2.1 Higher topological complexity of RAA groups

We use the notation set forth in § 1 regarding RAA groups. The higher topological complexity ${{\sf {TC}}}_r(H_\Gamma )$ was computed in [Reference González, Gutiérrez and Yuzvinsky20] in terms of the structure of cliques in $\Gamma$. The answer is spelled out in theorem 2.4 below in terms of the simplified expression in [Reference Farber and Oprea16].

The maximum sum in definition 2.3 giving the value of $z_r(\Gamma )$ might not be realized by cliques whose cardinality is maximal.

6.1 Open covers

We recall certain techniques involving open covers of spaces. These techniques go back at least to [Reference Ostrand25, Reference Schwarz29] and more recently have been used in [Reference Dranishnikov7, Reference Farber, Grant, Lupton and Oprea13, Reference Oprea and Strom24]. We shall use them in our study of the category and higher topological complexity of polyhedral products.

An open covering $U_0,\,\ldots ,\,U_{k+m}$ having the property of lemma 6.1, i.e. that any $(k+1)$ subsets cover $X$, is called a $(k+1)$-cover. We then have:

6.2 LS category of products

As a reference for our later use of open cover properties (proof of lemma 6.7), we give an easy proof of the standard product inequality for LS category:

Remark 6.6 The exact same proof as above applies to show

\[ {{\sf{cat}}}\left(\prod_{i=1}^{m} X_i\right) \leq \sum_{i=1}^{m} {{\sf{cat}}}(X_i). \]

Here, each open categorical cover is extended to one of the form $U_0,\, \ldots ,\, U_s$, where $s=\sum _{i=1}^{m} {{\sf {cat}}}(X_i)$.

6.3 LS category of polyhedral products

Let $\{(X_i,\,A_i)\}_{i=1}^{m}$ be a family of pairs of spaces, and $K$ be a simplicial complex on vertices $[m]:=\{1,\,2,\,\ldots ,\,m\}$. For a subset $\sigma \subseteq [m]$, let

\[ X^{\sigma} = \prod_{i=1}^{m} Y_i, \quad \textrm{where} \quad Y_i =\begin{cases} X_i & \textrm{if}\ i \in \sigma; \\ A_i & \textrm{if}\ i \not\in\sigma. \end{cases} \]

The polyhedral product $\mathcal {Z}(\{X_i,\,A_i\},\,K)$ is the union of all $X^{\sigma }$ with $\sigma$ running over the simplicesFootnote ² of $K$,

(6.1)\begin{equation} \mathcal{Z}(\{X_i,A_i\},K) = \cup_{{}{\sigma \in K}} X^{\sigma}. \end{equation}

We are only interested in the case where all $A_i = *$, and use the notation $\underline {X}^{K}=\mathcal {Z}(\{X_i,\,A_i\},\,K)$. If in addition all $X_i = X$, we simply write $X^{K}$. The connection to our earlier work, and motivation for the results given below is apparent once it is realized that any $K(H_\Gamma ,\,1)$ is the polyhedral product $(S^{1})^{K}$, where $K$ is the flag complex with $1$-skeleton given by $\Gamma$.

In [Reference Félix and Tanré18], the category of $X^{K}$ was computed to be ${{\sf {cat}}}(X)\cdot (1+ \mathrm {dim}(K))$ provided that $X$ is LS-logarithmic. Here, we shall generalize this result to polyhedral products $\underline {X}^{K}$. We start with the following result that is reminiscent of [Reference González, Gutiérrez and Yuzvinsky20, theorem 5.3].

Lemma 6.7 Let $\{(X_i,\,*)\}_{i=1}^{m}$ be a family of based spaces and $K$ be a simplicial complex on vertices $[m]$. Then

\[ {{\sf{cat}}}(\underline{X}^{K}) \leq \max_{[i_1,\ldots,i_n]}\left\{{{\sf{cat}}}(X_{i_1})+\cdots + {{\sf{cat}}}(X_{i_n})\rule{0mm}{4mm}\right\}, \]

where the maximum is taken over all Footnote ³ simplices $[i_1,\,\ldots ,\,i_n]$ of $K$.

Proof. Let ${}_iU_0,\,\ldots ,\,{}_iU_{k_i}$ denote an open cover of the space $X_i$ realizing ${{{\sf {cat}}}(X_i)=k_i}$. Fix homotopies ${}_iH_j$ contracting ${}_iU_j$ to a point in $X_i$. (These may be taken to be base pointed by [Reference Cornea, Lupton, Oprea and Tanré6, lemma 1.25].) Let $\sigma =[i_1,\,\ldots ,\,i_n]$ be a maximal simplex giving a ‘subproduct’

\[ X^{\sigma} = \prod_{j=1}^{n} X_{i_j} \subseteq \underline{X}^{K}. \]

Here, an element of $X^{\sigma }$ is thought of as element of $\underline {X}^{K}$ by completing with base points in the coordinates corresponding to vertices not in $\sigma$. By the proof of proposition 6.5 and remark 6.6, we see that $X^{\sigma }$ is covered by $W_0^{\sigma },\,\ldots ,\,W_s^{\sigma }$, where $s=\sum _{j=1}^{n} k_{i_j}$ and $W_j^{\sigma } = {}_{i_1}U_j \times \cdots \times {}_{i_n}U_j$ for extended covers ${}_{i_j}U_0,\, \ldots ,\, {}_{i_j}U_s$ for each space appearing in the product $X^{\sigma }$. Note that, by remark 6.2, the contracting homotopies for the extended covers are simply restrictions of the ${}_{i_j}H_r$. This is, in fact, the key point. If $\tau =[r_1,\,\ldots ,\,r_p]$ is another maximal simplex of $K$, then we obtain

