1·1 Weak category and weak topological complexity

We here recall some material from [ Reference Garcia-Calcines and Vandembroucq6 ], referring the reader to this paper for more details.

Recall that, given any map $f:E\rightarrow B$ , there is a Whitehead-type characterisation of sectional category, which is given as follows. Consider the n-sectional fatwedge of any map $f:E\rightarrow B,$ $\kappa _n:T^n(f)\rightarrow B^{n+1},$ inductively defined by starting with $\kappa _0=f:E\rightarrow B$ and considering the (fibrewise) join of $\kappa _{n-1}\times id_B$ and $id_{B^n}\times f.$ Then ${\mathrm{secat}}(f)\leqslant n$ if and only if there is, up to homotopy, a lift of the $(n+1)$ -diagonal map $\Delta_{n+1}=\Delta^B_{n+1}:B \to B^{n+1}$ .

For any $n\geqslant 0$ , we shall denote by $B^{\wedge n}$ the n-fold smash-product, by $q_{n}=q^B_{n}:B^{n}\to B^{\wedge n}$ the identification map and by $\bar \Delta_{n}=\bar \Delta^B_{n}:B\stackrel{\Delta_{n}}{\to} B^{n}\stackrel{q_{n}}{\to} B^{\wedge n}$ the reduced n-diagonal. Recall that the weak category of B, ${\mathrm{wcat}}(B)$ , introduced by Berstein and Hilton [ Reference Berstein and Hilton1 ], is the least integer n such that $\bar \Delta_{n+1}$ is homotopically trivial. This is a lower bound for the L-S category of B since ${\mathrm{cat}}(B)\leqslant n$ if and only if the diagonal $\Delta^B_{n+1}:B\to B^{n+1}$ lifts, up to homotopy, in the fatwedge $T^n(B)=\{(b_0,b_1,...,b_n)\in B^{n+1}:b_i=*,\hspace{3pt}\mbox{for some}\hspace{3pt}i\}$ and $B^{\wedge(n+1)}=B^{n+1}/T^n(B).$

Let $f: E\to B$ be a map and, for any integer n, let $C_{\kappa_n}$ be the homotopy cofibre of the n-sectional fat wedge of f, $\kappa _n:T^n(f)\rightarrow B^{n+1}$ . If $l_n:B^{n+1}\rightarrow C_{\kappa_n}$ denotes the induced map, then the weak sectional category of f, denoted by $\mbox{wsecat}(f)$ , is the least integer n (or $\infty$ ) such that the composition

\begin{align*} B \mathop{\longrightarrow}\limits^{\Delta_{n+1}} B^{n+1}\mathop{\longrightarrow}\limits^{l_n} C_{\kappa_n} \end{align*}

is homotopically trivial. This is a lower bound of sectional category whose most important properties are summarised in the following result.

Here ${\mathrm{nil}} \ker f^*$ denotes the nilpotency of the kernel of the morphism $f^*$ which is induced by f in cohomology (where the coefficients can be taken in any commutative ring). This is a classical lower bound for sectional category [ Reference Schwarz19 ]. As a consequence of Proposition 1·1(a), we note that ${\mathrm{wsecat}}(f)=\mbox{secat}(f)$ when $\mbox{secat}(f)={\mathrm{nil}} \ker f^*.$ Additional conditions which ensure the equality ${\mathrm{wsecat}}(f)=\mbox{secat}(f)$ are given in the following proposition established in [ Reference Garcia-Calcines and Vandembroucq6 ]:

Note that in [ Reference Garcia-Calcines and Vandembroucq6 ] this result was stated for $p\geqslant 2$ . Since we always have ${\mathrm{secat}}(f)\leqslant {\mathrm{cat}}(B)\leqslant N$ , the statement is actually also true for $p=1$ .

In this paper, we are more interested in the particular case of weak topological complexity, denoted by $\mbox{wTC}(X).$ This is a lower bound of topological complexity defined as $\mbox{wTC}(X)\,:\!=\,\mbox{wsecat}(\pi )$ where $\pi :X^I\rightarrow X\times X$ denotes the evaluation path fibration. As $\pi $ is the fibration associated with the diagonal map $\Delta :X\rightarrow X\times X,$ we also have that $\mbox{wTC}(X)=\mbox{wsecat}(\Delta).$ As a consequence of Proposition 1·1 we have the following equality:

\begin{align*}\mbox{wTC}(X)=\mbox{wcat}(C_{\Delta }(X)),\end{align*}

where $C_{\Delta }(X)$ stands for the homotopy cofibre of the diagonal map $\Delta :X\rightarrow X\times X$ . We note that the equality $\mbox{TC}(X)=\mbox{cat}(C_{\Delta }(X))$ holds for several classes of spaces (see [ Reference Garcia-Calcines and Vandembroucq7 , Reference González, Grant and Vandembroucq10 ]). However, it is not true in general as established by Dranishnikov ([ Reference Dranishnikov4 ]).

