On the second homotopy group of spaces of commuting elements in Lie groups
aa r X i v : . [ m a t h . A T ] S e p ON THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTINGELEMENTS IN LIE GROUPS
ALEJANDRO ADEM, JOS´E MANUEL G ´OMEZ, AND SIMON GRITSCHACHER
Abstract.
Let G be a compact connected Lie group and n > n -tuples in G , Hom( Z n , G ), and its quotient under the adjointaction, Rep( Z n , G ) := Hom( Z n , G ) /G . In this article we study and in many cases computethe homotopy groups π (Hom( Z n , G )). For G simply–connected and simple we show that π (Hom( Z , G )) ∼ = Z and π (Rep( Z , G )) ∼ = Z , and that on these groups the quotient mapHom( Z , G ) → Rep( Z , G ) induces multiplication by the Dynkin index of G . More generallywe show that if G is simple and Hom( Z , G ) ⊆ Hom( Z , G ) is the path–component of thetrivial homomorphism, then H (Hom( Z , G ) ; Z ) is an extension of the Schur multiplier of π ( G ) by Z . We apply our computations to prove that if B com G is the classifying spacefor commutativity at the identity component, then π ( B com G ) ∼ = Z ⊕ Z , and we constructexamples of non-trivial transitionally commutative structures on the trivial principal G -bundle over the sphere S . Introduction
Suppose that G is a compact connected Lie group. For an integer n > Z n , G )be the space of ordered commuting n -tuples in G endowed with the subspace topologyas a subset of G n . The group G acts by conjugation on Hom( Z n , G ) and the space ofrepresentations Rep( Z n , G ) := Hom( Z n , G ) /G can be identified with the moduli space ofisomorphism classes of flat connections on principal G -bundles over the torus ( S ) n .When n = 2 or n = 3 these moduli spaces appear naturally in quantum field theoriessuch as YangMills and ChernSimons theories. Motivated by this a systematic study of thespaces Rep( Z n , G ) was initiated by Borel, Friedman and Morgan in [11] and also by Kacand Smilga in [27]. In both of these papers the authors showed that the representationspaces Rep( Z n , G ) for n = 2 , G . If G is simply–connected and simple, then Rep( Z , G )can be furthermore identified with the moduli space of semistable principal bundles overan elliptic curve with structure group the complexification of G , and this is known to be aweighted projective space [9, 19, 30, 33]. On the other hand, in [1] Adem and Cohen starteda systematic study of the spaces of homomorphisms Hom( Z n , G ) from the point of view ofhomotopy theory. Since then a variety of authors have studied these spaces using techniques JMG acknowledges the support provided by the Max Planck Institute for Mathematics and Mincienciasthrough grant number FP44842-013-2018 of the Fondo Nacional de Financiamiento para la Ciencia, laTecnolog´ıa y la Innovaci´on. SG received funding from the European Union’s Horizon 2020 research andinnovation programme under the Marie Sklodowska-Curie grant agreement No. 846448. This project wasalso supported by the Danish National Research Foundation through the Copenhagen Centre for Geometryand Topology (DNRF151). AA acknowledges support from NSERC prior to October 2019. from geometry and homotopy theory. See for example [2, 8, 12, 23, 26, 38, 42, 46], amongothers.The spaces of ordered commuting pairs Hom( Z , G ) turn out to have a suprisingly com-plicated structure; their integral homology is not known for G = SU ( m ) when m >
2, andtorsion can appear at primes which divide the order of the Weyl group. From the pointof view of algebraic geometry, Hom( Z , G ) can be identified with Λ( G ), the inertia stackof G with the adjoint action. In this context Rep( Z , G ) can be regarded as the coarsemoduli space of the stack, and as the local isotropy groups are not finite, the geometry canbe rather intricate. In this article we describe the second homology group of the principalpath–component Hom( Z , G ) as an extension of the Schur multiplier of π ( G ) . Theorem 5.22.
Suppose that G is a semisimple compact connected Lie group. Then thereis an extension → Z s → H (Hom( Z , G ) ; Z ) → H ( π ( G ) ; Z ) → , where s > is the number of simple factors in the Lie algebra of G . Suppose that G is simply–connected and simple. As Rep( Z , G ) is a weighted projectivespace one has π (Rep( Z , G )) ∼ = Z . The preceding theorem will be deduced from a calculationof π (Hom( Z , G )). One of the fundamental results in Lie group theory is that π ( G ) = 0,and π ( G ) ∼ = Z . There is a canonical 3-dimensional integral cohomology class that can berealized through a group homomorphism SU (2) → G . In this paper we obtain the rathersurprising result that for commuting pairs there is a canonical class which now appears indimension two. Theorem 5.21.
Let G be a simply–connected and simple compact Lie group. Then π (Hom( Z , G )) ∼ = Z , and on this group the quotient map Hom( Z , G ) → Rep( Z , G ) induces multiplication by theDynkin index lcm { n ∨ , . . . , n ∨ r } where n ∨ , . . . , n ∨ r > are the coroot integers of G . The Dynkin index of G can be defined as the greatest common divisor of the degrees of π ( G ) → π ( SU ( N )) for all representations G → SU ( N ). The values of the Dynkin index of G are explicitly computed in [29, Proposition 4.7] and [31, Proposition 2.6] and agree withthe expressions in terms of coroot integers which are tabulated below.In Theorem 7.1 we show that any embedding SU (2) → G corresponding to a long root of G induces an isomorphism π (Hom( Z , SU (2))) ∼ = π (Hom( Z , G )). G SU ( m ) Spin ( k ) Sp ( l ) E E E F G coroot integers 1 1 , , , , , , , , , , , , , , Table 1.
The set of coroot integers and their least common multiple for thesimple simply–connected compact Lie groups ( m > k > l > N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 3
We note that Theorem 5.21 effectively computes π (Hom( Z , G ) ) when G is the groupof complex or real points of any reductive algebraic group. Indeed, the main result of [38]asserts that Hom( Z , G ) deformation retracts onto Hom( Z , K ) where K is a maximalcompact subgroup of G . So we may assume that G is a compact connected Lie group. Fromthe standard classification theorems we know that G ∼ = ˜ G/L , where ˜ G = ( S ) k × G ×· · ·× G s is a product of a torus and simply–connected simple compact Lie groups G , . . . , G s , and L is a finite subgroup in the center of ˜ G . By [22, Lemma 2.2] the quotient map ˜ G → G inducesa covering map Hom( Z , ˜ G ) → Hom( Z , G ) . From Theorem 5.21 we then deduce that π (Hom( Z , G ) ) ∼ = π (Hom( Z , G )) × · · · × π (Hom( Z , G s )) ∼ = Z s . Corollary 1.1.
Let G be the component of the identity of the group of complex or real pointsof a reductive algebraic group, which we assume is defined over R in the latter case. Then π (Hom( Z , G ) ) ∼ = Z s , where s > is the number of simple factors in the Lie algebra of G . For the groups SU ( m ) and Sp ( k ) we extend our computations to commuting n -tuples for n > Theorem 3.6.
Suppose that G is SU ( m ) or Sp ( m ) with m > . For every n > thequotient map Hom( Z n , G ) → Rep( Z n , G ) induces an isomorphism π (Hom( Z n , G )) ∼ = π (Rep( Z n , G )) . We then calculate π (Rep( Z n , G )) and obtain the following result. Theorem 4.1.
Let n > . Then (i) for every m > there is an isomorphism π (Hom( Z n , SU ( m ))) ∼ = Z ( n ) , and the standard inclusion SU ( m ) → SU ( m + 1) induces an isomorphism π (Hom( Z n , SU ( m ))) ∼ = −→ π (Hom( Z n , SU ( m + 1))) , (ii) for every k > there is an isomorphism π (Hom( Z n , Sp ( k ))) ∼ = Z ( n ) ⊕ ( Z / n − − n − ( n ) , and the standard inclusion Sp ( k ) → Sp ( k + 1) induces an isomorphism π (Hom( Z n , Sp ( k ))) ∼ = −→ π (Hom( Z n , Sp ( k + 1))) . By the same reasoning as above, using the covering space S × SU ( m ) → U ( m ) we observethat for every m, n > SU ( m ) → U ( m ) induces an isomorphism π (Hom( Z n , SU ( m ))) ∼ = π (Hom( Z n , U ( m )))so that the preceding theorem also covers the case of the unitary groups.For spaces of commuting pairs in Spin ( m ) we also explore the behavior of π with respectto stabilization. A. ADEM, J. M. G ´OMEZ, AND S. GRITSCHACHER
Theorem 6.3.