\[ W_j^{\tau} = {}_{r_1}U_j \times \cdots \times {}_{r_p}U_j \]

for $j =0,\,\ldots ,\, z=\sum _{i=1}^{p} k_{r_i}$. Here, we use the fact that if $m >p$, then the extended open cover $U_0,\,\ldots ,\,U_m$ of lemma 6.1 is an extension of $U_0,\,\ldots ,\,U_p$. Now, if $s > z$, then for $s \geq j > z$, let $W_j^{\tau } = \varnothing$. Then, because the open covers and homotopies are fixed for each $X_i$, if $W_j^{\sigma } \cap W_j^{\tau } \not = \varnothing$, the homotopies and open subsets agree on the intersection. Hence, they can be pieced together to contract $W_j^{\sigma } \cup W_j^{\tau }$ to a point in $\underline {X}^{K}$. This can then be done for all maximal simplices, so we define $W_j^{K} = \cup _\sigma W_j^{\sigma }$, where $\sigma$ is maximal. Here, we have $j=0,\,\ldots ,\, \max _{\sigma =[i_1,\ldots ,i_n]}\{k_{i_1}+\cdots + k_{i_n}\}$, where the maximum is taken over maximal simplices. Notice that by taking unions, we only require the maximal simplex with the greatest number of open sets.

Proof of theorem 1.4. By lemma 6.7, we need only show

\[ \max_{\sigma=[i_1,\ldots,i_n]}\{{{\sf{cat}}}(X_{i_1})+\cdots + {{\sf{cat}}}(X_{i_n})\}\leq {{\sf{cat}}}(\underline{X}^{K}). \]

Let ${}{\sigma _0}$ be a simplex realizing the maximum. But $X^{{}{\sigma _0}} \subseteq \underline {X}^{K}$ is a subproduct with a retraction $\underline {X}^{K} \to X^{{}{\sigma _0}}$ obtained by taking all other coordinates to base points. Since retracts have smaller category, we are done by the logarithmicity hypothesis.

Example 6.8 We illustrate how theorem 1.4 can be used to make (4.1) explicit in particular cases. Let $\{\Gamma _i\}_{i=1}^{m}$ be a family of graphs whose sets of vertices are pairwise disjoint. The set of vertices of the graph join $\Gamma :=\Gamma _1\ast \cdots \ast \Gamma _m$ is the union of the vertices of all the graphs $\Gamma _i$ where, besides the adjacencies in the individual graphs, each vertex of $\Gamma _i$ is taken to be adjacent in $\Gamma$ to each vertex of $\Gamma _j$ whenever $i\neq j$. Therefore, cliques $C_i$ of $\Gamma _i$ ($1\leq i\leq m$) form a new clique $C_1\ast \cdots \ast C_m$ (of size $\sum _i |C_i|$) of $\Gamma$. Hence, the size of the largest clique in $\Gamma$ is the sum of the sizes of the largest cliques in the individual graphs,

(6.2)\begin{equation} c(\Gamma)=\sum_{i=1}^{m}c(\Gamma_i). \end{equation}

In terms of the RAA groups, we have disjoint subgroups so that elements in different subgroups commute; hence, $H_\Gamma =H_{\Gamma _1} \times \cdots \times H_{\Gamma _m}$, and (6.2) gives

\[ \textrm{{cd}}(H_{\Gamma})={\textrm{cd}}(H_{\Gamma_1} \times \cdots \times H_{\Gamma_m}) = \sum_{i=1}^{m} \textrm{{cd}}(H_{\Gamma_i}). \]

The point to stress is that, in terms of classifying spaces, the above discussion shows that any collection $\underline {H_\Gamma }:=\{(K(H_{\Gamma _i},\,1),\,\ast )\}_{i=1}^{m}$ is LS-logarithmic. Note that the associated polyhedral product $\underline {H_\Gamma }^{K}:=\mathcal {Z}(\{K(H_{\Gamma _i},\,1),\,\ast \},\,K)$ classifies an RAA group, so its category is given by (4.1). Theorem 1.4 then yields an explicit formula for ${{\sf {cat}}}(\hspace {.4mm}\underline {H_\Gamma }^{K})$ in terms of the simplicial complex $K$ and the cliques of the graphs $\Gamma _i$.

6.4 TC of polyhedral products

For reference purposes, we record the following easy consequence of theorem 2.1 and lemma 6.7:

Corollary 6.9 For an integer $r\geq 2$, a family $\{(X_i,\,*)\}_{i=1}^{m}$ of based spaces and a simplicial complex $K$ on $[m]$ vertices, we have

\[ {{\sf{TC}}}_{{}{r}}(\underline{X}^{K}) \leq {}{r}\left(\max_{[i_1,\ldots,i_n]}\,\left\{{{\sf{cat}}}(X_{i_1})+\cdots + {{\sf{cat}}}(X_{i_n})\rule{0mm}{4mm}\right\}\right), \]

where the maximum is taken over all simplices $[i_1,\,\ldots ,\,i_n]$ of $K$.

Proof of theorem 1.5. In view of corollary 6.9, it suffices to show

\[ \max_{\sigma=[i_1,\ldots,i_n]}\left\{\rule{0mm}{4mm}{{\sf{TC}}}_{{}{r}}(X_{i_1})+\cdots + {{\sf{TC}}}_{{}{r}}(X_{i_n})\right\}\leq {{\sf{TC}}}_{{}{r}}(\underline{X}^{K}). \]

Let $\sigma _{{}{0}}$ be a simplex realizing the maximum. As in the proof of theorem 1.4, $X^{\sigma _{{}{0}}}$ is a retract of $\underline {X}^{K}$, so ${{\sf {TC}}}_{{}{r}}(X^{\sigma _{{}{0}}}) \leq {{\sf {TC}}}_{{}{r}}(\underline {X}^{K})$, and we are done by the logarithmicity hypothesis.