As a direct consequence of Proposition 1·2, we have the following corollary:

Remark 1·4. As pointed out for the general case of sectional category, we also have a cohomological condition ensuring the equality $\mbox{TC}(X)=\mbox{wTC}(X).$ Indeed, if $\tilde H^*$ stands for the reduced cohomology with coefficients in a field $\Bbbk$ , then ${\mathrm{nil}} \ker \Delta ^*$ is exactly ${\mathrm{zcl}}_{\Bbbk}(X)$ , the zero divisor cup length introduced by Farber in [ Reference Farber5 ]. Then, from Proposition 1·1(b) we have that $\mbox{TC}(X)=\mbox{wTC}(X)$ as far as $\mbox{TC}(X)={\mathrm{zcl}}_{\Bbbk}(X) .$ In [ Reference Garcia-Calcines and Vandembroucq6 , Reference Garcia-Calcines and Vandembroucq7 ], we have shown that there exist spaces X for which $\mbox{TC}(X)={\mathrm{wTC}}(X)>{\mathrm{zcl}}_{\Bbbk}(X) .$ More precisely, the space $X=S^3\cup _{\alpha }e^7$ , where $\alpha\in \pi_6(S^3)$ is the Blakers–Massey element (that is, X is the 7-skeleton of $\mbox{Sp}(2)$ ), satisfies

\begin{align*}\mathrm{zcl}_{\Bbbk}(X)=2 \quad {\mathrm{wTC}}(X)={\mathrm{TC}}(X)=3.\end{align*}

In the sequel, we will also use the following result (see [ Reference Garcia-Calcines and Vandembroucq6 ]).

1·2. Hopf invariants

In [ Reference Berstein and Hilton1 ], Berstein and Hilton have introduced two notions of (generalised) Hopf invariants to study the increment of the category/weak-category upon cell attachments. We will refer to these invariants as the ${\mathrm{cat}}$ -Hopf invariants and crude Hopf invariants, respectively. For the sake of simplicity, we will consider the following conditions, which are sufficient for our purpose. Let $X=K\cup_{\alpha }CS$ where K is a $(p-1)$ -connected CW complex with $p\geqslant 2$ , ${\mathrm{cat}}(K)\leqslant k$ and $\dim (K) \leqslant (k+1)p-2$ . We also assume that S is a finite wedge of spheres of the same dimension d satisfying $d\geqslant \dim(K)$ .

Given an attaching map $\alpha$ , the definition of the Berstein–Hilton Hopf invariants depends on the choice of a homotopy lifting $\phi:K\to T^k(K)$ of the diagonal map $\Delta^K_{k+1}:K\to K^{k+1}$ . However, under the condition $\dim (K) \leqslant (k+1)p-2$ , there is a unique such lifting and the Hopf invariants depend only on the homotopy class of $\alpha$ . We denote by $H(\alpha)\in [(CS,S),(K^{k+1},T^k(K))]\cong [S,F_k(K)]$ the ${\mathrm{cat}}$ -Hopf invariant of $\alpha$ (which can be seen as the obstruction to extend to X the lifting $\phi$ of $\Delta^K_{k+1}$ – see [ Reference Berstein and Hilton1 ] or [ Reference Cornea, Lupton, Oprea and Tanré3 ] for the precise definition). Here $F_k(K)$ denotes the homotopy fibre of the inclusion $T^k(K)\hookrightarrow K^{k+1}$ . The crude Hopf invariant of $\alpha$ is given by $\bar H(\alpha)\,:\!=\,(q_{k+1})_*H(\alpha)\in [\Sigma S,K^{\wedge(k+1)}]$ , where $(q_{k+1})_*$ is the morphism induced by the identification map $(K^{k+1},T^k(K))\to (K^{\wedge(k+1)},\ast)$ .

Under the conditions above, it follows from [ Reference Berstein and Hilton1 , corollary 3·9 and theorem 3·19] that

1. (i) ${\mathrm{cat}}(X) \leqslant k $ if, and only if, $H(\alpha)=0$ ,

2. (ii) ${\mathrm{wcat}}(X) \leqslant k $ if, and only if, $\bar H(\alpha)=0$ .

As it will be useful in this work, we detail the second statement through the following lemma. We include a proof which will lead to a generalisation in a subsequent section (Lemma 3·3).