For all m > the standard map Spin ( m ) → Spin ( m + 1) induces anisomorphism π (Hom( Z , Spin ( m ))) ∼ = −→ π (Hom( Z , Spin ( m + 1))) . Along the way, in Theorem 6.1 we prove an integral homology stability result for themoduli spaces Rep( Z , Spin ( m )).A motivation for the computations provided in this article is the construction of non-trivialtransitionally commutative structures (TC structures) on a trivial principal G -bundle. Fora Lie group G the classifying bundle for commutativity, E com G → B com G , is a principal G –bundle constructed out of the spaces of ordered commuting n -tuples in G for all n > B com G classifies principal G -bundles thatcome equipped with a TC structure. A TC structure on a principal G -bundle p : E → X isa choice of a lifting ˜ f : X → B com G , up to homotopy, of the classifying map f : X → BG of the bundle p : E → X . Therefore, the same underlying bundle can admit different TCstructures. In what follows we focus on the associated bundle for the component of theidentity E com G → B com G → BG .In the classical setting, the computation π ( G ) ∼ = Z implies that π ( BG ) ∼ = Z and thiscan be used to construct non–trivial principal G –bundles over the 4–sphere S . Our mainresult in the commutative context (see Corollary 8.3) is the following computation for anysimply–connected simple compact Lie group G : π ( E com G ) ∼ = Z and π ( B com G ) ∼ = π ( E com G ) ⊕ π ( BG ) ∼ = Z ⊕ Z . We provide explicit generators for these groups. Moreover we construct examples of non-trivial TC structures on the trivial principal G -bundle over the sphere S . As a by-productof this construction we obtain geometric representatives for the generators of the reducedcommutative K -theory group e K com ( S ).This article is organized as follows. In Section 2 we present a cohomology computationwith the goal of determining the rank of π (Hom( Z n , G ) ). In Section 3 we set up the spectralsequence for the homology of the homotopy orbit space EG × G Hom( Z n , G ) , which will bethe main device used for our computations. In Section 4 we calculate π of the space ofordered commuting n -tuples in G = SU ( m ) and G = Sp ( k ). Section 5 begins with a reviewof some known facts concerning centralizers of pairs of commuting elements. Then we provesome of the main technical results of the paper, eventually arriving at the calculation of π (Hom( Z , G )). In Section 6 we study the stability behavior for spaces of commuting pairsin the Spin groups. In Section 7 we study the distinguished role that the group SU (2) playsin the computation of the groups π (Hom( Z , G )). Finally, in Section 8 we provide geometricinterpretations for the results obtained in this article in terms of transitionally commutativeprincipal G -bundles. Acknowledgements
We thank Shrawan Kumar and Burt Totaro for their valuable feedback.
N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 5 Cohomological computations
Let G be a compact connected Lie group, let T G be a fixed maximal torus, and let W = N ( T ) /T be the Weyl group. We begin by recalling some known facts concerningthe path–components and the fundamental group of Hom( Z n , G ) which will be used in thesequel.In general, Hom( Z n , G ) is not path–connected. If π ( G ) is torsionfree, however, thenHom( Z , G ) is path–connected, because the centralizer of any element of G is connected (see[15, IX § Z , G )is path–connected if and only if π ( G ) is torsionfree, and for n > Z n , G )is path–connected if and only if H ∗ ( G ; Z ) is torsionfree. For example, this is the case for G = SU ( m ), G = U ( l ) and G = Sp ( k ).When Hom( Z n , G ) is not path–connected, we denote by Hom( Z n , G ) the path–connectedcomponent that contains the trivial homomorphism : Z n → G , equivalently the tuple = (1 G , . . . , G ). Then Hom( Z n , G ) consists precisely of those n -tuples that are containedin some maximal torus of G . For simply–connected G this follows, because the rank of thecentralizer of a commuting n -tuple is locally constant (see [11, Corollary 2.3.2]). For general G it follows by the same argument applied to the universal cover of G using [22, Lemma2.2]. Consequently, Rep( Z n , G ) := Hom( Z n , G ) /G ∼ = T n /W , where on the right hand side the Weyl group acts diagonally on T n . All of our statementswill concern the path-components Hom( Z n , G ) and Rep( Z n , G ) .The fundamental group of Hom( Z n , G ) is also well-understood. By [23, Theorem 1.1] theinclusion Hom( Z n , G ) ⊆ G n induces an isomorphism π (Hom( Z n , G ) ) ∼ = π ( G ) n . In particular, Hom( Z n , G ) is simply–connected if G is. In this situation also Rep( Z n , G ) issimply–connected by [12, Theorem 1.1]. We will use these facts frequently without mention.Now we turn our attention to the rational homology of Hom( Z n , G ) . As explained belowthe rational cohomology ring of Hom( Z n , G ) admits a well known description as a ring of W -invariants. A closed formula for the Poincar´e series of Hom( Z n , G ) was derived in [42](see Remark 2.3). For a fixed homological degree i ∈ N , however, determination of therank of H i (Hom( Z n , G ) ; Q ) can pose a combinatorial challenge. In this section we presenta calculation of H (Hom( Z n , G ) ; Q ). This determines the rank of π (Hom( Z n , G ) ) as anabelian group.To describe the rational cohomology of Hom( Z n , G ) consider the (surjective) map G × T n → Hom( Z n , G ) ( g, t , . . . , t n ) ( gt g − , . . . , gt n g − ) . It is invariant under the action of the normalizer N ( T ) on G by right translation and on T n diagonally by conjugation. Since G × N ( T ) T n ∼ = G/T × W T n , we obtain an induced map G/T × W T n → Hom( Z n , G ) . A. ADEM, J. M. G ´OMEZ, AND S. GRITSCHACHER
Baird showed in [8, Theorem 4.3] that if k is a field in which | W | is invertible, then this mapinduces an isomorphism H ∗ (Hom( Z n , G ) ; k ) ∼ = H ∗ ( G/T × W T n ; k ) ∼ = H ∗ ( G/T × T n ; k ) W . Theorem 2.1.
Suppose that the compact connected Lie group G is simple. Then H (Hom( Z n , G ) ; Q ) ∼ = Q ( n ) . Proof.
Because of the universal coefficient theorem, we may as well use complex coefficientsthroughout this proof. Let W act diagonally on H ( T ; C ) ⊗ H ( T ; C ). As a first step we aregoing to prove that ( H ( T ; C ) ⊗ H ( T ; C )) W ∼ = C . Let t be the Lie algebra of T and t C itscomplexification. There is an isomorphism of W -representations H ( T ; C ) ∼ = t ∗ C , where t ∗ C denotes the dual of t C . Therefore ( H ( T ; C ) ⊗ H ( T ; C )) W ∼ = ( t ∗ C ⊗ t ∗ C ) W . To complete thefirst step we need to prove that ( t ∗ C ⊗ t ∗ C ) W is a complex vector space of dimension 1. Notethat dim C ( t ∗ C ⊗ t ∗ C ) W is the number of times that the trivial representation appears in the W -representation t ∗ C ⊗ t ∗ C . This number is given bydim C ( t ∗ C ⊗ t ∗ C ) W = D χ t ∗ C , E = 1 | W | X w ∈ W χ t ∗ C ( w ) . In the above equation χ t ∗ C denotes the character of t ∗ C and h· , ·i the usual inner product ofcharacters. As t ∗ C is (the complexification of) a real W -representation we have χ t ∗ C ( w ) ∈ R .Moreover, t ∗ C is irreducible by [21, Proposition 14.31] since g C , the complexification of theLie algebra of G , is simple. Therefore,dim C ( t ∗ C ⊗ t ∗ C ) W = D χ t ∗ C , E = (cid:10) χ t ∗ C , χ t ∗ C (cid:11) = 1 . Next we prove that H (Hom( Z n , G ) ; C ) ∼ = C ( n ). With complex coefficients there areisomorphisms H (Hom( Z n , G ) ; C ) ∼ = H ( G/T × W T n ; C ) ∼ = H ( G/T × T n ; C ) W . Using the K¨unneth theorem we obtain a W -equivariant isomorphism H ( G/T × T n ; C ) ∼ = H ( G/T ; C ) ⊕ ( H ( G/T ; C ) ⊗ H ( T n ; C )) ⊕ H ( T n ; C ) . The flag variety
G/T is simply–connected and thus H ( G/T ; C ) = 0. Furthermore, as anungraded ring, H ∗ ( G/T ; C ) is the regular W -representation and the only copy of the trivialrepresentation is in degree 0. This implies that H ( G/T ; C ) W = 0. Therefore, H ( G/T × T n ; C ) W ∼ = H ( T n ; C ) W . By [15, IX § T /W is homeomorphic to
A/π ( G ), where A is theclosure of a Weyl alcove. Now π ( G ) is finite as G is simple, and A is contractible, hence H ( T ; C ) W ∼ = H ( T /W ; C ) ∼ = H ( A/π ( G ); C ) ∼ = H ( A ; C ) π ( G ) = 0. Using this and theK¨unneth theorem we obtain an isomorphism H ( T n ; C ) W ∼ = ( n ) M ( H ( T ; C ) ⊗ H ( T ; C )) W ∼ = C ( n ) . N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 7
Putting everything together we conclude that H (Hom( Z n , G ) ; C ) ∼ = C ( n ). The universalcoefficient theorem now finishes the proof. (cid:3) Corollary 2.2.
Let G be a simple compact connected Lie group. Then π (Hom( Z n , G ) ) hasrank (cid:0) n (cid:1) as an abelian group.Proof. As explained in the introduction π (Hom( Z n , G ) ) ∼ = π (Hom( Z n , ˜ G ) ), where ˜ G isthe universal covering group of G . It is compact as G is simple. Now Hom( Z n , ˜ G ) issimply–connected by [23, Theorem 1.1]. Therefore, by Hurewicz’ theorem, we obtain π (Hom( Z n , ˜ G ) ) ∼ = H (Hom( Z n , ˜ G ) ; Z ) . The assertion follows from Theorem 2.1. (cid:3)
Remark 2.3.
Let G be a simple compact connected Lie group. We can view the Weylgroup W as a reflection group on t and every element w ∈ W as a linear transformation w : t → t . Let d , d , . . . , d r be the characteristic degrees of W . By [42, Theorem 1.1] thePoincar´e series of Hom( Z n , G ) is given by P (Hom( Z n , G ) )( t ) = Q ri =1 (1 − t d i ) | W | X w ∈ W det(1 + tw ) n det(1 − t w ) ! . Our Theorem 2.1 shows that the coefficient of t in this polynomial is precisely (cid:0) n (cid:1) . Moreover,as G is simple π ( G ) is finite. Then π (Hom( Z n , G ) ) ∼ = π ( G ) n is finite too, so P (Hom( Z n , G ) )( t ) = 1 + (cid:18) n (cid:19) t + t q ( t ) , where q ( t ) is some polynomial with non-negative integer coefficients.3. The Bredon spectral sequence for
Hom( Z n , G ) In this section we will analyze the Bredon spectral sequence associated to the G -spaceHom( Z n , G ) .3.1. The set-up.
We begin with an observation:
Lemma 3.1.
