# On spaces of minimal higher topological complexity.

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

Let TCn denote the n-th topological complexity. For nilpotent CW space of
finite type, Grant–Lupton–Oprea proved the following: If TCn(X)=n-1 then X is
homotopy equivalent to an odd-dimensional sphere. In my previous version I
stated that I can get rid of the nilpotence condition. Currently I cannot do
it (the proof of Theorem 4.4 in the previous version is wrong) but I have made
some progress there, and I present it here.

2020 Mathematics Subject Classification: Primary 55M30, Secondary 55S40.

Competing interests: Declaration of interests: none

##  1\. Introduction

The $n$-th topological complexity $\operatorname{TC}_{n}(X)$ of a topological
space $X$ is the sectional category of a fibrational substitute of the
diagonal map $\Delta:X\to X^{n}$. This concept appeared in [R2010], see
[BGRT2014] or [GLO2013, Section 2] for greater details. Note that the
inequality $\operatorname{TC}_{n}(X)\geq n-1$ holds for all $X$ and $n$, and
so it seems reasonable to describe the spaces $X$ with
$\operatorname{TC}_{n}(X)=n-1$. It is easy to see
$\operatorname{TC}_{n}(S^{2k+1})=n-1$ for $n\geq 2$ and $k\geq 0$, [BGRT2014].
This is quite natural to conjecture that the converse is also true: If
$TC_{n}(X)=n-1$ then $X$ is homotopy equivalent to an odd-dimensional sphere.
Currently we have two theorems below.

######  1.1 Theorem.

[GLO2013, Theorem 3.4] Let X be a connected CW space of finite type. If $X$ is
simply connected that $X$ is homotopy equivalent to some sphere $S^{2r+1}$ of
odd dimension, $r\geq 1$. If $X$ is not simply connected then $X$ is either
acyclic space or homology sphere. Additionally, if $X$ is a nilpotent space or
a co-H-space then $X$ is a homotopy circle.

We have the following slight improvement of Theorem 1.1.

######  1.2 Theorem (Corollary 4.4).

Let X be a connected CW space of finite type. If
$\operatorname{TC}_{n}(X)=n-1,n\geq 2$, then $X$ is either homotopy equivalent
to some sphere $S^{2r+1}$ of odd dimension, $r\geq 0$ or to homology circle
with $\pi_{1}(X)={\mathbb{Z}}$.

We emphasize again: It is an open question whether the space $X$ with
$\operatorname{TC}_{n}(X)=n-1,n\geq 3$ and $\pi_{1}(X)={\mathbb{Z}}$ is a
homotopy circle.

We use the sign $\simeq$ for isomorphism of groups and homotopy equivalences
of spaces.

##  2\. Zero Divisors Cup Length

######  2.1 Definition.

Fix a natural number $n\geq 2$. Let $X$ be a connected CW space and
$d=d_{n}:X\to X^{n}$ the diagonal map, $d(x)=(x,\cdots,x)$. The zero divisor
ideal for $X$ is the kernel of the homomorphism

$d^{*}:H^{*}(X^{n})\to H^{*}(X).$

A non-zero element $x\in X^{n}$ is called a zero divisor element if
$d^{*}(x)=0$. A zero divisor cup length of $X$ denoted by
$\operatorname{zcl}(X)=\operatorname{zcl}_{n}(X)$ is a maximal number $k$ such
that there are classes $u_{i}\in H^{*}(X^{n};A_{i}),i=1,\ldots k$ where the
$A_{i}$’s are the local coefficient systems, with the following property:

$\displaystyle d^{*}u_{i}$ $\displaystyle=0\text{ for all }i=1,\ldots,k,\text{
and }$ $\displaystyle 0\neq$ $\displaystyle u_{1}\smile\cdots\smile u_{k}\in
H^{*}(X^{n};A_{1}\otimes\cdots\otimes A_{k}).$

######  2.2 Theorem.

$\operatorname{TC}_{n}(X)\geq\operatorname{zcl}_{n}(X)$ for all local
coefficient system .

