Finiteness of homotopy groups related to the non-abelian tensor product

TL;DR

This paper investigates conditions under which homotopy groups are finite, using properties of the non-abelian tensor product, and provides quantitative and application-based insights into classical theorems.

Contribution

It establishes new finiteness criteria for homotopy groups based on non-abelian tensor products and extends classical theorems with quantitative versions.

Findings

Finiteness conditions for homotopy groups in terms of tensors

A quantitative version of the Blakers-Massey theorem

Applications to homotopy pushouts and Eilenberg-MacLane spaces

Abstract

By using finiteness related result of non-abelian tensor product we prove finiteness conditions for the homotopy groups $\pi_n(X)$ in terms of the number of tensors. In particular, we establish a quantitative version of the classical Blakers-Massey triad connectivity theorem. Moreover, we study others finiteness conditions and equivalence properties that arise from the non-abelian tensor square. In the end, we give applications to homotopy pushout, especially in the case of Eilenberg-MacLane spaces.

Equations

$${\pi }_3(X,A,B)\cong {\pi }_2(A,C)\otimes {\pi }_2(B,C),$$

$${\pi }_{p+1}(A,A\cap B)\otimes {\pi }_{q+1}(B,A\cap B).$$

$$G\times H\to G,(g,h)\mapsto g^h;H\times G\to H,(h,g)\mapsto h^g$$

$$g^{\left(h^{g_1}\right)}={\left({\left(g^{g_1^{-1}}\right)}^h\right)}^{g_1}\text{mbox}and{h}^{\left(g^{h_1}\right)}={\left({\left(h^{h_1^{-1}}\right)}^g\right)}^{h_1}.$$

$$\begin{array}[]{ll}{\eta}(G,H)=\langle G\cup H^{\varphi}\ |&[g,{h}^{\varphi}]^{g_{1}}=[{g}^{g_{1}},({h}^{g_{1}})^{\varphi}],\;[g,{h}^{\varphi}]^{h^{\varphi}_{1}}=[{g}^{h_{1}},({h}^{h_{1}})^{\varphi}],\\ &\ \forall g,g_{1}\in G,\;h,h_{1}\in H\rangle.\end{array}$$

$$\eta (G,H)=\left([G,H^{\phi }]\cdot G\right)\cdot H^{\phi },$$

$${\pi }_3(SK(G,1))\cong J_2(G),$$

$$\cdots \to {\pi }_n(B,C)\to {\pi }_n(X,A)\to {\pi }_n(X,A,B)\to {\pi }_{n-1}(B,C)\to \cdots .$$

$${\pi }_{p+q+1}(X,A,B)\cong {\pi }_{p+1}(A,A\cap B)\otimes {\pi }_{q+1}(B,A\cap B),$$

$${\pi }_n(B)\to {\pi }_n(B,C)\to {\pi }_{n-1}(C)$$

$${\pi }_n(A)\to {\pi }_n(X)\to {\pi }_n(X,A)$$

$${\pi }_n(B,C)\to {\pi }_n(X,A)\to {\pi }_n(X,A,B).$$

$$\cdots \to {\pi }_2(X)\to {\pi }_3(SX)\to {\pi }_2(\Omega SX,X)\to \cdots$$

$$0\to J_2(G)\to [G,G^{\phi }]\to G^{\prime }\to 1,$$

$$1\to \Delta (G)\to J_2(G)\to H_2(G)\to 1,$$

$$G^{ab}=T\times F,$$

$$\Delta (T)\times \Delta (F)\times (T{\otimes }_{\mathbb{Z}}F),$$

$$1\to J_2(G)\to [G,G^{\phi }]\to G^{\prime }\to 1,$$

$$1\to \tilde{\Delta }(G)\to J_2(G)\to {\pi }_4(S^{2}X)\to 1.$$

$$C\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$F(X)\to F(g)\to F(a)$$

$$F(a)\to A\to X,$$

$$K(G,1)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

This work was partially supported by FAPDF - Brazil, Grant: 0193.001344/2016.

1. Introduction

In [8] Brown and Loday presented a topological significance for the non-abelian tensor product of groups. The non-abelian tensor product is used to describe the third relative homotopy group of a triad as a non-abelian tensor product of the second homotopy groups of appropriate subspaces. More specifically, in [8, Corollary 3.2], the third triad homotopy group is

[TABLE]

where $X$ is a pointed space and $\{A,B\}$ is an open cover of $X$ such that $A,B$ and $C=A\cap B$ are connected and $(A,C),(B,C)$ are $1$-connected.

