Higher generation by abelian subgroups in Lie groups
Omar Antolín-Camarena, Simon Gritschacher, Bernardo Villarreal
aa r X i v : . [ m a t h . A T ] S e p Higher generation by abelian subgroups in Lie groups
Omar Antol´ın-Camarena, Simon Gritschacher and Bernardo VillarrealSeptember 28, 2020
Abstract
To a compact Lie group G one can associate a space E (2 , G ) akin to the poset of cosets ofabelian subgroups of a discrete group. The space E (2 , G ) was introduced by Adem, F. Cohenand Torres-Giese, and subsequently studied by Adem and G´omez, and other authors. In thisshort note, we prove that G is abelian if and only if π i ( E (2 , G )) = 0 for i = 1 , ,
4. This is a Liegroup analogue of the fact that the poset of cosets of abelian subgroups of a discrete group issimply–connected if and only if the group is abelian.
Suppose that G is a discrete group and F is a family of subgroups of G . One can associate to F asimplicial complex C ( F , G ) whose n -simplices are the chains of cosets g A ⊂ g A ⊂ · · · ⊂ g n A n where g i ∈ G and A i ∈ F for all 0 i n . It is the order complex of what is commonly called thecoset poset associated to the pair ( F , G ). A natural question to ask is how the topological propertiesof C ( F , G ) are related to the algebraic properties of F and G . This question was studied by Abelsand Holz [1] in some generality, in particular with regards to the higher connectivity of C ( F , G ).For example, C ( F , G ) is connected if and only if F covers G , and C ( F , G ) is simply connected ifand only if G is isomorphic to the amalgamation of all A ∈ F along their intersections. In theirterminology, F is n -generating if π i ( C ( F , G )) = 0 for i n − A of all abelian subgroups of G . Then C ( A , G ) is connected, and it is easy to show that C ( A , G ) is simply connected if and only if G isabelian, see [14, Proposition 4.1]. When this is the case, C ( A , G ) is contractible. On the other hand,it may be surprising that this statement has a direct analogue in the world of Lie groups. It is theobjective of this note to formulate and prove this analogue.First, one has to clarify the meaning of C ( A , G ) when G itself carries a topology. The roleof the complex C ( A , G ) will be played by the geometric realization of a simplicial space, denotedby E (2 , G ) or E com G in the literature. It was introduced by Adem, Cohen and Torres-Giese [2]who studied basic properties of E (2 , G ) as part of a more general construction involving families ofnilpotent subgroups of G . For compact connected Lie groups G , further homological and homotopicalproperties of E (2 , G ) were described by Adem and G´omez [3]. In particular, E (2 , G ) can be relatedto the coset spaces G/A for closed abelian subgroups A ⊆ G , but the relationship is much moreintricate than in the discrete case. When G is discrete, then E (2 , G ) is homotopy equivalent to C ( A , G ).Our goal is then to establish a precise relationship between the vanishing of the homotopy groupsof E (2 , G ) and commutativity of G . To do this we promote the commutator map for G to a simplicialmap c : E (2 , G ) → B [ G, G ] , which will play a key role in the proof of our main result.1 heorem 1. For a compact Lie group G the following assertions are equivalent:(1) G is abelian(2) E (2 , G ) is contractible(3) c is null-homotopic(4) π i ( E (2 , G )) = 0 for i = 1 , , . There are two situations in which a stronger statement can be made than that of Theorem 1, bothof which are treated implicitly in our proof. Firstly, if G is an arbitrary discrete group, Proposition9 will show that the statement of Theorem 1 remains valid if (4) is replaced by π ( E (2 , G )) = 0. Fordiscrete groups the results of [1, Section I] imply that E (2 , G ) is homotopy equivalent to C ( A , G ).In this situation we obtain a new proof of the fact that C ( A , G ) is simply–connected if and only if G is abelian, and Theorem 1 may be viewed as a Lie group analogue thereof. Secondly, if G is acompact Lie group with abelian identity component, then Theorem 1 remains valid if (4) is replacedby E (2 , G ) is -connected . This is Proposition 18.It should be mentioned that the equivalence (1) ⇐⇒ (2) has a precursor in the work of Adem andG´omez [3] which concerns a variant of E (2 , G ) denoted E (2 , G ) . In general, the space of n -simplicesof E (2 , G ) is not connected, and E (2 , G ) is obtained by restricting to the basepoint component ineach simplicial degree. It is proved in [3, Corollary 7.5] that for connected G , E (2 , G ) is rationallyacyclic if and only if E (2 , G ) is contractible if and only if G is abelian. This statement fails to holdwhen G is disconnected (it fails for every non-abelian discrete group, for instance). In this case, onemust consider E (2 , G ) instead. While their proof relies on a well known description of the rationalcohomology of spaces of commuting elements in Lie groups, we obtain our result by a rather differentapproach – more homotopical than homological.Finally, it is worth mentioning that if not contractible, the spaces E (2 , G ) have an interestingyet difficult to understand homotopy type. For example, by [6] we have E (2 , SU (2)) ≃ S ∨ Σ RP and E (2 , O (2)) ≃ S ∨ S ∨ S , while for SU = colim n →∞ SU ( n ) it was shown in [10, Theorem 3.4] that E (2 , SU ) ≃ BSU × BSU h i × BSU h i × · · · , where BSU h n i is the (2 n − BSU . If G is an extraspecial p -group whoseFrattini quotient has rank 2 r >