Let G be a simply–connected compact Lie group. There is an isomorphism π (Hom( Z n , G ) ) ∼ = H ( EG × G Hom( Z n , G ) ; Z ) . Proof.
Since G is assumed to be simply–connected, BG is 3-connected, and Hom( Z n , G ) is simply–connected by [23, Theorem 1.1]. The long exact sequence of homotopy groupsassociated to the fibration sequenceHom( Z n , G ) → EG × G Hom( Z n , G ) → BG then implies that EG × G Hom( Z n , G ) is simply–connected, and that π (Hom( Z n , G ) ) ∼ = π ( EG × G Hom( Z n , G ) ). The result follows now from the Hurewicz theorem. (cid:3) A. ADEM, J. M. G ´OMEZ, AND S. GRITSCHACHER
As a consequence, the homotopy group π (Hom( Z n , G ) ) can be computed from the Borelequivariant homology of Hom( Z n , G ) . Now if X is a G -CW-complex, then the skeletalfiltration of X gives rise to an Atiyah-Hirzebruch style spectral sequence E p,q = H Gp ( X ; H q ) = ⇒ H p + q ( EG × G X ; Z ) , where H Gp ( X ; H q ) denotes Bredon homology with respect to the covariant coefficient system G/K
7→ H q ( G/K ) := H q ( BK ; Z ) , for K a closed subgroup of G . For Bredon homology in the setting of topological groups werefer the reader to the paper by Willson [50]. The spectral sequence is a special case of [50,Theorem 3.1]. We will refer to it as the Bredon spectral sequence .If G is a compact Lie group, then Hom( Z n , G ) may be viewed as an affine G -variety, andthus the underlying space admits a G -CW-structure by [37, Theorem 1.3]. In fact, if G is simply–connected, it is not too difficult to construct a G -CW-structure on Hom( Z n , G ) directly, and we will do this in Lemma 5.4 (see also Remark 5.5). Thus, taking X =Hom( Z n , G ) the spectral sequence we will study takes the form(1) E p,q = H Gp (Hom( Z n , G ) ; H q ) = ⇒ H p + q ( EG × G Hom( Z n , G ) ; Z ) . Notice that when q = 0 then H q = Z is the constant coefficient system. Therefore, E p, = H Gp (Hom( Z n , G ) ; Z ) ∼ = H p (Rep( Z n , G ) ; Z ) . We will be interested not only in calculating the homotopy group π (Hom( Z n , G ) ), butalso in describing the effect of the quotient map π : Hom( Z n , G ) → Rep( Z n , G ) on π . By Lemma 3.1, this map can be identified with the one induced on H by theprojection EG × G Hom( Z n , G ) → Hom( Z n , G ) /G = Rep( Z n , G ) from the homotopy orbit to the strict orbit. In the spectral sequence this corresponds to thecomposite H ( EG × G Hom( Z n , G ) ; Z ) → E ∞ , ֒ → E , ∼ = H (Rep( Z n , G ) ; Z ) . The case n = 2 is an interesting special case; it is known that if G is simply–connectedand simple, then Rep( Z , G ) can be identified naturally with the moduli space of semistableprincipal bundles over an elliptic curve with structure group the complexification of G , andthis moduli space is a weighted projective space CP ( n ∨ ), where n ∨ = ( n ∨ , . . . , n ∨ r ) is thetuple of coroot integers of G (see Section 5.1 for the definition of coroot integers and Section5.3 for the definition of a weighted projective space). This follows from the work of variousauthors [9, 19, 30, 33, 36, 39].Our proofs will be self-contained; in Proposition 5.10 we will prove independently, and byelementary topological methods, that Rep( Z , G ) is homotopy equivalent to CP ( n ∨ ). N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 9
The homology of weighted projective spaces was computed by Kawasaki [28] from whichone obtains, for simple simply–connected G of rank r , H p (Rep( Z , G ); Z ) ∼ = ( Z if p r is even , First consequences.
Going back to the general situation, we will prove that E , vanishes in certain cases. For this we will use the following lemma. Lemma 3.2.
Suppose that G is a compact Lie group and X is a path–connected G -CWcomplex that has a basepoint x fixed by G . Assume that M is a covariant coefficient systemsuch that M ( G/G ) = 0 and for every morphism
G/H → G/K in the orbit category theinduced map M ( G/H ) → M ( G/K ) is surjective. Then H G ( X ; M ) = 0 .Proof. After passing to an appropriate subdivision we may assume without loss of generalitythat the G -CW complex structure has a 0-dimensional cell corresponding to the basepoint x . As this basepoint is fixed by G this cell must be of the form G/G .By definition H Gp ( X ; M ) = H p ( C ∗ ( X ; M )). Here the group C i ( X ; M ) of cellular i -chainsis defined by C i ( X ; M ) = M σ ∈ S i ( X ) M ( G/G σ ) , where S i ( X ) is an indexing set of all i -dimensional G -cells of X , and G σ is the isotropy groupof σ ∈ S i ( X ). To prove the lemma we are going to show that the differential d : M τ ∈ S ( X ) M ( G/G τ ) → M σ ∈ S ( X ) M ( G/G σ )is surjective. Let us first recall the definition of d . Suppose that τ ∈ S ( X ) and let f τ : G/G τ × ∂D → X (0) be the attaching map of τ . Write ∂D = { , } . Suppose that f τ Z Z E , E , E , E , d Figure 1.
A portion of the E -page of the Bredon spectral sequence in thecase of commuting pairs in a simple, simply–connected, compact Lie group ofrank > restricts to G -maps f τ, : G/G τ × { } → G/G σ × D and f τ, : G/G τ × { } → G/G σ × D for σ i ∈ S ( X ), i = 0 ,
1. Given y ∈ M ( G/G τ ), then d ( y ) = M ( f τ, )( y ) − M ( f τ, )( y ) . Now suppose that σ ∈ S ( X ) and z ∈ M ( G/G σ ). We must show that z ∈ Im( d ). As X is assumed to be path–connected, we can find σ , . . . , σ n ∈ S ( X ) and τ , . . . , τ n ∈ S ( X )such that the attaching map f τ k restricts to f τ k , : G/G τ k × { } → G/G σ k − × D and f τ k , : G/G τ k × { } → G/G σ k × D , and such that σ n = σ and σ is the G -cell correspondingto the G -fixed basepoint x . By hypothesis M ( f τ k ,i ) is surjective for every k = 1 , . . . , n and i = 0 ,
1. Therefore, we find y n ∈ M ( G/G τ n ) such that M ( f τ n , )( y n ) = z n := z , and d ( y n ) = z n − z n − for some z n − ∈ M ( G/G σ n − ). Continuing like this we find for every k = 1 , . . . , n elements y k ∈ M ( G/G τ k ) and z k − ∈ M ( G/G σ k − ) such that M ( f τ k , )( y k ) = z k and d ( y k ) = z k − z k − . Since M ( G/G σ ) = M ( G/G ) = 0, we have that z = 0 and thus d ( y ) = z . Then d ( y n + y n − + · · · + y ) = ( z n − z n − ) + ( z n − − z n − ) + · · · + ( z − z ) + z = z n = z. This proves that d is surjective. (cid:3) In addition, we must make an assumption about the components of the isotropy groupsof Hom( Z n , G ) under the conjugation action by G . Suppose that x = ( x , . . . , x n ) ∈ Hom( Z n , G ) is an n -tuple of commuting elements. Then the isotropy group of x is G x = Z G ( x ) , i.e., the centralizer of the subset { x , . . . , x n } ⊆ G . The following fact is explained in[5, Example 2.4]. Lemma 3.3.
Let G be SU ( m ) or Sp ( m ) for some m > , and let n > . Then for every x ∈ Hom( Z n , G ) the centralizer Z G ( x ) is connected. In general, Z G ( x ) may not be connected. For example, when G = Spin ( m ) with m > x, y ) ∈ Hom( Z , G ) such that π ( Z G ( x, y )) ∼ = Z /
2. The following result isa special case of [11, Corollary 7.5.3] and a proof will be given in Section 5.1.
Lemma 3.4.
Let G be a simply–connected and simple compact Lie group. For every ( x, y ) ∈ Hom( Z , G ) the group π ( Z G ( x, y )) is a finite cyclic group (of order at most ). We can now prove:
Proposition 3.5.