For $r\geq 2$, we say that $X$ is ${{\sf {TC}}}_{{}{r}}$-logarithmic if ${{\sf {TC}}}_{{}{r}}(X^{s})=s\cdot {{\sf {TC}}}_{{}{r}}(X)$ for all $s \geq 2$.

Corollary 6.10 For a family $\{(X,\,*)\}_{i=1}^{m}$ of $m$-copies of the same based space $X$ and a simplicial complex $K$ on $[m]$ vertices, if ${{\sf {TC}}}_{{}{r}}(X)={}{r}\,{{\sf {cat}}}(X)$ and $X$ is both LS- and ${{\sf {TC}}}_{{}{r}}$-logarithmic for some $r\geq 2$, then

\[ {{\sf{TC}}}_{{}{r}}(X^{K})={{\sf{TC}}}_{{}{r}}(X)\cdot(1+\mathrm{dim}(K))={}{r}\,{{\sf{cat}}}(X^{K}). \]

Example 6.8 gives the LS-logarithmicity hypothesis in theorem 1.5 for families of RAA groups; the ${{\sf {TC}}}$-logarithmicity hypothesis also holds:

Proposition 6.12 For any $r\geq 2$ and any family $\{\Gamma _i\}_{i=1}^{m}$ of simplicial graphs,

\[ {{\sf{TC}}}_r(H_{{\Gamma_1} \ast \cdots \ast {\Gamma_m}}) ={{\sf{TC}}}_r(H_{\Gamma_1})+\cdots+{{\sf{TC}}}_r(H_{\Gamma_m}). \]

Proof. It suffices to consider the case $m=2$. Let $\Gamma$ and ${\Gamma '}$ be simplicial graphs and choose $C_1,\, \ldots ,\, C_r$ and $D_1,\,\ldots ,\,D_r$ cliques in $\Gamma$ and $\Gamma '$ respectively with $\cap _{i=1}^{r} C_i = \varnothing$ and $\cap _{i=1}^{r} D_i = \varnothing$ that realize ${{\sf {TC}}}_r(H_\Gamma )$ and ${{\sf {TC}}}_r({H_\Gamma '})$. Let $E_i = C_i * D_i$ (where $*$ again denotes the graph join). Clearly, $\cap _{i=1}^{r} E_i = \varnothing$, for all $C_i$ are pairwise disjoint from all $D_j$ and the total intersections of the $C_i$ and $D_i$ are individually empty. Then, using theorem 2.4, we obtain

\begin{align*} \sum_{i=1}^{r} |C_i| + \sum_{i=1}^{r} |D_i| & = \sum_{i=1}^{r} |E_i| \leq {{\sf{TC}}}_r(H_{\Gamma \ast {\Gamma'}}) \\ & \leq {{\sf{TC}}}_r(H_\Gamma) + {{\sf{TC}}}(H_{\Gamma'}) = \sum_{i=1}^{r} |C_i| + \sum_{i=1}^{r} |D_i|. \end{align*}

Hence, all inequalities are equalities and we are done.

The validity of the first hypothesis in theorem 1.5 for families of RAA groups depends on the actual structure of the graphs $\Gamma _i$. For instance, if a graph $\Gamma$ has two disjoint cliques of the maximal cardinality (as is the case for $D_v(\Gamma )$ in the first part of theorem 4.1), then ${{\sf {TC}}}_{{}{r}}(H_\Gamma )={}{r}\cdot {\textrm {cd}}(H_\Gamma )$, in view of theorem 2.4. Thus:

Corollary 6.13 If $\{\Gamma _i\}_{i=1}^{n}$ is a family of simplicial graphs such that each $\Gamma _i$ has two disjoint cliques of the maximal cardinality, then for any simplicial complex $K$ and any $r\geq 2$,

\[ {{\sf{TC}}}_{{}{r}}\left(\hspace{.3mm}\underline{H_\Gamma}^{K}\right) = \max_{[i_1,\ldots,i_n]}\left\{{{\sf{TC}}}_{{}{r}}({H_{\Gamma_{i_1}}})+\cdots + {{\sf{TC}}}_{{}{r}}(H_{\Gamma_{i_n}})\rule{0mm}{4mm}\right\}, \]

where the maximum is taken over all maximal simplices $[i_1,\,\ldots ,\,i_n]$ of $K$. Here, we follow the notation in example 6.8, in particular $\Gamma =\Gamma _1\ast \cdots \ast \Gamma _m$.

It would be interesting to know whether one or both of the logarithmicity hypotheses could be waived in theorem 1.5. The hypothesis ${{\sf {TC}}}_r=r\cdot {{\sf {cat}}}$ for the polyhedral product factors however, plays a crucial role even for ${{\sf {TC}}}_2$, as will be clear in the next and final section.

We close the section with a general lower estimate (used later in the paper) for the topological complexity of polyhedral products. We need:

Corollary 6.15 Let $\sigma _1,\,\ldots ,\,\sigma _r$ be an $r$-tuple of simplices in $K$ such that at least two simplices are disjoint. Then, for any family $\{(X_i,\,\ast )\}_{i=1}^{m}$,

\[ {{\sf{cat}}}({}{{X}^{\sigma_1}\times {X}^{\sigma_2}\times\cdots\times{X}^{\sigma_r}}) \leq {{\sf{TC}}}_r({}{\underline{X}}^{K}). \]

Here ${X}^{\sigma _i}$ stands for the product $\prod _{j\in \sigma _i}X_j$ (see the proof of corollary 6.7).