4, then the universal cover of E (2 , G ) is homotopy equivalent toa bouquet of r -dimensional spheres [15]. If G is a transitively commutative group, then E (2 , G ) ishomotopy equivalent to a bouquet of circles by [2, Proposition 8.8]. Other interesting properties of E (2 , G ) are proved in [4, 17, 18, 19]. Acknowledgments
SG received funding from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sklodowska-Curie grant agreement No. 846448. BV acknowledges support fromthe European Research Council (ERC) under the European Union’s Horizon 2020 research andinnovation programme (grant agreement No. 682922). BV was also partially supported by Mexico’sCONACYT ‘Programa de Becas de Posgrado y apoyos a la calidad, en la Modalidad de EstanciaPosdoctoral en el Extranjero.’ This project was also supported by the Danish National ResearchFoundation through the Copenhagen Centre for Geometry and Topology (DNRF151).2
The simplicial space of affinely commuting elements
Let G be a group. We begin by recalling the simplicial bar construction for G , since it will form thebasis for our constructions in the current and the following sections. The simplicial bar constructionfor the classifying space of G is the simplicial space B • G with n -simplices B n ( G ) := G n , face maps ∂ i : B n ( G ) → B n − ( G ) , ∂ i ( g , . . . , g n ) := ( g , . . . , g i g i +1 , . . . , g n ) if 0 < i < n ( g , . . . , g n ) if i = 0( g , . . . , g n − ) if i = n, and degeneracy maps s i : B n ( G ) → B n +1 ( G ) given by inserting the identity element 1 ∈ G in the( i + 1)-st position. Similarly, one defines a simplicial space E • G with n -simplices E n ( G ) := G n +1 , face maps ∂ i : E n ( G ) → E n − ( G ) , ∂ i ( g , . . . , g n ) := ( g , . . . , ˆ g i , . . . , g n ) , and degeneracy maps s i : E n ( G ) → E n +1 ( G ) given by duplicating the i -th coordinate. For every n > G acts on E n ( G ) diagonally by left translation, and this extends to an action onthe simplicial space E • G . The quotient map p : E • G → E • G/G ∼ = B • G can be identified with thesimplicial map given on n -simplices by E n ( G ) → B n ( G ) , ( g , . . . , g n ) ( g − g , . . . , g − n − g n ) . Now let us assume for a moment that G is a discrete group. Let A be the set of abelian subgroupsof G partially ordered by inclusion. We may form the union S A ∈A B • A inside B • G and considerthe pullback of simplicial sets E • (2 , G ) (cid:15) (cid:15) / / E • G p (cid:15) (cid:15) S A ∈A B • A / / B • G The pullback, which we denote by E • (2 , G ), can be identified with the simplicial subset of E • G consisting of those simplices ( g , . . . , g n ) ∈ E n ( G ) for which ( g − g , . . . , g − n − g n ) ∈ B n ( A ) for someabelian subgroup A ⊆ G . As EG is contractible, the geometric realization E (2 , G ) is the homotopyfiber of the inclusion S A ∈A BA → BG . It is therefore a measure for how well S A ∈A BA approximates BG . In other words, it is a measure for the group’s failure to be commutative.By the results of [1, Section I], E (2 , G ) is homotopy equivalent to C ( A , G ). The same simplicialconstruction, however, can be carried out for an arbitrary topological group. First, observe: Lemma 1.
Let G be a group. The following conditions on a finite subset S = { s , . . . , s n } of G areequivalent:(1) The elements s − s , s − s , . . . , s − n − s n pairwise commute.(2) The group h S − S i := h s − i s j | s i , s j ∈ S i is abelian.(3) The set S is contained in a single left coset of some abelian subgroup of G . roof. Condition (2) follows from (1), because each generator of h S − S i can be written as a productof the elements in (1). Condition (2) implies (3), because S ⊂ s h S − S i . The proof is completed byshowing (3) = ⇒ (1), which is immediate. Definition 2.
We say that a finite subset { g , . . . , g n } ⊂ G is affinely commutative if it satisfies anyof the equivalent conditions listed in Lemma 1.Let G be a topological group. For each n > E n (2 , G ) := { ( g , . . . , g n ) ∈ G n +1 | { g , . . . , g n } is affinely commutative } , with the topology induced from G n +1 . These spaces form a sub-simplicial space of E • G as it canbe readily seen that if { g , . . . , g n } is affinely commutative, then so are { g , . . . , ˆ g i , . . . , g n } as wellas { g , . . . , g i , g i , . . . , g n } for any 0 i n . We denote its geometric realization by E (2 , G ) := | E • (2 , G ) | . Remark 3.
