On the higher topological complexity of manifolds with abelian fundamental group

N. Cadavid-Aguilar Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Av. Instituto Politécnico Nacional 2508, San Pedro Zacatenco, Ciudad de México 07000 ncadavia@gmail.com , D. Cohen Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803 cohen@math.lsu.edu www.math.lsu.edu/˜cohen , J. González Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Av. Instituto Politécnico Nacional 2508, San Pedro Zacatenco, Ciudad de México 07000 jesus.glz-espino@cinvestav.mx , S. Hughes Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematical Institute, Endenicher Allee 60, 53115 Bonn, Germany sam.hughes.maths@gmail.com; hughes@math.uni-bonn.de samhughesmaths.github.io and L. Vandembroucq Centro de Matemática, Universidade do Minho, Braga, Portugal lucile@math.uminho.pt

Abstract.

We study the higher (or sequential) topological complexity $\operatorname{TC}_{s}$ of manifolds with abelian fundamental group. We give sufficient conditions for $\operatorname{TC}_{s}$ to be non-maximal in both the orientable and non-orientable cases. In combination with cohomological lower bounds, we also obtain some exact values for certain families of manifolds.

Introduction

For a path-connected space $X$ , the $s$ -th higher topological complexity $\operatorname{TC}_{s}(X)$ is the sectional category of the fibration $e_{s}\colon PX\to X^{s}$ , that is

\operatorname{TC}_{s}(X)=\operatorname{secat}(e_{s}\colon PX\to X^{s}),

where $PX$ denotes the space of paths in $X$ and

e_{s}(\gamma)=\Bigl{(}\gamma(0),\gamma\bigl{(}\frac{1}{s-1}\bigr{)},\ldots,\gamma\bigl{(}\frac{s-2}{s-1}\bigr{)},\gamma(1)\Bigr{)}

is the usual $s$ -th evaluation map. That is, in the reduced version used here, $\operatorname{TC}_{s}(X)$ is one less than the minimal number of open sets covering $X^{s}$ , over each of which the fibration $e_{s}$ admits a section.

Topological complexity $\operatorname{TC}(X)=\operatorname{TC}_{2}(X)$ was introduced by Farber in [13] and the ‘higher’ invariants were introduced by Rudyak in [18]. The invariants were developed and motivated by applications for motion planning problems in robotics. More precisely, viewing $X$ as the space of configurations of a mechanical system, the integer $\operatorname{TC}_{s}(X)$ provides a topological measure of the complexity of planning motion in $X$ from an initial configuration to a terminal configuration, passing through $s-2$ specified intermediate configurations.

Despite a huge body of research into these invariants, there are very few complete computations of $\operatorname{TC}_{s}(X)$ . Examples for which the full spectrum of invariants is known include products of spheres, surfaces, path-connected topological groups whose Lusternik-Schnirelmann category is known, closed simply-connected symplectic manifolds, classifying spaces of hyperbolic groups and some (additional) polyhedral product type spaces, see [2, 16, 17, 1]. In a number of these examples, the higher topological complexities attain the maximal values possible.

If $X$ is not simply connected, this maximal value is $\operatorname{TC}_{s}(X)\leq s\dim(X)$ , where $\dim(X)$ is the homotopy dimension of $X$ , see [2]. Work of Cohen–Vandembroucq [6] explored the non-maximality of $\operatorname{TC}_{2}(M)$ when $M$ is a manifold with abelian fundamental group. In this paper we extend these ideas to $\operatorname{TC}_{s}(M)$ for $s\geq 2$ .

Espinosa Baro, Farber, Mescher, and Oprea [12] have recently characterized the maximality of $\operatorname{TC}_{s}$ of a finite-dimensional CW-complex $X$ in terms of a canonical cohomology class generalizing the ‘Costa–Farber class’ introduced in [7] (see Section 1). Restricting our attention to a manifold $M$ with an abelian fundamental group $\pi$ and following the strategy of [6], we first express this characterization in terms of a homology class of the group $\pi^{s-1}$ (see Proposition 2.3 and Corollary 2.4). This permits us to establish the non-maximality of $\operatorname{TC}_{s}(M)$ in some cases. For example, when $M$ is orientable, we obtain the following result (see Section 3):

Theorem A.

Let $M$ be an orientable $n$ -dimensional connected closed manifold. In each of the following cases, we have $\operatorname{TC}_{s}(M)<sn$ :

(1)

$\pi_{1}(M)={\mathbb{Z}}^{r}$ with $(s-1)r<s\dim(M)$ ;
(2)

$\pi_{1}(M)={\mathbb{Z}}_{q}$ ;
(3)

$\pi_{1}(M)={\mathbb{Z}}^{r}\times{\mathbb{Z}}_{q}$ with $r<\dim(M)$ .

Computations of cohomological lower bounds of the $s$ -th topological complexity of the real projective spaces and lens spaces have attracted much interest [5, 9, 14, 8]. In Section 4, we show how these results provide lower bounds of $\operatorname{TC}_{s}$ for larger families of manifolds (see Proposition 4.2 and Theorem 4.7). Then Theorem A enables us to obtain the following exact values:

Theorem B.

Let $M$ be an orientable $n$ -dimensional connected closed manifold with maximal Lusternik–Schnirelmann category, that is, $\operatorname{\mathrm{cat}}(M)=n$ .

(1)

If $n\equiv 1$ mod $4$ and $\pi_{1}(M)={\mathbb{Z}}_{2}$ , then $\operatorname{TC}_{s}(M)=sn-1$ for $s$ sufficiently large.
(2)

If $n\equiv 1$ mod $2$ and $\pi_{1}(M)={\mathbb{Z}}_{p}$ where $p\geq 3$ is a prime, then $\operatorname{TC}_{s}(M)=sn-1$ for $s$ sufficiently large.

See Corollaries 4.4 and 4.9 for a more explicit description of the condition “ $s$ sufficiently large”.

The case of non-orientable manifolds is much more complicated. By [6, Theorem 1.2(1)], the topological complexity of a non-orientable manifold with abelian fundamental group is always non-maximal. However, it is well-known that there exist such non-orientable manifolds with maximal $\operatorname{TC}_{s}$ for $s\geq 3$ . For instance, for the real projective plane $P^{2}$ , we have $\operatorname{TC}_{3}(P^{2})=6=3\dim(P^{2})$ , see [16]. Furthermore, it has been shown in [5] and [9] that, when $n$ is even, the real projective space $P^{n}$ satisfies $\operatorname{TC}_{s}(P^{n})=sn$ for $s$ sufficiently large. For a fixed even integer $n$ , the sequence $(\operatorname{TC}_{s}(P^{n}))_{s\geq 2}$ forms an increasing sequence starting at $\operatorname{TC}_{2}(P^{n})$ , equal to the immersion dimension of $P^{n}$ ([15]), and stabilizing to $sn$ when $s$ is sufficiently large. As explained in [5], it would be very interesting to better understand this sequence. Our methods, developed in Section 5, permit us to obtain new information in this direction. In particular, in combination with Davis’ results [9], when $n=2^{r}-2$ , we have:

\operatorname{TC}_{s}(P^{n})\leq sn-1\ \mbox{for even }s\leq n,\quad\operatorname{TC}_{n}(P^{n})=n^{2}-1,\quad\mbox{and}\quad\operatorname{TC}_{s}(P^{n})=sn\ \mbox{for }s>n.

As before, this result can be extended to a larger family of manifolds:

Theorem C.

Let $M$ be a non-orientable $n$ -dimensional connected closed manifold with $\pi_{1}(M)=\mathbb{Z}_{2}$ and $n=2^{r}-2$ where $r\geq 3$ . Then, for any even $s\leq n$ , we have $\operatorname{TC}_{s}(M)<sn$ . If moreover $\operatorname{\mathrm{cat}}{M}=n$ , then $\operatorname{TC}_{n}(M)=n^{2}-1$ and $\operatorname{TC}_{s}(M)=sn$ for $s>n$ .

Notation and conventions

For a topological space $Y$ , we use $\dim(Y)$ to denote the homotopy dimension of $Y$ . The integral homology of $Y$ is denoted by $H_{*}(Y)$ , and the reduced homology by $\widetilde{H}_{*}(Y)$ . If $\pi=\pi_{1}(Y)$ , we denote the cohomology of $Y$ with coefficients in the local system determined by the ${\mathbb{Z}}[\pi]$ -module $V$ by $H^{*}(Y;V)$ .

For an element $a$ of a group $\pi$ , we often denote the inverse of $a$ by $\overline{a}$ .

We use the reduced version of sectional category throughout, so that for a fibration $p\colon E\to B$ , when finite, $\operatorname{secat}(p\colon E\to B)$ is one less than the minimal number of open sets covering $B$ , over each of which the fibration admits a section.

1. A $\operatorname{TC}_{s}$ canonical class

Canonical cohomology classes for higher topological complexity were recently introduced and studied by Espinosa Baro, Farber, Mescher, and Oprea, see [12]. In this brief preliminary section, with this work as a general reference ([12, §§5–6] in particular), we recall and discuss aspects of these classes which will be of subsequent use.

Let $X$ be a CW-complex. The standard dimensional upper bound for higher topological complexity is

(1.1)

\operatorname{TC}_{s}(X)\leq s\dim(X).

Although (1.1) can be improved in terms of the connectivity of $X$ , we are interested in the improvements coming from obstruction-theory techniques in cases where $X$ is not simply connected. A fundamental concept in this context is the notion of homological obstruction as considered in Schwarz’ monograph [19]. Recall that the fiber of $e_{s}\colon PX\to X^{s}$ is $\Omega X^{s-1}=(\Omega X)^{s-1}$ . In [12], the homological obstruction for sectioning $e_{s}$ over the 1-dimensional skeleton of $X^{s}$ is identified with a canonical twisted class,

(1.2)

\mathfrak{v}_{X,s}\in H^{1}(X^{s};I_{s}(\pi^{s-1}))=H^{1}(X^{s};\widetilde{H}_{0}(\Omega X^{s-1})),

where $\pi:=\pi_{1}(X)$ and $I_{s}(\pi^{s-1})$ denotes the augmentation ideal of $\pi^{s-1}$ , viewed as a ${\mathbb{Z}}[\pi^{s}]$ -submodule of ${\mathbb{Z}}[\pi^{s-1}]$ . Here the action of $\pi^{s}$ on $I_{s}(\pi^{s-1})$ , which corresponds to the monodromy associated with the fibration $e_{s}$ , is given by

(a_{1},\ldots,a_{s})\cdot(b_{1},\ldots,b_{s-1})=(a_{1}b_{1}\overline{a_{2}},a_{2}b_{2}\overline{a_{3}},\ldots,a_{s-1}b_{s-1}\overline{a_{s}}).

The class $\mathfrak{v}_{X,s}$ can also be described as the cohomology class induced by the crossed homomorphism $\nu_{X,s}\colon\pi^{s}\to I_{s}(\pi^{s-1})$ given by

\nu_{X,s}(a_{1},\ldots,a_{s})=\left(a_{1}\overline{a_{2}},a_{2}\overline{a_{3}},\ldots,a_{s-1}\overline{a_{s}}\right)-1_{s-1},

where $1_{s-1}$ is the unit element of $\pi^{s-1}$ . Obstruction-theoretic arguments lead then to the following result:

Theorem 1.1 ([12]).

Let $X$ be a CW-complex of dimension $n\geq 2$ . Then $\operatorname{TC}_{s}(X)<sn$ if and only if the $sn$ -th cup-power $\mathfrak{v}_{X,s}^{sn}=0$ .

Here, $\mathfrak{v}_{X,s}^{sn}$ lies in the cohomology of $X^{s}$ with coefficients in the $sn$ -th tensor power of $I_{s}(\pi^{s-1})$ endowed with the diagonal action of $\pi_{s}$ , denoted by $I_{s}^{sn}(\pi^{s-1})$ .

