composition

$K_{2}:(x,t)\mapsto
H(x)(t)=a_{x}\phi_{1}^{x}(t)+b_{x}\phi_{2}^{x}(t)\xmapsto{\widetilde{r}\hskip
0.28453pt\widetilde{q}(\mu_{x})^{-1}}\phi_{1}^{x}(t)+\phi_{2}^{x}(t).$

In view of the comments made after Lemma 7.6, the definition of $K_{2}$ is
independent of the choice of $\mu_{x}$. Hence, $K_{2}$ is well-defined, and
being a composition of continuous maps, it is continuous. For each $i$ and all
$x\in A_{i}$, $K_{i}(x,0)=2x=\partial_{2}(x)$ and $K_{i}(x,1)=2x_{0}$ by
Remark 3.2. Hence, we conclude that the maps $K_{i}$ are the required null-
homotopies. ∎

We now proceed as in [DJ, Section 4.2]. In Alexander-Spanier cohomology, we
have the following statement.

######  7.10 Theorem.

Suppose that $\alpha_{i}^{*}\in H^{k_{i}}\left(SP^{2}(X);R\right)$ for some
ring $R$ and $k_{i}\geq 1$ for $i=1,2$. Let $\alpha_{i}\in H^{k_{i}}(X;R)$ be
the image of $\alpha_{i}^{*}$ under the induced map $\partial_{2}^{*}$ such
that $\alpha_{1}\smile\alpha_{2}\neq 0$. Then
$i\operatorname{\mathsf{cat}}(X)\geq 2.$

The proof is the same as that of [DJ, Theorem 4.7] for the particular case
$n=2$.

######  7.11 Remark.

We note that the reason for using Alexander-Spanier cohomology is that in the
proof, we need to use the long exact sequence of the pairs
$(SP^{2}(X),A_{i})$. But here, the sets $A_{i}$ need not be open or closed.
So, $(SP^{2}(X),A_{i})$ may not be a good pair in the sense of [Ha] to use
singular cohomology.

######  7.12 Corollary.

If $\max\\{c\ell_{{\mathbb{R}}}(X),c\ell_{{\mathbb{Q}}}(X)\\}\geq 2$ for a
finite CW complex $X$, then $i\operatorname{\mathsf{cat}}(X)\geq 2$.

###### Proof.

Since the Alexander-Spanier cohomology groups are the same as the singular
cohomology groups for locally finite CW complexes [Sp], this statement follows
directly from Theorem 7.10 and Proposition 2.3. We note that the finiteness
hypothesis on the complex is needed to use Proposition 2.3. ∎

###  7.C. For $i\operatorname{\mathsf{TC}}$

We recall that a deformation of a subset $A\subset X$ to another subset
$D\subset X$ is a continuous map $H:A\times I\to X$ such that
$H|_{A\times\\{0\\}}:A\to X$ is the inclusion and $H(A\times\\{1\\})\subset
D$.

######  7.13 Lemma.

If $i\operatorname{\mathsf{TC}}(X)<2$, then there exist sets $B_{1}$ and
$B_{2}$ that cover $X\times X$ such that the set $B_{i}$ deforms to the
diagonal $\Delta X$ inside $SP^{2}(X\times X)$ for $i=1,2$.

###### Proof.

If $i\operatorname{\mathsf{TC}}(X)<2$, then there exists a continuous map
$s:X\times X\to{\mathcal{P}}({\mathcal{B}}_{n}(X))$ such that
$s(x,y)(0)=\delta_{x}$ and $s(x,y)(1)=\delta_{y}$ for all $(x,y)\in X\times
X$. Define

$B_{1}=\left\\{(x,y)\in X\times X\hskip 2.84526pt\middle|\hskip
2.84526pts(x,y)\text{ has a resolver of support }1\right\\}.$

Let $B_{2}=(X\times X)\setminus B_{1}$. Now, just like in Lemma 7.9, we define
the continuous maps $K_{i}:B_{i}\times I\to SP^{2}(X\times X)$ as the
following compositions.