The space E (2 , G ) was studied by Adem, Cohen and Torres-Giese in [2], where theconstruction was based on a different but isomorphic model of E • G . Namely, let E • G denote thesimplicial space with n -simplices G n +1 , face maps ∂ i ( g , . . . , g n ) := ( g , . . . , g i g i +1 , . . . , g n ) for 0 i < n and ∂ n ( g , . . . , g n ) = ( g , g , . . . , g n − ), and degeneracy maps s i given by inserting the identityelement 1 ∈ G in the ( i + 1)-st position. Then the map E • G → E • G given on n -simplices by( g , . . . , g n ) ( g , g − g , . . . , g − n − g n )is an isomorphism. Under this isomorphism E • (2 , G ) becomes the simplicial space considered in [2].We will need below a description of the fundamental group of E (2 , G ) when G is discrete. In thiscase, E (2 , G ) is the realization of a simplicial set and a standard presentation of its fundamentalgroup can be given, see for example [8, Proposition 2.7, p. 126]. To this end, we introduce for each( g, h ) ∈ G a formal variable x g,h and set X := { x g,h | ( g, h ) ∈ G } . Let us choose the 0-simplex1 ∈ G as the basepoint for E (2 , G ). Lemma 4.
Let G be discrete. Then, the fundamental group of E (2 , G ) admits the presentation π ( E (2 , G ) ,
1) = (cid:10) X (cid:12)(cid:12) { x g, , x ,g | g ∈ G } ∪ { x g,h x h,k x k,g | { g, h, k } ⊂ G is affinely commutative } (cid:11) . Specifically, the generator x g,h is represented by the loop in E (2 , G ) obtained by concatenatingthe straight paths from e to g to h to 1, following the 1-simplices (1 , g ), ( g, h ) and ( h, In this section we introduce our key tool, a natural map c : E (2 , G ) → B [ G, G ] whose homotopyclass will inform about contractibility of E (2 , G ). The construction of c will be possible, because ofthe following simple but crucial observation. Lemma 5.
Let { g, h, k } ⊂ G be an affinely commutative set. Then [ g, h ][ h, k ] = [ g, k ] .Proof. By hypothesis, [ g − h, h − k ] = [ h − g, k − h ] = 1, and thus[ g, h ][ h, k ] = g − ( h − g )( k − h ) k = g − ( k − h )( h − g ) k = g − k − gk. Lemma 5 is precisely what is needed to verify the following.4 orollary 6.
The maps c n : E n (2 , G ) → [ G, G ] n ( g , . . . , g n ) ([ g , g ] , . . . , [ g n − , g n ]) , defined for all n > , assemble into a map of simplicial spaces c • : E • (2 , G ) → B • [ G, G ] . Definition 7.
Upon geometric realization c • defines a map c : E (2 , G ) → B [ G, G ] , which we refer to as the commutator map .The rest of this section is devoted to establishing some basic properties of c . Let c : G × G → G , ( x, y ) [ x, y ]be the algebraic commutator map for G . Note that c factors through a map ˜ c : G ∧ G → [ G, G ],since [ g, h ] = 1 if either g = 1 or h = 1. Definition 8.
A topological group G is called homotopy abelian if the algebraic commutator map c is null-homotopic.The following proposition summarizes the main features of the commutator map c that the proofof Theorem 1 will rely on. Proposition 9.
Let G be either a discrete group or a compact Lie group, let c : E (2 , G ) → B [ G, G ] be the commutator map, and let c ∗ : π ( E (2 , G )) → π ( B [ G, G ]) be the map induced by c on fundamental groups.(1) If G is discrete, then c ∗ satisfies c ∗ ( x g,h ) = [ g, h ] for all x g,h ∈ π ( E (2 , G )) .(2) If either c is null-homotopic or E (2 , G ) is (2 dim G + 1) -connected, then G is homotopy abelian.(3) The map c ∗ is surjective, and it is trivial if and only if [ G, G ] is a connected Lie group.Proof. First, assume that G is discrete. As pointed out at the end of Section 2, the generator x g,h ∈ π ( E (2 , G )) is represented by the path obtained by concatenating the 1-simplices (1 , g ), ( g, h )and ( h, c takes these 1-simplices to [1 , g ] = 1, [ g, h ] and [ h,
1] = 1in [
G, G ], respectively.Next we prove (2). Since G is either discrete or a Lie group, the simplicial space E • (2 , G ) isproper (cf. [3, Appendix]), hence the fat and thin realizations are naturally homotopy equivalent: k E • (2 , G ) k ≃ E (2 , G ). If X is the geometric realization of a semi-simplicial space, we denote by F k X the k -th term in the skeletal filtration of X . Then, F k E • G k = F k E • (2 , G ) k ∼ = G ∗ G is thetopological join, and F k B • G k ∼ = SG is the unreduced suspension. There is a commutative diagram G ∗ G c ′ / / (cid:15) (cid:15) S [ G, G ] (cid:15) (cid:15) k E • (2 , G ) k c / / k B • [ G, G ] k (1)5here c ′ ([ t, g, h ]) = [ t, [ g, h ]] for t ∈ [0 ,