The construction of the class $\mathfrak{v}_{X,s}$ generalizes the $\operatorname{TC}$ canonical class of [7] and provides for $\operatorname{TC}_{s}$ an analogue of the classical Berstein-Schwarz class $\mathfrak{b}_{X}\in H^{1}(X;I(\pi))$ . Note that, in this case, $I(\pi)$ is the augmentation ideal of $\pi$ endowed with the left ${\mathbb{Z}}[\pi]$ -module structure induced by the multiplication of $\pi$ . As is well-known, the Lusternik–Schnirelmann category of $X$ , $\operatorname{\mathrm{cat}}(X)$ , satisfies $\operatorname{\mathrm{cat}}(X)=\dim X$ if and only if $\mathfrak{b}_{X}^{\dim(X)}\neq 0$ (see [3], [19] and [11] for a proof including the case $\dim(X)=2$ ).

Remark 1.2.

We conclude this section with a brief remark regarding the functoriality of these classes. For $\pi=\pi_{1}(X)$ , applying the above result to the classifying space $B\pi$ yields a crossed homomorphism and associated cohomology class, which we denote by $\nu_{\pi,s}$ and $\mathfrak{v}_{\pi,s}$ respectively.

Recall that if $f:X\to Y$ is a map, and $A$ is a ${\mathbb{Z}}[\pi_{1}(Y)]$ -module, then $f^{*}(A)$ denotes the ${\mathbb{Z}}[\pi_{1}(X)]$ -module whose underlying abelian group is $A$ and the action of $g\in\pi_{1}(X)$ on $a\in A$ is given by $g\cdot a:=\pi_{1}(f)(g)\cdot a$ . Taking $f\colon X\to Y=B\pi$ to be a classifying map, the isomorphism ${I}_{s}(\pi^{s-1})\cong(f^{s})^{*}{I}_{s}(\pi^{s-1})$ yields

{\mathfrak{v}}_{X,s}=(f^{s})^{*}{\mathfrak{v}}_{\pi,s}.

Similar considerations apply to the Berstein-Schwarz class $\mathfrak{b}_{\pi}\in H^{1}(\pi;I(\pi))$ (resp., $\mathfrak{b}_{X}\in H^{1}(X;I(\pi))$ ), induced by the crossed homomorphism $\beta_{\pi}\colon\pi\to I(\pi)$ , $\alpha\mapsto\alpha-1$ . Namely, the isomorphism $I(\pi)\cong f^{*}I(\pi)$ yields $\mathfrak{b}_{X}=f^{*}\mathfrak{b}_{\pi}$ .

2. Abelian fundamental group

In this section we extend to higher topological complexity some results of [6] which will be useful for our computations. The arguments are therefore similar to those of [6] as well as some of [10].

Assume from now on that $\pi=\pi_{1}(X)$ is abelian. We consider the group homomorphism ${}^{s}\chi:\pi^{s}\to\pi^{s-1}$ given by

{}^{s}\chi(a_{1},\dots,a_{s})=(a_{1}\overline{a_{2}},a_{2}\overline{a_{3}},\dots,a_{s-1}\overline{a_{s}}).

Note that the ${\mathbb{Z}}[\pi^{s}]$ -module ${}^{s}\chi^{*}(I(\pi^{s-1}))$ is exactly the ${\mathbb{Z}}[\pi^{s}]$ -module ${I}_{s}(\pi^{s-1})$ . With the notation regarding canonical classes, Berstein-Schwarz classes, and crossed homomorphisms of the previous section, we also have, for any $(a_{1},\dots,a_{s})\in\pi^{s}$ ,

\nu_{\pi,s}(a_{1},\dots,a_{s})=\beta_{\pi^{s-1}}(a_{1}\overline{a_{2}},a_{2}\overline{a_{3}},\dots,a_{s-1}\overline{a_{s}})=\beta_{\pi^{s-1}}({{}^{s}\chi}(a_{1},\dots,a_{s})).

We then have ${\mathfrak{v}}_{\pi,s}={{}^{s}\chi}^{*}{\mathfrak{b}}_{\pi^{s-1}}$ in $H^{1}(\pi^{s};{I}_{s}(\pi^{s-1}))=H^{1}(B\pi^{s};{I}_{s}(\pi^{s-1}))$ , and, for any $k$ ,

{\mathfrak{v}}^{k}_{X,s}=(\gamma^{s})^{*}{\mathfrak{v}}^{k}_{\pi,s}=(\gamma^{s})^{*}{(^{s}\chi)}^{*}{\mathfrak{b}}^{k}_{\pi^{s-1}}\,\,\mbox{ in }H^{k}(X^{s};{I}_{s}^{k}(\pi^{s-1}))

where $\gamma:X\to B\pi$ is a classifying map.

In order to establish our results, it is useful to consider the cofiber of the diagonal map $\Delta_{s}=\Delta_{s}^{X}:X\to X^{s}$ . We denote it by $C_{\Delta_{s}}(X)$ . We will more generally use the notation $\Delta_{s}^{Z}:Z\to Z^{s}$ to denote the $s$ -diagonal of a set $Z$ and suppress the superscript when the context is clear.

Proposition 2.1.

Let $X$ be an $n$ -dimensional CW-complex with $n\geq 2$ . Suppose that $\pi=\pi_{1}(X)$ is abelian and let $\gamma:X\to B\pi$ be a classifying map. Then for any $s\geq 2$ we have

(1)

$\mathfrak{v}_{X,s}=q^{*}\mathfrak{b}_{C_{\Delta_{s}}(X)}$ in $H^{1}(X^{s};I_{s}(\pi^{s-1}))$ where $q:X^{s}\to C_{\Delta_{s}}(X)$ is the identification map.
(2)

$\operatorname{TC}_{s}(X)<sn$ if and only if $\operatorname{\mathrm{cat}}(C_{\Delta_{s}}(X))<sn$ .
(3)

If $\operatorname{TC}_{s}(X)<sn$ then, for any ${\mathbb{Z}}[\pi^{s-1}]$ -module $A$ and for any homology class $\mathsf{c}\in H_{sn}(X^{s};(^{s}\chi\gamma^{s})^{*}A)$ , the class $\mathfrak{c}=\gamma^{s}_{*}(\mathsf{c})\in H_{sn}(\pi^{s};(^{s}\chi)^{*}A)$ satisfies ${}^{s}\chi_{*}(\mathfrak{c})=0$ .

Proof.

First observe that the homomorphism ${}^{s}\chi{\circ\Delta_{s}^{\pi}}$ is trivial. Consequently, the map $B^{s}\chi\circ B{{\Delta_{s}^{\pi}}}$ obtained after applying the functor $B$ is also trivial. By identifying $B\pi^{s}$ with $(B\pi)^{s}$ and $B{\Delta_{s}^{\pi}}$ with $\Delta_{s}^{B\pi}$ , we have a commutative diagram of the following form

where $\xi$ is induced by the quotient property. Since $\pi$ is abelian, we have an exact sequence $1\to\pi\stackrel{{\scriptstyle\Delta_{s}^{\pi}}}{{\longrightarrow}}{\pi^{s}}\stackrel{{{}^{s}\chi}}{{\longrightarrow}}{\pi^{s-1}}\to 1$ and, using the Van Kampen theorem, we can see that $\pi_{1}(C_{\Delta_{s}}(X))=\pi^{s-1}$ and that $\pi_{1}(\xi)$ is an isomorphism. Consequently $\xi$ is a classifying map and the Berstein-Schwarz class of $C_{\Delta_{s}}(X)$ is given by ${\mathfrak{b}}_{C_{\Delta}}=\xi^{*}{\mathfrak{b}}_{\pi^{s-1}}$ . By the commutativity of the diagram we then get $q^{*}(I(\pi^{s-1}))\cong{{}^{s}\chi}^{*}(I(\pi^{s-1}))\cong I_{s}(\pi^{s-1})$ and $q^{*}{\mathfrak{b}}_{C_{\Delta_{s}}(X)}=\mathfrak{v}_{X,s}$ as claimed in the first item.

The equality established above implies that $q^{*}{\mathfrak{b}}^{sn}_{C_{\Delta_{s}}(X)}=\mathfrak{v}^{sn}_{X,s}$ . For dimensional reasons, the map $q^{*}:H^{sn}(C_{\Delta_{s}}(X);I(\pi^{s-1}))\to H^{sn}(X^{s};I_{s}(\pi^{s-1}))$ is an isomorphism. We therefore have ${\mathfrak{b}}^{sn}_{C_{\Delta_{s}}(X)}=0$ if and only if $\mathfrak{v}^{sn}_{X,s}=0$ , which implies the second item.

We now prove the last item. Let $\mathsf{c}\in H_{sn}(X^{s};(^{s}\chi\gamma^{s})^{*}A)$ be a nonzero class and let $\mathfrak{c}=\gamma^{s}_{*}(\mathsf{c})$ . We have ${}^{s}\chi_{*}(\mathfrak{c})=\xi_{*}q_{*}(\mathsf{c})$ . Note that $q_{*}(\mathsf{c})$ is a homology class of degree $sn$ . Since $\operatorname{TC}_{s}(X)<sn$ we have $\operatorname{\mathrm{cat}}(C_{\Delta_{s}}(X))<sn$ . Therefore the classifying map $\xi$ factors up to homotopy through an $(sn-1)$ -dimensional space. Consequently, $\xi_{*}q_{*}(\mathsf{c})=0$ and the result follows. ∎

Remark 2.2.

In the situation of Proposition 2.1, if $A$ is a trivial ${\mathbb{Z}}[\pi^{s-1}]$ -module and $\mathsf{c}\in H_{sn}(X^{s};A)$ is an element such that the class $\mathfrak{c}=\gamma^{s}_{*}(\mathsf{c})\in H_{sn}(\pi^{s};A)$ satisfies ${}^{s}\chi_{*}(\mathfrak{c})\neq 0$ then $\operatorname{TC}_{s}(X)=sn$ .

Item (3) of Proposition 2.1 is sharp under reasonably general conditions. Let $M$ be an $n$ -dimensional connected closed manifold with fundamental group $\pi=\pi_{1}(M)$ and let $\omega=\omega_{M}:\pi\to\{\pm 1\}$ be the homomorphism determined by the first Stiefel-Whitney class of $M$ . Recall that the orientation module of $M$ , denoted by $\widetilde{{\mathbb{Z}}}=\widetilde{{\mathbb{Z}}}_{M}$ , is the abelian group ${\mathbb{Z}}$ given with a structure of ${\mathbb{Z}}[\pi]$ -module determined by $a\cdot t=\omega(a)t$ for $a\in\pi$ , $t\in{\mathbb{Z}}$ . Note that $\widetilde{{\mathbb{Z}}}_{M^{s}}=\widetilde{{\mathbb{Z}}}_{M}^{\otimes s}$ , which additively is ${\mathbb{Z}}$ with $\pi^{s}$ action given by $(a_{1},a_{2},\dots,a_{s})t=\omega(a_{1})\omega(a_{2})\cdots\omega(a_{s})t$ .

Proposition 2.3.

Let $M$ be an $n$ -dimensional connected closed manifold with $n\geq 2$ and $\pi=\pi_{1}(M)$ abelian. Assume there is a ${\mathbb{Z}}[\pi^{s-1}]$ -module $A$ such that the ${\mathbb{Z}}[\pi^{s}]$ -modules ${}^{s}\chi^{*}(A)$ and $\widetilde{{\mathbb{Z}}}^{\otimes s}$ are isomorphic. Then the following two conditions are equivalent:

(1)

The class $\mathfrak{m}:=\gamma_{*}([M])\in H_{n}(\pi;\widetilde{{\mathbb{Z}}})$ satisfies ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})=0$ in $H_{sn}(\pi^{s-1};A)$ .
(2)

$\operatorname{TC}_{s}(X)<sn$ .