Lemma 1·6. Let K be a $(p-1)$ -connected CW complex with $p\geqslant 2$ , ${\mathrm{cat}}(K)\leqslant k$ and $\dim (K) \leqslant (k+1)p-2$ . Let $d\geqslant \dim (K)$ and $\alpha: S\to K$ be a map where S is a finite wedge of spheres $S^d$ . Then, for $X=K\cup_{\alpha }CS$ , there exists a homotopy commutative diagram

where $\delta$ is the connecting map in the Puppe sequence of the cofibration $S\stackrel{\alpha}{\to}K\to X$ . Moreover ${\mathrm{wcat}}(X)\leqslant k$ if, and only if, $\bar H(\alpha)=0$ .

Proof. The diagram can be deduced from [ Reference Berstein and Hilton1 , proposition 3·7(ii)], see also [ Reference Cornea, Lupton, Oprea and Tanré3 , lemma 6·33]. It is clear that $\bar H(\alpha)=0$ implies that ${\mathrm{wcat}}(X)\leqslant k$ . Conversely, let us suppose that ${\mathrm{wcat}}(X)\leqslant k$ . Then $\bar{\Delta}_{k+1}\simeq \ast$ . Note that the inclusion $K^{\wedge (k+1)}\hookrightarrow X^{\wedge (k+1)}$ is a $(kp+d)$ -equivalence. Since $\dim(X)=d+1< kp+d$ , we deduce from $\bar{\Delta}_{k+1}\simeq \ast$ that $\bar H(\alpha)\delta\simeq \ast$ . Considering the Puppe sequence of the cofibration $S\stackrel{\alpha}{\to}K\to X$ we then get a map $\Sigma K \to K^{\wedge (k+1)}$ making homotopy commutative the following diagram:

Since $K^{\wedge (k+1)}$ is $(k+1)p-1$ -connected and $\dim (K) \leqslant (k+1)p-2$ we see that this map is homotopically trivial and consequently $\bar H(\alpha)=0$ .

We will also use the following observations:

Remark 1·7. Let us consider the commutative diagram corresponding to the identification map $(K^{k+1},T^k(K))\to (K^{\wedge(k+1)},\ast)$ . If we denote by $\bar q_{k+1}: F_k(K) \to \Omega K^{\wedge(k+1)}$ the induced map at the level of the homotopy fibers, then, through adjunction, $\bar H(\alpha)$ can be identified to the following composite:

\begin{align*}\Sigma S \stackrel{\Sigma H(\alpha)}{\longrightarrow} \Sigma F_k(K) \stackrel{\Sigma \bar q_{k+1}}{\longrightarrow } \Sigma \Omega K^{\wedge(k+1)} \stackrel{ev}{\longrightarrow } K^{\wedge(k+1)},\end{align*}

where ev is the evaluation map. When $K=S^p$ or more generally any ( $p-1$ )-connected CW-complex with $p\geqslant 2$ , we can use a Blakers–Massey argument to see that the map

\begin{align*}\Sigma F_k(K) \stackrel{\Sigma \bar q_{k+1}}{\longrightarrow} \Sigma \Omega K^{\wedge(k+1)} \stackrel{ev}{\longrightarrow} K^{\wedge(k+1)}\end{align*}

is a $p(k+2)-1$ equivalence.

Remark 1·8. In the specific case of a map $\alpha: S \to S^p$ ( $p\geqslant 2$ ), where S is a finite wedge of spheres $S^d$ with $d < 3p-2$ , the cat-Hopf invariant $H(\alpha)$ exhibits stability and is entirely determined by its projection $H_0(\alpha): S \to S^{2p-1}$ onto the bottom sphere of

\begin{align*}F_1(S^p)=\Omega S^p \ast \Omega S^p \simeq S^{2p-1} \vee S^{3p-2} \vee S^{3p-2} \vee \cdots\end{align*}

In this case we also have $\bar H(\alpha)=\Sigma H_0(\alpha)$ .

1·3. Cofibre of the diagonal map of spheres

Recall that we denote the homotopy cofiber of the diagonal map $\Delta: X \rightarrow X \times X$ for any space X as $C_{\Delta}(X)$ . In this subsection, we will recall important facts on $C_{\Delta}(S^n)$ and its reduced diagonal $\bar{\Delta}=\bar{\Delta}_2: C_{\Delta}(S^n) \rightarrow C_{\Delta}(S^n) \wedge\ C_{\Delta}(S^n)$ .

As shown in [ Reference Garcia-Calcines and Vandembroucq6 , proposition 28]), $C_{\Delta}(S^n)$ is homotopy equivalent to the adjunction space $S^n \cup _{[\iota _n,\iota _n]} e^{2n}$ , where $[\iota _n,\iota _n]: S^{2n-1} \rightarrow S^n$ represents the Whitehead bracket of the identity map $\iota _n: S^n \rightarrow S^n$ with itself. Taking this fact into consideration, we can easily obtain our next result:

Observe that the suspension $\Sigma [\iota_n, \iota_n]$ is homotopical trivial so (a) is easily satisfied, whereas (b) is a direct consequence of (a) and the properties of the smash product and the suspension functors.