1] and g, h ∈ G . Up to homotopy, c ′ can be identified withthe map Σ˜ c : Σ G ∧ G → Σ[ G, G ]. If c is null-homotopic, diagram (1) implies that the compositeΣ( G ∧ G ) Σ˜ c −−→ Σ[ G, G ] → B [ G, G ]is null-homotopic as well. We also get that this composite is null-homotopic if k E • (2 , G ) k is(2 dim G + 1)-connected, since then the map G ∗ G → k E • (2 , G ) k appearing in the diagram isnull-homotopic by a standard obstruction theory argument. As a map between path connectedspaces is null-homotopic if and only if it is based null-homotopic, the adjoint map G ∧ G ˜ c −→ [ G, G ] ≃ −→ Ω B [ G, G ] , and hence ˜ c , are null-homotopic. Since the algebraic commutator map c : G × G → G factors through˜ c , it is null-homotopic as well. This finishes the proof of (2).Now we prove (3). If G is discrete it follows directly from statement (1). Assume G is a compactLie group and let G δ denote G equipped with the discrete topology. Let d : G δ → G be the canonicalmap. The commutator map c for G and the commutator map c δ for G δ are related by a commutativediagram E (2 , G δ ) E (2 ,d ) (cid:15) (cid:15) c δ / / B [ G δ , G δ ] Bd (cid:15) (cid:15) E (2 , G ) c / / B [ G, G ] . Recall that for Lie groups, the commutator subgroup [
G, G ] is defined to be the closure of thealgebraic commutator subgroup. But the commutator subgroup of a compact Lie group is alwaysclosed (see [12, Theorem 6.11]), so [
G, G ] δ = [ G δ , G δ ].The diagram induces a commutative diagram on fundamental groups. Now consider the com-posite homomorphism π ( E (2 , G δ )) c δ ∗ −→ [ G δ , G δ ] π ( Bd ) −−−−→ π ( B [ G, G ]) (2)obtained by going through the top right corner of the diagram. Under the isomorphism π ( B [ G, G ]) ∼ = π ([ G, G ]) and the identification of [ G δ , G δ ] with [ G, G ] δ the map π ( Bd ) corresponds to the canon-ical surjection [ G, G ] δ → π ([ G, G ]). Moreover, by part (1) the map c δ ∗ is surjective. Together thisimplies that (2) is surjective, and by commutativity of the diagram c ∗ must be surjective, as well. Inparticular, if c ∗ is trivial, then π ([ G, G ]) = 1. Conversely, if [
G, G ] is path connected, then B [ G, G ]is simply connected, hence c ∗ is trivial. Remark 10.
Perhaps surprisingly, there exist homotopy abelian compact Lie groups G for which c is not null-homotopic. Hence, the converse of part (2) of Proposition 9 fails to hold. An exampleillustrating this is the central extension1 → S → ( S × Q ) / Z / → Z / × Z / → , where Q is the quaternion group of order eight. The quotient G = ( S × Q ) / Z / Z / h ( − , − i ⊂ S × Q . It is indeed homotopy abelian; the commutatorsubgroup is [ G, G ] = { [(1 , , [( − , } ∼ = Z /
2, which is a discrete subgroup of the path-connectedgroup S , thus making the algebraic commutator map null-homotopic. But by part (3) of Proposition9, c cannot be trivial on fundamental groups, since [ G, G ] is not connected.
Remark 11.
Let j : B [ G, G ] → BG be the map induced by the inclusion [ G, G ] ⊆ G . There isanother description, up to homotopy, of the composition j c : E (2 , G ) → BG . We shall not needit to prove our main theorem; but it seems worth mentioning, because it is not obvious from the6efinition. Let C n ( G ) ⊆ G n denote the subspace of n -tuples of commuting elements in G . Then C • ( G ) ⊆ B • G is a sub-simplicial space, whose realization we denote by B (2 , G ). The compositemap E (2 , G ) ⊆ EG p −→ BG factors through the inclusion i : B (2 , G ) → BG . By abuse of notation, we write p : E (2 , G ) → B (2 , G ) for the projection. Note that there is an automorphism φ − : B (2 , G ) → B (2 , G ) inducedby the map G → G , g g − . We claim that the diagram E (2 , G ) p / / c (cid:15) (cid:15) B (2 , G ) φ − / / B (2 , G ) i (cid:15) (cid:15) B [ G, G ] j / / BG commutes up to homotopy. Indeed, it is tedious but straightforward to verify that the collection ofmaps { h i } i n defined by h i : E n (2 , G ) → B n +1 ( G ) = G n +1 ( g , . . . , g n ) ([ g , g ] , . . . , [ g i − , g i ] , g − i , g − i +1 g i , . . . , g − n g n − )is a simplicial homotopy between j c and iφ − p in the sense of [13, Definition 9.1]. The proof of Theorem 1 will require a couple of propositions, the first of which is a characterizationof homotopy abelian compact Lie groups.