Here we denote by $\mathfrak{m}^{\times s}\in H_{sn}(\pi^{s};\widetilde{{\mathbb{Z}}}^{\otimes s})$ the image of the fundamental class of $M^{s}$ under the homomorphism induced by $\gamma^{s}:M^{s}\to B\pi^{s}$ . Note that we also denote by $\widetilde{{\mathbb{Z}}}$ the local system over $B\pi$ arising from the isomorphism $\pi_{1}(\gamma)$ induced by the classifying map $\gamma\colon M\to B\pi$ .

Proof.

From the naturality of the cap-product and the assumption that $A$ is a ${\mathbb{Z}}[\pi^{s-1}]$ -module satisfying ${}^{s}\chi^{*}(A)\cong\widetilde{{\mathbb{Z}}}^{\otimes s}$ we get the following diagram.

The cap-product on the first line is an isomorphism by Poincaré duality. The bottom vertical map in the third column corresponds to the morphism

\chi_{*}:H_{*}(B\pi^{s};{I}_{s}^{sn}(\pi^{s-1})\otimes\widetilde{{\mathbb{Z}}}^{\otimes s})\to H_{*}(B\pi^{s-1};{I}^{sn}(\pi^{s-1})\otimes A)

in degree $0$ . It is induced by the obvious isomophism between the underlying ${\mathbb{Z}}$ -modules $I^{sn}(\pi^{s-1})\otimes\widetilde{{\mathbb{Z}}}^{\otimes s}$ and $I^{sn}(\pi^{s-1})\otimes A$ and is an isomorphism on the coinvariants because ${}^{s}\chi$ is surjective.

Let $[M]\in H_{n}(M;\widetilde{{\mathbb{Z}}})$ be the fundamental class. Since the third column of the diagram is comprised of isomorphisms, we have

(^{s}\chi)_{*}(\gamma^{s})_{*}([M^{s}])\cap\mathfrak{b}^{sn}_{\pi^{s-1}}=0\mbox{ if and only if }[M^{s}]\cap(\gamma^{s})^{*}(^{s}\chi)^{*}\mathfrak{b}^{sn}_{\pi^{s-1}}=0.

This is equivalent to saying

(^{s}\chi)_{*}(\mathfrak{m}^{\times s})\cap\mathfrak{b}^{sn}_{\pi^{s-1}}=0\mbox{ if and only if }[M^{s}]\cap\mathfrak{v}^{sn}_{M,s}=0.

The hypothesis ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})=0$ yields $[M^{s}]\cap\mathfrak{v}^{sn}_{M,s}=0$ . By Poincaré duality, we can then conclude that $\mathfrak{v}^{sn}_{M,s}=0$ and consequently $\operatorname{TC}_{s}(M)<sn$ . ∎

Corollary 2.4.

Let $M$ be an orientable $n$ -dimensional manifold with $n\geq 2$ and abelian fundamental group $\pi=\pi_{1}(M)$ . The class $\mathfrak{m}=\gamma_{*}([M])\in H_{n}(\pi;{\mathbb{Z}})$ satisfies ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})=0$ in $H_{sn}(\pi^{s-1};{\mathbb{Z}})$ if and only if $\operatorname{TC}_{s}(X)<sn$ .

Proof.

Since $M$ is orientable, the orientation module $\widetilde{{\mathbb{Z}}}$ is just ${\mathbb{Z}}$ with trivial action. Taking $A={\mathbb{Z}}$ also with trivial action, we have ${}^{s}\chi^{*}(A)\cong{{\mathbb{Z}}}^{\otimes s}$ and the result follows from Proposition 2.3. ∎

Remark 2.5.

When $M$ is non-orientable, the ${\mathbb{Z}}[\pi^{s-1}]$ -module $A=\widetilde{{\mathbb{Z}}}^{\otimes s-1}$ satisfies the assumptions of Proposition 2.3 for $s=2$ ([6]) but fails to do so for all $s>2$ . For instance, set $s=3$ and suppose that $b\in\pi$ is an element for which the orientation character $\omega:\pi\to\{\pm 1\}$ satisfies $\omega(b)=-1$ . Then, for $a,c\in\pi$ and $t\in{\mathbb{Z}}$ we have

(a,b,c)\cdot t=\omega(a)\omega(b)\omega(c)t=-\omega(a)\omega(c)t

while

{}^{3}\chi(a,b,c)\cdot t=(a\overline{b},b\overline{c})\cdot t=\omega(a\overline{b})\omega(b\overline{c})t=\omega(a)\omega(c)t.

This shows that the map ${}^{3}\chi$ does not induce a homomorphism from $H_{*}(\pi^{3};\widetilde{{\mathbb{Z}}}^{\otimes 3})$ to $H_{*}(\pi^{2};\widetilde{{\mathbb{Z}}}^{\otimes 2})$ . Note that, in Proposition 2.3, $A$ must be, as an abelian group, isomorphic to ${\mathbb{Z}}$ . Furthermore, since ${}^{s}\chi$ is surjective, the ${\mathbb{Z}}[\pi^{s-1}]$ -module structure on $A$ is forced by the hypothesis ${}^{s}\chi^{*}(A)\cong\widetilde{{\mathbb{Z}}}^{\otimes s}$ and this condition is impossible when $s$ is odd. For instance, again set $s=3$ , choose $b\in\pi$ as above and assume ${}^{3}\chi^{*}(A)\cong\widetilde{{\mathbb{Z}}}^{\otimes 3}$ . The equalities ${}^{3}\chi(b,1,1)={}^{3}\chi(1,\overline{b},\overline{b})=(b,1)$ then lead to the impossible

t=\omega(\overline{b})\omega(\overline{b})t={}^{3}\chi(1,\overline{b},\overline{b})\cdot t=(b,1)\cdot t={}^{3}\chi(b,1,1)\cdot t=\omega(b)t=-t.

Nonetheless, when $s=2\sigma$ , $\sigma\geq 1$ , the ${\mathbb{Z}}[\pi^{s-1}]$ -module $A=\widetilde{{\mathbb{Z}}}\otimes({\mathbb{Z}}\otimes\widetilde{{\mathbb{Z}}})^{\sigma-1}$ does satisfy ${}^{s}\chi^{*}(A)\cong\widetilde{{\mathbb{Z}}}^{\otimes s}$ , and we explore its usage in Section 5.

3. Some calculations in the orientable case

Let $M$ be an orientable connected closed manifold with $\pi=\pi_{1}(M)$ abelian. In this section we will use Corollary 2.4 to establish the non-maximality $\operatorname{TC}_{s}(M)<s\dim(M)$ for some families of manifolds with abelian fundamental groups.

Let $\gamma:M\to B\pi$ be a classifying map and let $\mathfrak{m}=\gamma_{*}([M])\in H_{n}(\pi;{\mathbb{Z}})$ . Since $M$ is orientable, we will suppress the ${\mathbb{Z}}$ -coefficients from the notation. In all cases, we will see that ${}^{s}\chi_{*}(\mathfrak{m}^{\otimes s})=0$ in $H_{sn}(\pi^{s-1})$ .

In our first result, we suppose that $\pi$ is a free abelian group. This case has already been considered in [12] in the more general context of finite CW-complexes. Here, restricting to closed manifolds, we obtain a slightly stronger statement than [12, Corollary 6.14].

Proposition 3.1.

Let $M$ be an orientable $n$ -dimensional connected closed manifold with $\pi_{1}(M)={\mathbb{Z}}^{r}$ and let $s\geq 2$ . If $sn>(s-1)r$ then $\operatorname{TC}_{s}(M)<sn$ .

Proof.

Let $\pi={\mathbb{Z}}^{r}$ , let $\gamma:M\to B\pi$ be a classifying map and let $\mathfrak{m}=\gamma_{*}([M])\in H_{n}(\pi))$ . For degree reasons, we can see that ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})=0$ in $H_{sn}(\pi^{s-1})$ . Indeed $B\pi=(S^{1})^{r}$ and $H_{k}(\pi)=0$ for $k>r$ . Consequently $H_{sn}(\pi^{s-1})=0$ if $sn>(s-1)r$ . ∎

In general, observe that the homomorphism ${}^{s}\chi:\pi^{s}\to\pi^{s-1}$ given by

{}^{s}\chi(a_{1},\dots,a_{s})=(a_{1}\overline{a_{2}},a_{2}\overline{a_{3}},\dots,a_{s-1}\overline{a_{s}})

can be decomposed as

(3.1)

^{s}\chi=(\underbrace{\chi\times\cdots\times\chi}_{s-1})\circ({\rm Id}\times\underbrace{\Delta\times\cdots\times\Delta}_{s-2}\times{\rm Id})

where $\Delta=\Delta_{2}^{\pi}:\pi\to\pi\times\pi$ is the diagonal map and $\chi={{}^{2}\chi}$ . Denote by $j:\pi\to\pi$ the inversion. Since $\chi$ can be seen as the multiplication of $\pi$ , $\mu:\pi\times\pi\to\pi$ , precomposed with ${\rm Id}\times j$ , we have, for classes $\mathfrak{a},\mathfrak{b}\in H_{*}(\pi)$ ,

\chi_{*}(\mathfrak{a}\times\mathfrak{b})=\mathfrak{a}\wedge j_{*}(\mathfrak{b})

where $\wedge$ is the Pontryagin product, that is, the product induced by $\mu$ in homology, see [4, V.5].

In the results below, we consider the cyclic group ${\mathbb{Z}}_{q}=\langle v~|~v^{q}=1\rangle$ and work at the chain level. Recall the classical resolution of ${\mathbb{Z}}$ as a trivial ${\mathbb{Z}}[{\mathbb{Z}}_{q}]$ -module given by

(3.2)

where $N_{q}(v)=1+v+\cdots+v^{q-1}$ .

In the following lemma, we recall the morphisms induced by the diagonal $\Delta$ , the multiplication $\mu$ and the inversion $j$ on the level of resolutions (see [4, page 108] and [6, §3.2]). Let $[k]$ denote the generator of degree $k$ in (3.2), and write $B_{i,j}$ for the binomial coefficient $\binom{\,i+j\,}{i}$ .

Lemma 3.2.

At the level of the resolution (3.2),

(a)

$\Delta$ is given on generators by $[p]\mapsto\sum_{k+l=p}\Delta_{kl}[p]$ where

\Delta_{kl}[p]=\left\{\begin{array}[]{ll}[k]\otimes[l]&k\mbox{ even;}\\ \mathopen{[}k\mathclose{]}\otimes v[l]&k\mbox{ odd, }l\mbox{ even;}\\ \sum_{0\leq i<j\leq q-1}v^{i}[k]\otimes v^{j}[l]&k\mbox{ odd, }l\mbox{ odd.}\end{array}\right.

(b)

$\mu$ is given on generators by the formulæ

	$\displaystyle[2i]\otimes[2j]$	$\displaystyle\mapsto B_{i,j}\,[2(i+j)];$
	$\displaystyle[2i]\otimes[2j+1]$	$\displaystyle\mapsto B_{i,j}\,[2(i+j)+1];$
	$\displaystyle[2i+1]\otimes[2j]$	$\displaystyle\mapsto B_{i,j}\,[2(i+j)+1];$
	$\displaystyle[2i+1]\otimes[2j+1]$	$\displaystyle\mapsto 0.$

$(c)$

$j$ is given on generators by

$[i]\to N_{q-1}^{k}(v)[i]\qquad\mbox{if }i\in\{2k,2k-1\}.$

We denote by $C_{\bullet}({\mathbb{Z}}_{q})$ the ${\mathbb{Z}}$ -chain complex obtained by tensoring the resolution (3.2) with ${\mathbb{Z}}$ over ${\mathbb{Z}}_{q}$ .

(3.3)

Recall that the homology of this chain complex gives $H_{*}({\mathbb{Z}}_{q})=H_{*}({\mathbb{Z}}_{q};{\mathbb{Z}})$ . In positive degrees, $H_{+}({\mathbb{Z}}_{q})$ is concentrated in odd degrees.