In [12], Ellis gave a finiteness criterion for the triad homotopy group in terms of the finiteness of the involved groups (see also [2, 3]). More generally, finiteness conditions to $\pi_{n}(X)$ when excision theorem holds is given by the finiteness of $\pi_{n}(A)$, $\pi_{n}(B)$ and $\pi_{n-1}(C)$. However, when the excision property does not hold, the failure is mesured by the triad homotopy groups $\pi_{n}(X,A,B)$ with $n\geq 3$. Therefore, finiteness of $\pi_{n}(X)$ also depends on the triad homotopy groups $\pi_{n}(X,A,B)$. Thus, in order to give a bound for $\pi_{n}(X)$, it is needed to study finiteness conditions on $\pi_{n}(X,A,B)$. In [9], another application of the non-abelian tensor product is done by Brown and Loday, where they extended the classical Blakers-Massey triad connectivity theorem, which states that if $A,B,$ and $A\cap B$ are connected, $\{A,B\}$ is an open cover of $X$, $(A,A\cap B)$ is p-connected, and $(B,A\cap B)$ is q-connected, then $\pi_{p+q+1}(X,A,B)$ is isomorphic to the non-abelian tensor product

[TABLE]

The hypothesis $p,q\geq 2$ is broadened to $p,q\geq 1$, and the hypothesis $\pi_{1}(A\cap B)=0$ is removed.

For the convenience of the reader we repeat the relevant definitions (cf. [2, 3, 16]). Let $G$ and $H$ be groups each of which acts upon the other (on the right),

[TABLE]

and on itself by conjugation, in such a way that for all $g,g_{1}\in G$ and $h,h_{1}\in H$,

[TABLE]

In this situation we say that $G$ and $H$ act compatibly on each other. Let $H^{\varphi}$ be an extra copy of $H$, isomorphic via $\varphi:H\rightarrow H^{\varphi},\;h\mapsto h^{\varphi}$, for all $h\in H$. Consider the group $\eta(G,H)$ defined in [16] as

[TABLE]

It is a well known fact (see [16, Proposition 2.2]) that the subgroup $[G,H^{\varphi}]$ of $\eta(G,H)$ is canonically isomorphic with the non-abelian tensor product $G\otimes H$, as defined by Brown and Loday in their seminal paper [8], the isomorphism being induced by $g\otimes h\mapsto[g,h^{\varphi}]$ (see also Ellis and Leonard [13]). It is clear that the subgroup $[G,H^{\varphi}]$ is normal in $\eta(G,H)$ and one has the decomposition

[TABLE]

where the dots mean (internal) semidirect products. We observe that when $G=H$ and all actions are conjugations, $\eta(G,H)$ becomes the group $\nu(G)$ introduced in [20]. Recall that an element $\alpha\in\eta(G,H)$ is called a tensor if $\alpha=[a,b^{\varphi}]$ for suitable $a\in G$ and $b\in H$. We write $\operatorname{T_{\otimes}}(G,H)$ to denote the set of all tensors (in $\eta(G,H)$). When $G=H$ and all actions are by conjugation, we simply write $\operatorname{T_{\otimes}}(G)$ instead of $\operatorname{T_{\otimes}}(G,G)$. A number of structural results for the non-abelian tensor product of groups (and related constructions) in terms of the set of tensors where presented in [2, 3, 4, 5, 21]).

Our contribution is to give finiteness conditions and bounds to the triad homotopy groups $\pi_{n}(X,A,B)$, furthermore finiteness condition and bound is given to $\pi_{n}(X)$. We establish the following related results.

Theorem A**.**

Let $X$ be a union of open subspaces $A$, $B$ such that $A,B$ and $C=A\cap B$ are path-connected, and the pairs $(A,C)$ and $(B,C)$ are respectively $p$-connected and $q$-connected. Suppose that $\pi_{n}(A)$, $\pi_{n}(B)$, $\pi_{n-1}(C)$ and the set of tensors $\operatorname{T_{\otimes}}(\pi_{p+1}(A,C),\pi_{q+1}(B,C))$ are finite, where $n=p+q+1$. Then $\pi_{n}(X)$ is a finite group with $\{a,b,c,m\}$-bounded order, where $|\pi_{n}(A)|=a$, $|\pi_{n}(B)|=b$, $|\pi_{n-1}(C)|=c$ and $m=|\operatorname{T_{\otimes}}(\pi_{p+1}(A,C),\pi_{q+1}(B,C))|$.