Proposition 12.
Let G be a compact Lie group. Then G is homotopy abelian if and only if π ( G ) is abelian and G is a central extension of π ( G ) by a torus.Proof. Suppose that G is homotopy abelian. Let G ⊆ G be the component of the identity and let T ⊆ G be a maximal torus. As the commutator map c : G × G → G is null-homotopic, it factorsthrough G and its restriction to G is null-homotopic, too. It follows that G is homotopy abelian.A result of Araki, James and Thomas [7] asserts that a compact, connected, homotopy abelian Liegroup is abelian. Hence, G = T . Thus G fits into an extension1 → T → G p −→ π ( G ) → . It is clear that π ( G ) is abelian, and so it remains to show that T is central.Note that Aut( T ) ∼ = Aut( H ( T ; Z )) is discrete. For g ∈ G let conj g ∈ Aut( T ) denote the innerautomorphism t g − tg . The map g conj g must be constant on connected components and thusfactors through a representation ρ : π ( G ) → Aut( H ( T ; Z )) . To show that T is central, it is enough to show that ρ is constant. Fix g ∈ G . Let c ( − , g ) : T → T denote the composition T t ( t,g ) −−−−−→ G × G c −→ T , which has image in T = G because c (1 , g ) = 1. The composite map is null-homotopic, becausethe commutator map is null-homotopic. On the other hand, c ( − , g ) can be identified with thecomposition T t ( t − ,t ) −−−−−−→ T × T id × conj g −−−−−−→ T × T · −→ T , T . As this map is null-homotopic, the induced map on H ( T ; Z ) is zero. This implies that, for any x ∈ H ( T ; Z ), we must have0 = − x + ρ ( p ( g ))( x ) , hence ρ ( p ( g )) = id . This finishes the proof that T is central.Conversely, suppose that G is a central extension of π ( G ) by a torus T and assume that π ( G )is abelian. The central extension is classified by a 2-cocycle ω : π ( G ) × π ( G ) → T . As an abstractgroup, G is isomorphic to T × π ( G ) with group law( t, x )( s, y ) := ( tsω ( x, y ) , xy ) , see [11, Remark 18.1.14]. A short computation shows that, when π ( G ) is abelian, the commutatorof any two elements ( t, x ) and ( s, y ) of T × π ( G ) reads[( t, x ) , ( s, y )] = ( ω ( x, y ) ω ( y, x ) − , . Thus, the commutator map c : G × G → G factors through π ( G ) × π ( G ) and has image in T . Since T is connected, c is null-homotopic. Remark 13.
The central extension ( S × Q ) / Z / Proposition 14.
Let G be a compact Lie group. If π ( E (2 , G )) = 0 , then the component of theidentity G ⊆ G is abelian, hence G is an extension of π ( G ) by a torus. The proof of the proposition requires some preparation. Let C n ( G ) ⊆ G n denote the subspaceof n -tuples of commuting elements in G . Lemma 15.
The realization of the sub-simplicial space C • +1 ( G ) ⊆ E • (2 , G ) is contractible.Proof. Implicit in the statement is the claim that C • +1 ( G ) is a sub-simplicial space of E • (2 , G ). Sincein E • (2 , G ) the faces and degeneracies delete and duplicate coordinates (as they do in the simplicialmodel of EG described in Section 2) it is easy to check that C • +1 ( G ) is indeed a sub-simplicialspace.To prove it is contractible we can straightforwardly adapt one of the usual proofs that EG iscontractible: the simplicial model of EG can be augmented by adding a unique ( − s − ( g , . . . , g n ) = (1 , g , . . . , g n )for any n ≥ −
1. This extra degeneracy preserves C • +1 ( G ) and thus also shows that its geometricrealization is contactible.We now define a homotopy equivalent model for E (2 , G ) which will turn out convenient. Considerthe simplicial space ¯ E • (2 , G ) with n -simplices¯ E n (2 , G ) := E n (2 , G ) /C n +1 ( G )and simplicial structure the one induced by E • (2 , G ).As C • +1 ( G ) → E • (2 , G ) is a levelwise cofibration of good simplicial spaces the map of realizations | C • +1 ( G ) | → E (2 , G ) is a cofibration. By Lemma 15 | C • +1 ( G ) | is contractible, so the map E (2 , G ) → E (2 , G ) / | C • +1 ( G ) | is a homotopy equivalence. Since geometric realization commutes with takingcofibers, the levelwise quotient maps induce a homotopy equivalence E (2 , G ) ≃ −→ ¯ E (2 , G ) . Just like E (2 , G ), the assignment G ¯ E (2 , G ) is natural for homomorphisms of groups, and so isthe equivalence E (2 , G ) ≃ ¯ E (2 , G ). 8 emark 16. In the introduction we mentioned the space E (2 , G ) , which is the geometric realizationof the sub-simplicial space E • (2 , G ) ⊆ E • (2 , G ) consisting of the connected component of (1 , . . . , E n (2 , G ) := E n (2 , G ) /C n +1 ( G ) . Indeed, the extra degeneracy used in the proof of Lemma 15preserves the sub-simplicial space C • +1 ( G ) consisting in degree n of the connected component of C n +1 ( G ) containing (1 , . . . , c : E (2 , G ) → B [ G, G ] factors through ¯ E (2 , G ). To keep the notationsimple we denote the resulting map c : ¯ E (2 , G ) → B [ G, G ] by the same letter. Observe that ¯ E (2 , G )is a reduced simplicial space, and the space of 1-simplices is G /C ( G ). Therefore, the simplicial1-skeleton is Σ G /C ( G ) and the commutator map restricted to the 1-skeleton c | : Σ G /C ( G ) → Σ[ G, G ]is simply the suspension of the map induced by the algebraic commutator map c : G → [ G, G ] ⊂ G . Lemma 17.