Proof. Without loss of generality we can take $\sigma _1 \cap \sigma _2 = \varnothing$. Consider the projection $p_1\colon {}{\underline {X}}^{K} \to {}{{X}}^{\sigma _1}$ with homotopy fibre $F$. The inclusion $i_1\colon {}{{X}}^{\sigma _1} \to {}{\underline {X}}^{K}$ satisfies $p_1 \circ i_1 = \mathrm {id}$, so ${}{{X}}^{\sigma _1}$ is a retract of ${}{\underline {X}}^{K}$. Hence, we may consider $i_1$ to be a homotopy section of the homotopy fibration $p_1$. By theorem 6.14,

(6.3)\begin{equation} {{\sf{cat}}}({}{{X}}^{\sigma_1} \times F \times ({}{\underline{X}}^{K})^{r-2}) \leq {{\sf{TC}}}_r({}{\underline{X}}^{K}). \end{equation}

The inclusion $i_2\colon {}{{X}}^{\sigma _2} \to {}{\underline {X}}^{K}$ (with associated retraction $p_2\colon {}{\underline {X}}^{K} \to {}{{X}}^{\sigma _2}$) has $p_1 \circ i_2 = *$, since $\sigma _1 \cap \sigma _2 = \varnothing$, so $i_2$ factors up to homotopy through the fibre $F$. Moreover, composing the resulting homotopy class ${X}^{\sigma _2}\to F$ with the fibre inclusion and $p_2$ gives the identity on ${}{{X}}^{\sigma _2}$ (for $p_2\circ i_2=\text {id}$), so ${}{{X}}^{\sigma _2}$ is a homotopy retract of $F$. Of course, it is also the case that ${}{{X}}^{\sigma _3} \times \cdots \times {}{{X}}^{\sigma _r}$ is a retract of $({}{\underline {X}}^{K})^{r-2}$ as well, so ${}{{X}}^{\sigma _1} \times \cdots \times {}{{X}}^{\sigma _r}$ is a homotopy retract of ${}{{X}}^{\sigma _1} \times F \times ({}{\underline {X}}^{K})^{r-2}$. Thus

\[ {{\sf{cat}}}({}{{X}}^{\sigma_1} \times \cdots \times {}{{X}}^{\sigma_r}) \leq {{\sf{cat}}}({}{{X}}^{\sigma_1} \times F \times ({}{\underline{X}}^{K})^{r-2}) \leq {{\sf{TC}}}_r({}{\underline{X}}^{K}), \]

which yields the result.

7.1 Proof of proposition 1.8

Recall from [Reference Bahri, Bendersky, Cohen and Gitler2, theorem 2.35] that the mod-2 cohomology ring of $\underline {P}^{K}$ is

\[ H^{*}(\underline{P}^{K})\cong \bigotimes_{i=1}^{m}H^{*}({P}^{n_i})/I_K. \]

Here, $I_K$ is the generalized Stanley–Reisner ideal generated by all elements $x_{r_1}\otimes x_{r_2}\otimes \cdots \otimes x_{r_t},$ satisfying $x_{r_{i}}\in \overline {H}^{ * }({P}^{n_{r_i}})$ and $[r_{1},\,\ldots ,\,r_{t}]\notin {K}$. For each $i\in [m]$, let $u_{i}\in H^{*}(\underline {P}^{K})$ be the pullback of the generator of $H^{1}({P}^{n_i})$ under the canonical projection $\underline {P}^{K}\to {P}^{n_i}$ onto the $i$-th coordinate. In these terms, $I_K$ is generated by the powers $u_{i}^{n_{i}+1}$ and the products $u_{r_1}\cdots u_{r_t}$ with $[r_1,\,\ldots ,\,r_t]\notin {K}$. In particular, a graded basis for $H^{*}(\underline {P}^{K})$ is given by the monomials $u_{1}^{e_1}\cdots u_{m}^{e_m}$ having $0\leq e_{i}\leq n_i$ and $[i: e_{i}>0]\in {K}$. Note that $u_{i}\otimes 1+1\otimes u_{i}\in [H^{*}(\underline {P}^{K})]^{\otimes 2}$ is a zero divisor for each $i\in \{1,\,\ldots ,\,m\}$.

Let $\sigma _{1},\,\sigma _{2}\in {K}$, say with $\sigma _{1}\setminus \sigma _{2}=[i_1,\,\ldots ,\,i_{r}]$, $\sigma _{2}\setminus \sigma _{1}=[j_1,\,\ldots ,\,j_{s}]$, and $\sigma _{1}\cap \sigma _{2}=[k_{1},\,\ldots ,\,k_{w}]$. It suffices to show that, in $[H^{*}(\underline {P}^{K})]^{\otimes 2}$,

(7.3)\begin{equation} \left(\prod_{i\in\sigma_{1}{\bigtriangleup}\sigma_2}(u_{i}\otimes 1+1\otimes u_{i})^{n_i}\right)\left(\,\prod_{i\in\sigma_{1}\cap\sigma_2}(u_{i}\otimes 1+1\otimes u_{i})^{{{\sf{zcl}}}(P^{n_i})}\right)\neq 0. \end{equation}