Suppose that either n = 2 , or that n > and G is SU ( m ) or Sp ( m ) forsome m > . Then E , = H G (Hom( Z n , G ) ; H ) = 0 .Proof. For ease of notation we denote Hom( Z n , G ) simply by X . Let H and K be isotropygroups of X and assume that gHg − K for some g ∈ G . If we can show that the map H ( G/H ) = H ( BH ; Z ) → H ( BK ; Z ) = H ( G/K )induced by conjugation and inclusion is surjective, then Lemma 3.2 finishes the proof, for x = (1 G , . . . , G ) ∈ X is fixed by the conjugation action and H ( G/G ) = H ( BG ; Z ) = 0since BG is 3-connected. N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 11
Factoring the map through the isomorphism induced by H → gHg − , it suffices to provesurjectivity in the case where H K and H ( BH ; Z ) → H ( BK ; Z ) is induced by theinclusion of H into K .To see this we consider first the inclusion of the identity components H K . There isa torus T H which is maximal for both H and K as both groups have maximal rank.By [16, Theorem V 7.1] the maps π ( T ) → π ( H ) and π ( T ) → π ( K ) induced by theinclusions are both surjective. This in turn implies that the map π ( H ) → π ( K ) inducedby the inclusion is surjective. From the natural isomorphisms H ( BH ; Z ) ∼ = π ( BH ) ∼ = π ( H ) , and the corresponding ones for K , we derive that H ( BH ; Z ) → H ( BK ; Z ) is surjective.Surjectivity of H ( BH ; Z ) → H ( BK ; Z ) follows then, because there is a commutativediagram with exact rows H ( BH ; Z ) π ( H ) / / (cid:15) (cid:15) H ( BH ; Z ) (cid:15) (cid:15) / / H ( BK ; Z ) π ( K ) / / H ( BK ; Z ) / / BK → BK → Bπ ( K ) , and the corresponding one for H . In more detail, consider the Serre spectral sequence˜ E p,q = H p ( π ( K ); H q ( BK ; Z )) = ⇒ H p + q ( BK ; Z ) . As BK is path–connected and simply–connected, H ( BK ; Z ) ∼ = Z is the trivial π ( K )-module and H ( BK ; Z ) = 0. In particular, both ˜ E , and ˜ E , vanish.Now observe that π ( K ) is a cyclic group; if n = 2, then this is Lemma 3.4, and if G iseither SU ( m ) or Sp ( m ), then K is connected by Lemma 3.3. As a consequence, we havethat H ( π ( K ); Z ) = 0 and, therefore, ˜ E , = 0. Furthermore, we can identify ˜ E , with thecoinvariants H ( BK ; Z ) π ( K ) . The vanishing of ˜ E , implies that ˜ E , = ˜ E , . It follows that˜ E , = H ( BK ; Z ) π ( K ) / Im( d : ˜ E , → ˜ E , ) . For degree reasons, the differentials from or to ˜ E r , are zero when r >
4, which implies that˜ E ∞ , = ˜ E , . As a result, there is an exact sequence0 → Im( d : ˜ E , → ˜ E , ) → H ( BK ; Z ) π ( K ) → H ( BK ; Z ) → , and a corresponding exact sequence for H . The diagram is obtained by naturality of theSerre spectral sequence. (cid:3) As a consequence of Proposition 3.5 we obtain the following theorem, which was stated inthe introduction in terms of π rather than H (the two statements being equivalent by theHurewicz theorem). Theorem 3.6.
Suppose that G is SU ( m ) or Sp ( m ) with m > . For every n > thequotient map π : Hom( Z n , G ) → Rep( Z n , G ) induces an isomorphism H (Hom( Z n , G ); Z ) ∼ = H (Rep( Z n , G ); Z ) . Proof.
Let X := Hom( Z n , G ) and consider the Bredon spectral sequence E p,q = H Gp ( X ; H q ) = ⇒ H p + q ( EG × G X ; Z ) . As pointed out before, the map we aim to show is an isomorphism is identified with thecomposite H ( EG × G X ; Z ) → E ∞ , ֒ → E , . By Lemma 3.3, G acts on X with connected isotropy groups, hence E p, = H Gp ( X ; H ) = 0for all p >
0. This implies that d : E , → E , is trivial, so that E ∞ , ∼ = E , . Now E , vanishes by Proposition 3.5, whence the only possibly non-zero term of total degreetwo on the E ∞ -page is E ∞ , . Consequently, the projection H ( EG × G X ; Z ) → E ∞ , is anisomorphism. (cid:3) As a result, the computation of π (Hom( Z n , G )) for G = SU ( m ) or G = Sp ( m ) is reducedto the calculation of H (Rep( Z n , G ); Z ). This will be adressed in Section 4.For the case of commuting pairs we record an intermediate result in the following lemma. Lemma 3.7.
Let G be a simply–connected and simple compact Lie group. Then π (Hom( Z , G )) ∼ = Z ⊕ E , , where E , = H G (Hom( Z , G ); H ) is a finite group. Moreover, the image of π ∗ : π (Hom( Z , G )) → π (Rep( Z , G )) is a finite index subgroup d Z ⊆ Z where d equals the order of E , = H G (Hom( Z , G ); H ) .Proof. In the Bredon spectral sequence for Hom( Z , G ) we have that E , ∼ = Z (see Figure1), and E , = 0 by Proposition 3.5. Because EG × G Hom( Z , G ) is simply–connected, wemust have E ∞ , = 0. The only way this can happen is by d : E , → E , being surjective.As all isotropy groups of Hom( Z , G ) are compact, the coefficient system H takes values infinite abelian groups. Since Hom( Z , G ) is compact, E , = H G (Hom( Z , G ); H ) is finitelygenerated. Together this implies that E , is finite. Therefore, the subgroup E ∞ , E , isker( d ) ∼ = d Z , where d is the order of the cyclic group E , .Finally, E ∞ , ∼ = E , because E , = 0. Since E ∞ , = ker( d ) ∼ = Z is free, we have that π (Hom( Z , G )) ∼ = E ∞ , ⊕ E , . (cid:3) The groups H Gp (Hom( Z , G ); H ), p > Commuting n -tuples in SU ( m ) and Sp ( k )In this section we focus on the cases G = SU ( m ) and G = Sp ( k ). Since Hom( Z n , G ) andRep( Z n , G ) are path–connected for all n >
1, we will omit the subscript in this section. Theorem 4.1.
Let n > . Then N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 13 (i) for every m > there is an isomorphism π (Hom( Z n , SU ( m ))) ∼ = Z ( n ) , and the standard inclusion SU ( m ) → SU ( m + 1) induces an isomorphism π (Hom( Z n , SU ( m ))) ∼ = −→ π (Hom( Z n , SU ( m + 1))) , (ii) for every k > there is an isomorphism π (Hom( Z n , Sp ( k ))) ∼ = Z ( n ) ⊕ ( Z / n − − n − ( n ) , and the standard inclusion Sp ( k ) → Sp ( k + 1) induces an isomorphism π (Hom( Z n , Sp ( k ))) ∼ = −→ π (Hom( Z n , Sp ( k + 1))) . Proof.
Let us first consider the quaternionic unitary groups Sp ( k ) with k >
1. Because ofTheorem 3.6 it suffices to calculate H (Rep( Z n , Sp ( k )); Z ). By [2, Proposition 6.3] there is ahomeomorphism Rep( Z n , Sp ( k )) ∼ = SP k (( S ) n / Z / SP k denotes the k -th symmetricpower. Here Z / S ⊆ C and diagonally on ( S ) n . Underthis identification the inclusion Sp ( k ) → Sp ( k + 1) given by block sum with a 1 × SP k (( S ) n / Z / → SP k +1 (( S ) n / Z / S ) n / Z / H (( S ) n / Z / Z ) = 0 and H (( S ) n / Z / Z ) ∼ = Z ( n ) ⊕ ( Z / n − − n − ( n ) . This together with the theorem of Dold and Thom imply that SP ∞ (( S ) n / Z /
2) is a simply–connected space, and H ( SP ∞ (( S ) n / Z / Z ) ∼ = π ( SP ∞ (( S ) n / Z / ∼ = Z ( n ) ⊕ ( Z / n − − n − ( n ) . By Steenrod’s splitting [47, Section 22] the map SP k (( S ) n / Z / → SP k +1 (( S ) n / Z /
2) issplit injective in homology for every k >
1. As the groups H ( SP k (( S ) n / Z / Z ) for k = 1and k = ∞ agree, each of the maps H ( SP k (( S ) n / Z / Z ) → H ( SP k +1 (( S ) n / Z / Z )must be an isomorphism. This proves part (ii) of the theorem.Consider now the special unitary group SU ( m ). By [32, Proposition 4.4] the spaceRep( Z n , SU ( m )) is homotopy equivalent to the universal cover of Rep( Z n , U ( m )). Thereis a homeomorphism Rep( Z n , U ( m )) ∼ = SP m (( S ) n ), so that we obtain π (Rep( Z n , SU ( m ))) ∼ = π ( SP m (( S ) n )) . By the Dold-Thom theorem, there is an isomorphism π ( SP ∞ (( S ) n )) ∼ = H (( S ) n ; Z ) ∼ = Z ( n ) . To prove part (i) of the theorem it suffices to show that for every m > SP m (( S ) n ) → SP ∞ (( S ) n ) induces an isomorphism of second homotopy groups. But thisfollows at once from the unpublished preprint [48, Section 8], where it is shown that for a connected based CW complex X and every k > π k ( SP m ( X )) → π k ( SP m +1 ( X ))is an isomorphism whenever m > k . (cid:3) Remark 4.2. As SU (2) ∼ = Sp (1) also the case of SU (2) is covered by Theorem 4.1.In the case of commuting pairs the inclusion SU (2) ֒ → SU (3) induces an isomorphism π (Hom( Z , SU (2))) ∼ = π (Hom( Z , SU (3))). This follows from Theorem 3.6 and the factthat the induced map of representation spaces Rep( Z , SU (2)) → Rep( Z , SU (3)) can beidentified with the standard inclusion CP ⊆ CP as shown in [32, Lemma 5.4].5. Commuting pairs in a simple and simply–connected group
Throughout this section G will denote a simply–connected and simple compact Lie groupof rank r >
1, unless stated otherwise. In this case Hom( Z , G ) is path–connected andsimply–connected, hence we will drop the subscript from the notation. One objective ofthis section is to prove Theorem 5.21, which asserts that π (Hom( Z , G )) ∼ = Z and on π themap Hom( Z , G ) → Rep( Z , G ) induces multiplication by the Dynkin index of G .In view of Lemma 3.7 the proof amounts to a calculation of H Gp (Hom( Z , G ); H ) for p = 0 ,
1. Note that, by Lemma 3.4, π ( H ) is abelian as H ranges over the isotropy groupsof Hom( Z , G ). Thus, H ( G/Z G ( x, y )) = H ( BZ G ( x, y ); Z ) ∼ = π ( Z G ( x, y )) , for all ( x, y ) ∈ Hom( Z , G ). This justifies introducing the coefficient system π : G/Z G ( x, y ) π ( Z G ( x, y )) , which we will use from now on instead of H .5.1. The component group π ( Z G ( x, y )) . In order to calculate the homology groups H G ∗ (Hom( Z , G ); π ) we must acquire a good understanding of the coefficient system π .Thus, our first goal is to describe the group of components of the centralizer Z G ( x, y ) of apair ( x, y ) ∈ Hom( Z , G ). The description will be in terms of the fundamental group of thederived group DZ G ( x ) (Lemma 5.2). This description can be found in [11], but since theproofs may not be easy to find we include them here for convenience.We begin by explaining some of our notation regarding root systems and the fundamentalalcove. Let g C be the complexified Lie algebra of G . As a Cartan subalgebra of g C wechoose t C , the complexification of the Lie algebra t of the maximal torus T . We will workwith the real roots of the root system associated with ( g C , t C ). Thus the roots are R -valuedfunctionals α : t → R such that the weight space g α = { Z ∈ g C | [ H, Z ] = 2 πiα ( H ) Z for all H ∈ t C } is non-trivial. Choose a W -invariant inner product h· , ·i on t . For each root α ∈ t ∗ wecan find a unique element h α ∈ t such that α ( H ) = h H, h α i for every H ∈ t . Define α ∨ = 2 h α / h h α , h α i ∈ t . The element α ∨ is called the coroot associated to the root α and For the particular case X = ( S ) n this stabilization result can also be proved directly by constructing asuitable CW-structure on SP m (( S ) n ). However, to avoid digression, we chose not to include a proof of thisfact. N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 15 is independent of the choice of inner product h· , ·i . Let ∆ = { α , . . . , α r } be a fixed set ofsimple roots for ( g C , t C ) and α the lowest root. We can write − α ∨ in the form − α ∨ = n ∨ α ∨ + n ∨ α ∨ + · · · + n ∨ r α ∨ r for unique integers n ∨ , n ∨ , . . . , n ∨ r > coroot integers of G . It will beconvenient to set n ∨ := 1. The coroot integers n ∨ , . . . , n ∨ r can be found, for instance, in theappendix of [11] and they are also listed in Table 1.Within t we find the coroot lattice Q ∨ defined as the Z -span of { α ∨ , . . . , α ∨ r } ⊆ t , andthe integral lattice Λ := ker(exp : t → T ) which contains Q ∨ . In general, these two latticesdetermine the fundamental group of G by the well known formula π ( G ) ∼ = Λ /Q ∨ , see [15, IX § G is simply–connected, we have thatΛ = Q ∨ .Let A = A (∆) ⊆ t be the closed alcove contained in the closed Weyl chamber determinedby ∆ and that contains 0 ∈ t . The alcove A is an r -dimensional simplex supported by thehyperplanes { α j = 0 } for 1 j r and the affine hyperplane { α = − } . (If G weresemisimple, then A would be the product of the alcoves of its simple factors.) As explainedin [15, IX § G is compact and simply–connected, the exponential mapinduces a homeomorphism A ∼ = −→ T /W .