After looping the commutator map c : ¯ E (2 , SU (2)) → BSU (2) has a section up tohomotopy, and this section s : SU (2) → Ω ¯ E (2 , SU (2)) is natural with respect to homomorphisms f : SU (2) → G in the sense that the diagram Ω ¯ E (2 , SU (2)) Ω ¯ E (2 ,f ) / / Ω c (cid:15) (cid:15) Ω ¯ E (2 , G ) Ω c (cid:15) (cid:15) SU (2) s D D f / / [ G, G ] with the dotted arrow filled in commutes up to homotopy.Proof. In the diagram we have implicitly used the canonical homotopy equivalence [
G, G ] ≃ Ω B [ G, G ]adjoint to the inclusion Σ[
G, G ] → B [ G, G ]. By adjunction it is enough to construct a map s ′ : Σ SU (2) → ¯ E (2 , SU (2)) making the following diagram commute:¯ E (2 , SU (2)) ¯ E (2 ,f ) / / c (cid:15) (cid:15) ¯ E (2 , G ) c (cid:15) (cid:15) Σ SU (2) incl / / s ′ BSU (2) Bf / / B [ G, G ]The desired section s may then be defined as the adjunct of s ′ . As the simplicial 1-skeleton of¯ E (2 , SU (2)) is Σ SU (2) /C ( SU (2)) it suffices to construct a section of the map c | : Σ SU (2) /C ( SU (2)) → Σ SU (2) , and s ′ may be defined as the composite of this section with the inclusion into ¯ E (2 , SU (2)).It is shown in [5, Section VI 1(a)] that the restriction of the algebraic commutator map to thenon-commuting pairs in SU (2), c | : SU (2) − C ( SU (2)) → SU (2) − { } , is a locally trivial bundle with fiber c − ( − c | = Σ( c | ) + , where ( c | ) + is the map inducedby c | on one-point compactifications. As SU (2) − { } is contractible there is a homeomorphism ofthe total space of the fiber bundle with ( SU (2) − { } ) × c − ( −
1) under which c | corresponds to theprojection onto the first factor. Now c − ( −
1) is compact, since it is a closed subset of the compactspace SU (2) . Thus, there is a homeomorphism[( SU (2) − { } ) × c − ( − + ∼ = SU (2) ∧ c − ( − + , c − ( − + denotes c − ( −
1) with a disjoint basepoint added. Under this homeomorphism ( c | ) + can be identified with the map SU (2) ∧ c − ( − + → SU (2) induced by the projection c − ( − + → S . A choice of basepoint of c − ( −
1) gives a section SU (2) ∧ S → SU (2) ∧ c − ( − + , and itssuspension yields a section for c | . Proof of Proposition 14.
Let G denote the component of the identity of G . We must show that G is abelian. Clearly, this follows if we can show that [ G, G ] is abelian. For [ G , G ] is a subgroup of[ G, G ] , and the commutator group of a connected compact Lie group is semisimple.Thus, assume for contradiction that [ G, G ] is non-abelian. It is well known that the universalcover of a compact connected Lie group K decomposes as a product of simply–connected simple Liegroups { K i } i =1 ,...,k and a copy of R m , giving π ( K ) = π ( K ) ⊕ · · · ⊕ π ( K k ). For K = [ G, G ] wemust have k >
1, since [
G, G ] is assumed non-abelian. In [9, Chapter III Proposition 10.2] it is shownthat in a simply–connected simple Lie group K i one can find a subgroup isomorphic to SU (2) suchthat the inclusion SU (2) → K i induces an isomorphism in π ( − ). Thus we find a homomorphism f : SU (2) → [ G, G ] such that π ( f ) : π ( SU (2)) → π ([ G, G ] ) is injective. To reach a contradictionit suffices to show that the map π ( SU (2)) → π ([ G, G ]) obtained by composition with the inclusion[
G, G ] ⊆ [ G, G ] is zero.Application of π ( − ) to the homotopy commutative diagram in Lemma 17 yields a commutativediagram π (Ω ¯ E (2 , SU (2))) π ( f ′ ) / / π (Ω ¯ E (2 , G )) π (Ω c ) (cid:15) (cid:15) π ( SU (2)) π ( s ) O O π ( f ) / / π ([ G, G ])where f ′ := Ω ¯ E (2 , f ). Since E (2 , G ) is homotopy equivalent with ¯ E (2 , G ), we have that π (Ω ¯ E (2 , G )) ∼ = π ( ¯ E (2 , G )) ∼ = π ( E (2 , G )) . By assumption this group is zero, hence π ( f ) = 0.The final item needed to prove Theorem 1 is the following proposition. Proposition 18.