An application of Theorem A is that $\pi_{3}(K(C_{r^{\infty}},2)\vee K(C_{s^{\infty}},2))$ is trivial, where $``\vee"$ is the wedge sum, $r$ and $s$ are primes, and $K(G,n)$ is an Eilenberg-MacLane space (i.e., a topological space having just one nontrivial homotopy group $\pi_{n}(K(G,n))\cong G$). See also Remark 2.1 and Corollary 2.2 below.

In [8], Brown and Loday shown that the third homotopy group of the suspension of an Eilenberg-MacLane space $K(G,1)$ satisfies

[TABLE]

where $J_{2}(G)$ denotes the kernel of the derived map $\kappa:[G,G^{\varphi}]\to G^{\prime}$, given by $[g,h^{\varphi}]\mapsto[g,h]$ (cf. [21, Chapter 2 and 3]). Many authors had studied bounds to the order of $\pi_{3}(SK(G,1))$ (cf. [1, 6, 7, 17]). Now, we can deduce a finiteness criterion for $\pi_{3}(SX)$ in terms of $\pi_{2}(X)$ and the number of tensors $\operatorname{T_{\otimes}}(G)$, where $\pi_{1}(X)\cong G$ and $SX$ is the suspension of a space $X$ (see Remark 2.4, below).

Theorem B**.**

Let $X$ be a connected space and $\pi_{1}(X)=G$. Suppose that the set of tensors $\operatorname{T_{\otimes}}(G)$ has exactly $m$ tensors in $\nu(G)$ and $\pi_{2}(X)$ is finite with $|\pi_{2}(X)|=a$. Then $\pi_{3}(SX)$ is a finite group with $\{a,m\}$-bounded order.

It is well known that the finiteness of the non-abelian tensor square $[G,G^{\varphi}]$ does not imply the finiteness of the group $G$ (see Remark 2.7(b), below). In [18], Parvizi and Niroomand prove that if $G$ is a finitely generated subgroup and the non-abelian tensor square $[G,G^{\varphi}]$ is finite, then $G$ is finite. We obtain equivalence conditions (see the following theorem) and a topological related result (see Corollary 3.3).

Theorem C**.**

Let $G$ be a finitely generated group. The following properties are equivalents.

• (a)

The group $G$ is finite;

• (b)

The set of tensors $\operatorname{T_{\otimes}}(G)$ is finite;

• (c)

The non-abelian tensor $[G,G^{\varphi}]$ is finite;

• (d)

The derived subgroup $G^{\prime}$ is locally finite and the kernel $J_{2}(G)\cong\pi_{3}(SK(G,1))$ is periodic;

• (e)

The derived subgroup $G^{\prime}$ is locally finite and the Diagonal subgroup $\Delta(G)$ is periodic;

• (f)

The derived subgroup $G^{\prime}$ is locally finite and the subgroup $\tilde{\Delta}(G)=\langle[g,h^{\varphi}][h,g^{\varphi}]\mid g,h\in G\rangle$ is periodic;

• (g)

The non-abelian tensor square $[G,G^{\varphi}]$ is locally finite.

In algebraic topology, non-abelian tensor product arises from a homotopy pushout, see [8, 10]. The homotopy pushout (or homotopy amalgameted sum) is well known for the application in the classical Seifert-van Kampen theorem as well as Higher Homotopy Seifert-van Kampen theorem in the case of a covering by two open sets. Our contribution is to apply the idea of Theorem A and related construction on tensor product in order to obtain finiteness results related to homotopy pushout. For instance, see Proposition 4.2 for an application to homotopy pushout of Eilenberg-Maclane spaces.

The paper is organized as follows. In the next section we describe finiteness criteria for the group $\pi_{n}(X)$ in terms of the number of tensors. In particular, we establish a quantitative version of the classical Blakers-Massey triad connectivity theorem. In the third section we examine some finite necessary conditions for the group $G$ in terms of certain torsion elements of the non-abelian tensor square $[G,G^{\varphi}]$. In the final section, as an application we obtain finiteness criteria for the homotopy pushout that depends on the number of tensor of non-abelian tensor product of groups.

2. Finiteness Conditions

Let $(X,A,B)$ be a triad, that is, $A$ and $B$ are subspaces of $X$, containing the base-point in $C=A\cap B$, such that the triad homotopy group $\pi_{n}(X,A,B)$ for $n\geq 3$ fit into a long exact sequence

[TABLE]

Let $X$ be a pointed space and $\{A,B\}$ an open cover of $X$ such that $A,B$ and $C=A\cap B$ are connected and $(A,C),(B,C)$ are 1-connected, see [22].