In particular, every x ∈ G is conjugate to an element of the form exp(˜ x ) for a uniquelydetermined ˜ x ∈ A . Thus, for the purpose of describing Z G ( x ) we may assume that x = exp(˜ x )for some ˜ x ∈ A . In this case we choose T Z G ( x ) as a maximal torus for Z G ( x ).Let ˜∆ := ∆ ∪ { α } be the extended set of simple roots. Abusing notation slightly, we willsometimes regard ˜∆ simply as the set of indices { , , . . . , r } of the extended set of simpleroots. Let ˜∆( x ) ⊆ ˜∆ be the proper subset defined by(2) ˜∆( x ) := { α ∈ ˜∆ | ˜ x lies in the wall of A determined by α } . Let Q ∨ ( x ) Q ∨ be the sublattice of the coroot lattice spanned by { α ∨ i | i ∈ ˜∆( x ) } . Then˜∆( x ) is a system of simple roots for Z G ( x ) relative to T , and Q ∨ ( x ) is the correspondingcoroot lattice, see for example [16, V.2 Proposition 2.3(ii)]. As Z G ( x ) has the same integrallattice as G we have(3) π ( Z G ( x )) ∼ = Λ /Q ∨ ( x ) ∼ = Q ∨ /Q ∨ ( x ) . The following lemma is a combination of Corollary 3.1.3 and Proposition 7.6.1 of [11]. Aproof of the first part may be found in [27, Theorem 1].
Lemma 5.1.
The fundamental group of DZ G ( x ) is isomorphic to the torsion subgroup of Q ∨ /Q ∨ ( x ) and is a cyclic group of order n ∨ ( x ) := gcd { n ∨ i | i ∈ ˜∆ \ ˜∆( x ) } . A representative in Q ∨ for the generator of π ( DZ G ( x )) is ζ ( x ) := 1 n ∨ ( x ) X i ∈ ˜∆ \ ˜∆( x ) n ∨ i α ∨ i . Moreover, if x ′ = exp(˜ x ′ ) is such that ˜ x ′ ∈ A and Z G ( x ) Z G ( x ′ ) , then the inclusion Z G ( x ) → Z G ( x ′ ) induces an injection π ( DZ G ( x )) → π ( DZ G ( x ′ )) sending ζ ( x ) n ∨ ( x ′ ) n ∨ ( x ) ζ ( x ′ ) . Proof.
By [15, IX § DZ G ( x ) → Z G ( x ) induces an isomorphismof π ( DZ G ( x )) onto the torsion subgroup of π ( Z G ( x )) ∼ = Q ∨ /Q ∨ ( x ).Writing the coroot lattice as Q ∨ = ( L i ∈ ˜∆ Z h α ∨ i i ) / Z h P i ∈ ˜∆ n ∨ i α ∨ i i , we may write Q ∨ /Q ∨ ( x )in the form Q ∨ /Q ∨ ( x ) ∼ = M i ∈ ˜∆ \ ˜∆( x ) Z h α ∨ i i / Z h X i ∈ ˜∆ \ ˜∆( x ) n ∨ i α ∨ i i = M i ∈ ˜∆ \ ˜∆( x ) Z h α ∨ i i / Z h n ∨ ( x ) ζ ( x ) i . Since ζ ( x ) is a linear combination of { α ∨ i | i ∈ ˜∆ \ ˜∆( x ) } with coprime coefficients, it can becompleted to a basis of L i ∈ ˜∆ \ ˜∆( x ) Z h α ∨ i i . In this basis it becomes obvious that Q ∨ /Q ∨ ( x ) ∼ = Z | ˜∆ \ ˜∆( x ) |− ⊕ Z /n ∨ ( x ), where Z /n ∨ ( x ) is generated by the image of ζ ( x ).For the second part notice that Z G ( x ) Z G ( x ′ ) implies that every wall of A containing˜ x also contains ˜ x ′ , hence ˜∆( x ) ⊆ ˜∆( x ′ ). Therefore, Q ∨ ( x ) Q ∨ ( x ′ ) and by naturalityof the isomorphism (3) the map π ( Z G ( x )) → π ( Z G ( x ′ )) corresponds to the projection Q ∨ /Q ∨ ( x ) → Q ∨ /Q ∨ ( x ′ ). On the torsion subgroups this projection maps ζ ( x ) n ∨ ( x ) X i ∈ ˜∆ \ ˜∆( x ′ ) n ∨ i α ∨ i = n ∨ ( x ′ ) n ∨ ( x ) ζ ( x ′ ) , which is an injection Z /n ∨ ( x ) → Z /n ∨ ( x ′ ). (cid:3) Let ( x, y ) ∈ Hom( Z , G ) and let ˜ y denote a lift of y ∈ Z G ( x ) in the universal coveringgroup ^ Z G ( x ). The component group π ( Z G ( x, y )) may be described as follows. Lemma 5.2.
For z ∈ Z G ( x, y ) let [ z ] ∈ π ( Z G ( x, y )) denote the path component determinedby z . Let ˜ z be any lift of z in ^ Z G ( x ) . Then the map δ y : π ( Z G ( x, y )) → π ( DZ G ( x )) Z ( ^ DZ G ( x )) defined by δ y ([ z ]) = [˜ y, ˜ z ] is an injective group homomorphism.Proof. We follow [11, Section 7.3]. Consider the universal covering sequence1 → Q ∨ /Q ∨ ( x ) → ^ Z G ( x ) → Z G ( x ) → . The sequence is acted upon by the cyclic group h ˜ y i through conjugation by ˜ y on ^ Z G ( x ) andby y on Z G ( x ), leaving invariant the central subgroup Q ∨ /Q ∨ ( x ). Passing to fixed pointsand noting that Z G ( x ) h ˜ y i ∼ = Z G ( x, y ) yields the exact sequence1 → Q ∨ /Q ∨ ( x ) → ^ Z G ( x ) h ˜ y i → Z G ( x, y ) δ −→ Q ∨ /Q ∨ ( x ) , where the connecting homomorphism δ is defined by δ ( z ) = ˜ z ˜ y ˜ z − = [˜ y, ˜ z ]. N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 17
Now ^ Z G ( x ) h ˜ y i is connected, by [15, IX § y in the simply–connected group ^ Z G ( x ). Therefore, ^ Z G ( x ) h ˜ y i maps to the identity componentof Z G ( x, y ). Since Q ∨ /Q ∨ ( x ) is discrete, and by exactness, δ descends to an injective map δ y : π ( Z G ( x, y )) → Q ∨ /Q ∨ ( x ).To finish the proof, note that π ( Z G ( x, y )) is finite, since Z G ( x, y ) is compact, so δ y factorsthrough the torsion subgroup of Q ∨ /Q ∨ ( x ). By Lemma 5.1, the latter is identified with π ( DZ G ( x )). (cid:3) Lemma 3.4 is now immediate:
Proof of Lemma 3.4.
Let ( x, y ) ∈ Hom( Z , G ). By Lemmas 5.1 and 5.2, π ( Z G ( x, y )) is asubgroup of a cyclic group of order n ∨ ( x ). A look at the coroot diagrams in the appendixof [11] (or at Table 1) shows that 1 n ∨ ( x )
6. But then π ( Z G ( x, y )) must also be cyclic,and of order at most six. (cid:3) For later use we record a further consequence of the preceding lemmas, a special case of[11, Corollary 7.6.2].