$K_{1}:(x,y,t)\mapsto(x,y,t,y)\mapsto(s(x,y)(t),y)\mapsto 2(s(x,y)(t),y),$
$K_{2}:(x,y,t)\mapsto(x,y,t,y)\mapsto(s(x,y)(t),y)=\left(a\phi_{1}(t)+b\phi_{2}(t),y\right)$
$\mapsto\left(\phi_{1}(t)+\phi_{2}(t),y\right)\mapsto\left(\phi_{1}(t),y\right)+\left(\phi_{2}(t),y\right),$

where we fixed a resolver $a\phi_{1}+b\phi_{2}$ of $s(x,y)$. Again, $K_{2}$ is
independent of the choice of resolvers of $s(x,y)$ in
${\mathcal{B}}_{2}(P(X))$. So, its continuity follows from the arguments of
Lemma 7.9 and the functoriality of $SP^{2}$. Remark 3.2 gives that
$K_{i}(x,y,0)=2(x,y)$ and $K_{i}(x,y,1)=2(y,y)$. Thus, each $K_{i}$ defines
the required deformation. ∎

The proof of the following theorem in Alexander-Spanier cohomology is the same
as that of [DJ, Theorem 4.12] for the particular case $n=2$.

######  7.14 Theorem.

Suppose that $\beta_{i}^{*}\in H^{k_{i}}\left(SP^{2}(X\times X);R\right)$ for
some ring $R$ and $k_{i}\geq 1$ for $i=1,2$. Let $\beta_{i}\in
H^{k_{i}}(X\times X;R)$ be the image of $\beta_{i}^{*}$ under the homomorphism
induced by the inclusion. If $\beta_{i}$ are zero-divisors such that
$\beta_{1}\smile\beta_{2}\neq 0$, then $i\operatorname{\mathsf{TC}}(X)\geq 2$.

######  7.15 Corollary.

If $\max\\{zc\ell_{{\mathbb{R}}}(X),zc\ell_{{\mathbb{Q}}}(X)\\}\geq 2$ for a
finite CW complex $X$, then $i\operatorname{\mathsf{TC}}(X)\geq 2$.

###### Proof.

Since the Alexander-Spanier cohomology groups are the same as the singular
cohomology groups for locally finite CW complexes [Sp], this statement follows
directly from Theorem 7.14 and Proposition 2.3. ∎

###  7.D. For $i\operatorname{\mathsf{TC}}_{m}$

To obtain $2$ as a lower bound for $i\operatorname{\mathsf{TC}}_{m}$ for all
spaces $X$ and $m\geq 2$, we proceed in a way similar to [J, Section 4.B].
However, we will have to make some necessary modifications.

For a fixed $m\geq 2$, let $\zeta:P(SP^{2}(X))\to(SP^{2}(X))^{m}$ be the
fibration

$\zeta:\phi\mapsto\left(\phi(t_{1}),\phi(t_{2}),\ldots,\phi(t_{m})\right)$

for $t_{i}=(i-1)/(m-1)$. The diagonal map $\partial_{2}:X\to SP^{2}(X)$ gives
the product $\partial_{2}^{m}:X^{m}\to(SP^{2}(X))^{m}$ defined as
$\partial_{2}^{m}:\left(x_{1},\ldots,x_{m}\right)\mapsto(2x_{1},\ldots,2x_{m})$.
We now consider the following pullback diagram.

(7.a)
${\mathcal{D}_{m}}$${P(SP^{2}(X))}$${X^{m}}$${(SP^{2}(X))^{m}}$$\scriptstyle{a}$$\scriptstyle{\sigma}$$\scriptstyle{\zeta}$$\scriptstyle{\partial_{2}^{m}}$

We note that in this setting, one can replace the set $X^{m}$ on the left with
any of its subset $C_{i}$ and obtain the above pullback diagram with the
corresponding pullback space $\mathcal{D}_{m,i}$ and the canonical projection
fibration $\sigma_{i}:\mathcal{D}_{m,i}\to C_{i}$.

######  7.16 Lemma.

If $i\operatorname{\mathsf{TC}}_{m}(X)<2$, then there exist sets $C_{1}$ and
$C_{2}$ that cover $X^{m}$ and over each of which the fibration
$\sigma_{i}:\mathcal{D}_{m,i}\to C_{i}$ has a section.