Expanding the first factor on the left-hand side of (7.3), we get

(7.4)\begin{align} &\left(\sum_{\ell_{1}=0}^{n_{i_1}}\binom{n_{i_1}}{\ell_1}\right.\left.u_{i_1}^{n_{i_1}-\ell_1}\otimes u_{i_1}^{\ell_1}\rule{0mm}{7.5mm}\right) \cdots\left(\sum_{\ell_{r}=0}^{n_{i_r}}\binom{n_{i_r}}{\ell_r}u_{i_{r}}^{n_{i_r}-\ell_{r}}\otimes u_{i_{r}}^{\ell_{r}}\right)\nonumber\\ &\qquad\cdot\left(\sum_{q_{1}=0}^{n_{j_1}}\right.\left.\binom{n_{j_1}}{q_1}u_{j_1}^{n_{j_1}-q_1}\otimes u_{j_1}^{q_1}\rule{0mm}{7.5mm}\right) \cdots\left(\sum_{q_{s}=0}^{n_{j_s}}\binom{n_{j_s}}{q_s}u_{j_s}^{n_{j_s}-q_{s}}\otimes u_{j_{s}}^{q_{s}}\right) \nonumber\\ &\qquad\qquad=\left(\sum_{\substack{0\leq \ell_{t}\leq n_{i_t}\\ 1\leq t\leq r}}\binom{n_{i_1}}{\ell_1}\cdots\binom{n_{i_r}}{\ell_r}u_{i_1}^{n_{i_1}-\ell_1}\cdots u_{i_r}^{n_{i_r}-\ell_r}\otimes u_{i_1}^{\ell_1}\cdots u_{i_r}^{\ell_r}\right) \end{align}

(7.5)\begin{align} &\cdot\left(\sum_{\substack{0\leq q_{t}\leq n_{j_t}\\ 1\leq t\leq s}}\binom{n_{j_1}}{q_1}\cdots\binom{n_{j_s}}{q_s}u_{j_1}^{n_{j_1}-q_1}\cdots u_{j_s}^{n_{j_s}-q_s}\otimes u_{j_1}^{q_1}\cdots u_{j_s}^{q_s}\right). \end{align}

On the other hand, it is elementary to see that ${{\sf {zcl}}}(P^{n_i})=2^{\theta _i}-1$ where $2^{\theta _{i}-1}\leq n_i<2^{\theta _{i}}$. In particular $n_i\leq 2^{\theta _i}-1<2n_{i}$. With this in mind, the second factor on the right-hand side of (7.3) becomes

(7.6)\begin{align} &\left(\sum_{h_{1}=2^{\theta_{k_1}}-1-n_{k_1}}^{n_{k_1}}u_{k_1}^{2^{\theta_{k_1}}-1-h_1}\otimes u_{k_1}^{h_1}\rule{0mm}{9mm}\right)\cdots\left(\sum_{h_{w}=2^{\theta_{k_w}}-1-n_{k_w}}^{n_{k_w}}u_{k_w}^{2^{\theta_{k_w}}-1-h_w}\otimes u_{k_w}^{h_w}\right) \nonumber\\ &\quad = \left(\sum_{\substack{2^{\theta_{k_t}}-1-n_{k_t}\leq h_{t}\leq n_{k_t}\\ 1\leq t\leq w}}u_{k_1}^{2^{\theta_{k_1}}-1-h_1}\cdots u_{k_w}^{2^{\theta_{k_w}}-1-h_w}\otimes u_{k_1}^{h_1}\cdots u_{k_w}^{h_w}\right). \end{align}

The conclusion then follows by observing that, in the product of the expressions above, the basis element

\[ u_{i_1}^{n_{i_1}}\cdots u_{i_r}^{n_{i_r}}u_{k_1}^{n_{k_1}}\cdots u_{k_w}^{n_{k_w}}\otimes u_{j_1}^{n_{j_1}}\cdots u_{j_s}^{n_{j_s}}u_{k_1}^{2^{\theta_{k_1}}-1-n_{k_1}}\cdots u_{k_w}^{2^{\theta_{k_w}}-1-n_{k_w}} \]

arises only from the product of:

• • $u_{i_1}^{n_{i_1}}\cdots u_{i_r}^{n_{i_r}}\otimes 1\,$ in (7.4), with $\ell _{t}=0$ for all $t\in \{1,\,\ldots ,\,r\}$;

• • $1\otimes u_{j_1}^{n_{j_1}}\cdots u_{j_s}^{n_{j_s}}\,$ in (7.5), with $q_{t}=n_{j_t}$ for all $t\in \{1,\,\ldots ,\,s\}$;

• • $u_{k_1}^{n_{k_1}}\cdots u_{k_w}^{n_{k_w}}\otimes u_{k_1}^{2^{\theta _{k_1}}-1-n_{k_1}}\cdots u_{k_w}^{2^{\theta _{k_w}}-1-n_{k_w}}\,$ in (7.6), with $h_{t}=2^{\theta _{k_t}}-1-n_{k_t}$ for all $t\in \{1,\,\ldots ,\,w\}$.

7.2 Axial and nonsingular maps

The rest of the paper is devoted to the proof of theorem 1.7. For this, we need the full force of the concept of axial map and its relationship to the topological complexity of real projective spaces. We review the needed details.

It is classical that a nonsingular map as above exists only for $k\geq n$, and that $k=n$ is possible only for $n\in \{1,\,3,\,7\}$, in which case the complex, quaternion, and octonion multiplications can be used. In the latter cases, the multiplication $\omega \cdot z$ yields a strong nonsingular map, while the multiplication $\omega \cdot \overline {z}$ yields a nonsingular map $f\colon \mathbb {R}^{n+1}\times \mathbb {R}^{n+1}\to \mathbb {R}^{n+1}$ satisfying

\[ f(x,x)=(\lambda_x,0,\ldots,0) \]

with $\lambda _x>0$ for $x\neq 0$. From the polyhedral product viewpoint, the latter condition allows us to detect the ${{\sf {cat}}}$/${{\sf {TC}}}$ mixed behaviour illustrated in example 7.2 – see the discussion following the next definition.