Lemma 5.3.
Let x = exp(˜ x ) and x ′ = exp(˜ x ′ ) for some ˜ x, ˜ x ′ ∈ A , and let y, y ′ ∈ T . Supposethat Z G ( x ) Z G ( x ′ ) and Z G ( x, y ) Z G ( x ′ , y ′ ) . Then the map π ( Z G ( x, y )) → π ( Z G ( x ′ , y ′ )) induced by the inclusion is injective.Proof. By naturality of the connecting homomorphism, there is a commutative diagram π ( Z G ( x, y )) δ y / / (cid:15) (cid:15) π ( DZ G ( x )) (cid:15) (cid:15) π ( Z G ( x ′ , y ′ )) δ y ′ / / π ( DZ G ( x ′ ))in which the left hand vertical map is induced by the inclusion Z G ( x, y ) Z G ( x ′ , y ′ ), and theone on the right is induced by the inclusion Z G ( x ) → Z G ( x ′ ). The assertion follows, because δ y is injective by Lemma 5.2, and π ( DZ G ( x )) → π ( DZ G ( x ′ )) is injective by Lemma 5.1. (cid:3) Equivariant cell structure.
We will now describe Hom( Z , G ) as a G -equivariantCW-complex. This will enable us to compute the p -localization of H G ∗ (Hom( Z , G ); π ) asthe homology of a certain G -subcomplex of Hom( Z , G ), see Corollary 5.7.The G -CW-structure on Hom( Z , G ) is obtained from the simplicial structure of the Weylalcove A as follows. Recall that Rep( Z , G ) ∼ = T /W and T /W ∼ = A . Let p i : Hom( Z , G ) → A , i = 1 , π : Hom( Z , G ) → Rep( Z , G ) and the projectiononto the i -th component T /W → A . Let F n , n = 0 , . . . , r , denote the set of n -dimensionalfaces of A . A has a standard CW-structure whose set of n -cells is F n . Let ( A × A ) ( n ) be the n -skeletonof A × A in the product CW-structure. DefineHom( Z , G ) ( n ) := ( p × p ) − (( A × A ) ( n ) ) . This defines an increasing sequence of G -spaces(4) Hom( Z , G ) (0) ⊆ Hom( Z , G ) (1) ⊆ · · · ⊆ Hom( Z , G ) (2 r ) = Hom( Z , G )such that Hom( Z , G ) ( n ) is obtained from Hom( Z , G ) ( n − by attaching a set of equivariant n -cells, as we now explain. Notation.
To simplify the notation, we identify A with a subset of T without making theexponential map explicit. For a face σ ⊆ A we let b ( σ ) ∈ σ denote its barycenter. Since thecentralizer Z G ( x ) of some x ∈ G equals the stabilizer of x under the conjugation action of G on itself, we may write G x instead of Z G ( x ). Moreover, since a face σ ⊆ A is pointwisefixed if and only if b ( σ ) is fixed, we shall write G σ for G b ( σ ) . Similarly, we write W σ for the isotropy group of b ( σ ) ∈ T under the action of W , G ( x,y ) for the centralizer Z G ( x, y ), G ( σ,wτ ) for G ( b ( σ ) ,wb ( τ )) where w ∈ W and σ, τ are faces of A .For each pair of faces σ, τ of A , let C ( σ, τ ) denote a complete set of representatives for thedouble cosets W σ \ W/W τ . For n > J n of the G - n -cells is J n = G ( σ,τ ) ∈ F i × F j i + j = n C ( σ, τ ) . The G - n -cells { e nα | α ∈ J n } are built from the faces of the alcove A in the followingfashion. Given α = ( σ, τ, w ) ∈ J n , the closed G - n -cell e nα ⊆ Hom( Z , G ) is of the form e nα = φ nα ( G/G ( σ,wτ ) × σ × τ ) where the characteristic map φ nα is given by φ nα : G/G ( σ,wτ ) × σ × τ → Hom( Z , G )( gG ( σ,wτ ) , x, y ) ( x, wy ) g . Here the superscript g indicates simultaneous conjugation by g .In the definition of φ nα it must be checked that the right hand side is independent ofthe choice of representative for the coset gG ( σ,wτ ) . To see this notice that G σ G x and G wτ = G ˜ wτ G ˜ wy = G wy where ˜ w ∈ N G ( T ) is a lift of w ∈ W . Therefore,(5) G ( σ,wτ ) = G σ ∩ G wτ G x ∩ G wy = G ( x,wy ) , showing that φ nα is well defined. Lemma 5.4.
The filtration (4) is a G -CW-structure on Hom( Z , G ) whose set of G - n -cellsis { e nα | α ∈ J n } .Proof. Set Hom( Z , G ) ( − := ∅ and assume n >
0. Let P denote the pushout of G ( σ,τ,w ) ∈ J n G/G ( σ,wτ ) × σ × τ ←− G ( σ,τ,w ) ∈ J n G/G ( σ,wτ ) × ∂ ( σ × τ ) ⊔ Jn f nα −−−−→ Hom( Z , G ) ( n − where the attaching maps { f nα | α ∈ J n } arise as the restriction of the characteristic maps { φ nα | α ∈ J n } to the boundary of the complex σ × τ . We must show that the map h : P → Hom( Z , G ) ( n ) N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 19 induced by the characteristic maps and the inclusion Hom( Z , G ) ( n − ֒ → Hom( Z , G ) ( n ) isa homeomorphism. In fact, as P is compact and Hom( Z , G ) ( n ) is Hausdorff, it is enough toshow h is a bijection.Clearly, the image of h contains Hom( Z , G ) ( n − ⊆ Hom( Z , G ) ( n ) . Now suppose that( z , z ) ∈ Hom( Z , G ) ( n ) \ Hom( Z , G ) ( n − . Since G is assumed simply–connected, thereexists g ∈ G such that ( z , z ) = ( x, w ′ y ) g for some w ′ ∈ W and uniquely determined x, y ∈ A . Furthermore, there are unique faces σ ∈ F i and τ ∈ F n − i such that ( x, y ) is inthe relative interior of σ × τ . Now suppose that w ∈ C ( σ, τ ) represents the double cosetdetermined by w ′ . Then w ′ = awb for a ∈ W σ and b ∈ W τ . Thus, ( x, w ′ y ) = ( x, wy ) ˜ a where ˜ a ∈ N G ( T ) is a lift of a . Now φ n ( σ,τ,w ) : ( g ˜ aG ( σ,wτ ) , x, y ) ( z , z ) showing that h issurjective.Now suppose that p, p ′ ∈ P and h ( p ) = h ( p ′ ). If either p or p ′ is represented by an elementof Hom( Z , G ) ( n − then so is the other. It follows that p = p ′ as Hom( Z , G ) ( n − ֒ → Hom( Z , G ) ( n ) is injective.If neither p nor p ′ lifts to Hom( Z , G ) ( n − , then h ( p ) and h ( p ′ ) each lie in the image ofa charactersitic map. This means that there are ( σ, τ, w ) , ( σ ′ , τ ′ , w ′ ) ∈ J n and g, g ′ ∈ G such that h ( p ) = ( x, wy ) g and h ( p ′ ) = ( x ′ , w ′ y ′ ) g ′ for some ( x, y ) ∈ int( σ × τ ) and ( x ′ , y ′ ) ∈ int( σ ′ × τ ′ ) (where int denotes the relative interior of a cell). Then( x, wy ) ≡ ( x ′ , w ′ y ′ ) modulo W , which implies that x = x ′ and y = y ′ (by projecting to A × A ), and further that σ = σ ′ and τ = τ ′ as every point of A × A lies in the relative interior of a unique cell. Let w ′′ ∈ W be such that ( x, wy ) = ( w ′′ x, w ′′ w ′ y ). Then w ′′ ∈ W σ and w − w ′′ w ′ ∈ W τ . This impliesthat w ∈ C ( σ, τ ) and w ′ ∈ C ( σ, τ ) represent the same double coset, hence w = w ′ . Finally,( x, wy ) g = ( x, wy ) g ′ implies that g ≡ g ′ modulo G ( σ,wτ ) , and therefore p = p ′ . It follows that h is injective. (cid:3) Remark 5.5.
In the same way, one can construct a G -CW-structure on Hom( Z k , G ) forany k >
1. The G - n -cells are then indexed over k -tuples ( σ , . . . , σ k ) ∈ F i × · · · × F i k such that P j i j = n and a complete set of representatives C ( σ , . . . , σ k ) for the W -orbitsof the diagonal W -set W/W σ × · · · × W/W σ k . (When k = 2 the latter reduces to a setof representatives for the double cosets W σ \ W/W σ .) This, in fact, gives the indexing setof the n -cells in a (non-equivariant) CW-structure on T k /W . A proof analogous to that ofLemma 5.4 then shows that this CW-structure can be lifted to a G -equivariant CW-structureon Hom( Z k , G ) .Let π ( G ( x,y ) ) ( p ) = π ( G ( x,y ) ) ⊗ Z Z ( p ) denote the localization of the abelian group π ( G ( x,y ) )(see Lemma 3.4) at p . Relative to the CW-structure just described we have: Lemma 5.6.