###### Proof.

Since $i\operatorname{\mathsf{TC}}_{m}(X)<2$, there exists a $2$-intertwined
$m$-navigation algorithm $s_{m}:X^{m}\to{\mathcal{P}}({\mathcal{B}}_{2}(X))$
on $X$. Define

$C_{1}=\left\\{\overline{x}=(x_{1},\ldots,x_{m})\in X^{m}\hskip
2.84526pt\middle|\hskip 2.84526pts_{m}(\overline{x})\text{ has a resolver of
support }1\right\\}.$

Let $C_{2}=X^{m}\setminus C_{1}$. We define $K_{i}:C_{i}\times I\to
SP^{2}(X^{m})$ as follows.

$K_{1}:(\overline{x},t)\mapsto s_{m}(\overline{x})(t)\mapsto 2\hskip
0.85358pts_{m}(\overline{x})(t),$ $K_{2}:(\overline{x},t)\mapsto
s_{m}(\overline{x})(t)=a\phi_{1}(t)+b\phi_{2}(t)\mapsto\phi_{1}(t)+\phi_{2}(t),$

where we fixed a resolver $a\phi_{1}+b\phi_{2}$ of $s_{m}(\overline{x})$.
Since $K_{i}$ are continuous, we obtain continuous maps $H_{i}:C_{i}\to
P(SP^{2}(X))$ such that $\zeta H_{i}(\overline{x})=(2x_{1},\ldots,2x_{m})$.
Thus, by the universal property of the above pullback diagram, with the roles
of $X^{m}$, $\mathcal{D}_{m}$, and $\sigma$ replaced, respectively, with
$C_{i}$, $\mathcal{D}_{m,i}$, and $\sigma_{i}$ for $i=1,2$, we conclude that
there exists a section $\tau_{i}:C_{i}\to\mathcal{D}_{m,i}$ of the fibration
$\sigma_{i}:\mathcal{D}_{m,i}\to C_{i}$. ∎

For a fixed $m\geq 2$, let $\Delta_{2}:SP^{2}(X)\to(SP^{2}(X))^{m}$ be the
diagonal map. Then, we follow ideas from [J, Theorem 4.4] to prove the
following theorem in Alexander-Spanier cohomology.

######  7.17 Theorem.

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

$i\operatorname{\mathsf{TC}}_{m}(X)\geq 2.$

###### Proof.

Since $P(SP^{2}(X))$ deforms to $SP^{2}(X)$, we can find a homotopy
equivalence $g:SP^{2}(X)\to P(SP^{2}(X))$. Now, if possible, let
$i\operatorname{\mathsf{TC}}_{m}(X)<2$. Then by Lemma 7.16, there exists a
cover $\\{C_{1},C_{2}\\}$ of $X^{m}$ and maps
$\tau_{i}:C_{i}\to\mathcal{D}_{m,i}$ such that
$\sigma_{i}\tau_{i}=\mathbbm{1}_{C_{i}}$. Let us fix some $i\in\\{1,2\\}$ and
consider the following diagram.

(7.b)
${H^{k_{i}}(C_{i};R)}$${H^{k_{i}}(\mathcal{D}_{m,i};R)}$${H^{k_{i}}((SP^{2}(X))^{m};R)}$${H^{k_{i}}(P(SP^{2}(X));R)}$${H^{k_{i}}(SP^{2}(X);R)}$$\scriptstyle{\sigma_{i}^{*}}$$\scriptstyle{{\tau_{i}^{*}\vphantom{1}}}$$\scriptstyle{\zeta^{*}}$$\scriptstyle{(\partial_{2}^{m})^{*}}$$\scriptstyle{\Delta_{2}^{*}}$$\scriptstyle{a^{*}}$$\scriptstyle{g*}$

Since $g^{*}$ is an isomorphism,
$\Delta_{2}^{*}(\alpha_{i}^{*})=0\implies\zeta^{*}(\alpha_{i}^{*})=0$ in the
lower triangle. The commutativity of the top square then gives

$(\partial_{2}^{m})^{*}(\alpha_{i}^{*})=\tau_{i}^{*}a^{*}\zeta^{*}(\alpha_{i}^{*})=0.$

From the long exact sequence of the pair $((SP^{2}(X))^{m},C_{i})$ in
Alexander-Spanier cohomology,