As in [6], we denote by $\wedge:C_{\bullet}({\mathbb{Z}}_{q})\otimes C_{\bullet}({\mathbb{Z}}_{q})\to C_{\bullet}({\mathbb{Z}}_{q})$ the Pontryagin product, which is given by the formulæ $(b)$ of Lemma 3.2:

(3.4)

\left\{\begin{array}[]{l}~[2i]\wedge[2k]=B_{i,k}[2i+2k],\quad\quad[2i+1]\wedge[2k+1]=0,\\ \\ ~[2i]\wedge[2k+1]=[2k+1]\wedge[2i]=B_{i,k}[2i+2k+1].\end{array}\right.

We denote by $\mathbf{j}:C_{\bullet}({\mathbb{Z}}_{q})\to C_{\bullet}({\mathbb{Z}}_{q})$ the morphism induced by the inversion, which is from Lemma 3.2 $(c)$ given by

(3.5)

\mathbf{j}([i])=(q-1)^{k}[i]\qquad\mbox{if }i\in\{2k,2k-1\}.

In these terms, the chain map $\chi_{\bullet}={{}^{2}\chi}_{\bullet}$ induced by $\chi={{}^{2}\chi}$ can be described as the composite

We will also use the diagonal approximation of $C_{\bullet}({\mathbb{Z}}_{q})$ , obtained from Lemma 3.2 $(a)$ :

(3.6)

\begin{array}[]{rcl}\Delta_{\bullet}:C_{\bullet}({\mathbb{Z}}_{q})&\to&C_{\bullet}({\mathbb{Z}}_{q})\otimes C_{\bullet}({\mathbb{Z}}_{q})\\[2.0pt] {[p]}&\mapsto&\sum\limits_{k+l=p}\alpha_{kl}[k]\otimes[l].\end{array}

Here $\alpha_{kl}=1$ if $kl$ is even and $\alpha_{kl}=(q-1)q/2$ if $kl$ is odd.

Proposition 3.3.

Let $M$ be an orientable $n$ -dimensional connected closed manifold with $\pi_{1}(M)={\mathbb{Z}}_{q}$ . Then, for any $s\geq 2$ , we have $\operatorname{TC}_{s}(M)<sn$ .

Proof.

Let $\gamma:M\to B\pi$ be a classifying map, where $\pi={\mathbb{Z}}_{q}$ , and let $\mathfrak{m}=\gamma_{*}([M])\in H_{n}(\pi;{\mathbb{Z}}))$ . We will see that ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})=0$ in $H_{sn}(\pi^{s-1};{\mathbb{Z}})$ . If $\dim M$ is even, this is immediate since $H_{+}(\pi)$ is concentrated in odd degrees, which implies $\mathfrak{m}=0$ . We then suppose that $\dim M=2p+1$ . A cycle $\mathbf{m}\in C_{2p+1}({\mathbb{Z}}_{q})$ representing the class $\mathfrak{m}$ is of the form $\mathbf{m}=\lambda[2p+1]$ for some $\lambda\in{\mathbb{Z}}$ . In order to compute ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})$ we use the decomposition (3.1) and analyze the element ${}^{s}\chi_{\bullet}(\mathbf{m}^{\otimes s})$ which is given by

(\chi_{\bullet})^{\otimes s-1}(\mathbf{m}\otimes\underbrace{\Delta_{\bullet}\mathbf{m}\otimes\cdots\otimes\Delta_{\bullet}\mathbf{m}}_{s-2}\otimes\mathbf{m}).

The element $\mathbf{m}\otimes\Delta_{\bullet}\mathbf{m}\otimes\cdots\otimes\Delta_{\bullet}\mathbf{m}\otimes\mathbf{m}$ is given by a ${\mathbb{Z}}$ -linear combination of elements of the form

[2p+1]\otimes[k_{1}]\otimes[l_{1}]\otimes\cdots\otimes[k_{s-2}]\otimes[l_{s-2}]\otimes[2p+1]

where $k_{i}+l_{i}=2p+1$ for any $1\leq i\leq s-2$ . Setting $l_{0}=k_{s-1}=2p+1$ , there will be necessarily some $i\in\{0,\dots,s-2\}$ such that $l_{i}$ and $k_{i+1}$ are both odd. Applying $(\chi_{\bullet})^{\otimes s-1}$ to the element above yields

([2p+1]\wedge\mathbf{j}[k_{1}])\otimes([l_{1}]\wedge\mathbf{j}[k_{2}])\otimes\cdots\otimes([l_{s-2}]\wedge\mathbf{j}[k_{s-1}]).

If $l_{i}$ and $k_{i+1}$ are both odd, the corresponding factor $([l_{i}]\wedge\mathbf{j}[k_{i+1}])$ vanishes since $\mathbf{j}[k_{i+1}])$ is a multiple of $[k_{i+1}]$ and the Pontryagin product of two odd degree elements is zero. Consequently, we obtain ${}^{s}\chi_{\bullet}(\mathbf{m}^{\otimes s})=0$ and ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})=0$ . ∎

Proposition 3.4.

Let $M$ be an orientable $n$ -dimensional connected closed manifold with $\pi_{1}(M)={\mathbb{Z}}^{r}\times{\mathbb{Z}}_{q}$ such that $r<n$ . Then, for any $s\geq 2$ , we have $\operatorname{TC}_{s}(M)<sn$ .

Proof.

Let $\gamma:M\to B\pi$ be a classifying map, where $\pi={\mathbb{Z}}^{r}\times{\mathbb{Z}}_{q}$ , and let $\mathfrak{m}=\gamma_{*}([M])\in H_{n}(\pi;{\mathbb{Z}}))$ . By the Künneth formula, we have $H_{*}({\mathbb{Z}}^{r}\times{\mathbb{Z}}_{q})=H_{*}({\mathbb{Z}}^{r})\otimes H_{*}({\mathbb{Z}}_{q})$ . Since $n>r$ , we can write $\mathfrak{m}=\sum\sigma_{i}\otimes\alpha_{i}$ where $\alpha_{i}\in H_{+}({\mathbb{Z}}_{q})$ and $\sigma_{i}\in H_{*}({\mathbb{Z}}^{r})=\bigwedge(x_{1},\dots,x_{r})$ with each $x_{j}$ of degree $1$ . Since $H_{+}({\mathbb{Z}}_{q})$ is concentrated in odd degrees, a cycle $\mathbf{m}$ representing $\mathfrak{m}$ can be described as a sum of terms of the form $\lambda\sigma\otimes[2p+1]$ where $\lambda\in{\mathbb{Z}}$ , $p\geq 0$ and $\sigma\in\bigwedge(x_{1},\cdots,x_{r})$ is a class regarded as a cycle. The element $\mathbf{m}\otimes\Delta_{\bullet}\mathbf{m}\otimes\cdots\otimes\Delta_{\bullet}\mathbf{m}\otimes\mathbf{m}$ is therefore given by a ${\mathbb{Z}}$ -linear combination of elements of the form

(3.7)

\sigma_{0}\otimes[l_{0}]\otimes\sigma_{1}\otimes[k_{1}]\otimes\tilde{\sigma}_{1}\otimes[l_{1}]\otimes\cdots\otimes\sigma_{s-2}\otimes\otimes[k_{s-2}]\otimes\tilde{\sigma}_{s-2}\otimes[l_{s-2}]\otimes\sigma_{s-1}[k_{s-1}]

where $l_{0}$ , $k_{i}+l_{i}$ for $1\leq i\leq s-2$ , and $k_{s-1}$ are all odd and the elements $\sigma_{i},\tilde{\sigma}_{i}$ belong to $\bigwedge(x_{1},\cdots,x_{r})$ . The calculation of $\chi_{\bullet}$ on (say) $\sigma_{0}\otimes[l_{0}]\otimes\sigma_{1}\otimes[k_{1}]$ is made componentwise and gives rise to factors of the form

\pm(\sigma_{0}\wedge\sigma_{1})\otimes([l_{0}]\wedge{\mathbf{j}}[k_{1}]).

As in the proof of Proposition 3.3, there will be necessarily, in the expression (3.7), some $i\in\{0,\dots,s-2\}$ such that $l_{i}$ and $k_{i+1}$ are both odd. After applying $\chi_{\bullet}$ , the corresponding factor will be $0$ . Consequently, we obtain ${}^{s}\chi_{\bullet}(\mathbf{m}^{\otimes s})=0$ and ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})=0$ . We can hence conclude that $\operatorname{TC}_{s}(M)<sn$ . ∎

Limiting examples

Examples 4.1 and 4.2 from [6] show that the conditions in Propositions 3.1 and 3.4 are sharp. We now show that Proposition 3.3 cannot be extended to manifolds whose fundamental group is of the form ${\mathbb{Z}}_{p}\times{\mathbb{Z}}_{p}$ where $p$ is a prime.

Example 3.5.

A manifold $N$ with $\pi_{1}(N)={\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ and $\operatorname{TC}_{3}(N)=3\dim(N)$ .

Set $\pi={\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ and consider $C_{\bullet}(\pi)=C_{\bullet}({\mathbb{Z}}_{3})\otimes C_{\bullet}({\mathbb{Z}}_{3})$ . We will write $[ik]$ instead of $[i]\otimes[k]$ . We first consider the cycle $\mathbf{m}=[05]+[50]$ and denote by $\mathfrak{m}$ its homology class. We will see that ${}^{3}\chi_{*}(\mathfrak{m}^{\times 3})\neq 0$ . By the Universal Coefficient Theorem, it is actually sufficient to see that ${}^{3}\chi_{*}(\mathfrak{m}^{\times 3}_{{\mathbb{Z}}_{3}})\neq 0$ where $\mathfrak{m}_{{\mathbb{Z}}_{3}}$ corresponds to $\mathfrak{m}$ in $H_{5}(\pi;{\mathbb{Z}}_{3})$ . As $H_{*}(\pi;{\mathbb{Z}}_{3})\cong H_{*}({\mathbb{Z}}_{3};{\mathbb{Z}}_{3})\otimes H_{*}({\mathbb{Z}}_{3};{\mathbb{Z}}_{3})$ and $H_{*}({\mathbb{Z}}_{3};{\mathbb{Z}}_{3})={\mathbb{Z}}_{3}[k]$ for all $k\geq 0$ , we will continue to write $\mathfrak{m}_{{\mathbb{Z}}_{3}}=[05]+[50]$ .

Using the diagonal approximation associated to the resolution (3.2) described in Lemma 3.2 (or, tensoring the diagonal (3.6) by ${\mathbb{Z}}_{q}$ ) we can check that the homology diagonal of $H_{*}({\mathbb{Z}}_{3};{\mathbb{Z}}_{3})$ satisfies

\Delta_{*}[0]=[0]\otimes[0]\qquad\Delta_{*}[5]=\sum\limits_{k+l=5}[k]\otimes[l].

Consequently, the homology diagonal of $H_{*}(\pi;{\mathbb{Z}}_{3})$ satisfies:

\Delta_{*}[05]=\sum\limits_{k+l=5}[0k]\otimes[0l]\qquad\Delta_{*}[50]=\sum\limits_{k+l=5}[k0]\otimes[l0].

We have to compute:

(\chi_{*}\otimes\chi_{*})\left(([05]+[50])\otimes(\Delta_{*}[05]+\Delta_{*}[50])\otimes([05]+[50])\right).

A term of the form $\chi_{*}([kl]\otimes[k^{\prime}l^{\prime}])$ is given in $H_{*}(\pi;{\mathbb{Z}}_{3})=H_{*}({\mathbb{Z}}_{3};{\mathbb{Z}}_{3})\otimes H_{*}({\mathbb{Z}}_{3};{\mathbb{Z}}_{3})$ by a componentwise calculation:

\chi_{*}([kl]\otimes[k^{\prime}l^{\prime}])=(-1)^{lk^{\prime}}([k]\wedge j_{*}[k^{\prime}])\otimes([l]\wedge j_{*}[l^{\prime}]).