Let us delve into the analysis of the reduced diagonal map

\begin{align*}\bar{\Delta }:C_{\Delta }(S^n)\rightarrow C_{\Delta }(S^n)\wedge C_{\Delta }(S^n)\end{align*}

and its implications. By utilising Lemma 1·6 mentioned earlier, with $K=S^n$ and $\alpha =[\iota _n,\iota _n]$ , we can decompose the reduced diagonal as follows:

As observed in Remark 1·8, $\bar{H}([\iota _n,\iota _n])$ is precisely the suspension of the map

\begin{align*}H_0([\iota _n,\iota _n]):S^{2n-1}\rightarrow S^{2n-1}.\end{align*}

Moreover, it is well–known that $H_0([\iota _n,\iota _n])=\pm h([\iota _n,\iota _n])\iota _{2n-1},$ where $h([\iota _n,\iota _n])\in \mathbb{Z}$ denotes the classical Hopf invariant of $[\iota _n,\iota _n]$ , and that $h([\iota _n,\iota _n])=0$ when n is odd, and $h([\iota _n,\iota _n])=2$ when n is even. Hence, $\bar{\Delta }:C_{\Delta} (S^n)\rightarrow C_{\Delta }(S^n)\wedge C_{\Delta }(S^n)$ is homotopically trivial when n is odd. However, when n is even, $\bar{H}([\iota _n,\iota _n]) :S^{2n}\rightarrow S^{2n}$ is a map whose degree is (up to sign) 2. Applying the suspension functor to the diagram above we can obtain another homotopy commutative diagram

where $\pm 2Id :S^{2n+1}\rightarrow S^{2n+1}$ denotes the map having degree $\pm $ 2, $S^{2n+1}\vee S^{3n+1}\vee S^{3n+2}\vee S^{4n+1}\rightarrow S^{2n+1}$ is a homotopy retraction of $S^{2n+1}\rightarrow S^{2n+1}\vee S^{3n+1}\vee S^{3n+2}$ and $S^{2n+1}\rightarrow S^{2n+1}\vee S^{n+1}$ a homotopy section of $S^{2n+1}\vee S^{n+1}\rightarrow S^{2n+1}$ .

2·1 Odd dimensional case

To begin with, let us recall that for any space X we can construct the homotopy cofiber of the diagonal map given by

\begin{align*}X\stackrel{\Delta }{\longrightarrow}{X\times X}\stackrel{\rho _X}{\longrightarrow }C_{\Delta }(X),\end{align*}

where $\rho _X$ denotes the canonical map. It is worth noting that for any sphere $S^n$ , the projections $p_1:X\times S^n\rightarrow X$ and $p_2:X\times S^n\rightarrow S^n$ induce a natural map $\pi :C_{\Delta }(X\times S^n)\rightarrow C_{\Delta }(X)\times C_{\Delta }(S^n).$ In fact, we can regard $\pi $ as the map induced by the homeomorphism $id_X\times T\times id_{S^n}:X\times S^n\times X\times S^n \stackrel{\cong }{\longrightarrow } X\times X\times S^n\times S^n$ , where T is defined as $T(x,s)\,:\!=\,(s,x)$ . In particular, we have the following commutative diagram

(^*)

The forthcoming lemma concerning the previously defined map $\pi$ is of great importance for our purposes. Its proof relies on the following well-known facts:

1. (i) if $X\stackrel{f}{\longrightarrow }Y\stackrel{p}{\longrightarrow}C_f$ is a homotopy cofibre sequence and f admits a homotopy retraction, then there exists $\sigma :\Sigma C_f\rightarrow\Sigma Y$ a homotopy section of $\Sigma p$ ;

2. (ii) for any two connected spaces X, Y, there is a natural homotopy equivalence $\Sigma (X\times Y)\stackrel{\simeq}{\to} \Sigma X\vee \Sigma Y\vee \Sigma(X\wedge Y).$ In particular, there exists a homotopy section for the suspension of the projection map $X\times Y\rightarrow X\wedge Y.$

Proof. For any space Z, we take $\sigma_Z:\Sigma C_{\Delta}(Z) \to \Sigma (Z\times Z)$ a homotopy section of $\Sigma\rho_Z.$ Then, by the following decomposition of the map $\Sigma (\rho_X\wedge \rho_{S^n})$ :

we see that $\Sigma (\rho_X\wedge \rho_{S^n})$ admits a homotopy section. As we have the homotopy equivalence $\Sigma (\rho_X \times \rho_{S^n})\simeq \Sigma\rho_X \vee \Sigma\rho_{S^n}\vee \Sigma (\rho_X\wedge \rho_{S^n})$ we easily obtain a homotopy section for $\Sigma (\rho_X \times \rho_{S^n}).$ We complete the proof by applying the suspension functor to the diagram (*) above.