$\cdots\to
H^{k_{i}}((SP^{2}(X))^{m},C_{i};R)\xrightarrow{j_{i}^{*}}H^{k_{i}}((SP^{2}(X))^{m};R)\xrightarrow{(\partial_{2}^{m})^{*}}H^{k_{i}}(C_{i};R)\to\cdots,$

there exists $\overline{\alpha_{i}^{*}}\in H^{k_{i}}((SP^{2}(X))^{m},C_{i};R)$
such that $j_{i}^{*}(\overline{\alpha_{i}^{*}})=\alpha_{i}^{*}$. Further, let
$(\partial_{2}^{m})^{*}(\overline{\alpha_{i}^{*}})=\overline{\alpha}_{i}\in
H^{k_{i}}(X^{m},C_{i};R)$. For $j^{*}=j_{1}^{*}\oplus
j_{2}^{*},(j^{\prime})^{*}=(j^{\prime})^{*}_{1}\oplus(j^{\prime})^{*}_{2}$,
and $k=k_{1}+k_{2}$, we get the following commutative diagram.

(7.c) ${H^{k}(X^{m};R)}$${H^{k}\left(X^{m},C_{1}\cup
C_{2};R\right)}$${H^{k}\left((SP^{2}(X))^{m};R\right)}$${H^{k}\left((SP^{2}(X))^{m},C_{1}\cup
C_{2};R\right)}$$\scriptstyle{(j^{\prime})^{*}}$$\scriptstyle{(\partial_{2}^{m})^{*}}$$\scriptstyle{(\partial_{2}^{m})^{*}}$$\scriptstyle{j^{*}}$

The cup product $\overline{\alpha_{1}^{*}}\smile\overline{\alpha_{2}^{*}}$ in
the bottom-right goes to $\alpha_{1}\smile\alpha_{2}\neq 0$ in the top-left.
But in the process, it factors through
$\overline{\alpha}_{1}\smile\overline{\alpha}_{2}\in H^{k}(X^{m},X^{m};R)=0$.
This is a contradiction. Hence, we must have
$i\operatorname{\mathsf{TC}}_{m}(X)\geq 2$. ∎

As explained at the end of Section 7.A, this lower bound cannot be improved in
general using our methods. However, if Lemma 7.16 is proved for each $n\geq 3$
by some other method, then our proof of Theorem 7.17 will still work by
accordingly replacing the role of the two cohomology classes $\alpha_{i}^{*}$
with more number of classes.

##  8\. Computations

###  8.A. For $i\operatorname{\mathsf{cat}}$

Computations of $d\operatorname{\mathsf{cat}}(X)$ for various spaces $X$ were
done in [DJ, Section 6.1] using their rational cup-lengths
$c\ell_{{\mathbb{Q}}}(X)$. Here, we will use Corollary 7.12 to get various
examples of closed manifolds $X$ for which
$i\operatorname{\mathsf{cat}}(X)=2$.

######  8.1 Example.

Using rational and real cup-lengths and the $\operatorname{\mathsf{cat}}$
values, we get:

  * •

$i\operatorname{\mathsf{cat}}(\Sigma_{g})=2$ for each closed orientable
surface $\Sigma_{g}$ of genus $g\geq 1$.

  * •

$i\operatorname{\mathsf{cat}}({\mathbb{C}}P^{2})=2$.

  * •

$i\operatorname{\mathsf{cat}}(S^{k_{1}}\times S^{k_{2}})=2$ for each
$k_{i}\geq 1$.

  * •

If $(M^{4},\omega)$ is a closed, simply connected $4$-dimensional symplectic
manifold such that $[\omega]^{2}\neq 0$ for the cohomology class $[\omega]\in
H^{2}(M^{4};{\mathbb{R}})$ corresponding to the symplectic $2$-form $\omega$,
then $i\operatorname{\mathsf{cat}}(M)=2$.