Taking into account the formulas for the inversion and for the Pontryagin product (induced in ${\mathbb{Z}}_{3}$ -homology by the formulas (3.5) and (3.4) given above) we have

\chi_{*}([04]\otimes[05]])=([0]\wedge j_{*}[0])\otimes([4]\wedge j_{*}[5])=[0]\otimes([4]\wedge(-[5]))=-[0]\otimes(6[9])=-6[09]

which vanishes since we are working with coefficients in ${\mathbb{Z}}_{3}$ . We can thus check that

(\chi_{*}\otimes\chi_{*})\left([50]\otimes([01]\otimes[04])\otimes([05]+[50])\right)=[51]\otimes[54]

and that this is the only term belonging to ${\mathbb{Z}}_{3}[51]\otimes H_{*}(\pi;{\mathbb{Z}}_{3})$ in the expansion of ${}^{3}\chi_{*}(\mathfrak{m}^{\times 3}_{{\mathbb{Z}}_{3}})$ . Since $[51]\otimes[54]$ does not vanish in ${\mathbb{Z}}_{3}[51]\otimes H_{*}(\pi;{\mathbb{Z}}_{3})$ , we can conclude that ${}^{3}\chi_{*}(\mathfrak{m}^{\times 3}_{{\mathbb{Z}}_{3}})\neq 0$ . Consequently ${}^{3}\chi_{*}(\mathfrak{m}^{\times 3})\neq 0$ .

We can next follow the same strategy as in [6] to show that there exists a manifold $N$ with fundamental group $\pi={\mathbb{Z}}_{3}\times{\mathbb{Z}}_{3}$ and maximal $\operatorname{TC}_{3}$ . More precisely, considering the lens spaces $L_{3}^{5}=S^{5}/{\mathbb{Z}}_{3}$ and $L_{3}^{\infty}=S^{\infty}/{\mathbb{Z}}_{3}=B{\mathbb{Z}}_{3}$ , we can realize the class $[05]+[50]\in H_{*}(\pi)$ as the image of the fundamental class of $M=L_{3}^{5}\#L_{3}^{5}$ under the map induced by

f\colon M\xrightarrow{\ \text{pinch}\ }L_{3}^{5}\vee L_{3}^{5}{\lhook\joinrel\longrightarrow}L_{3}^{\infty}\vee L_{3}^{\infty}=B(\pi).

We can then use surgery to replace $M$ by a manifold $N$ with $\pi_{1}(N)=\pi$ and $f$ by a classifying map $\gamma:N\to B\pi$ . In this way, $\mathfrak{m}=\gamma_{*}([N])$ and, from ${}^{3}\chi_{*}(\mathfrak{m}^{\times 3})\neq 0$ and Proposition 2.1 (3), we can deduce that $\operatorname{TC}_{3}(N)=3n$ .

In [6], it has been shown that the regular topological complexity $\operatorname{TC}=\operatorname{TC}_{2}$ of a non-orientable manifold with abelian fundamental group is never maximal. This is not longer true for $\operatorname{TC}_{s}$ with $s\geq 3$ . For instance, for the real projective plane $P^{2}$ , $s$ -zero-divisor cuplength considerations imply that $\operatorname{TC}_{3}(P^{2})=6$ , see [9] and the discussion in §4 below. With the approach of this paper, we pursue more general maximality results of this nature next.

4. Cohomological lower bounds

In this section, we use cohomological lower bounds on $\operatorname{TC}_{s}$ given by the $s$ -zero-divisor cup length or $\operatorname{TC}_{s}$ -weights as well as specific calculations from [9, 8] to obtain lower bounds on the higher topological complexity of families of manifolds with finite cyclic fundamental group and maximal LS-category. In some cases, exact values are given by using our results from Section 3.

Let $\Bbbk$ be a field. Recall that, for a space $X$ , the ( $\Bbbk$ -coefficients) $s$ -zero-divisor cup length, $\operatorname{\mathrm{zcl}}_{s}(X)=\operatorname{\mathrm{zcl}}_{s}(X;\Bbbk)$ , is the maximum of the set

\left\{\ell\ |\ u_{1}\dots u_{\ell}\neq 0,u_{i}\in\mathbf{Z}_{s}(X;\Bbbk)\right\}

where

\mathbf{Z}_{s}(X;\Bbbk)=\ker\left(\bigotimes_{i=1}^{s}H^{\ast}(X;\Bbbk)\stackrel{{\scriptstyle\cup}}{{\to}}H^{\ast}(X;\Bbbk)\right).

We have $\operatorname{\mathrm{zcl}}_{s}(X)\leq\operatorname{TC}_{s}(X)$ , see [2].

In some cases, a better lower bound can be obtained through the notion of $\operatorname{TC}_{s}$ -weight. Recall (see [14, §2]) that if $p:E\to B$ is a fibration and $u\in\widetilde{H}^{*}(B;\Bbbk)$ is a nontrivial class, the weight of $u$ associated to $p$ , $\operatorname{\mathrm{wgt}}_{p}(u)$ , is the largest integer $k$ such that $f^{*}(u)=0$ for any map $f:Y\to B$ satisfying $\operatorname{secat}(f^{*}(p))<k$ . If $u\neq 0$ , then $\operatorname{\mathrm{wgt}}_{p}(u)>0$ if and only if $p^{*}(u)=0$ and $\operatorname{secat}(p)\geq\operatorname{\mathrm{wgt}}_{p}(u)$ . Moreover, if $u_{1},\dots,u_{l}\in\widetilde{H}^{*}(B;\Bbbk)$ satisfy $u_{1}\cup\cdots\cup u_{l}\neq 0$ then

\operatorname{\mathrm{wgt}}_{p}(u_{1}\cup\cdots\cup u_{l})\geq\operatorname{\mathrm{wgt}}_{p}(u_{1})+\cdots+\operatorname{\mathrm{wgt}}_{p}(u_{l}).

For a space $X$ , the $\operatorname{TC}_{s}$ -weight, denoted by $\operatorname{\mathrm{wgt}}_{s}$ , is the weight associated to the fibration $e_{s}:PX\to X^{s}$ . Taking coefficients in $\Bbbk$ , the morphism $e_{s}^{*}$ can be identified with the $s$ -fold cup-product and we can define the ( $\Bbbk$ -coefficients) weighted $s$ -zero divisor cup length, $\operatorname{\operatorname{\mathrm{zcl}}^{\underline{w}}_{s}}(X)=\operatorname{\operatorname{\mathrm{zcl}}^{\underline{w}}_{s}}(X;\Bbbk)$ , to be the maximum of the set

\left\{\sum_{i=1}^{\ell}\operatorname{\mathrm{wgt}}_{s}(u_{i})\ |\ u_{1}\dots u_{\ell}{\neq 0,u_{i}}\in\mathbf{Z}_{s}(X;\Bbbk)\ \right\}.

We have $\operatorname{TC}_{s}(X)\geq\operatorname{\operatorname{\mathrm{zcl}}^{\underline{w}}_{s}}(X)$ . We also note that if $f:Y\to X$ is a map and $u\in H^{*}(X^{s};\Bbbk)$ satisfies $(f^{s})^{*}(u)\neq 0$ then $\operatorname{\mathrm{wgt}}_{s}(f^{*}(u))\geq\operatorname{\mathrm{wgt}}_{s}(u)$ .

4.1. Manifolds with $\pi_{1}(M)={\mathbb{Z}}_{2}$ and $\operatorname{\mathrm{cat}}(M)=\dim M$

The ${\mathbb{Z}}_{2}$ -coefficient $s$ -zero-divisor cuplength of the real projective space $P^{n}$ has been studied extensively, see [5] and [9]. We will see in Proposition 4.2 below how to use these results to obtain information on $\operatorname{TC}_{s}(M)$ when $\pi_{1}(M)={\mathbb{Z}}_{2}$ and $\operatorname{\mathrm{cat}}(M)=\dim M$ . We first recall some results from [9].

For an integer $n>0$ with binary expansion $\cdots d_{2}d_{1}d_{0}$ , i.e., $n=\sum_{i\geq 0}d_{i}2^{i}$ , with digits $d_{i}\in\{0,1\}$ , let

•

$\nu(n)$ denote the exponent in the maximal 2-power dividing $n$ , i.e., $\nu(n)$ is the minimal $i$ with $d_{i}=1$ ;
•

$S(n):=\{i>0:\cdots d_{i+1}d_{i}d_{i-1}\cdots=\cdots 011\cdots\}$ , the set of binary positions $i$ starting (from left to right) a block of consecutive 1’s of length at least 2;
•

$Z_{i}(n):=\sum_{j=0}^{i}(1-d_{j})2^{j}$ , the complement of the binary expansion of $n$ mod $2^{i+1}$ .

Building on [5], Davis [9] proves that, for $s\geq 3$ , the ${\mathbb{Z}}_{2}$ -coefficient $s$ -zero-divisors cuplength of the $n$ -dimensional real projective space $P^{n}$ is given by

\operatorname{\mathrm{zcl}}_{s}(P^{n})=sn-m_{n,s}

where $m_{n,s}=\max\{2^{\nu(n+1)}-1,2^{i+1}-1-sZ_{i}(n):i\in S(n)\}$ . In particular, for even $n$ (so that $P^{n}$ is non-orientable), $P^{n}$ has maximal possible $\operatorname{TC}_{s}(P^{n})$ , that is, $\operatorname{TC}_{s}(P^{n})=sn$ , whenever

(4.1)

s\geq\max\left\{3,\left\lceil\frac{2^{i+1}-1}{Z_{i}(n)}\right\rceil:i\in S(n)\right\}.

We specialize two cases of the condition (4.1):

Example 4.1.

(a)

When $n$ is even and its binary expansion has no blocks of two or more consecutive 1’s, we have $S(n)=\emptyset$ and the inequality (4.1) reduces to $s\geq 3$ . Note that this condition for the maximality of $\operatorname{TC}_{s}(P^{n})$ is sharp, since $\operatorname{TC}_{2}(P^{n})<2n$ ([15], [7, Theorem 1]).
(b)

For $n=2^{r+1}-2$ , we have $S(n)=\{r\}$ , $m_{n,s}=2^{r+1}-1-s$ and (4.1) becomes $s\geq 2^{r+1}-1$ . Note that, when $s=n=2^{r+1}-2$ , we have $\operatorname{TC}_{n}(P^{n})\in\{sn,sn-1\}$ . We will see in Section 5 that $\operatorname{TC}_{n}(P^{n})=sn-1$ so that the condition $s\geq 2^{r+1}-1$ for the maximality of $\operatorname{TC}_{s}(P^{n})$ is sharp again.

Thanks to the following result, Davis’ computations of $\operatorname{\mathrm{zcl}}_{s}(P^{n})$ have impact on more general manifolds.

Proposition 4.2.

Let $M$ be an $n$ -dimensional connected closed manifold with $\operatorname{\mathrm{cat}}(M)=n$ and $\pi_{1}(M)={\mathbb{Z}}_{2}$ . Then, for any $s\geq 2$ , $\operatorname{TC}_{s}(M)\geq\operatorname{\mathrm{zcl}}_{s}(P^{n};{\mathbb{Z}}_{2})$ .

Proof.