Let G be a compact Lie group and assume that the component of the identity G is abelian. If E (2 , G ) is -connected, then c is null-homotopic.Proof. Since π ( E (2 , G )) = 0 by assumption, we deduce from Proposition 9 part (3) that [ G, G ]is connected. Then [
G, G ] ⊆ G , and since [ G, G ] is also closed it is a torus. Therefore, B [ G, G ]is an Eilenberg-MacLane space of type K ( Z r ,
2) for some r >
0, and the homotopy class of thecommutator map c : E (2 , G ) → B [ G, G ] ≃ K ( Z r , H ( E (2 , G ); Z r ). Since E (2 , G ) is assumed 2-connected, wehave that H ( E (2 , G ); Z r ) = 0. Hence c is null-homotopic, as desired.We can now prove the main result of this paper. Theorem 1.
For a compact Lie group G the following assertions are equivalent:(1) G is abelian(2) E (2 , G ) is contractible(3) c is null-homotopic(4) π i ( E (2 , G )) = 0 for i = 1 , , . roof. Clearly, if G is abelian, then E (2 , G ) is contractible, because in this case every subset { g , . . . , g n } ⊆ G is affinely commutative, so E (2 , G ) = EG and EG is contractible. If E (2 , G )is contractible, then it is obvious that c is null-homotopic, and that π i ( E (2 , G )) = 0 for i = 1 , ⇒ (1), and (4) = ⇒ (3).Suppose that c is null-homotopic. Then Proposition 9 part (2) and Proposition 12 imply that G is a central extension of π ( G ) by a torus. In addition the map c ∗ of fundamental groups istrivial, hence [ G, G ] is a connected Lie group by Proposition 9 part (3). Then it is a subgroup of G = T , hence a torus. It will suffice to show that [ G, G ] is finitely generated, since a torus is finitelygenerated only if it is the trivial group. But as pointed out in the proof of Proposition 12, [
G, G ] isgenerated by the image of the map π ( G ) × π ( G ) → T ( x, y ) ω ( x, y ) ω ( y, x ) − , which is finite, since π ( G ) is finite. This shows that (3) = ⇒ (1).We will now show that (4) = ⇒ (3). By assumption π ( E (2 , G )) = 0, so Proposition 14 impliesthat the identity component G ⊆ G is abelian. Proposition 18 now finishes the proof.There is an intriguing relationship of E (2 , G ) with bundle theory. In [3] it is explained howthe i -th homotopy group of E (2 , G ) can be interpreted as the set of “transitionally commutativestructures” on the trivial principal G -bundle over S i . We refer to [3] for more background. If G isnon-abelian, then π i ( E (2 , G )) = 0 for some i ∈ { , , } . Therefore, our theorem has the followingcorollary. Corollary 19.
Let G be a non-abelian compact Lie group. Then the trivial principal G -bundle overat least one of S , S or S admits two distinct transitionally commutative structures. Remark 20.
The compactness condition in Theorem 1 is necessary. For example, SL (2 , R ) is ahomotopy abelian Lie group as it deformation retracts onto SO (2), but it is not abelian. On the otherhand, a result of Pettet and Suoto [16, Corollary 1.2] implies that E (2 , SL (2 , R )) ≃ E (2 , SO (2)) = ESO (2), which is contractible.
The results in this paper would be well explained by a splitting up to homotopy of the loopedcommutator map Ω c , and hence a splitting of spacesΩ E (2 , G ) ≃ [ G, G ] × Ω X , for some space X . Indeed, if such a splitting exists, then any of the equivalent conditions listed inTheorem 1 readily implies that [ G, G ] = 1. Note that a connected and simply-connected compactLie group with trivial π is necessarily trivial.The splitting exists for G = SU (2) as proved in Lemma 17. It also exists for G = O (2) and for G = SU . For example, we proved in [6, Theorem 1.5] that E (2 , O (2)) ≃ Σ( S × S ). For any group G there is a homotopy fiber sequence G ∗ G → Σ G → BG by Ganea’s theorem. After looping, theunit map G → ΩΣ G splits the homotopy fiber sequence, henceΩΣ G ≃ G × Ω( G ∗ G ) . In particular, there is a homotopy equivalenceΩ E (2 , O (2)) ≃ S × S × Ω(( S × S ) ∗ ( S × S )) . S = SO (2) = [ O (2) , O (2)]. To prove that Ω c : Ω E (2 , O (2)) → SO (2) splits up tohomotopy it is enough to show that Ω c is surjective on fundamental groups. Because the inclusion SO (2) → O (2) induces an isomorphism π ( BSO (2)) ∼ = π ( BO (2)), one can equivalently show, usingRemark 11, that iφ − p : E (2 , O (2)) → BO (2) is surjective on π . Surjectivity follows from resultsin [17] as we will now explain. The authors construct a map f : S → B (2 , O (2)) such that if isnull-homotopic but iφ − f is a generator of π ( BO (2)) ∼ = Z ([17, Proposition 3.5]). Thus we canfind a lift of f up to homotopy ˜ f : S → E (2 , O (2)) such that iφ − p ˜ f is a generator of π ( BO (2)).We leave it to the reader to show that Ω c : Ω E (2 , SU ) → SU has a splitting up to homotopyusing [10, Theorem 3.4] and Remark 11.There are too few examples known to build a firm opinion, but the results of this paper suggestthat the following question warrants further study. Question 21.