######  8.2 Remark.

Let $M$ and $N$ be two closed $k$-dimensional manifolds of which at least one
is orientable. We know that
$\widetilde{H}_{i}(M\\#N)=\widetilde{H}_{i}(M)\oplus\widetilde{H}_{i}(N)$ for
all $i<k$. Let $(m,n)\in H^{i}(M\\#N)$ and $(m^{\prime},n^{\prime})\in
H^{j}(M\\#N)$. Then,

$(m,n)\smile(m^{\prime},n^{\prime})=\begin{cases}\left(m\smile
m^{\prime},n\smile n^{\prime}\right)&:i+j<k\\\ m\smile m^{\prime}+n\smile
n^{\prime}&:i+j=k\end{cases}$

Thus, for ${\mathbb{F}}\in\\{{\mathbb{Q}},{\mathbb{R}}\\}$,
$\max\\{c\ell_{{\mathbb{F}}}(M),c\ell_{{\mathbb{F}}}(N)\\}\leq
c\ell_{{\mathbb{F}}}(M\\#N)$.

######  8.3 Example.

Using Corollary 7.12 with the corresponding ${\mathbb{F}}$ cup-lengths, Remark
8.2, and the connected sum formula
$\operatorname{\mathsf{cat}}(M\\#N)=\max\\{\operatorname{\mathsf{cat}}(M),\operatorname{\mathsf{cat}}(N)\\}$
from [DS, Proposition 11], we get the following conclusions.

  * •

For $3\leq n\leq 5$, let $K^{n}$ denote a closed, simply connected
$n$-dimensional manifold. Note that $\operatorname{\mathsf{cat}}(K^{n})\leq 2$
due to [CLOT, Theorem 1.50]. Thus, the $i\operatorname{\mathsf{cat}}$ value
for each of the following five classes of manifolds is $2$.

$K^{3}\\#(S^{1}\times S^{2}),\hskip 2.84526ptK^{4}\\#\hskip
1.42262pt{\mathbb{C}}P^{2},\hskip 2.84526ptK^{4}\\#M^{4},\hskip
2.84526ptK^{4}\\#(S^{2}\times S^{2}),\hskip 2.84526ptK^{5}\\#(S^{3}\times
S^{2}).$

  * •

$i\operatorname{\mathsf{cat}}(\Sigma_{g}\\#L^{2})=2$ for any closed
$2$-dimensional manifold $L^{2}$ and $g\geq 1$.

######  8.4 Remark.

Let $N_{g}$ denote the closed non-orientable surface of genus $g\geq 1$. For
each $g\geq 2$, $d\operatorname{\mathsf{cat}}(N_{g})=2$ is computed using the
covering map inequality from [DJ, Theorem 3.7]. Since we do not know whether
such an inequality holds in the case of $i\operatorname{\mathsf{cat}}$ as
well, we instead use the fact that that $N_{g}=\Sigma_{1}\\#N_{g-2}$ for
$g\geq 3$. So, we can take $L^{2}=N_{g-2}$ in Example 8.3 and get
$i\operatorname{\mathsf{cat}}(N_{g})=2$ for all $g\geq 3$.

Since $i\operatorname{\mathsf{cat}}(N_{1})=1$, this only leaves open the case
of $N_{2}$, the Klein bottle.

######  8.5 Example.

Let $L$ be a smooth simply connected manifold of dimension $4$, other than
$S^{4}$. Then $L$ is homotopic to $M\\#N$ where
$M\in\\{{\mathbb{C}}P^{2},S^{2}\times S^{2},-{\mathbb{C}}P^{2}\\}$. By Remark
8.2 and the fact that $\operatorname{\mathsf{cat}}(L)=2$ [DS, Section 7], we
get $i\operatorname{\mathsf{cat}}(L)=2$.

###  8.B. For $i\operatorname{\mathsf{TC}}_{m}$

For the case of $i\operatorname{\mathsf{TC}}_{2}=i\operatorname{\mathsf{TC}}$,
we will use Corollary 7.15 and [Far1] to get some examples of locally finite
CW complexes $X$ for which $i\operatorname{\mathsf{TC}}(X)=2$.

######  8.6 Example.

Using rational zero-divisior cup-lengths and $\operatorname{\mathsf{TC}}$
values, we get:

  * •

$i\operatorname{\mathsf{TC}}(S^{2n-1})=1$ and
$i\operatorname{\mathsf{TC}}(S^{2n})=2$ for each $n$.