By using the relative homotopy long exact sequences and the third triad homotopy group, Ellis and McDermott obtain interesting bound to the order of $\pi_{3}(X)$ (cf. [14, Proposition 5]). We have obtained (as a consequence of Theorem A) a more general version. In contrast to their bound, we do not require that $\pi_{2}(A,C)$ and $\pi_{2}(B,C)$ be finite groups and also there is no need to estimate $\pi_{2}(X,C)$.

Following the same setting, we apply the extended Blakers-Massey triad connectivity [9, Theorem 4.2]

[TABLE]

in order to present finiteness condition and bound to $\pi_{n}(X)$ in terms of set of tensors $\operatorname{T_{\otimes}}(\pi_{p+1}(A,C),\pi_{q+1}(B,C)),$ where $n=p+q+1$.

Proof of Theorem A.

Consider the relative homotopy long exact sequences, as in [22],

[TABLE]

[TABLE]

[TABLE]

By the first exact sequence, we deduce that $\pi_{n}(B,C)$ is a finite group with $\{b,c\}$-bounded order. According to Brown-Loday’s result [9, Theorem 4.2], the group $\pi_{n}(X,A,B)$ is isomorphic to the non-abelian tensor product $M\otimes N$, where $M=\pi_{p+1}(A,C)$ and $N=\pi_{q+1}(B,C)$. As $|T_{\otimes}(\pi_{p+1}(A,C),\pi_{q+1}(B,C))|=m$, we have $\pi_{n}(X,A,B)$ is finite with $m$-bounded order ([3, Theorem B]). From this we conclude that the group $\pi_{n}(X,A)$ is finite with $\{b,m\}$-bounded order. In the same manner we can see that $\pi_{n}(X)$ is finite with $\{a,b,c,m\}$-bounded order. The proof is complete. ∎

Remark 2.1**.**

A direct application of Theorem A is that $\pi_{3}(K(G,2)\vee K(H,2))$ is a finite group with $m$-bounded order, when $|\operatorname{T_{\otimes}}(G,H)|=m$. In particular, $\pi_{3}(K(C_{r^{\infty}},2)\vee K(C_{s^{\infty}},2))$ is trivial, where $s$ and $r$ are primes and $K(G,n)$ is Eilenberg-MacLane space (a topological space having just one nontrivial homotopy group $\pi_{n}(K(G,n))\cong G$).

By using the same idea of the previous remark we have the following corollary.

Corollary 2.2**.**

Let $A$ and $B$ be $p$-connected and $q$-connected locally contractible spaces, respectively. Suppose that $|\pi_{n}(A)|=a$, $|\pi_{n}(B)|=b$ and $|\operatorname{T_{\otimes}}(\pi_{p+1}(A),\pi_{q+1}(B))|=m$, where $n=p+q+1$. Then $\pi_{n}(A\vee B)$ is finite with $\{a,b,m\}$-bounded order.

The previous corollary is an example that we can use the non-abelian tensor product to overcome the failure of the excision property, which reflects in $\pi_{n}(A\vee B)$ being different of $\pi_{n}(A)\oplus\pi_{n}(B)$ in general, where $n\geq 2$.

The following result provide a finiteness criterion to the group $G$ in terms of the number of tensors in the non-abelian tensor square $[G,G^{\varphi}]$.

Corollary 2.3**.**

Let $X$ be connected space and $\pi_{1}(X)=G$. Suppose that the first homology group of $X$, $H_{1}(X,\mathbb{Z})$, is finitely generated and the set of tensors $\operatorname{T_{\otimes}}(G)\subseteq\nu(G)$ has exactly $m$ tensors. Then $H_{1}(X,\mathbb{Z})$ and $\pi_{1}(X)=G$ are finite with $m$-bounded orders.

Proof.

Recall that $X$ is a connected space and $\pi_{1}(X)=G$. Suppose that the first homology of $X$, $H_{1}(X,\mathbb{Z})$, is finitely generated and the set of tensors $\operatorname{T_{\otimes}}(G)\subseteq\nu(G)$ has exactly $m$ tensors. We need to prove that $H_{1}(X,\mathbb{Z})$ and $\pi_{1}(X)=G$ are finite with $m$-bounded order.