Let G be a compact Lie group. Does the commutator map c : E (2 , G ) → B [ G, G ] split up to homotopy after looping? One way of establishing a splitting is by showing that the restriction c | of the commutator mapto the simplicial 1-skeleton of ¯ E (2 , G ) has a splitting up to homotopy. This was carried out for G = SU (2) in Lemma 17. However, one can show that c | splits neither for G = O (2) nor for G = SO (3). For example, for G = SO (3) we have H ( SO (3); Z ) ∼ = Z / H ( SO (3) /C ( SO (3)); Z ) ∼ = Z using [20, Theorem 1.2]. This motivates the following question. Question 22.
For which groups G does the commutator map c | : Σ G /C ( G ) → Σ[ G, G ] split up to homotopy? References [1] H. Abels and S. Holz. Higher generation by subgroups.
J. Algebra , , no. 2 (1993) 310–341.[2] A. Adem, F. Cohen and E. Torres-Giese. Commuting elements, simplicial spaces and filtrations ofclassifying spaces. Math. Proc. Cambridge Philos. Soc. (2012), 91–114.[3] A. Adem and J. M. G´omez. A classifying space for commutativity in Lie groups.
Algebr. Geom. Topol. (2015) 493–535.[4] A. Adem, J. M. G´omez, J. Lind and U. Tillman. Infinite loop spaces and nilpotent K-theory. Algebr.Geom. Topol. (2017) 869-893.[5] S. Akbulut and J. McCarthy. Cassons Invariant for Oriented Homology Spheres . Mathematical Notes36, Princeton University Press, 1990.[6] O. Antol´ın-Camarena, S. Gritschacher and B. Villarreal. Classifying spaces for commu-tativity of low-dimensional Lie groups.
Math. Proc. Cambridge Phil. Soc. (2019), 1–46. doi:10.1017/S0305004119000240 .[7] S. Araki, I. M. James and E. Thomas. Homotopy-abelian Lie groups.
Bull. Amer. Math. Soc. , (1960),no. 4, 324–326.[8] M. Aschbacher, R. Kessar and B. Oliver. Fusion Systems in Algebra and Topology. London MathematicalSociety Lecture Note Series . Cambridge University Press, 2011.
9] R. Bott and H. Samelson. Applications of the Theory of Morse to symmetric spaces.
Amer. J. Math. , (1958), 964–1029.[10] S. Gritschacher. The spectrum for commutative complex K -theory. Algebr. Geom. Topol. (2018)1205–1249.[11] J. Hilgert and K.-H. Neeb. Structure and geometry of Lie groups.
Springer Monographs in Mathematics,Springer, New York 2012, 744 pp.[12] K. Hofmann and S. Morris.
The structure of compact groups.
The geometry of iterated loop spaces.
Vol. 271. Springer, 2006.[14] C. Okay. Colimits of abelian groups.
J. Algebra , (2015) 1–12.[15] C. Okay. Spherical posets from commuting elements. J. Group Theory (2018), 593–628.[16] A. Pettet and J. Suoto. Commuting tuples in reductive groups and their maximal compact subgroups. Geom. Topol. , (2013), no. 5, 2513–2593.[17] D. Ramras and B. Villarreal. Commutative cocycles and stable bundles over surfaces. Forum Math. (2019), no. 6, 1395–1451.[18] M. Stafa. Polyhedral products, flag complexes and monodromy representations. Topology Appl. (2018), 12–30.[19] E. Torres-Giese. Higher commutativity and nilpotency in finite groups,
Bull. London Math. Soc. , ,no. 6 (2012), 1259–1273.[20] E. Torres-Giese and D. Sjerve. Fundamental groups of commuting elements in Lie groups. Bull. LondonMath. Soc. , , no. 1 (2008), 65–76. Omar Antol´ın-CamarenaInstituto de Matem´aticas, UNAM, Mexico City, Mexico
E-mail address : [email protected] Simon GritschacherDepartment of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark
E-mail address : [email protected] Bernardo VillarrealInstituto de Matem´aticas, UNAM, Mexico City, Mexico
E-mail address : [email protected]@matem.unam.mx