###### Proof.

This is a special case of [Sva1962, Theorem 4]. ∎

##  3\. Berstein–Schwarz Class

Consider a pointed connected $CW$ space $(X,x_{0})$. Put
$\pi=\pi_{1}(X,x_{0})$, ${\mathbb{Z}}[\pi]$ the group ring of $\pi$, and
$I(\pi)$ the augmentation ideal of ${\mathbb{Z}}[\pi]$, i.e. the kernel of the
augmentation map $\varepsilon:{\mathbb{Z}}[\pi]\to{\mathbb{Z}}$. The
cohomological dimension of $\pi$ is denoted by $\operatorname{cd}\pi$.

Let $p:\widetilde{X}\to X$ be the universal covering map for $X$ and
$\widetilde{X}_{0}=p^{-1}(x_{0})$. The exactness of the sequence

$0\to H_{1}(\widetilde{X},\widetilde{X}_{0})\to H_{0}(\widetilde{X}_{0})\to
H_{0}(\widetilde{X})$

yields an isomorphism $H_{1}(\widetilde{X},\widetilde{X}_{0})\cong I(\pi)$.
So, we get isomorphisms

$H^{1}(X,x_{0};I(\pi))\cong\operatorname{Hom}_{{\mathbb{Z}}[\pi]}(H_{1}(\widetilde{X},\widetilde{X}_{0}),I(\pi))\cong\operatorname{Hom}_{{\mathbb{Z}}[\pi]}(I(\pi),I(\pi)).$

We denote by $\overline{\mathfrak{b}}\in H^{1}(X,x_{0};I(\pi))$ the element
that corresponds to the identity $1_{I(\pi)}$ in the right-hand side. (Here we
use local coefficient system for the group $I(\pi)$.) Finally, we set

(3.1) $\mathfrak{b}=\mathfrak{b}_{X}=j^{*}\overline{\mathfrak{b}}\in
H^{1}(X;I(\pi))$

where $j:X=(X,\emptyset)\to(X,x_{0})$ is the inclusion of pairs. We call
$\mathfrak{b}_{X}$ the Berstein-Schwarz class of $X$.

######  3.1 Remark.

To justify the name ”Berstein–Schwarz class”, note that the class appeared in
[Sva1962, p.99] (implicitly) and [Ber1976, (3.7)].

We put $I(\pi)^{\otimes n}=I(\pi)\otimes\cdots\otimes I(\pi)$ ($n$ times) and
consider the cup power $\mathfrak{b}_{X}^{n}\in H^{n}(X;I(\pi)^{\otimes n})$.

Let $B\pi$ denote the classifying space for $\pi$ and put
$\mathfrak{b}_{\pi}:=\mathfrak{b}_{B\pi}$.

######  3.2 Theorem.

If $\operatorname{cd}\pi=m\leq\infty$ then $\mathfrak{b}^{n}_{\pi}\neq 0$ for
all $n\leq m$.

###### Proof.

See e.g. [Sva1962, Prop. 34] or [DR2009, Theorem 3.4]. ∎

##  4\. More on Minimal Higher Topological Complexity

Let $X$ be a connected CW space of finite type.

Put $1^{\otimes k}=1\otimes\ldots\otimes 1$ ($k$ factors). Put
$\pi=\pi_{1}(X)$. Let $\mathfrak{b}=\mathfrak{b}_{X}$ be the Berstein-Schwarz
class (3.1)

######  4.1 Lemma.

If $0\neq\mathfrak{b}^{2}\in H^{2}(X;I(\pi)^{\otimes 2})$ then

(4.1) $\prod_{i=1}^{n-1}(1^{\otimes i}\otimes\mathfrak{b}\otimes
1^{\otimes(n-i-1)}-\mathfrak{b}\otimes 1^{\otimes(n-1)})^{\otimes 2}\neq 0.$