By Theorem A, the non-abelian tensor square $[G,G^{\varphi}]$ is finite with $m$-bounded order. Consequently, the derived subgroup $G^{\prime}$ is finite with $m$-bounded order. Since $G^{ab}$ is finitely generated, it follows that the abelianization $G^{ab}$ is isomorphic to an subgroup of the non-abelian tensor square $[G,G^{\varphi}]$ ([2, Theorem C (a)]). Therefore the abelianization $G^{ab}$ is finite with $m$-bounded order. Consequently, $G$ is finite with $m$-bounded order. The proof is complete. ∎

In the case of the suspension triad $(SX;C_{+}X,C_{-}X)$, see [8, 22], we can obtained finiteness criteria and bounds to the order of $\pi_{3}(SX)$, where $C_{-}X$ and $C_{+}X$ are the two cones of $X$ in $SX$.

Proof of Theorem B.

Consider the long exact sequence

[TABLE]

where $\Omega SX$ is the space of loops in $SX$, maps from the circle $S^{1}$ to $SX$, equipped with the compact–open topology. The result follows by applying Theorem A on $[G,G^{\varphi}]\cong\pi_{2}(\Omega SX,X)$. ∎

Remark 2.4**.**

In the above result, it is worth noting that $\pi_{2}(X)$ does not need to be trivial, therefore the result is more general compared to the bounds for $\pi_{3}(SK(G,1))$ when $X=K(G,1)$. However, in [8], proved that if $\pi_{1}(X)=G$ and $\pi_{2}(X)$ is trivial, then $\pi_{3}(SX)\cong J_{2}(G)=\ker(\kappa)$, where $\kappa:[G,G^{\varphi}]\to G^{\prime}$, given by $[g,h^{\varphi}]\mapsto[g,h]$.

Combining the above bounds to the order of the non-abelian tensor square and [8, Proposition 4.10] we can obtained, under appropriate conditions in the set of tensors $\operatorname{T_{\otimes}}(G)$, some finiteness criteria and bounds to the orders of $\pi_{3}(SK(G,1))$ and $\pi_{2}^{S}(K(G,1))$ the second stable homotopy of Eilenberg-MacLane space.

Corollary 2.5**.**

Let $G$ be a group. Suppose that the set $\operatorname{T_{\otimes}}(G)$ has exactly $m$ tensors in $\nu(G)$. Then

• (a)

The second stable homotopy group $\pi_{2}^{S}(K(G,1))$ is finite with $m$-bounded order;

• (b)

$\pi_{3}(SK(G,1))$* is finite with $m$-bounded order.*

A particular case of the previous corollary is when $G$ is a Prüfer group $C_{p^{\infty}}$ so $\pi_{2}^{S}(K(G,1))$ and $\pi_{3}(SK(G,1))$ are trivial groups.

Next we give a sufficient and necessary conditions to the suspension of Eilenberg-Maclane space $K(G,1)$ be a finite group.

Proposition 2.6**.**

Let G be a BFC-group such that $G^{ab}$ is finitely generated. Then, $\pi_{3}(SK(G,1))$ is finite if and only if $\pi_{1}(K(G,1))$ is finite.

Proof.

Assume that $J_{2}(G)\cong\pi_{3}(SK(G,1))$ is finite. We have the following short exact sequence

[TABLE]

since $G^{\prime}$ is finite by Neumann’s Theorem [19, 14.5.11], we have that the non-abelian tensor square $[G,G^{\varphi}]$ is finite. Since $G^{ab}$ is finitely generated, it follows that the abelianization $G^{ab}$ is isomorphic to a subgroup of the non-abelian tensor square $[G,G^{\varphi}]$ ([2, Theorem C (a)]). Moreover, we can deduce that the abelianization $G^{ab}$ is a subgroup of the diagonal subgroup $\Delta(G)$. From this we deduce that the abelianization $G^{ab}$ is finite and so, $G$ is finite.

Conversely, suppose that $\pi_{1}(K(G,1))$ is finite. Consequently, the non-abelian tensor square $[G,G^{\varphi}]$ is finite and so, $\pi_{3}(SK(G,1))$ is finite. The proof is complete. ∎

Remark 2.7**.**

(a)  Assume that $G$ is a BFC-group and the abelianization $G^{ab}$ is finitely generated. Consider the following short exact sequence (cf. [21, Section 2]),

[TABLE]

*where $J_{2}(G)$ is isomorphic to $\pi_{3}(SK(G,1))$ and $H_{2}(G)$ is the second homology group. Recall that the Diagonal subgroup $\Delta(G)$ is given by $\Delta(G)=\langle[g,g^{\varphi}]\mid g\in G\rangle$. The subgroup $\Delta(G)$ is finite if and only if $\pi_{1}(K(G,1))$ is finite (if and only if the group $\pi_{3}(SK(G,1))$ is finite). See [8, Section 2] for more details.