Let $\gamma:M\to P^{\infty}=B{\mathbb{Z}}_{2}$ be a classifying map and let $x\in H_{1}(B{\mathbb{Z}}_{2};{\mathbb{Z}}_{2})={\mathbb{Z}}_{2}$ be the generator. For dimensional reasons, $\gamma$ factors as $M\stackrel{{\scriptstyle c}}{{\to}}P^{n}\hookrightarrow P^{\infty}$ . Let $x_{M}=\gamma^{*}(x)=c^{*}(x)\in H^{1}(M;{\mathbb{Z}}_{2})$ . The hypothesis that $\operatorname{\mathrm{cat}}(M)=n$ implies that $x_{M}\neq 0$ , see [3]. Consequently $c^{*}:H^{*}(P^{n};{\mathbb{Z}}_{2})\to H^{*}(M;{\mathbb{Z}}_{2})$ as well as $(c^{s})^{*}:H^{*}((P^{n})^{s};{\mathbb{Z}}_{2})\to H^{*}(M^{s};{\mathbb{Z}}_{2})$ are monomorphisms. As the image by $(c^{s})^{*}$ of a $s$ -zero-divisor of $P^{n}$ over ${\mathbb{Z}}_{2}$ gives rise to a $s$ -zero-divisor of $M$ over ${\mathbb{Z}}_{2}$ , we can conclude that $\operatorname{\mathrm{zcl}}_{s}(M;{\mathbb{Z}}_{2})\geq\operatorname{\mathrm{zcl}}_{s}(P^{n};{\mathbb{Z}}_{2})$ and the result follows. ∎

From Example 4.1 and the discussion above, we directly obtain:

Corollary 4.3.

Let $M$ be an $n$ -dimensional connected closed manifold with $\operatorname{\mathrm{cat}}(M)=n$ and $\pi_{1}(M)={\mathbb{Z}}_{2}$ . If $n$ is even and $s$ satisfies the inequality (4.1), then $\operatorname{TC}_{s}(M)=sn$ . In particular:

(a)

If $n$ is even and its binary expansion of $n$ contains no consecutive digits equal to $1$ , then $\operatorname{TC}_{s}(M)=sn$ for any $s\geq 3$ .
(b)

If $n=2^{r+1}-2$ , then $\operatorname{TC}_{s}(M)=sn$ for any $s\geq 2^{r+1}-1$ .

By using Davis’ computations in combination with Proposition 3.3, we can also state:

Corollary 4.4.

Let $M$ be an orientable connected closed manifold satisfying the conditions $n=\dim(M)=\operatorname{\mathrm{cat}}(M)$ and $\pi_{1}(M)={\mathbb{Z}}_{2}$ . If $n\equiv 1\bmod 4$ , then $\operatorname{TC}_{s}(M)=sn-1$ for $s$ satisfying the inequality (4.1).

Proof.

In this case $m_{n,s}=1$ so that $\operatorname{TC}_{s}(M)\geq sn-1$ for $s$ satisfying the inequality (4.1). The other direction follows from Proposition 3.3. ∎

Remark 4.5.

Observe that the orientability hypothesis, together with the condition $\dim(M)=\operatorname{\mathrm{cat}}(M)$ , implies that $n$ is odd. Indeed, if $n$ were even, the image of the fundamental class of $M$ by $\gamma_{*}:H_{n}(M)\to H_{n}(B{\mathbb{Z}}_{2})=0$ would vanish. But this fact would force, by Poincaré duality, the $n$ th power of the Berstein-Schwarz class of $M$ to vanish, contradicting the equality $\dim(M)=\operatorname{\mathrm{cat}}(M)$ .

4.2. Odd dimensional manifolds with $\pi_{1}(M)={\mathbb{Z}}_{p}$ and $\operatorname{\mathrm{cat}}(M)=\dim M$

Throughout this section we consider a prime $p\geq 3$ . Recall that the classifying space $B{\mathbb{Z}}_{p}$ can be identified with the infinite dimensional lens space $L_{p}^{\infty}$ . In order to have an analogue of Proposition 4.2, we first note the following result:

Lemma 4.6.

Let $M$ be an orientable connected closed $(2n+1)$ -manifold satisfying the conditions $\operatorname{\mathrm{cat}}(M)=2n+1$ and $\pi_{1}(M)={\mathbb{Z}}_{p}$ . If $\gamma\colon M\to B{\mathbb{Z}}_{p}$ is a classifying map, then $\gamma_{\ast}([M])\in H_{2n+1}({\mathbb{Z}}_{p};{\mathbb{Z}}_{p})$ is non-zero.

Proof.

Considering the Berstein–Schwarz class $\mathfrak{b}_{M}\in H^{1}(M;I(\pi))$ , we have $\operatorname{\mathrm{cat}}(M)=\dim(M)=2n+1$ if and only if $\mathfrak{b}_{M}^{2n+1}\neq 0$ . By Poincaré duality, the second statement is equivalent to $\mathfrak{b}_{M}^{2n+1}([M])\neq 0$ . Taking cap products we obtain

[M]\cap\mathfrak{b}_{M}^{2n+1}=[M]\cap\gamma^{\ast}(\mathfrak{b}^{2n+1}_{{\mathbb{Z}}_{p}})\neq 0.

Since $\gamma$ induces an isomorphism at the level of fundamental groups, naturality of the cap-products yields

\gamma_{\ast}([M])\cap\mathfrak{b}^{2n+1}_{{\mathbb{Z}}_{p}}\neq 0.

Hence, $\gamma_{\ast}([M])\neq 0$ in $H_{2n+1}({\mathbb{Z}}_{p};{\mathbb{Z}})={\mathbb{Z}}_{p}$ . Since $H_{2n+1}({\mathbb{Z}}_{p};{\mathbb{Z}})=H_{2n+1}({\mathbb{Z}}_{p};{\mathbb{Z}}_{p})={\mathbb{Z}}_{p}$ we can conclude that $\gamma_{\ast}([M])\neq 0$ in $H_{2n+1}({\mathbb{Z}}_{p};{\mathbb{Z}}_{p})$ . ∎

Theorem 4.7.

Let $M$ be a closed orientable $(2n+1)$ -manifold with $\operatorname{\mathrm{cat}}(M)=2n+1$ and $\pi_{1}(M)={\mathbb{Z}}_{p}$ where $p\geq 3$ is a prime. Then, $\operatorname{TC}_{s}(M)\geq\operatorname{\operatorname{\mathrm{zcl}}^{\underline{w}}_{s}}(M;{\mathbb{Z}}_{p})\geq\operatorname{\operatorname{\mathrm{zcl}}^{\underline{w}}_{s}}(L_{p}^{2n+1};{\mathbb{Z}}_{p})$ .

Proof.

Let $\gamma\colon M\to B{\mathbb{Z}}_{p}$ be a classifying map. We have a commutative triangle

(4.2)

where the inclusion is simply the $(2n+1)$ -skeleton of the infinite dimensional lens space $L^{\infty}_{p}\simeq B{\mathbb{Z}}_{p}$ . Since $\operatorname{\mathrm{cat}}(M)=2n+1$ , we know by the previous lemma that $\gamma_{\ast}([M])\neq 0$ in $H_{2n+1}({\mathbb{Z}}_{p};{\mathbb{Z}}_{p})$ . Consequently $\phi_{\ast}([M])\neq 0$ in $H_{2n+1}(L^{2n+1};{\mathbb{Z}}_{p})$ . Recall that

H^{\ast}(L^{2n+1}_{p};{\mathbb{Z}}_{p})={\mathbb{Z}}_{p}[x,y]/(y^{n+1},x^{2})

where $|x|=1$ , $|y|=2$ . In particular, $H^{2n+1}(L^{2n+1}_{p};{\mathbb{Z}}_{p})={\mathbb{Z}}_{p}xy^{n}$ . We first check that $\phi^{*}:H^{*}(L_{p}^{2n+1};{\mathbb{Z}}_{p})\to H^{*}(M;{\mathbb{Z}}_{p})$ is a monomorphism. It suffices to show that $\phi^{\ast}(x)$ and $\phi^{\ast}(y),\dots,\phi^{\ast}(y^{n})$ are non-trivial in $H^{\ast}(M;{\mathbb{Z}}_{p})$ . Consider the following cap-product diagram:

Both horizontal morphisms are isomorphisms by Poincaré duality. As $\phi_{\ast}([M])\neq 0$ in $H_{2n+1}(L^{2n+1};{\mathbb{Z}}_{p})$ , $\phi_{*}([M]$ is not divisible by $p$ in $H_{2n+1}(L^{2n+1};{\mathbb{Z}})={\mathbb{Z}}$ . Consequently, $\phi_{*}([M])\cap xy^{n}\neq 0$ and using the diagram we have that $[M]\cap\phi^{\ast}(xy^{n})\neq 0$ . We thus have $\phi^{*}(xy^{n})\neq 0$ in $H^{*}(M;{\mathbb{Z}}_{p})$ and hence $\phi^{\ast}(x)$ and $\phi^{\ast}(y^{i})$ for $i=1,\dots,n$ are non-zero. Therefore $\phi^{*}$ is a monomorphism. By Künneth formula, we obtain that $(\phi^{s})^{*}$ is also a monomorphism. Let $u\neq 0\in\mathbf{Z}_{s}(L_{p}^{2n+1};{\mathbb{Z}}_{p})$ . Then $(\phi^{s})^{*}(u)\in\mathbf{Z}_{s}(M;{\mathbb{Z}}_{p})$ and $(\phi^{s})^{*}(u)\neq 0$ . Consequently $\operatorname{\mathrm{wgt}}_{s}((\phi^{s})^{*}(u))\geq\operatorname{\mathrm{wgt}}_{s}(u)$ and the results follows by definition of $\operatorname{\operatorname{\mathrm{zcl}}^{\underline{w}}_{s}}$ . ∎

There are extensive computations of $\operatorname{\operatorname{\mathrm{zcl}}^{\underline{w}}_{s}}(L^{n}_{p};{\mathbb{Z}}_{p})$ , see [14, §5] and [8, Section 5]. By using the previous theorem, we can use the information coming from these computations for a larger class of manifolds.

Corollary 4.8.

Let $s\geq 2$ and let $M$ be a closed orientable $(2n+1)$ -manifold with $\operatorname{\mathrm{cat}}(M)=2n+1$ and $\pi_{1}(M)={\mathbb{Z}}_{p}$ where $p\geq 3$ is prime. Then,

\operatorname{TC}_{s}(M)\geq\begin{cases}s\cdot(\ell+\ell^{\prime}+1)-1&\text{if $s$ is even};\\ (s-1)\cdot(\ell+\ell^{\prime})+s+2n-1&\text{if $s$ is odd}\end{cases}

where $0\leq\ell,\ell^{\prime}\leq n$ are any integers such that $m$ does not divide $\binom{\ell+\ell^{\prime}}{\ell}^{\lfloor s/2\rfloor}$ .

Proof.

Here we apply the computation [8, Theorem 5.2] and Theorem 4.7. ∎

In many situations, for example if $s$ is much larger than the dimension of $M$ , we obtain an exact computation.

Corollary 4.9.

Let $s\geq 2$ and let $M$ be a closed orientable $(2n+1)$ -manifold with $\operatorname{\mathrm{cat}}(M)=2n+1$ and $\pi_{1}(M)={\mathbb{Z}}_{p}$ where $p\geq 3$ is a prime. If $s$ does not divide $\binom{2n}{n}^{\lfloor s/2\rfloor}$ , then

\operatorname{TC}_{s}(M)=s(2n+1)-1.

Proof.

The lower bound follows from Corollary 4.8 (see also [8, Theorem 5.3]) and the upper bound is Proposition 3.3. ∎

5. Some calculations in the non-orientable case

We now address the (non-)maximality of $\operatorname{TC}_{s}(M)$ for non-orientable manifolds having $\pi_{1}(M)={\mathbb{Z}}_{2}$ . The case $s=2$ is well understood ([7, Theorem 1]), so we assume $s\geq 3$ from now on. For such cases the non-maximality of $\operatorname{TC}_{s}(M)$ demands further restrictions on $s$ . The aim of this section is to establish the following result.

Theorem 5.1.

Let $M$ be a non-orientable $n$ -dimensional manifold with $\pi_{1}(M)=\mathbb{Z}_{2}$ and $n=2^{r+1}-2$ . Then, for any even $s$ no greater than $2^{r+1}-2$ , we have $\operatorname{TC}_{s}(M)<sn$ .

By Corollary 4.3(b), we know that, for $n=2^{r+1}-2$ , $\operatorname{TC}_{s}(M)=sn$ for $s\geq 2^{r+1}-1$ , so that in this case the upper limiting restriction on $s$ in Theorem 5.1 is in fact sharp and we have:

Corollary 5.2.