We are now in a position to state and prove our first main result in this section, specifically when the dimension of the sphere is odd:

Theorem 2·2. Suppose X is a $(p-1)$ -connected CW-complex ( $p\geqslant 2$ ) satisfying ${\mathrm{wTC}}(X)=k\geqslant 2$ . If $\dim(X)\leqslant pk-1,$ then we have

\begin{align*}{\mathrm{wTC}}(X\times S^n)\geqslant {\mathrm{wTC}}(X)+1\end{align*}

for all n. If, in addition, ${\mathrm{wTC}}(X)={\mathrm{TC}}(X)$ , then ${\mathrm{TC}}(X\times S^n)\geqslant {\mathrm{TC}}(X)+1$ and hence, ${\mathrm{TC}}(X\times S^n)= {\mathrm{TC}}(X)+1$ when n is odd.

Proof. By utilising the identity ${\mathrm{wTC}}(X)={\mathrm{wcat}}(C_{\Delta}(X))$ , we can establish that the kth reduced diagonal $\bar{\Delta}_k: C_{\Delta}(X) \to C_{\Delta}(X)^{\wedge k}$ is not trivial. Furthermore, due to the inequality $\dim(X) \leqslant pk-1$ , the map $\bar{\Delta}_k$ is stably non-trivial, that is, $\Sigma \bar{\Delta}_k \not\simeq *$ (equivalently, $\Sigma ^m\bar{\Delta}_k \not\simeq *$ , for all m). Indeed, we have $\dim(C_{\Delta}(X)) = 2\dim(X)$ and $C_{\Delta}(X)^{\wedge k}$ is $(pk-1)$ -connected, allowing us to apply the Freudenthal suspension theorem.

In order to establish the non-triviality of the $(k+1)$ th reduced diagonal $\bar{\Delta}_{k+1}:C_{\Delta}(X\times S^n)\to C_{\Delta}(X\times S^n)^{\wedge k+1}$ , it is enough to demonstrate that its suspension is not homotopically trivial. To accomplish this, we first construct the following commutative diagram:

Next, we apply the suspension functor to the diagram above, resulting in a new diagram where the curved arrows represent homotopy sections. Those are given by Lemma 2·1 and the fact that, for any pair of connected spaces X,Y, the suspension of the projection map $X\times Y\rightarrow X\wedge Y$ always has a homotopy section:

By using a simple diagram chase argument, where the homotopy sections must be considered, we can verify that the top map $\Sigma \bar{\Delta}_{k+1}$ is not homotopy trivial as long as the bottom map, $\Sigma \bar{\Delta}_{k}\wedge id$ , is not homotopy trivial.

On the other hand, as we have that $\Sigma C_{\Delta}(S^{n})\simeq S^{n+1}\vee S^{2n+1}$ are homotopy equivalent (see Proposition 1·9 (a)) such a bottom map, which is equivalent to $\bar{\Delta}_{k}\wedge \Sigma id:C_{\Delta}(X)\wedge \Sigma C_{\Delta}(S^n)\rightarrow C_{\Delta}(X)^{\wedge k}\wedge \Sigma C_{\Delta}(S^n)$ can be, in turn, naturally identified with $\Sigma^{n+1}\bar{\Delta}_{k}\vee\Sigma^{2n+1}\bar{\Delta}_{k}$ . Since $\bar{\Delta}_{k}:C_{\Delta}(X)\rightarrow C_{\Delta}(X)^{\wedge k}$ is stably non trivial, we conclude the result. The final statement follows from the product formula ${\mathrm{TC}}(X\times S^n)\leqslant {\mathrm{TC}}(X)+{\mathrm{TC}}(S^n)$ which yields ${\mathrm{TC}}(X\times S^n)\leqslant {\mathrm{TC}}(X)+1$ when n is odd.

2·2. Even dimensional case

We first recall that, when n is even, the product formula ${\mathrm{TC}}(X\times S^n)\leqslant {\mathrm{TC}}(X)+{\mathrm{TC}}(S^n)$ reads ${\mathrm{TC}}(X\times S^n)\leqslant {\mathrm{TC}}(X)+2$ . To examine and analyse this case, it is necessary to introduce a new notion for maps beforehand, which has certain similarities with both the conilpotency of a topological space and the weak sectional category of a map.