(b) Note that the hypothesis of the finitely generated abelianization is really needed. For instance, the Prufer group $G=C_{p^{\infty}}$ is an infinite group such that the non-abelian tensor square $[G,G^{\varphi}]$ is trivial and so, finite. In particular, $J_{2}(G)\cong\pi_{3}(SK(G,1))$ is also trivial.

A direct application of Corollary 2.3 and Corollary 2.5 for the suspension of Eilenberg-Maclane space $K(G,1)$ and the second stable homotopy of $K(G,1)$ is the following finiteness condition.

Corollary 2.8**.**

Let $G$ be a group. Then, $\pi_{3}(SK(G,1))$ and $G^{\prime}$ are finite if and only if $\operatorname{T_{\otimes}}(G)$ is finite. Moreover, suppose that $G$ is perfect then, $\pi_{2}^{S}(K(G,1))$ and $G$ are finite if and only if $\operatorname{T_{\otimes}}(G)$ is finite.

3. Torsion elements in the non-abelian tensor square

This section is will be devoted to obtain some finiteness conditions for the group $G$ in terms of the torsion elements in the non-abelian tensor square. Specifically, our proofs involve looking at the description of the diagonal subgroup $\Delta(G)\leqslant[G,G^{\varphi}]$. Such a description has previously been used by the authors [2, 21].

It is well known that the finiteness of the non-abelian tensor square $[G,G^{\varphi}]$ does not imply the finiteness of the group $G$ (see Remark 2.7, above). In [18], Parvizi and Niroomand prove that if $G$ is a finitely generated subgroup and the non-abelian tensor square $[G,G^{\varphi}]$ is finite, then $G$ is finite. Later, in [2], the authors prove that if $G$ is a finitely generated locally graded group and the exponent of the non-abelian tensor square $\exp([G,G^{\varphi}])$ is finite, then $G$ is finite. The next result can be viewed as a generalization of the above result.

Theorem C**.**

Let $G$ be a finitely generated group. The following properties are equivalents.

• (a)

The group $G$ is finite;

• (b)

The set of tensors $\operatorname{T_{\otimes}}(G)$ is finite;

• (c)

The non-abelian tensor $[G,G^{\varphi}]$ is finite;

• (d)

The derived subgroup $G^{\prime}$ is locally finite and the kernel $J_{2}(G)\cong\pi_{3}(SK(G,1))$ is periodic;

• (e)

The derived subgroup $G^{\prime}$ is locally finite and the Diagonal subgroup $\Delta(G)$ is periodic;

• (f)

The derived subgroup $G^{\prime}$ is locally finite and the subgroup $\tilde{\Delta}(G)$ is periodic;

• (g)

The non-abelian tensor square $[G,G^{\varphi}]$ is locally finite.

Proof.

$(a)\Rightarrow(b)$ and $(c)\Rightarrow(d)\Rightarrow(e)\Rightarrow(f)$ are directly.

$(b)\Rightarrow(c).$ Suppose that $\operatorname{T_{\otimes}}(G)$. By [2, Theorem A], the non-abelian tensor square $[G,G^{\varphi}]$ is finite.

$(f)\Rightarrow(g)$. By Schmidt’s Theorem [19, 14.3.1], it suffices to prove that the abelianization $G^{ab}$ is finite.

Since $G^{ab}$ is finitely generated, we deduce that

[TABLE]

where $T$ is the torsion part and $F$ the free part of $G^{ab}$ (cf. [19, 4.2.10]). If the abelianization $G^{ab}$ is not periodic, then there exists an element of infinite order $x\in G$ such that $xG^{ab}\in F$. In particular, $[x,x^{\varphi}][x,x^{\varphi}]=[x,x^{\varphi}]^{2}$ is an infinite element in $\tilde{\Delta}(G)$. Consequently, $F$ is trivial and $G^{ab}=T$ is finite.

$(g)\Rightarrow(a).$ First we prove that the abelianization $G^{ab}$ is finite. Arguing as in the above paragraph, we deduce that $G^{ab}=T\times F,$ where $T$ is the torsion part and $F$ the free part of $G^{ab}$. From [21, Remark 5] we conclude that $\Delta(G^{ab})$ is isomorphic to

[TABLE]

where $T\otimes_{\mathbb{Z}}F$ is the usual tensor product of $\mathbb{Z}$-modules. In particular, the free part of $\Delta(G^{ab})$ is precisely $\Delta(F)$. Now, the canonical projection $G\twoheadrightarrow{G^{ab}}$ induces an epimorphism $q:\Delta(G)\twoheadrightarrow{\Delta(G^{ab})}$. Since $\Delta(G)$ is locally finite, it follows that $\Delta(G^{ab})$ is also locally finite. Consequently, $F$ is trivial and thus $G^{ab}$ is periodic and, consequently, finite.