If the manifold $M$ in Theorem 5.1 has $\operatorname{\mathrm{cat}}{M}=n$ , then $\operatorname{TC}_{s}(M)=sn-1$ for $s=2^{r+1}-2$ and $\operatorname{TC}_{s}(M)=sn$ for $s\geq 2^{r+1}-1$ .

Proof.

The equality $\operatorname{TC}_{s}(M)=sn-1$ for $s=2^{r+1}-2$ follows from $m_{2^{r+1}-2,2^{r+1}-2}=1$ (see Example 4.1(b)), Proposition 4.2 and Theorem 5.1. ∎

Corollary 5.2 should be compared to the fact that $\operatorname{TC}_{s}(P^{2^{r}})$ is maximal for $s\geq 3$ , but $\operatorname{TC}_{2}(P^{2^{r}})=\operatorname{Imm}(P^{2^{r}})=2^{r+1}-1$ ([15]). Worth noting is the fact that the case $M=P^{6}$ in Corollary 5.2 (with $r=2$ ) upgrades the observation in [5, (7.4)] that $\delta_{6}(6)\leq 1$ to an equality, giving evidence for what would be regular behavior of the higher topological complexity of projective spaces $P^{m}$ with $m=2^{a}+2^{a+1}$ .

Suitable analogues of Theorem 5.1 should hold for more general values of $n$ , but the complexity of calculations seems to be a major obstacle towards obtaining corresponding proofs.

We now start working towards the proof of Theorem 5.1. From now on $\pi:={\mathbb{Z}}_{2}$ and $s=2\sigma$ with $1\leq\sigma\leq 2^{r}-1=n/2$ . Set $\widehat{{\mathbb{Z}}}:=\widetilde{{\mathbb{Z}}}\otimes({\mathbb{Z}}\otimes\widetilde{{\mathbb{Z}}})^{\sigma-1}$ , the ${\mathbb{Z}}[\pi^{s-1}]$ -module of Remark 2.5. By Proposition 2.3, it suffices to establish the triviality of

(5.1)

^{s}\chi_{*}(\mathfrak{m}^{\times s})\in H_{sn}(\pi^{s-1};\widehat{\mathbb{Z}})

where

(5.2)

^{s}\chi_{*}\colon H_{*}(\pi^{s};\widetilde{\mathbb{Z}}^{\otimes s})\to H_{*}(\pi^{s-1};\widehat{\mathbb{Z}}).

As in §3, our starting point is the free ${\mathbb{Z}}[{\mathbb{Z}}_{q}]$ -resolution (3.2) of ${\mathbb{Z}}$ with $q=2$ . Recall that $[k]$ denotes the generator of degree $k$ . In addition to the chain complex $C_{\bullet}(\pi)$ of (3.3), we will also need the complex $\widetilde{C}_{\bullet}(\pi)$ ,

(5.3)

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obtained by tensoring (3.2) with $\widetilde{{\mathbb{Z}}}$ over $\pi$ . Abusing notation, we continue using $[k]$ for the generators of both $C_{\bullet}(\pi)$ and $\widetilde{C}_{\bullet}(\pi)$ .

The homology groups in (5.2) can be computed from the complexes $\widetilde{C}_{\bullet}(\pi)^{\otimes s}$ and

(5.4)

\mathcal{D}_{\bullet}:=\widetilde{C}_{\bullet}(\pi)\otimes({C}_{\bullet}(\pi)\otimes\widetilde{C}_{\bullet}(\pi))^{\sigma-1}.

In both cases, we will use the shorthand $[i_{1},\ldots,i_{\ell}]$ for a tensor product $[i_{1}]\otimes\cdots\otimes[i_{\ell}]$ . The Künneth formula and the fact that the homology of $\widetilde{C}_{\bullet}(\pi)$ is 2-torsion (in all degrees) gives:

Lemma 5.3.

The element ${}^{s}\chi_{*}(\mathfrak{m}^{\times s})$ in (5.1) is torsion. Indeed, both groups in (5.2) are torsion.

Let $\mathcal{H}_{\bullet}$ denote the quotient of $\mathcal{D}_{\bullet}$ resulting from killing all boundaries, and consider the obvious monomorphism $\iota:H_{\bullet}(\pi^{s-1};\widehat{\mathbb{Z}})\hookrightarrow\mathcal{H}_{\bullet}$ . The triviality of the element in (5.1) follows from Lemma 5.3 and the following key result, whose proof is addressed in the rest of the section through a direct analysis of (5.1) and (5.2).

Proposition 5.4.

The class $\iota(^{s}\chi_{*}(\mathfrak{m}^{\times s}))$ is an element of the torsion-free (graded) subgroup of $\mathcal{H}_{\bullet}$ .

In computing the homology groups in (5.2) using the complexes $\widetilde{C}_{\bullet}(\pi)^{\otimes s}$ and $\mathcal{D}_{\bullet}$ , we will use $\widetilde{\pi}$ for a factor where $\widetilde{C}_{\bullet}(\pi)$ is meant to be taken, reserving the notation $\pi$ for factors where $C_{\bullet}(\pi)$ is meant to be taken. For instance, the diagonal morphism $\Delta:\pi\to\pi\times\pi$ and the group-multiplication morphism $\mu:\pi\times\pi\to\pi$ extend to morphisms $\Delta:\widetilde{\pi}\to\widetilde{\pi}\times\pi$ , $\Delta:\widetilde{\pi}\to\pi\times\widetilde{\pi}$ and $\mu:\widetilde{\pi}\times\widetilde{\pi}\to\widetilde{\pi}$ that are compatible with the implied module structures. In these terms, since $\pi={\mathbb{Z}}_{2}=\langle v\mid v^{2}=1\rangle$ in the present case, the inversion morphism plays no role and the map ${}^{s}\chi$ factors as

(5.5)		$\displaystyle\widetilde{\pi}^{s}\stackrel{{\scriptstyle 1\times(\Delta\times\Delta)^{\sigma-1}\times 1}}{{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow}}{}$	$\displaystyle\widetilde{\pi}\times(\widetilde{\pi}\times\pi\times\pi\times\widetilde{\pi})^{\sigma-1}\times\widetilde{\pi}$
(5.5)		$\displaystyle=$	$\displaystyle(\widetilde{\pi}\times\widetilde{\pi})\times(\pi\times\pi\times\widetilde{\pi}\times\widetilde{\pi})^{\sigma-1}\stackrel{{\scriptstyle\mu\times(\mu\times\mu)^{\sigma-1}}}{{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow}}\widetilde{\pi}\times(\pi\times\widetilde{\pi})^{\sigma-1}.$

Recalling that $B_{i,j}$ denotes the binomial coefficient $\binom{\,i+j\,}{i}$ , the formulæ of Lemma 3.2 $(b)$ written with the shorthand in use in this section read

(5.6)	$\displaystyle{}[2i,2j]$	$\displaystyle\mapsto B_{i,j}\,[2(i+j)];$
	$\displaystyle{}[2i,2j+1]$	$\displaystyle\mapsto B_{i,j}\,[2(i+j)+1];$
	$\displaystyle{}[2i+1,2j]$	$\displaystyle\mapsto B_{i,j}\,[2(i+j)+1];$
	$\displaystyle{}[2i+1,2j+1]$	$\displaystyle\mapsto 0.$

We note also that, since $\pi={\mathbb{Z}}_{2}$ , the formula of Lemma 3.2(a) giving $\Delta$ on generators at the level of resolutions can be written

(5.7)

[k]\to\sum_{p+q=k}[p]\otimes v^{\mathrm{odd}(p)}\hskip-1.70717pt\cdot[q],

where $v$ generates $\pi$ and ${\mathrm{odd}(p)}=1$ if $p$ is odd and $0$ otherwise.

The class $\mathfrak{m}\in H_{*}(\pi;\widetilde{\mathbb{Z}})$ is either trivial or, else, represented by the cycle $[n]$ in (5.3) —recall $n$ is even. For the purposes of proving Theorem 5.1, we may safely assume the latter possibility. Then, $\mathfrak{m}^{\times s}$ is represented in $\widetilde{C}_{\bullet}(\pi)^{\otimes s}$ by the corresponding tensor product $[n,n,\ldots,n]$ . We chase the latter element under the first map of the composite (5.5) to get, in view of (5.7),

\displaystyle[\widetilde{n},\widetilde{n},\ldots,\widetilde{n}]\mapsto[\widetilde{n}]\otimes\left(\hskip-2.84526pt\left(\,\sum_{p+q=n}[\widetilde{p}]\otimes v^{\text{odd}(p)}\cdot[q]\right)\otimes\left(\,\sum_{p+q=n}[p]\otimes v^{\text{odd}(p)}\cdot[\widetilde{q}]\right)\hskip-2.84526pt\right)^{\hskip-2.84526pt\otimes\sigma-1}\hskip-5.69054pt\otimes[\widetilde{n}].

Note that in the latter expression we are extending in the obvious way the convention above regarding the use of $\pi$ and $\widetilde{\pi}$ . Then, after tensoring with the needed coefficients (thus dropping the $\sim$ indicators), this becomes

	$\displaystyle[{n},{n},\ldots,{n}]\mapsto$	$\displaystyle[{n}]\otimes\left(\hskip-2.84526pt\left(\,\sum_{p+q=n}[p,q]\right)\otimes\left(\,\sum_{p+q=n}(-1)^{p}[p,q]\right)\hskip-2.84526pt\right)^{\hskip-2.84526pt\otimes\sigma-1}\hskip-5.69054pt\otimes[{n}]$
	$\displaystyle=$	$\displaystyle\sum_{\genfrac{}{}{0.0pt}{}{p_{i}+q_{i}=n}{1\leq i\leq s-2}}(-1)^{{}^{\sum_{j=1}^{\sigma-1}p_{2j}}}[n,p_{1},q_{1},p_{2},q_{2},\cdots,p_{s-2},q_{s-2},n].$

Since $n$ is even, the parity of each $p_{i}$ agrees with the one of the corresponding $q_{i}$ , so the last expression can be rewritten as

\displaystyle\sum(-1)^{{}^{\sum_{j=1}^{\sigma-1}\delta_{2j}}}[n,2p_{1}+\delta_{1},2q_{1}+\delta_{1},2p_{2}+\delta_{2},2q_{2}+\delta_{2},\cdots\hskip-0.85358pt,2p_{s-2}+\delta_{s-2},2q_{s-2}+\delta_{s-2},n],

where the sum now runs over $1\leq i\leq s-2$ , $\delta_{i}\in\{0,1\}$ and $p_{i}+q_{i}=n/2-\delta_{i}$ . Using the formulae (5.6), we finally obtain the image of $[n,\ldots,n]$ under the entire composition in (5.5). This image may be expressed as

(5.8)

[n,\ldots,n]\mapsto\sum(-1)^{{}^{\sum_{j=1}^{\sigma-1}\delta_{2j}}}B_{\frac{n}{2},p_{1}}B_{q_{1},p_{2}}\cdots B_{q_{s-3},p_{s-2}}B_{q_{s-2},\frac{n}{2}}\cdot\mathbf{G},

where

\mathbf{G}=[n+2p_{1}+\delta_{1},2(q_{1}+p_{2})+\delta_{1}+\delta_{2},\ldots,2(q_{s-3}+p_{s-2})+\delta_{s-3}+\delta_{s-2},n+2q_{s-2}+\delta_{s-2}]

and the sum runs over the same indices as above, except now that no two consecutive $\delta_{j}$ and $\delta_{j+1}$ can simultaneously equal 1, in view of (5.6). In what follows we set $m={n}/{2}$ .