It is elementary to see that a nonsingular map $f\colon \mathbb {R}^{n+1}\times \mathbb {R}^{n+1}\to \mathbb {R}^{k+1}$ induces and is induced by an axial map $\alpha \colon {P}^{n}\times {P}^{n}\longrightarrow {P}^{k}$, and that, in these conditions, $f$ is strong if and only if $\alpha$ is strong. In turn, a strong axial map ${P}^{n}\times {P}^{n}\to {P}^{k}$ can be constructed out of a smooth immersion ${P}^{n}\looparrowright \mathbb {R}^{k}$ ([Reference Sanderson28, theorem 2.1]; note that $k>n$ is forced). Conversely, for $k>n$, the existence of a (not necessarily strong) axial map ${P}^{n}\times {P}^{n}\to {P}^{k}$ implies the existence of a smooth immersion ${P}^{n}\looparrowright \mathbb {R}^{k}$ ([Reference Adem, Gitler and James1]). The ‘strong’ requirement, which is then free when $k>n$, is central in what follows. (The situation $n=k$ is special in that there cannot be a smooth immersion ${P}^{n}\looparrowright \mathbb {R}^{n}$, and yet an axial map ${P}^{n}\times {P}^{n}\to {P}^{n}$ exists if and only if $n\in \{1,\,3,\,7\}$.)

These facts imply that, for $n\neq 1,\,3,\,7$, ${{\sf {TC}}}({P}^{n})= \operatorname {Imm}({P}^{n})$, the Euclidean immersion dimension of $P^{n}$ ([Reference Farber, Tabachnikov and Yuzvinsky17, theorem 7.1]), which also agrees with the smallest integer $k$ greater than $n$ for which there is a strong axial map $P^{n}\times P^{n}\to P^{k}$. Furthermore, if $k>n$, an axial map $P^{n}\times P^{n}\to P^{k}$ is known to be null-homotopic on the diagonal. The combination of such a behaviour with the strong condition for axial maps yields the main ingredient in the proof of theorem 1.7. Actually, the strong property for axial maps is directly responsible for the mixed ${{\sf {cat}}}$/${{\sf {TC}}}$ behaviour – and resulting apparent sharpness – of theorem 1.7 (see (7.9) below).

The following fact is adapted from [Reference Farber, Tabachnikov and Yuzvinsky17, lemma 5.7]. We leave the proof as a standard algebraic topology exercise for the reader.

7.3 Proof of theorem 1.7

For each $i\in \{1,\,\ldots ,\,m\}$, fix once and for all a nonsingular map $f_{i}:\mathbb {R}^{n_{i}+1}\times \mathbb {R}^{n_{i}+1}\to \mathbb {R}^{{{\sf {TC}}}({P}^{n_i})+1}$ such that

(7.7)\begin{equation} f_i(x,x)=(\lambda_{x},0,\ldots,0),\quad \lambda_{x}\geq 0, \end{equation}

for all $x\in \mathbb {R}^{n_i+1}$, with $\lambda _x>0$ for $x\neq 0$. Furthermore, if $n_i\neq 1,\,3,\,7$, we also require $f_i$ to be strong:

(7.8)\begin{equation} f_i({\ast},x)=x=f_i(x,\ast), \mbox{ for all } x\in\mathbb{R}^{n_{i}+1}. \end{equation}

For $k\in \{0,\,1,\,\ldots ,\,{{\sf {TC}}}({P}^{n_i})\}$, let $f_{ik}$ denote the $(k+1)$-st coordinate map of $f_{i}$, and consider the subsets $V_{ik}\subset P^{n_i}\times P^{n_i}$ defined by

\begin{align*} V_{i0}&=\{(A,B)\colon f_{i0}(a,b)\neq 0\hspace{1mm}\text{for some}\hspace{1mm}(a,b)\in A\times B\}\\ V_{ik}&=\{(A,B)\colon A\neq B\hspace{1mm}\text{and}\hspace{1mm}f_{ik}(a,b)\neq 0\hspace{1mm}\text{for some}\hspace{1mm}(a,b)\in A\times B\}, \mbox{ for } k>0. \end{align*}

Observe that $\{V_{i0},\,V_{i1},\,\ldots ,\,V_{i{{\sf {TC}}}({P}^{n_i})}\}$ is an open cover of ${P}^{n_i}\times {P}^{n_i}$, with the diagonal of ${P}^{n_i}$ contained in $V_{i0}$, in view of (7.7).

We replace the covering $\{V_{ik}\}_k$ of each $P^{n_i}$ by one which consists of pairwise disjoint sets. Put $U_{ik}:=V_{ik}\setminus (V_{i0}\cup \cdots \cup V_{i(k-1)})$ (so $U_{i0}=V_{i0}$). We say that an ordered pair of lines $(A,\,B)$ in $\mathbb {R}^{n_{i}+1}$ produces $k$ zeroes ($k\in \{0,\,1,\,\ldots ,\,{{\sf {TC}}}({P}^{n_i})\}$) when $(A,\,B)\in U_{ik}$. The justification of the latter convention comes by observing that $(A,\,B)\in U_{ik}$ if and only if

\[ f_{i0}(a,b)=\cdots=f_{i(k-1)}(a,b)=0\neq f_{ik}(a,b) \]

for some (and therefore any) vectors $a\in A$ and $b\in B$. The number of zeroes produced by $(A,\,B)$ is denoted $Z(A,\,B)$.