  * •

$i\operatorname{\mathsf{TC}}(S^{2k_{1}-1}\times S^{2k_{2}-1})=2$ for any
$k_{i}\geq 1$.

  * •

$i\operatorname{\mathsf{TC}}(X)=2$ if $X$ is a finite graph with
$\beta_{1}(X)>1$.

For the $2$-torus $\Sigma_{1}$, since it is a topological group, we get from
Corollary 4.21 and Example 8.1 that
$i\operatorname{\mathsf{TC}}(\Sigma_{1})=i\operatorname{\mathsf{cat}}(\Sigma_{1})=2$.

For all the closed manifolds $X$ considered in the previous subsection, we see
that
$i\operatorname{\mathsf{cat}}(X)=d\operatorname{\mathsf{cat}}(X)=\operatorname{\mathsf{cat}}(X)$.
Also, for all the finite CW complexes $Y$ considered till this point in this
section, we see that
$i\operatorname{\mathsf{TC}}(Y)=d\operatorname{\mathsf{TC}}(Y)=\operatorname{\mathsf{TC}}(Y)$.

We now focus on computing $i\operatorname{\mathsf{TC}}_{m}$ values for certain
finite CW complexes for each $m\geq 2$. The technique is similar to that used
in [J, Section 7] for $d\operatorname{\mathsf{TC}}_{m}$. However, the proofs
need to be modified because we need cohomology classes in the ring of
$(SP^{2}(X))^{m}$ now as opposed to classes in the ring of $SP^{2}(X^{m})$ in
the case of $d\operatorname{\mathsf{TC}}_{m}$ for $n=2$. We prove the
following statement to explicitly demonstrate our technique, using which,
similar computations can be done to get more estimates for
$i\operatorname{\mathsf{TC}}_{m}(X)$ in the same way as the computations were
done in [J, Section 7.B] to get more estimates for
$d\operatorname{\mathsf{TC}}_{m}(X)$.

######  8.7 Proposition.

Let $X$ be a finite CW complex and $m\geq 3$. If $H^{d}(X;{\mathbb{F}})\neq 0$
for some $d\geq 1$ and ${\mathbb{F}}\in\\{{\mathbb{Q}},{\mathbb{R}}\\}$, then
$i\operatorname{\mathsf{TC}}_{m}(X)\geq 2$.

###### Proof.

For brevity, call $Y=SP^{2}(X)$. Let $v\in H^{d}(X;{\mathbb{F}})$ such that
$v\neq 0$. Due to Proposition 2.3, there exists $w\in H^{d}(Y;{\mathbb{F}})$
such that $\partial_{2}^{*}(w)=v$, where $\partial_{2}:X\to Y$ is the diagonal
embedding. For $1\leq i\leq m$, let $k_{i}:Y^{m}\to Y$ and
$\text{proj}_{i}:X^{m}\to X$ be the respective projections onto the $i^{th}$
coordinates. Let $\partial_{2}^{m}:X^{m}\to Y^{m}$ be the product map, and
$\Delta:X\to X^{m}$ and $\Delta_{2}:Y\to Y^{m}$ be the respective diagonal
maps. Then for each $i$, the following diagram commutes.

(8.a)
${Y}$${Y^{m}}$${Y}$${X}$${X^{m}}$${X}$$\scriptstyle{k_{i}}$$\scriptstyle{\Delta_{2}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\text{proj}_{i}}$$\scriptstyle{\partial_{2}^{m}}$$\scriptstyle{\Delta}$$\scriptstyle{\partial_{2}}$

For each $1\leq i\leq m-1$, we define maps
$\phi_{i}:H^{d}(X;{\mathbb{F}})\oplus H^{d}(X;{\mathbb{F}})\to
H^{d}(X^{m};{\mathbb{F}})$ as $\phi_{i}(x\oplus
y)=\operatorname{proj}_{i}^{*}(x)-\operatorname{proj}_{m}^{*}(y)$ and elements
$\alpha_{i}=\phi_{i}(v\oplus v)$ exactly like in [J, Proposition 7.1].
Similarly, we define $\psi_{i}:H^{d}(Y;{\mathbb{F}})\oplus
H^{d}(Y;{\mathbb{F}})\to H^{d}(Y^{m};{\mathbb{F}})$ as $\psi_{i}(a\oplus
b)=k_{i}^{*}(a)-k_{m}^{*}(b)$ and $\alpha_{i}^{*}=\psi_{i}(w\oplus w)$. Then
the following commutes.