It remains to prove that the derived subgroup $G^{\prime}$ is finite. Since $G$ is finitely generated and $G^{ab}$ is finite, it follows that the derived subgroup $G^{\prime}$ is finitely generated ([19, 1.6.11]). As $G^{\prime}$ is an homomorphic image of the non-abelian tensor square $[G,G^{\varphi}]$, we have $G^{\prime}$ is finite. From this we deduce that $G$ is finite. The proof is complete. ∎

Proposition 3.1**.**

Let $G$ be a polycyclic-by-finite group. Suppose that the non-abelian tensor $[G,G^{\varphi}]$ is periodic. Then $G$ is finite.

Proof.

Since the derived subgroup $G^{\prime}$ is an epimorphic image of the non-abelian tensor square $[G,G^{\varphi}]$, it follows that $G^{\prime}$ is also periodic. In particular, we can deduce that $G^{\prime}$ is finite. Now, arguing as in the proof of Theorem Theorem C, we deduce that $G^{ab}$ is finite. The proof is complete. ∎

It is well known that if $G$ is a group with exponent $\exp(G)\in\{2,3,4,6\}$, then $G$ is locally finite (Levi-van der Waerden, Sanov, see [19, Section 14.2] for more details). We also examine the finiteness of the group $G$, when the non-abelian tensor square $[G,G^{\varphi}]$ has small exponent.

Corollary 3.2**.**

Let $n\in\{2,3,4,6\}$ and $G$ a finitely generated group. Assume that the exponent of the non-abelian tensor square $\exp([G,G^{\varphi}])$ is exactly $n$. Then $G$ is finite.

Proof.

It suffices to see that the non-abelian tensor square $[G,G^{\varphi}]$ is locally finite (see [19, Section 14.2] for more details). By Theorem Theorem C, the group $G$ is finite. The proof is complete. ∎

The following corollary is a topological version of Theorem C.

Corollary 3.3**.**

Let $G$ be a finitely generated group and $X$ be a topological space such that $\pi_{1}(X)=G$ and $\pi_{2}(X)$ is trivial. Then the following properties are equivalents.

• (a)

The group $\pi_{1}(X)=G$ is finite;

• (b)

The derived subgroup $G^{\prime}$ is locally finite and $\pi_{3}(SX)$ is periodic;

• (c)

The derived subgroup $G^{\prime}$ is locally finite and $F(\pi_{3}(SX))\cong F(H_{2}(G))$, where $F(H)$ is the free part of a group $H$;

• (d)

The derived subgroup $G^{\prime}$ is locally finite and $F(\pi_{3}(SX))\cong F(\pi_{4}(S^{2}X))$, where $F(H)$ is the free part of a group $H$.

Proof.

Since $\pi_{2}(X)$ is trivial, it follows that $J_{2}(G)\cong\pi_{3}(SX)$ (cf. [8, Proposition 3.3]). Consider the short exact sequences, as in [8],

[TABLE]

[TABLE]

Therefore, the result follows by applying Theorem C. ∎

4. Application to homotopy pushout

We end this paper by proving some finiteness criteria for the homotopy pushout. Consider the following commutative square of spaces

[TABLE]

and denote $F(f),F(g)$ and $F(a)$ the homotopy fibre of $f,g$ and $a$, respectively, and let $F(X)$ be the homotopy fibre of $F(g)\rightarrow F(a)$. The previous square is called a homotopy pushout when the canonical map of squares from the double mapping cylinder $\mathrm{M}(f,g)$ to $X$ is a weak equivalence of spaces at the four corners, for more details see [8]. For the homotopy pushout, we have an analogous of Theorem A.

Proposition 4.1**.**

Let the following square of spaces be a homotopy pushout

[TABLE]

Suppose that $\pi_{3}(A)$, $\pi_{2}(F(g))$ and the set of tensors $\operatorname{T_{\otimes}}(G,H)$ are finite, where $G=\pi_{1}(F(f))$ and $H=\pi_{1}(F(g))$. Then $\pi_{3}(X)$ is a finite group with $\{n_{a},n_{b},m\}$-bounded order, where $|\pi_{3}(A)|=n_{a}$, $|\pi_{2}F(g)|=n_{b}$ and $|\operatorname{T_{\otimes}}(G,H)|=m$.