Let p be a prime. The subspace X G ( p ) ⊆ Hom( Z , G ) defined by X G ( p ) := { ( x, y ) ∈ Hom( Z , G ) | π ( G ( x,y ) ) ( p ) = 0 } is a G -subcomplex of Hom( Z , G ) .Proof. It is clear that X G ( p ) is a union of open G -cells. We must show that it is also a unionof closed G -cells. Let α = ( σ, τ, w ) ∈ J n and write ∂e nα := e nα ∩ Hom( Z , G ) ( n − , int( e nα ) := e nα \ ∂e nα . Suppose that int( e nα ) ⊆ X G ( p ). This means that π ( G ( σ,wτ ) ) ( p ) = 0. We are going to showthat ∂e nα ⊆ X G ( p ).Suppose that ( z , z ) ∈ ∂e nα . Then there is g ∈ G such that ( z , z ) = ( x, wy ) g where( x, y ) ∈ ∂ ( σ × τ ). In particular, π ( G ( z ,z ) ) ∼ = π ( G ( x,wy ) ).As noted in (5) we have that G σ G x and G ( σ,wτ ) G ( x,wy ) . By Lemma 5.3, the map π ( G ( σ,wτ ) ) → π ( G ( x,wy ) ) induced by the inclusion is injective. The map remains injectiveafter p -localization, hence π ( G ( x,wy ) ) ( p ) = 0. But this implies that ( z , z ) ∈ X G ( p ), whichfinishes the proof. (cid:3) Let us write ( π ) ( p ) for the coefficient system obtained by localizing π objectwise at p . Corollary 5.7.
Let P denote the set of primes which divide at least one coroot integer of G . Then there is an isomorphism H G ∗ (Hom( Z , G ); π ) ∼ = M p ∈P H G ∗ ( X G ( p ); ( π ) ( p ) ) . Proof.
We know from Lemma 3.4 that π takes values in finite abelian groups. Hence, itsplits as a direct sum of its localizations at the various primes.On the other hand, we derive from Lemmas 5.1 and 5.2 that ( π ) ( p ) is trivial unless p divides a coroot integer of G . Therefore, π ∼ = L p ∈P ( π ) ( p ) .The corollary is now a consequence of Lemma 5.6; by definition of X G ( p ), the inclusion ofthe chain complex for H ∗ ( X G ( p ); ( π ) ( p ) ) into the chain complex for H ∗ (Hom( Z , G ); ( π ) ( p ) )is an isomorphism. (cid:3) To understand why the decomposition of Corollary 5.7 is useful, we must take a closerlook at the coefficient system ( π ) ( p ) . Lemma 5.8.
Let p ∈ P . Suppose that G / ∈ { E , E } or that p > . Then H G ∗ ( X G ( p ); ( π ) ( p ) ) ∼ = H ∗ ( R G ( p ); Z /p ) where R G ( p ) := X G ( p ) /G .Proof. Let G and p be fixed. The assumption that either G / ∈ { E , E } or that p > O G denote the orbit category of G , whose objects are the homogeneous spaces G/H for closed subgroups H G and whose morphisms are the G -equivariant maps. For theconstruction of the chain complex C ∗ ( X G ( p ); π ) only a subcategory O ′ G of O G is relevant,namely the subcategory generated by the equivariant maps that enter into the definition ofthe differentials. To prove the lemma it suffices to show that under the stated assumptionsthe restriction of ( π ) ( p ) to O ′ G is naturally isomorphic to the constant coefficient system Z /p .The maps defining the differential C n ( X G ( p ); π ) → C n − ( X G ( p ); π ) arise by consideringthe composite of an attaching map f nα for an n -cell e nα and the collapse of the complementof an ( n − e n − β into a point [50]. In our situation this looks as follows. Let α =( σ, τ, w ) ∈ J n , let β = ( σ ′ , τ ′ , w ′ ) ∈ J n − , and let e nα respectively e n − β be the correspondingclosed cells of Hom( Z , G ). The pair ( e nα , e n − β ) can contribute to the differential only if N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 21 int( e n − β ) ∩ ∂e nα = ∅ . Let us assume that the intersection is non-empty. It follows, then, that σ ′ × τ ′ ⊆ ∂ ( σ × τ ) and that e n − β ⊆ ∂e nα .Now int( e n − β ) contains the homeomorphic image (under the characteristic map φ n − β of e n − β ) of the orbit G/G ( σ ′ ,w ′ τ ′ ) ×{ ( b ( σ ′ ) , b ( τ ′ )) } . The preimage of this orbit under the attachingmap f nα is G/G ( σ,wτ ) × { ( b ( σ ′ ) , b ( τ ′ )) } . The composite map G/G ( σ,wτ ) × { ( b ( σ ′ ) , b ( τ ′ )) } ( φ n − β ) − ◦ f nα −−−−−−−→ G/G ( σ ′ ,w ′ τ ′ ) × { ( b ( σ ′ ) , b ( τ ′ )) } is a G -equivariant map and is therefore determined by the image of eG ( σ,wτ ) . Recalling thedefinition of f nα and φ n − β we find that eG ( σ,wτ ) g − G ( σ ′ ,w ′ τ ′ ) where g is any element of G satisfying(6) ( b ( σ ′ ) , wb ( τ ′ )) g = ( b ( σ ′ ) , w ′ b ( τ ′ )) . (The existence of such g is implicit in the assumption that e n − β ⊆ ∂e nα .) In particular, wehave that g ∈ G σ ′ .Now let O ′ G be the subcategory of O G generated by the G -maps G/G ( σ,wτ ) → G/G ( σ ′ ,w ′ τ ′ ) eG ( σ,wτ ) g − G ( σ ′ ,w ′ τ ′ ) where g satisfies condition (6), and σ ′ ⊆ σ and τ ′ ⊆ τ . Then O ′ G includes all morphismsneeded to form the chain complex C ∗ ( X G ( p ); π ).We are going to show that the restriction of ( π ) ( p ) to O ′ G is naturally isomorphic to theconstant coefficient system Z /p . Consider a morphism in O ′ G and let π ( G ( σ,wτ ) ) → π ( G ( σ ′ ,w ′ τ ′ ) )be the map obtained by applying π to it. It can be factored as the map induced by theinclusion G ( σ,wτ ) → G ( σ ′ ,wτ ′ ) and the map induced by conjugation ( − ) g : G ( σ ′ ,wτ ′ ) → G ( σ ′ ,w ′ τ ′ ) .This yields the upper row in the following diagram:(7) π ( G ( σ,wτ ) ) / / δ wb ( τ ) (cid:15) (cid:15) π ( G ( σ ′ ,wτ ′ ) ) ( − ) g / / δ wb ( τ ′ ) (cid:15) (cid:15) π ( G ( σ ′ ,w ′ τ ′ ) ) δ w ′ b ( τ ′ ) (cid:15) (cid:15) π ( DG σ ) / / π ( DG σ ′ ) π ( DG σ ′ ) Z /p ǫ σ ζ ( b ( σ )) O O Z /p. ǫ σ ′ ζ ( b ( σ ′ )) O O The first map in the middle row is the one induced by the inclusion G σ → G σ ′ . Let us showthat the upper half of the diagram commutes. The left hand square commutes by naturalityof the connecting homomorphism. To see that the right hand square commutes, let ˜ g denotea lift of g in the universal cover g G σ ′ , and let z ∈ G ( σ ′ ,wτ ′ ) . By definition of the connectinghomomorphism (see Lemma 5.2), we obtain δ w ′ b ( τ ′ ) ( z g ) = [ ^ w ′ b ( τ ′ ) , e z g ] = [ ^ ( wb ( τ ′ )) g , e z g ] = [ ^ wb ( τ ′ ) , ˜ z ] ˜ g = δ wb ( τ ′ ) ( z ) . In the second equality we used (6), and in the last equality we used the fact that thecommutator is in the center of g G σ ′ .Next, we are going to show that the vertical arrows in the upper half of the diagrambecome isomorphisms after p -localization. Let ( x, y ) ∈ X G ( p ) with x ∈ A . Then, byLemma 5.2, π ( DG x ) contains p -torsion, hence p | n ∨ ( x ) as π ( DG x ) ∼ = Z /n ∨ ( x ). The setof coroot integers of G displayed in Table 1 is at the same time the set of possible valuesthat n ∨ ( x ) can attain as x ranges over the alcove A . If G / ∈ { E , E } or p >
2, then n ∨ ( x )does not contain repeated primes. Hence, π ( DG x ) ( p ) ∼ = Z /p . By Lemma 5.2, the map δ y : π ( G ( x,y ) ) → π ( DG x ) is injective, and it remains so after p -localization. It follows that π ( G ( x,y ) ) ( p ) ∼ = −→ π ( DG x ) ( p ) . To finish the proof we show that if σ ′ ⊆ σ and p | n ∨ ( b ( σ )), then the map π ( DG σ ) ( p ) ∼ = −→ π ( DG σ ′ ) ( p ) induced by the inclusion G σ G σ ′ is naturally isomorphic to the identity at Z /p . Thisis achieved by making an appropriate choice of generators. Recall from Lemma 5.1 that π ( DG σ ) is a cyclic group of order n ∨ ( b ( σ )) generated by ζ ( b ( σ )). Let ǫ σ = n ∨ ( b ( σ )) = 6 and p = 2 , n ∨ ( b ( σ )) = 6 and p = 3 , Z /p → π ( DG σ ) sending 1 ǫ σ ζ ( b ( σ )) is injective and becomes an isomorphismafter p -localization. Moreover, the map π ( DG σ ) → π ( DG σ ′ ) sends ǫ σ ζ ( b ( σ )) ǫ σ ′ ζ ( b ( σ ′ ))by the second part of Lemma 5.1. This demonstrates commutativity of the lower part ofdiagram (7). As a consequence, (the p -localization of) diagram (7) establishes a naturalisomorphism of ( π ) ( p ) with Z /p as asserted. (cid:3) Let us comment on what happens when G ∈ { E , E } and p = 2. Let us restrict thecoefficient system π (0) to the G -subcomplex X G (2) of Hom( Z , G ). According to Table 1 thecoefficient system ( π ) (2) can now evaluate to Z / Z /
4. Thus, in contrast to Lemma 5.8it need not be isomorphic to a constant coefficient system. To handle these cases we observethat the argument of Lemma 5.6 shows that X G (4) := { ( x, y ) ∈ X G (2) | π ( G ( x,y ) ) (2) ∼ = Z / } is a G -subcomplex of X G (2). Restricted to X G (4) the coefficient system ( π ) (2) is naturallyisomorphic to Z / Lemma 5.9.