For simplicity we write $X=\underline {P}^{K}$, the polyhedral product in theorem 1.7, but keep the notation ${P}^{\sigma }$ for the ‘subproducts’ in corollary 6.15. An element $(A_1,\,A_2)$ of $X^{2}$ will be thought of as a matrix of size $m\times 2$, i.e.

\begin{equation} \nonumber (A_1,A_2)= \begin{pmatrix} A_{11} & A_{12}\\ \vdots & \vdots\\ A_{m1} & A_{m2}\end{pmatrix}, \end{equation}

where each column belongs to $X$, say $(A_{1j},\,\ldots ,\,A_{mj})\in {P}^{\sigma _{j}}$ with $\sigma _{1},\,\sigma _{2}\in {K}$. Each row $(A_{i1},\,A_{i2})$ of the matrix $(A_1,\,A_2)$ lies in a unique set $U_{ik}$ with $k=Z(A_{i1},\,A_{i2})$. The number of zeroes determined by $(A_1,\,A_2)$, denoted by $Z(A_1,\,A_2)$, is the sum of zeroes produced by the rows of $(A_1,\,A_2)$,

\[ Z(A_1,A_2)=\sum_{i=1}^{m}Z(A_{i1},A_{i2}). \]

It is apparent that $Z(A_1,\,A_2)\leq \sum _{i\in \sigma _{1}\cup \sigma _{2}}{{\sf {TC}}}({P}^{n_i})$. However, in view of (7.8), the number of zeroes produced by rows of type either $(A_{i1},\,\ast )$ or $(\ast ,\, A_{i2})$ of $(A_1,\,A_2)$ calls for a more careful consideration. Namely, note that $Z(A_{i1},\,A_{i2})=Z(A_{i1},\,\ast )\leq n_i$ for $i\in \sigma _{1}\setminus \sigma _{2}$, because $f_{i}$ fulfils (7.8) when $n_i\neq 1,\,3,\,7$ (recall that $n_i={{\sf {TC}}}({P}^{n_i})$ if $n_i=1,\,3,\,7$). Likewise, if $i\in \sigma _{2}\setminus \sigma _{1}$, then $Z(A_{i1},\,A_{i2})=Z(\ast ,\,A_{i2})\leq n_i$. Therefore

(7.9)\begin{equation} Z(A_1,A_2)\leq\sum_{i\in \sigma_{1}{\bigtriangleup}\sigma_{2}}n_i+\sum_{i\in \sigma_1\cap\sigma_{2}}{{\sf{TC}}}({P}^{n_i}), \end{equation}

and we have proved:

The proof of theorem 1.7 will be complete once a local rule (i.e. a local section for the end-points evaluation map $X^{[0,1]}\to X\times X$) is constructed on each $W_{j}$. This will be achieved by splitting $W_{j}$ into topologically disjoint subsets (proposition 7.10 below), and then showing that each such subset admits a local rule.

A partition of an integer $j\in \{0,\,1,\,\ldots ,\, \mathcal {N}^{(n_1,\,\ldots ,\,n_m)}(K)\}$ is an ordered tuple $(j_1,\,\ldots ,\, j_{m})$ of integers satisfying $j=j_{1}+\cdots +j_{m}$ and $0\leq j_{i}\leq {{\sf {TC}}}({P}^{n_i})$ for each $i=1,\,\ldots ,\,m$. For such a partition we set

\begin{equation} \nonumber W_{(j_1,\ldots, j_{m})}=\{(A_1,A_2)\in X^{2}: Z(A_{i1},A_{i2})=j_i \hspace{1mm}\text{ for each }\hspace{1mm}i\in[m]\}.\end{equation}

We show that the resulting set-theoretic partition $W_{j}=\displaystyle {\bigsqcup} W_{(j_1,\ldots , j_{m})}$, where the union runs over the partitions $(j_1,\,\ldots ,\,j_m)$ of $j$, is topological, i.e. $W_{j}$ has the weak topology determined by the several $W_{(j_1,\ldots , j_{m})}$. In fact:

Proposition 7.10 Two different partitions $(j_1,\,\ldots ,\, j_{m})$ and $(r_1,\,\ldots ,\, r_{m})$ of the same $j\in \{0,\,1,\,\ldots ,\,\mathcal {N}^{(n_1,\,\ldots ,\,n_m)}(K)\}$ have

\[ W_{(j_1,\ldots, j_{m})}\cap\overline{W_{(r_1,\ldots, r_{m})}}=\varnothing=\overline{W_{(j_1,\ldots, j_{m})}}\cap W_{(r_1,\ldots, r_{m})}. \]

Proof. It suffices to show the first equality assuming $j_{\ell }< r_{\ell }$ for some $\ell \in [m]$. For elements $(A_{1},\,A_{2})\in W_{(j_1,\ldots ,j_{m})}$ and $(B_{1},\,B_{2})\in W_{(r_1,\ldots ,r_{m})}$ we have

\begin{align*} &(A_{\ell 1},A_{\ell 2})\in U_{\ell j_{\ell}}\Rightarrow f_{\ell j_{\ell}}(a_{\ell 1},a_{\ell 2})\neq 0 \text{ for any }(a_{\ell 1},a_{\ell 2})\in A_{\ell 1}\times A_{\ell 2}\\ &(B_{\ell 1},B_{\ell 2})\in U_{\ell r_{\ell}}\Rightarrow f_{\ell j_{\ell}}(b_{\ell 1},b_{\ell 2})=0 \text{ for any } (b_{\ell 1}, b_{\ell 2})\in B_{\ell 1}\times B_{\ell 2}. \end{align*}

The assertion then follows since the latter condition is inherited by elements of $\overline {W_{(r_1,\ldots ,r_{m})}}$.

In the rest of the paper we construct a local rule on each $W_{(j_1,\ldots ,j_m)}$. For $i\in \{1,\,\ldots ,\,m\}$, consider the canonical Riemannian structure on the unit $n_i$-sphere $S^{n_i}$. Since the antipodal involution ${S}^{n_i}\to {S}^{n_i}$ is a fixed-point free properly discontinuous isometry, ${P}^{n_i}$ inherits a canonical quotient Riemannian structure $g_i$. Let $L_{i}(\gamma )$ denote the resulting length of a smooth curve $\gamma$ in ${P}^{n_i}$, and let

\[ d_{i}(x,y)=\inf \{L_{i}(\gamma):\gamma\hspace{1mm}\text{is a geodesic on}\hspace{1mm}{P}^{n_i}\hspace{1mm}\text{from}\hspace{1mm}x\hspace{1mm}\text{to}\hspace{1mm}y\} \]

be the associated metric. Note that the curves $\lambda _{ik}(A,\,B)$ described in remark 7.7 are geodesic on ${P}^{n_i}$. Without loss of generality we can assume that each $g_i$ is normalized so that any geodesic $\gamma$ on ${P}^{n_i}$ satisfies $L_{i}(\gamma )\leq 1/2$.