Proof.

By [8, Theorem 3.1] we have that $\pi_{1}(F(X))\simeq\pi_{1}(F(f))\otimes\pi_{1}(F(g))$. Since $|T_{\otimes}(\pi_{1}(F(f)),\pi_{1}(F(g)))|=m$ then $\pi_{1}(F(X))$ is finite with $m$-bounded order ([3, Theorem B]). By using $|\pi_{3}(A)|=n_{a}$, $|\pi_{2}F(g)|=n_{b}$ and the long exact sequences of the fibrations

[TABLE]

[TABLE]

it follows that $\pi_{3}(X)$ is a finite group with $\{n_{a},n_{b},m\}$-bounded. ∎

Many authors had studied some finiteness conditions for the non-abelian tensor product of groups (cf. [2, 4, 11, 12, 15, 18]). For instance, in [15], Moravec proved that if $G,H$ are locally finite groups acting compatibly on each other, then so is $G\otimes H$. In [11], Donadze, Ladra and Thomas proved interesting finiteness criteria for non-abelian tensor product in terms of the involved groups. In [2, 3], the authors prove a finiteness criterion for the non-abelian tensor product of groups in terms of the number of tensors. In the case of homotopy pushout for Eilenberg-Maclane spaces is possible to obtain results direct from the study of the non-abelian tensor product of groups. We obtain the following related result.

Proposition 4.2**.**

Let $M,N$ be normal subgroups of a group $G$, and form the homotopy pushout

[TABLE]

• (a)

Suppose that the subgroups $M$ and $N$ are locally finite. Then $\pi_{2}(X)$ and $\pi_{3}(X)$ are locally finite.

• (b)

Suppose that $M$ is a non-abelian free group of finite rank and $N$ is a finite group. Then $\pi_{2}(X)$ and $\pi_{3}(X)$ are finite.

• (c)

Suppose that the set of tensors $\operatorname{T_{\otimes}}(M,N)\subseteq\eta(M,N)$ is finite. Then $\pi_{3}(X)$ is a finite group with $m$-bounded order, where $|\operatorname{T_{\otimes}}(M,N)|=m$.

Proof.

According to Brown and Loday’s result [8, Corollary 3.4], the group $\pi_{3}(X)$ is isomorphic to a subgroup of the non-abelian tensor product $[M,N^{\varphi}]$ and $\pi_{2}(X)$ is isomorphic to $(M\cap N)/[M,N]$. In particular, the group $\pi_{2}(X)$ is an homomorphic image of $M\cap N$.

(a). As $M\cap N$ is locally finite we have $\pi_{2}(X)$ is locally finite. Now, since $M$ and $N$ are locally finite, it follows that the non-abelian tensor product $[M,N^{\varphi}]$ is locally finite (Moravec, [15]). Consequently, the group $\pi_{3}(X)$ is locally finite.

(b). Since $M$ is finite and $N$ is a non-abelian free group, we deduce that $\pi_{2}(X)$ is finite. According to Donadze, Ladra and Thomas’ result [11, Corollary 4.7], we conclude that the non-abelian tensor product $[M,N^{\varphi}]$ is finite and so, $\pi_{3}(X)$ is finite.

(c). According to Theorem A, the non-abelian tensor product $[M,N^{\varphi}]$ is finite with $m$-bounded order. In particular, the group $\pi_{3}(X)$ is finite with $m$-bounded order. The proof is complete. ∎

Remark 4.3**.**

Note that Proposition 4.2 (c) in a certain sense cannot be improved. For instance, if $M=N=C_{p^{\infty}}$, then the group $\pi_{2}(X)\cong C_{p^{\infty}}$ is infinite and $\pi_{3}(X)$ is trivial.

As an interesting consequence of Proposition 4.2, we obtain that: if $G=MN$ such that $M\cap N$ and $[M,N^{\varphi}]$ are trivial, then $X$ is 3-connected, i.e., $\pi_{n}(X)$ is trivial for $n=1,2$ and $3$, see the following example.

Example 4.4**.**

When $G=C_{r^{\infty}}\times C_{s^{\infty}}$ in Proposition 4.2 with $r$ and $s$ primes (not necessarily distinct), we have that $X$ is $3$-connected. In fact, $\pi_{1}(X)$ is trivial by Van Kampen theorem, as well $\pi_{2}(X)=0$ as a consequence of the Van Kampen theorem for maps, and finally the triviality of $\pi_{3}(X)$ follows from the proposition above.