###### Proof.

Take a point $x_{0}\in X$ and define

$\varphi:X^{n-1}\to
X^{n},\quad\varphi(x_{1},\ldots,x_{n-1})\mapsto(x_{1},\ldots,x_{n-1},x_{0}).$

The induced map

$\displaystyle\varphi^{*}:$ $\displaystyle H^{*}(X^{n};I(\pi)^{\otimes n})\to
H^{*}(X^{n-1};I(\pi)^{\otimes(n-1)}),$ $\displaystyle\varphi^{*}$
$\displaystyle(u_{1}\otimes\cdots\otimes u_{n-1}\otimes
u_{n})=u_{1}\otimes\cdots\otimes u_{n-1}$

yields the map

(4.2) $\prod_{i=1}^{n-1}(1^{\otimes i}\otimes\mathfrak{b}\otimes
1^{\otimes(n-i)}-\mathfrak{b}\otimes 1^{\otimes(n-1)})^{\otimes
2}\mapsto\prod_{i=1}^{n-1}(1^{\otimes i}\otimes\mathfrak{b}\otimes
1^{\otimes(n-i)})^{2}.$

Since $\mathfrak{b}^{2}\neq 0$, and because of the Künneth theorem, the right-
hand part

$\prod_{i=1}^{n-1}(1^{\otimes i}\otimes\mathfrak{b}\otimes
1^{\otimes(n-i)})^{2}\in H(X^{2n-2};I(\pi)^{\otimes(2n-2)})$

of (4.2) is non-zero. Thus

$\prod_{i=1}^{n-1}(1^{\otimes i}\otimes\mathfrak{b}\otimes
1^{\otimes(n-i)}-\mathfrak{b}\otimes 1^{\otimes(n-1)})^{\otimes 2}\neq 0.$

∎

######  4.2 Corollary.

If $H^{2}(X;I(\pi)^{\otimes 2})\neq 0$ then $\operatorname{TC}_{n}(X)\geq
2n-2$.

###### Proof.

The non-zero product (4.1) contains $2n-2$ non-zero factors each which is
annihilated by $d^{*}$. Now the result follows from Theorem 2.2. ∎

######  4.3 Proposition.

If $\operatorname{TC}_{n}(X)=n-1$ then $\operatorname{cd}{\pi}\leq 1$.

###### Proof.

Because of Corollary 4.4 and contraposition, we conclude that: if
$\operatorname{TC}_{n}(X)=n-1$ then $H^{2}(X;I(\pi)^{\otimes 2})=0$. We can
assume that the classifying space $B\pi$ can be obtained from $X$ by adding
cells of dimensional $\geq 3$. Hence, $H^{2}(B\pi;I(\pi)^{\otimes 2})=0$, and
so $\mathfrak{b}_{\pi}^{2}=0$, and the result follows from Theorem 3.2. ∎

######  4.4 Corollary.

If $\operatorname{TC}_{n}(X)=n-1$ then either $X$ is homotopy equivalent to an
odd-dimensional sphere $S^{2n+1},n\geq 1$ or an integral homology circle with
$\pi_{1}(X)={\mathbb{Z}}$.

###### Proof.

By Proposition 4.3 we know that either $\operatorname{cd}(\pi)=0$ or
$\operatorname{cd}\pi=1$. If $\operatorname{cd}(\pi)=0$ then $\pi$ is trivial
group, and so $X$ is simply connected. In this case $X$ is homotopy equivalent
to an odd-dimensional sphere $S^{2n+1},n\geq 1$ by [GLO2013, Theorem 1.3].

If $\operatorname{cd}(\pi)=1$ then $\pi$ is a free group by [Sta1968,
Swan1969]. Because of [GLO2013, Theorem 3.4] we know that $X$ is either
integral homology sphere or acyclic space. First, $H_{1}(X)$ is a free
nontrivial abelian group and hence $X$ cannot be acyclic. Furthermore, since
$H_{1}(X)$ is a free abelian group and $X$ is an integral homology sphere, we
conclude that $X$ is an integral homology circle with
$\pi_{1}(X)={\mathbb{Z}}$. ∎

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