As a first step, we reparametrize the families of paths $\lambda _{ik}$. Explicitly, for $k\in \{0,\,1,\,\ldots ,\,{{\sf {TC}}}({P}^{n_i})\}$, consider the section $\tau _{ik}:U_{ik}\to ({P}^{n_i})^{[0,1]}$ of the end-points evaluation map $({P}^{n_i})^{[0,1]}\to {P}^{n_i}\times {P}^{n_i}$ given by

\[ \tau_{ik}(A_{1},A_{2})(t)=\begin{cases} A_1, & \hspace{1mm}\text{if }\hspace{1mm} d_{i1}+d_{i2}=0;\\ \lambda_{ik}(A_1,A_2)\left(\displaystyle\frac{t}{d_{i1}+d_{i2}}\right), & \hspace{1mm}\text{if }\hspace{1mm} 0\leq t\leq (d_{i1}+d_{i2})\neq 0;\\ A_{2}, & \hspace{1mm}\text{if }\hspace{1mm} 0\neq (d_{i1}+d_{i2})\leq t\leq 1, \end{cases} \]

for $(A_1,\,A_2)\in U_{ik}$, where $d_{ij}=d_{i}(A_j,\,\ast )$, $j=1,\,2$. Note that $\tau _{ik}$ is continuous on the open subset of $U_{ik}$ determined by the condition $d_{i1}+d_{i2}\neq 0$. The latter open subset of $U_{ik}$ equals in fact $U_{ik}$ unless $k=0$, so that $\tau _{ik}$ is continuous on the whole $U_{ik}$ for $k\in \{1,\,\ldots ,\,{{\sf {TC}}}({P}^{n_i})\}$. The continuity of $\tau _{i0}$ on $U_{i0}$ follows from the continuity of $\lambda _{i0}$ and the fact that $\lambda _{i0}(A,\,A)$ is the constant path for all $A\in {P}^{n_i}$. Note in addition that $\tau _{ik}$ reaches its end point at time $d_{i1}+d_{i2}$ (this is function-wise speaking). In particular $\tau _{ik}(A_1,\,\ast )$ reaches $\ast$ at time $d_{i1}$.

The map $\varphi :X^{2}\to (\prod _{i=1}^{m}{P}^{n_i})^{[0,1]}$ we are after is defined by

\[ \varphi(A_{1},A_{2})=(\varphi_{1}(A_{11},A_{12}),\ldots,\varphi_{m}(A_{m1},A_{m2})), \]

with $i$-th coordinate $\varphi _{i}(A_{i1},\,A_{i2})$, the path in ${P}^{n_i}$ from $A_{i1}$ to $A_{i2}$ given by

(7.10)\begin{equation} \varphi_{i}(A_{i1},A_{i2})(t)=\begin{cases} A_{i1}, & \hspace{1mm}\text{if }\hspace{1mm} 0\leq t\leq t_{A_{i1}};\\ \mu(A_{i1},A_{i2})(t-t_{A_{i1}}), & \hspace{1mm}\text{if }\hspace{1mm} t_{A_{i1}}\leq t\leq 1. \end{cases} \end{equation}

Here, $t_{A_{i1}}=1/2-d_{i}(A_{i1},\,\ast )$ and

(7.11)\begin{equation} \mu(A_{i1},A_{i2})=\begin{cases} \tau_{i0}(A_{i1},A_{i2}), & \hspace{1mm}\text{if }\hspace{1mm} (A_{i1},A_{i2})\in U_{i0};\\ \hspace{1mm}\vdots & \hspace{2mm}\vdots\\ \tau_{i{{\sf{TC}}}({P}^{n_i})}(A_{i1},A_{i2}), & \hspace{1mm}\text{if }\hspace{1mm} (A_{i1},A_{i2})\in U_{i{{\sf{TC}}}({P}^{n_i})}.\end{cases} \end{equation}

By construction, $\varphi$ is a global section of the end-points evaluation map

\[ \left(\prod_{i=1}^{m}{P}^{n_i}\right)^{[0,1]}\longrightarrow \,\left(\prod_{i=1}^{m}{P}^{n_i}\right)^{2}. \]

Although $\varphi$ is not globally continuous, the restriction of $\varphi$ to each $W_{(j_1,\ldots , j_{m})}$, where $(j_1,\,\ldots ,\,j_{m})$ is a partition of $j\in \{0,\,1,\,\ldots ,\, \mathcal {N}^{(n_1,\,\ldots ,\,n_m)}(K)\}$, is continuous because formulas (7.11) can be rewritten as

\[ \mu=\begin{cases} \tau_{i0}, & \hspace{1mm}\text{if }\hspace{1mm} j_{i}=0;\\ \hspace{1mm}\vdots & \hspace{2mm}\vdots\\ \tau_{i{{\sf{TC}}}({P}^{n_i})}, & \hspace{1mm}\text{if }\hspace{1mm} j_{i}={{\sf{TC}}}({P}^{n_i}). \end{cases} \]

So, the crux of the matter is to show (in proposition 7.14 below) that the restriction of $\varphi$ to each $W_{(j_1,\ldots ,j_m)}$ lands in $X^{[0,1]}$.