Remember that, for a given map $f:E\rightarrow B$ and any integer n, we use the notation $C_{\kappa_n}$ to represent the homotopy cofiber of the n-sectional fat wedge of f. Additionally, let $\Delta ^f_{n+1}:B\rightarrow C_{\kappa_n}$ denote the composite of the induced map $l_n$ into the cofiber with the diagonal map $\Delta _{n+1}^B:B\rightarrow B^{n+1}$ :

We will conveniently restrict ourselves to maps over a finite-dimensional CW-complex. Observe that, if B is a finite dimensional CW-complex and X is any space, then by the Freudenthal suspension theorem we have that

\begin{align*}[\Sigma ^NB,\Sigma ^NX]\cong [\Sigma ^{N+1}B,\Sigma ^{N+1}X]\cong [\Sigma ^{N+2}B,\Sigma ^{N+2}X]\cong ...\end{align*}

for N sufficiently large. In this case, $[\Sigma ^{\infty }B,\Sigma ^{\infty }X]\cong [\Sigma ^{N}B,\Sigma ^{N}X]$ .

Definition 2·3. Let $f:E\rightarrow B$ be a map where B is a finite dimensional CW-complex. The 2-sigma weak sectional category of a space f, denoted as $\sigma {\mathrm{wsecat}}(f;2)$ , is defined as the smallest non-negative integer n for which the stable map

\begin{align*}2\Sigma ^{\infty}(\Delta ^f_{n+1}):\Sigma ^{\infty}B\rightarrow \Sigma ^{\infty}C_{\kappa_n} \end{align*}

is trivial. If no such integer k exists, then $\sigma {\mathrm{wsecat}}(f;2)=\infty .$

It is evident from its definition that this homotopy invariant serves as a lower bound for the weak sectional category: $\sigma \mbox{wsecat}(f;2)\leqslant \mbox{wsecat}(f).$ Similarly to the approach taken with the weak sectional category [ Reference Garcia-Calcines and Vandembroucq6 ], this invariant can also be described in a more manageable equivalent form. Indeed, if we take the homotopy cofiber of f, ${E}\stackrel{f}{\longrightarrow }B\stackrel{j}{\longrightarrow }C_f$ , then we can consider both the reduced diagonal $\bar{\Delta}^B_{n+1}:B\rightarrow B^{\wedge (n+1)}$ and the induced map $j^{\wedge (n+1)}:B^{\wedge (n+1)}\rightarrow C_f^{\wedge (n+1)}$ . We can denote their composite as follows:

\begin{align*}\bar{\Delta }^f_{n+1}\,:\!=\,j^{\wedge (n+1)}\bar{\Delta}^B_{n+1}:B\rightarrow C_f^{\wedge (n+1)}.\end{align*}

Proposition 2·4. Let $f:E\rightarrow B$ be a map where B is a finite dimensional CW-complex. Then, $\sigma \mbox{wsecat}(f;2)\leqslant n$ if, and only if, the stable map

\begin{align*}2\Sigma ^{\infty }(\bar{\Delta }^f_{n+1}):\Sigma ^{\infty }B\rightarrow \Sigma ^{\infty }C_f^{\wedge (n+1)}\end{align*}

is trivial.

Proof. By following the analogous steps outlined for the weak sectional category in [ Reference Garcia-Calcines and Vandembroucq6 ], one simply needs to consider the following homotopy commutative diagram. The top square of the diagram represents a homotopy pushout, and as a result, the induced map $C_{\kappa _n}\rightarrow C_f^{\wedge (n+1)}$ between the corresponding homotopy cofibers becomes a homotopy equivalence:

Applying conveniently the suspension functor we obtain the result. We leave the details to the reader.

A special case of $\sigma \mbox{wsecat}({-};2)$ is $\sigma \mbox{wcat}({-};2)$ for finite dimensional CW-complexes.

Definition 2·5. Let B be a finite dimensional CW-complex. Then we define the 2-sigma weak category of B, denoted as $\sigma {\mathrm{wcat}}(B;2),$ as the sigma weak sectional category of the map $*\rightarrow B$ . In other words, it is the smallest non-negative integer n for which the stable map

\begin{align*}2\Sigma ^{\infty} \bar{\Delta }^B_{n+1}:\Sigma ^{\infty}B\rightarrow \Sigma ^{\infty} B^{\wedge (n+1)}\end{align*}

is trivial. If no such integer n exists, then $\sigma {\mathrm{wcat}}(B;2)=\infty .$

Recall that the conilpotency of a space B, usually denoted as $\mbox{conil}(B)$ , is defined as the least integer n (or infinity) such that the suspension of the $(n+1)$ th reduced diagonal map, $\Sigma \bar{\Delta }_{n+1}^B:\Sigma B\rightarrow \Sigma B^{\wedge (n+1)}$ , is homotopically trivial. By the very definition of $\sigma \mbox{wcat}({-};2),$ we have a chain of inequalities

\begin{align*}\sigma \mbox{wcat}(B;2)\leqslant \mbox{conil}(B)\leqslant {\mathrm{wcat}} (B)\leqslant {\mathrm{cat}} (B).\end{align*}