(8.b) ${w\oplus w}$${\alpha_{i}^{*}}$${w-w=0}$${H^{d}(Y;{\mathbb{F}})\oplus
H^{d}(Y;{\mathbb{F}})}$${H^{d}(Y^{m};{\mathbb{F}})}$${H^{d}(Y;{\mathbb{F}})}$${H^{d}(X;{\mathbb{F}})\oplus
H^{d}(X;{\mathbb{F}})}$${H^{d}(X^{m};{\mathbb{F}})}$${H^{d}(X;{\mathbb{F}})}$${v\oplus
v}$${\alpha_{i}}$${v-v=0}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\in}$$\scriptstyle{\psi_{i}}$$\scriptstyle{\partial_{2}^{*}\oplus\hskip
2.84526pt\partial_{2}^{*}\hskip
2.84526pt}$$\scriptstyle{\Delta_{2}^{*}}$$\scriptstyle{(\partial_{2}^{m})^{*}}$$\scriptstyle{\partial_{2}^{*}}$$\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 $H^{d}(Y;{\mathbb{F}})$ (resp.
$H^{d}(X;{\mathbb{F}})$) is an isomorphism. Due to [R, Proposition 3.5], we
have

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

Since $m\geq 3$, we can take $k_{i}=d$ and $\alpha_{i}^{*}\in
H^{k_{i}}((SP^{2}(X))^{m};{\mathbb{F}})$ for $i=1,2$ in the statement of
Theorem 7.17 to get $i\operatorname{\mathsf{TC}}_{m}(X)\geq 2$. ∎

Of course, if $H^{d}(X;{\mathbb{F}})\neq 0$ for some $d\geq 1$ and
${\mathbb{F}}\in\\{{\mathbb{Q}},{\mathbb{R}}\\}$, then
$i\operatorname{\mathsf{TC}}_{3}(X)\geq 2$ by Proposition 8.7, and using
Proposition 4.19, we get that $i\operatorname{\mathsf{TC}}_{m}(X)\geq 2$ for
all $m\geq 4$ as well. However, we proved Proposition 8.7 by specifying the
indexes $m\geq 3$ so that if the lower bound in Theorem 7.17 is improved, then
the technique of Proposition 8.7 remains useful more generally to obtain
better lower bounds for $i\operatorname{\mathsf{TC}}_{m}(X)$ for all $m\geq
3$, just like [J, Proposition 7.1] in the case of
$d\operatorname{\mathsf{TC}}_{m}$.

######  8.8 Remark.

We note that the hypothesis in Proposition 8.7, that
$H^{d}(X;{\mathbb{F}})\neq 0$ for some $d\geq 1$ and
${\mathbb{F}}\in\\{{\mathbb{Q}},{\mathbb{R}}\\}$, cannot be dropped for any
$m\geq 3$ due to the example of Higman’s group ${\mathcal{H}}$ for which
$i\operatorname{\mathsf{TC}}_{m}({\mathcal{H}})=1$ for all $m\geq 2$ by
Proposition 5.1.

######  8.9 Corollary.

If $X$ is a closed orientable manifold or a closed non-orientable surface of
genus $g>1$, then $i\operatorname{\mathsf{TC}}_{m}(X)\geq 2$ for all $m\geq
3$.

######  8.10 Example.

For each $n\geq 1$, $i\operatorname{\mathsf{TC}}_{3}(S^{2n-1})=2$ due to
Corollary 8.9 and the fact that $\operatorname{\mathsf{TC}}_{3}(S^{2n-1})=2$,
see [R, Section 4]. Hence, the non-decreasing sequence
$\\{i\operatorname{\mathsf{TC}}_{m}(S^{2n-1})\\}_{m\geq 2}$ is not constant
for any $n$.

## Acknowledgement

The author would like to thank Alexander Dranishnikov for his kind guidance
and various helpful discussions. The author is also grateful for his keen
interest in this project.

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