Having described a cycle representing the obstruction in (5.1), we next spell out the complex (5.4) where it lies. Degreewise, $\mathcal{D}_{\bullet}$ is $\mathbb{Z}$ -free with basis given by elements $[u_{1},v_{1},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}]$ for non-negative integers $u_{i}$ and $v_{i}$ , and with differential

	$\displaystyle\partial[u_{1},$	$\displaystyle v_{1},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}]=$
	$\displaystyle=$	$\displaystyle-2\hskip 2.27621pt\text{odd}\hskip 1.13809pt(u_{1})[u_{1}-1,v_{1},u_{2},v_{2},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}]$
		$\displaystyle\quad+(-1)^{u_{1}}\hskip 0.85358pt2\hskip 2.27621pt\text{even}\hskip 1.13809pt(v_{1})[u_{1},v_{1}-1,u_{2},v_{2},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}]$
		$\displaystyle\quad-(-1)^{u_{1}+v_{1}}\hskip 0.85358pt2\hskip 2.27621pt\text{odd}\hskip 1.13809pt(u_{2})[u_{1},v_{1},u_{2}-1,v_{2},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}]$
(5.9)			$\displaystyle\quad+(-1)^{u_{1}+v_{1}+u_{2}}\hskip 0.85358pt2\hskip 2.27621pt\text{even}\hskip 1.13809pt(v_{2})[u_{1},v_{1},u_{2},v_{2}-1,\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}]$
		$\displaystyle\quad\pm\cdots$
		$\displaystyle\quad-(-1)^{u_{1}+v_{1}+\cdots+u_{\sigma-2}+v_{\sigma-2}}\hskip 0.85358pt2\hskip 2.27621pt\text{odd}\hskip 1.13809pt(u_{\sigma-1})[u_{1},v_{1},u_{2},v_{2},\ldots,u_{\sigma-1}-1,v_{\sigma-1},u_{\sigma}]$
		$\displaystyle\quad+(-1)^{u_{1}+v_{1}+\cdots+u_{\sigma-2}+v_{\sigma-2}+u_{\sigma-1}}\hskip 0.85358pt2\hskip 2.27621pt\text{even}\hskip 1.13809pt(v_{\sigma-1})[u_{1},v_{1},u_{2},v_{2},\ldots,u_{\sigma-1},v_{\sigma-1}-1,u_{\sigma}]$
		$\displaystyle\quad-(-1)^{u_{1}+v_{1}+\cdots+u_{\sigma-1}+v_{\sigma-1}}\hskip 0.85358pt2\hskip 2.27621pt\text{odd}\hskip 1.13809pt(u_{\sigma})[u_{1},v_{1},u_{2},v_{2},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}-1],$

where a basis element with a negative entry is meant to be interpreted as zero.

All the information needed to prove Proposition 5.4 is of course contained in (5.8) and (5.9). The following constructions are meant to organize a proof argument.

Definition 5.5.

For a positive integer $k$ , let $p(k)$ denote the set of binary positions where the binary expansion of $k$ has digit 1. For instance, $p(5=4+1)=\{0,2\}$ and $p(42=32+8+2)=\{1,3,5\}$ .

A standard well known fact is:

Lemma 5.6.

A binomial coefficient $B_{a,b}$ is even if and only if $p(a)\cap p(b)\neq\varnothing$ .

The following result is the only place where the special assumptions in Theorem 5.1 (i.e., $s=2\sigma\leq n=2m=2^{r+1}-2$ with $r\geq 1$ ) are needed.

Proposition 5.7.

Any coefficient

(5.10)

B_{m,p_{1}}B_{q_{1},p_{2}}\cdots B_{q_{s-3},p_{s-2}}B_{q_{s-2},m}

in (5.8) with $p_{1}+q_{1}=m$ (i.e., with $\delta_{1}=0$ ) is even.

Proof.

We use without further notice the fact coming from Lemma 5.6 that any binomial coefficient $B_{m-j,i}$ is even whenever $0\leq j<i\leq m$ . Recall $m=2^{r}-1$ and

(5.11)

s=2\sigma\leq 2^{r+1}-2

with $r\geq 1$ . Assume for a contradiction that some coefficient (5.10) is odd (i.e., that all of its binomial-coefficient factors are odd) and has $\delta_{1}=0$ . Recall the forced conditions

(1)

$p_{i}+q_{i}=m-\delta_{i}$ with $p_{i},q_{i}\geq 0$ and $\delta_{i}\in\{0,1\}$ ;
(2)

$1\leq i<s-2$ and $\delta_{i}=1$ implies $\delta_{i+1}=0$ ,

for $1\leq i\leq s-2$ . Let $2\leq i_{1}<i_{2}<\cdots<i_{k}\leq s-2$ be all the indices $j$ (if any) with $\delta_{j}=1$ . Note that

(5.12)

0\leq k\leq\sigma-1,

in view of (2).

The coefficient $B_{m,p_{1}}$ is odd by hypothesis, so $p_{1}=0$ and $q_{1}=m$ —the latter equality holds in view of (1) since $\delta_{1}=0$ . Actually, the same argument can be used iteratively for $1\leq j<i_{1}$ (so $\delta_{j}=0$ ) with the binomial coefficients $B_{q_{j-1},p_{j}}$ (e.g. $q_{0}:=m$ ) to show that

p_{j}=0

and

q_{j}=m

Next, since $m=q_{i_{1}-1}$ , $B_{q_{i_{1}-1},p_{i_{1}}}$ is odd and $\delta_{i_{1}}=1$ , we get

p_{i_{1}}=0

and

q_{i_{1}}=m-1

and now the process repeats with a slight adjustment. For starters, $q_{i_{1}}=m-1$ , $B_{q_{i_{1}},p_{i_{1}+1}}$ is odd and $\delta_{i_{1}+1}=0$ is forced by (2), so that $p_{i_{1}+1}\leq 1$ and $q_{i_{1}+1}\geq m-1$ . We then iterate the latter argument: For $i_{1}<j<i_{2}$ , the assumption $\delta_{j}=0$ and the fact that $B_{q_{j-1},p_{j}}$ is odd with $q_{j-1}\geq m-1$ yield

p_{j}\leq 1

and

q_{j}\geq m-1

Of course, the last two inequalities now hold for all $1\leq j<i_{2}$ . The next round of iterations start with the fact that $B_{q_{i_{2}-1},p_{i_{2}}}$ is odd with $q_{i_{2}-1}\geq m-1$ and $\delta_{i_{2}}=1$ , to get

p_{i_{2}}\leq 1

and

q_{i_{2}}\geq m-2

and the process has a corresponding new obvious adjustment to yield

p_{j}\leq 2

and

q_{j}\geq m-2

for $1\leq j<i_{3}$ , whereas

p_{i_{3}}\leq 2

and

q_{i_{3}}\geq m-3

Just before the last adjustment we get

p_{j}\leq k-1

and

q_{j}\geq m-k+1

for $1\leq j<i_{k}$ , whereas

p_{i_{k}}\leq k-1

and

q_{i_{k}}\geq m-k

However, after this point the conditions $p_{j}\leq k$ and $q_{j}\geq m-k$ are kept for all $j\leq s-2$ . In particular, $q_{s-2}\geq m-k=2^{r}-1-k\geq 1$ , in view of (5.11) and (5.12). But then the final factor $B_{q_{s-2},m}$ of (5.10) is even, a contradiction. ∎

Proof of Proposition 5.4.

The right hand-side of (5.9) yields the defining relations in $\mathcal{H}_{\bullet}$ . Namely, for each tuple $(u_{1},v_{1},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma})$ of non-negative integers there is a defining relation

(5.13)

0=U_{1}+V_{1}+U_{2}+V_{2}+\cdots+U_{\sigma-1}+V_{\sigma-1}+U_{\sigma}

where

	$\displaystyle U_{i}:=$	$\displaystyle(-1)^{p_{i}}\,2\,\text{odd}(u_{i})\,[u_{1},v_{1},\ldots,u_{i-1},v_{i-1},u_{i}-1,v_{i},u_{i+1},v_{i+1},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}],$
	$\displaystyle V_{i}:=$	$\displaystyle(-1)^{q_{i}}\,2\,\text{even}(v_{i})\,[u_{1},v_{1},\ldots,u_{i-1},v_{i-1},u_{i},v_{i}-1,u_{i+1},v_{i+1},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}],$
	$\displaystyle p_{i}:=$	$\displaystyle 1+\sum_{1\leq j<i}(u_{j}+v_{j})\text{ \ \ and \ \ }q_{i}:=\sum_{1\leq j<i}(u_{j}+v_{j})+u_{i}.$

The tuple $(u_{1},v_{1},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma})$ and the basis element $[u_{1},v_{1},\ldots,u_{\sigma-1},v_{\sigma-1},u_{\sigma}]$ of $\mathcal{D}_{\bullet}$ are said to be even (respectively, odd) when $u_{1}$ is even (respectively, odd). In the odd case, (5.13) gives a way to write the double of the class in $\mathcal{H}_{\bullet}$ represented by an even basis element as a linear combination of the doubles of classes represented by odd basis elements. On the other hand, in the even case $U_{1}=0$ and the right hand-side of (5.13) is a linear combination of doubles of classes represented by even basis elements. A straightforward computation¹¹1The calculation is formally identical to the standard verification that $\partial^{2}=0$ for the boundary morphism $\partial$ in the singular complex of a given space. Details are left as an exercise for the reader. shows that the latter linear combination vanishes directly in $\mathcal{D}_{\bullet}$ when (the double of) each even summand is replaced by the corresponding linear combination of odd basis elements. Thus, relations (5.13) coming from even tuples are irrelevant. Since each even basis element appears in a single relation (5.13) coming from an odd tuple, we see that the subgroup of $\mathcal{H}_{\bullet}$ spanned by the classes represented by odd basis elements is torsion free. The proof is complete in view of Proposition 5.7. ∎

Acknowledgments

The research of L. Vandembroucq was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Project UID/00013: Centro de Matemática da Universidade do Minho (CMAT/UM). Sam Hughes was supported by a Humboldt Research Fellowship at Universität Bonn.

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On the higher topological complexity of manifolds with abelian fundamental group

Abstract.

Introduction

Theorem A.

Theorem B.

Theorem C.

Notation and conventions

1. A TCs\operatorname{TC}_{s} canonical class

Theorem 1.1 ([12]).

Remark 1.2.

2. Abelian fundamental group

Proposition 2.1.

Proof.

Remark 2.2.

Proposition 2.3.

Proof.

Corollary 2.4.

Proof.

Remark 2.5.

3. Some calculations in the orientable case

Proposition 3.1.

Proof.

Lemma 3.2.

Proposition 3.3.

Proof.

Proposition 3.4.

Proof.

Limiting examples

Example 3.5.

4. Cohomological lower bounds

4.1. Manifolds with π1​(M)=ℤ2\pi_{1}(M)={\mathbb{Z}}_{2} and cat⁡(M)=dimM\operatorname{\mathrm{cat}}(M)=\dim M

Example 4.1.

Proposition 4.2.

Proof.

Corollary 4.3.

Corollary 4.4.

Proof.

Remark 4.5.

4.2. Odd dimensional manifolds with π1​(M)=ℤp\pi_{1}(M)={\mathbb{Z}}_{p} and cat⁡(M)=dimM\operatorname{\mathrm{cat}}(M)=\dim M

Lemma 4.6.

Proof.

Theorem 4.7.

Proof.

Corollary 4.8.

Proof.

Corollary 4.9.

Proof.

5. Some calculations in the non-orientable case

Theorem 5.1.

Corollary 5.2.

Proof.

Lemma 5.3.

Proposition 5.4.

Definition 5.5.

Lemma 5.6.

Proposition 5.7.

Proof.

Proof of Proposition 5.4.

Acknowledgments

References

1. A $\operatorname{TC}_{s}$ canonical class

4.1. Manifolds with $\pi_{1}(M)={\mathbb{Z}}_{2}$ and $\operatorname{\mathrm{cat}}(M)=\dim M$

4.2. Odd dimensional manifolds with $\pi_{1}(M)={\mathbb{Z}}_{p}$ and $\operatorname{\mathrm{cat}}(M)=\dim M$