Corollary 2·6. If $f:E\rightarrow B$ is a map where B is a finite dimensional CW-complex, then

\begin{align*}\sigma {\mathrm{wsecat}}(f;2)\leqslant \sigma {\mathrm{wcat}}(B;2).\end{align*}

We also have an analogous result to that given for weak category as in Proposition 1·5. Indeed,

Proof. We take N sufficiently large so that $[\Sigma^{\infty}A,\Sigma^{\infty}A^{\wedge k}]\cong [\Sigma^{N}A,\Sigma^{N}A^{\wedge k}]$ and $[\Sigma^{\infty}B,\Sigma^{\infty}B^{\wedge k}]\cong [\Sigma^{N}B,\Sigma^{N}B^{\wedge k}]$ . Since $f^{\wedge k}$ is an $(r+p(k-1))$ -equivalence, we deduce that $\Sigma^Nf^{\wedge k}$ is an $(r+p(k-1)+N)$ -equivalence. Furthermore, $\dim(\Sigma^NA)=\dim(A)+N$ , and thus

\begin{align*}(\Sigma ^Nf^{\wedge k})_*:[\Sigma ^{N}A,\Sigma ^{N}A^{\wedge k}]\rightarrow [\Sigma ^{N}A,\Sigma ^{N}B^{\wedge k}]\end{align*}

is injective. Now, since $\sigma wcat(A;2)\geqslant k$ , we have $2\Sigma^N(\bar {\Delta}_k^A)\not \simeq *$ . Taking into account the injection above and the following commutative diagram

we conclude that $2\Sigma^N(\bar {\Delta}_k^B)\not \simeq *$ , implying $\sigma {\mathrm{wcat}}(B;2)\geqslant k$ .

Continuing the parallelism with the weak sectional category, we will now explore the relationship with the homotopy cofiber of the map when it admits a homotopy retraction.

Proposition 2·8. Let $f:E\rightarrow B$ be a map between finite dimensional CW-complexes. If f admits a homotopy retraction, then

\begin{align*}\sigma {\mathrm{wsecat}}(f;2)=\sigma {\mathrm{wcat}}(C_f;2).\end{align*}

Now suppose that $\sigma \mbox{wsecat}(f;2)\leqslant n$ . We choose N sufficiently large so that

\begin{align*}[\Sigma ^{\infty }B,\Sigma ^{\infty }C_f^{\wedge (n+1)}]\cong [\Sigma ^{N}B,\Sigma ^{N}C_f^{\wedge (n+1)}]\end{align*}

and $[\Sigma ^{\infty }C_f,\Sigma ^{\infty }C_f^{\wedge (n+1)}]\cong [\Sigma ^{N}C_f,\Sigma ^{N}C_f^{\wedge (n+1)}]$ . Taking into account the commutative diagram

we have that the following composite is homotopically trivial

\begin{align*}(2\Sigma ^N(\bar{\Delta }_{n+1}^{C_f}))(\Sigma ^Nj)=(\Sigma ^Nj^{\wedge (n+1)})(2\Sigma ^N(\bar{\Delta }_{n+1}^B))=2\Sigma ^N(\bar{\Delta }_{n+1}^f)\simeq *.\end{align*}

Therefore, if we denote as $\delta :C_f\rightarrow \Sigma E$ the induced map in the homotopy cofiber of $j:B\rightarrow C_f,$ then there is a map $\phi$ making commutative the following diagram

However, from the Barrat–Puppe sequence

\begin{align*}E\stackrel{f}{\longrightarrow}B\stackrel{j}{\longrightarrow }C_f\stackrel{\delta }{\longrightarrow }{\Sigma E}\stackrel{\Sigma f}{\longrightarrow }{\Sigma B}{\longrightarrow }\cdots\end{align*}

we deduce that $\delta \simeq (\Sigma r)(\Sigma f)\delta \simeq *,$ where r stands for the homotopy retraction of f. We conclude that $2\Sigma ^N(\bar{\Delta }^{n+1}_{C_f})\simeq *$ , that is, $\sigma \mbox{wcat}(C_f;2)\leqslant n.$

A particular case of $\sigma {\mathrm{wsecat}}({-};2)$ that is specially relevant in our work is as follows:

Definition 2·9. The 2-sigma weak topological complexity of a finite dimensional CW-complex X is defined as the 2-sigma weak sectional category of the diagonal map $\Delta \,:\,X\rightarrow X\times X:$

\begin{align*}\sigma {\mathrm{wTC}}(X;\,2)\,:\!=\,\sigma {\mathrm{wsecat}}(\Delta;\,2).\end{align*}

As a direct consequence of this definition and all the previously established results, we obtain the following corollary:

Corollary 2·10. Let X be a finite dimensional CW-complex. Then

\begin{align*}\sigma {\mathrm{wTC}}(X;2)=\sigma {\mathrm{wcat}}(C_{\Delta }(X);2)\leqslant {\mathrm{wcat}}(C_{\Delta }(X))={\mathrm{wTC}}(X).\end{align*}

And now we are ready to state and prove the main result of this subsection, considering the even-dimensional case.

Theorem 2·11. Suppose X is a finite dimensional $(p-1)$ -connected CW-complex $(p\geqslant 2)$ satisfying $\sigma {\mathrm{wTC}}(X;2)=k\geqslant 2$ . If $\dim(X)\leqslant pk-1,$ then we have

\begin{align*}{\mathrm{wTC}}(X\times S^n)\geqslant \sigma {\mathrm{wTC}}(X;2)+2\end{align*}

for all n even. If, in addition, $\sigma {\mathrm{wTC}}(X;2)={\mathrm{TC}}(X)$ , then ${\mathrm{TC}}(X\times S^n)={\mathrm{TC}}(X)+{\mathrm{TC}}(S^n)$ , for all n, independently of its parity.

Proof. Since $\sigma {\mathrm{wTC}} (X;2)=\sigma {\mathrm{wcat}} (C_{\Delta }(X);2)$ , we can deduce that the map

\begin{align*}2\Sigma ^{\infty }(\bar{\Delta }_k):\Sigma ^{\infty }C_{\Delta }(X)\rightarrow \Sigma ^{\infty }C_{\Delta }(X)^{\wedge k}\end{align*}

is nontrivial. Additionally, applying the Freudenthal suspension theorem, we obtain that, $2\Sigma ^m(\bar{\Delta }_k):\Sigma ^{m}C_{\Delta }(X)\rightarrow \Sigma ^mC_{\Delta }(X)^{\wedge k}$ is not homotopy trivial for all m, considering that $\dim (X)\leqslant pk-1$ .

To establish the inequality ${\mathrm{wTC}}(X\times S^n)\geqslant k+2$ , we can follow the same steps as in Theorem 2·2 for the case of odd dimensions. In this sense it suffices to demonstrate that the suspension of the map $\bar{\Delta }_{k+2}:C_{\Delta }(X\times S^n)\rightarrow C_{\Delta }(X\times S^n)^{\wedge (k+2)}$ is not homotopically trivial. To this aim we first construct a similar diagram

By applying the suspension functor to this diagram, we obtain

where the curved arrows are homotopy sections. By a completely analogous argument to the one made in the proof of Theorem 2·2, ultimately we must verify that the bottom map $\Sigma \bar{\Delta }_k\wedge \bar{\Delta }:\Sigma C_{\Delta}(X)\wedge C_{\Delta}(S^n)\rightarrow \Sigma (C_{\Delta}(X))^{\wedge k}\wedge C_{\Delta}(S^n)^{\wedge 2} $ , is non-trivial. This map can be considered, up to homeomorphism, as

\begin{align*}\bar{\Delta }_k\wedge \Sigma \bar{\Delta }:C_{\Delta}(X)\wedge \Sigma C_{\Delta}(S^n)\rightarrow (C_{\Delta}(X))^{\wedge k}\wedge \Sigma C_{\Delta}(S^n)^{\wedge 2}.\end{align*}

Moreover, it produces, according to the argument at the end of Section 2, the following homotopy commutative diagram, where the curved arrow on the left represents a homotopy section, and the curved arrow on the right represents a homotopy retraction:

Once again, the top arrow in the diagram is non-trivial in homotopy as long as the bottom arrow remains non-trivial in homotopy. However, the bottom arrow can be equivalently expressed as $\pm 2\Sigma ^{2n+1}(\bar{\Delta }_k):\Sigma ^{2n+1}C_{\Delta }(X)\rightarrow \Sigma ^{2n+1}C_{\Delta}(X)^{\wedge k},$ which, according to our assumptions, is not homotopy trivial. The remainder of the proof is straightforward. Note that the equality $\sigma {\mathrm{wTC}}(X;2)={\mathrm{TC}}(X)$ implies ${\mathrm{wTC}}(X)={\mathrm{TC}}(X)$ . Therefore the equality ${\mathrm{TC}}(X\times S^n)={\mathrm{TC}}(X)+{\mathrm{TC}}(S^n)$ for n odd follows from Theorem 2·2.