Let G = E or G = E . Then H G ∗ ( X G (4); ( π ) (2) ) ∼ = H ∗ ( R G (4); Z / , where R G (4) := X G (4) /G . N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 23
The quotient spaces R G ( p ) . Recall the G -CW-complex X G ( p ) = { ( x, y ) ∈ Hom( Z , G ) | π ( Z G ( x, y )) ( p ) = 0 } and its orbit space R G ( p ) = X G ( p ) /G . At this point, we have reduced the computation of H G ∗ (Hom( Z , G ); π ) to a computation of the non-equivariant homology of R G ( p ), with anexception when G is E or E . As R G ( p ) is the orbit space of a G -subcomplex of Hom( Z , G )it is a subcomplex in the induced CW-structure of T /W . It would be interesting to describethis subcomplex, but here we will content ourselves with a calculation of the homology of R G ( p ). This will be achieved by providing an explicit homotopy equivalence of R G ( p ) witha weighted projective space.Let w = ( w , . . . , w r ) be a tuple of positive integers. Consider the weighted action of thecircle group S ⊆ C on the unit sphere S r +1 ⊆ C r +1 defined by λ · ( z , . . . , z r ) = ( λ w z , . . . , λ w r z r ) ( λ ∈ S , ( z , . . . , z r ) ∈ S r +1 ) . The quotient space CP ( w ) := S r +1 / S w is called a weighted projective space . Here we use the subscript w to indicate the weighted S -action. The most familiar example arises when w = (1 , . . . ,
1) in which case CP ( w ) = CP r is the usual complex projective space.Recall that, relative to a fixed simply–connected simple compact Lie group G , we let P = { n ∨ , . . . , n ∨ r }\{ , , } ∩ { n ∨ , . . . , n ∨ r } denote the set of primes dividing a coroot integerof G . Define Z := { n ∨ , . . . , n ∨ r }\{ , } ∩ { n ∨ , . . . , n ∨ r } . Thus Z = P except when G is E or E in which case Z = P ∪ { } . The objective of thissubsection is to prove: Proposition 5.10.
Let n ∨ = ( n ∨ , . . . , n ∨ r ) be the tuple of coroot integers of G and let p ∈ Z be fixed. Let n ∨ ( p ) = ( n ∨ i , . . . , n ∨ i k ) denote the tuple obtained from n ∨ by removing thoseentries that are not divisible by p . Let ι p : CP ( n ∨ ( p )) → CP ( n ∨ ) be the map defined inhomogeneous coordinates by [ y , . . . , y k ] [ z , . . . , z r ] where z l = y j if l = i j for some j k , and z l = 0 otherwise. Then there is a commutative diagram R G ( p ) ≃ / / (cid:15) (cid:15) CP ( n ∨ ( p )) ι p (cid:15) (cid:15) Rep( Z , G ) ≃ / / CP ( n ∨ ) in which the horizontal arrows are homotopy equivalences and the left hand vertical arrow isthe subspace inclusion. Like any complete toric variety a weighted projective space is simply–connected (for thedefinition of weighted projective space as a toric variety see for example [20, p. 35], andfor the result on the fundamental group see [20, Section 3.2]). The integral cohomology of CP ( w ) was computed by Kawasaki in [28, Theorem 1]. The homology is then obtained fromthe universal coefficient theorem:(8) H k ( CP ( w ); Z ) ∼ = ( Z , if k r is even,0 , otherwise.In view of this result we have: Corollary 5.11.
Let p ∈ P and assume that G
6∈ { E , E } or that p > . Then H Gk ( X G ( p ); ( π ) ( p ) ) ∼ = ( Z /p , if k ℓ −
1) is even , , otherwise , where ℓ > is the number of coroot integers of G divisible by p .Proof. By Lemma 5.8, H G ∗ ( X G ( p ); ( π ) ( p ) ) ∼ = H ∗ ( R G ( p ); Z /p ). By Proposition 5.10, R G ( p )is homotopy equivalent to a weighted projective space of complex dimension ℓ −
1. Thehomology groups follow therefore from Kawasaki’s computation (8). (cid:3)
Our approach to Proposition 5.10 is as follows. The homotopy equivalence Rep( Z , G ) ≃ CP ( n ∨ ) will be obtained as a composite of maps,(9) Rep( Z , G ) ∼ = −→ A ⊗ I F ≃ −→ A ⊗ I ¯ F ∼ = −→ CP ( n ∨ ) . The two spaces in the middle are certain coends that will be defined below, and the homotopyequivalence is induced from a natural equivalence of diagrams F ∼ −→ ¯ F . The equivalence R G ( p ) ≃ CP ( n ∨ ( p )) will be obtained from this by restriction to subspaces, whence thediagram in Proposition 5.10 will commute by construction.In brief, the homotopy equivalence Rep( Z , G ) ≃ CP ( n ∨ ) is given by(10) ( x, y ) modulo G ( a , a t , . . . , a r t r ) / vuut r X i =0 a i modulo S n ∨ , where x ∈ A and ( a , . . . , a r ) ∈ ∆ r are the barycentric coordinates of x , and y ∈ T and( t , . . . , t r ) ∈ ( S ) r are the coordinates of y with respect to the one-parameter subgroups of T determined by the coroots α ∨ , . . . , α ∨ r . This will be clear once we have proved Lemma5.19.To describe the first one of the two homeomorphisms in (9) we will regard Rep( Z , G )and R G ( p ) as spaces over the alcove A . Fix p ∈ Z . For an integer m let A ( m ) ⊆ A be thesubspace of the fundamental alcove defined by x ∈ A ( m ) ⇐⇒ m | n ∨ ( x ) . This is again a simplex, in fact, a face of A as we explain below (Lemma 5.14). Note thatif ( x, y ) ∈ X G ( p ) and x ∈ A , then p | n ∨ ( x ), hence x ∈ A ( p ). Thus, there is a commutativediagram R G ( p ) / / q (cid:15) (cid:15) Rep( Z , G ) pr (cid:15) (cid:15) A ( p ) / / A N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 25 in which pr is the projection of Rep( Z , G ) onto the first component and q is its restrictionto the subspace R G ( p ). The two horizontal maps are the inclusions. We will describe thefibers of pr and q . Notation.
When K is a group, we write K/ Ad K for the quotient by the conjugation (oradjoint) action of K on itself. The conjugacy class of an element g ∈ K is denoted by( g ) ∈ K/ Ad K . Similarly, given ( x, y ) ∈ Hom( Z , G ) we write (( x, y )) ∈ Rep( Z , G ) for itsequivalence class under simultaneous conjugation. When ( x, y ) ∈ T , we will also write(( x, y )) ∈ T /W for its equivalence class under the diagonal action of W .Clearly, if x ∈ A , then the assignment ( y ) (( x, y )) defines a homeomorphism(11) Z G ( x ) / Ad Z G ( x ) ∼ = pr − ( x ) . On the other hand, if x ∈ A ( p ), then q − ( x ) = pr − ( x ) ∩ R G ( p ) corresponds to a subspaceof Z G ( x ) / Ad Z G ( x ) . To describe it we rewrite (11) using the finite covering ρ : ^ DZ G ( x ) × Z ( Z G ( x )) → Z G ( x )( g, t ) u ( g ) t , where Z ( Z G ( x )) is the identity component of the center of Z G ( x ), and u : ^ DZ G ( x ) → DZ G ( x ) is the universal covering (see [15, IX § C Z ( ^ DZ G ( x )) × Z ( Z G ( x )) such that ρ descends to an isomorphism(12) ^ DZ G ( x ) × C Z ( Z G ( x )) ∼ = −→ Z G ( x ) . Under this isomorphism (11) becomes(13) pr − ( x ) ∼ = (cid:16) ^ DZ G ( x ) / Ad ^ DZ G ( x ) (cid:17) × C Z ( Z G ( x )) . Here C acts on the adjoint orbits in the natural way: If K is any group, then an action ofthe center Z ( K ) on K/ Ad K is defined by(14) c · ( g ) = ( cg ) , ( c ∈ Z ( K ) , g ∈ K ) . We can now describe the fibers of q : R G ( p ) → A ( p ). Lemma 5.12.
Let p ∈ Z , and let x ∈ A ( p ) . Then, there is an element ξ ( x ) ∈ Z ( ^ DZ G ( x )) and a homeomorphism q − ( x ) ∼ = (cid:16) ^ DZ G ( x ) / Ad ^ DZ G ( x ) (cid:17) h ξ ( x ) i × C Z ( Z G ( x )) . Note that both C and h ξ ( x ) i act through subgroups of the center Z ( ^ DZ G ( x )). As thecenter is abelian, the two actions commute, and there is an induced action of C on the h ξ ( x ) i -fixed points. Proof.
Let us discuss the case where p is a prime. The case p = 4 is treated analogously, seeRemark 5.13. By definition of R G ( p ) and the isomorphism (11) we have that(15) q − ( x ) ∼ = { y ∈ Z G ( x ) | π ( Z G ( x, y )) ( p ) = 0 } / Ad Z G ( x ) . To reformulate the condition π ( Z G ( x, y )) ( p ) = 0 we will use the connecting homomorphismdefined in Lemma 5.2. To this end, let ξ ( x ) := ( any generator of π ( DZ G ( x )) ( p ) ∼ = Z /p if p > , the unique element of order 2 in π ( DZ G ( x )) (2) if p = 2viewed as an element in the center of ^ DZ G ( x ). Then we have that π ( Z G ( x, y )) ( p ) = 0 ⇐⇒ ξ ( x ) ∈ Im( δ y ) ⇐⇒ ∃ ˜ z ∈ ^ Z G ( x ) : [˜ y, ˜ z ] = ξ ( x ) . Since ^ Z G ( x ) ∼ = ^ DZ G ( x ) × ^ Z ( Z G ( x )) , the last condition is equivalent to y ∈ U ( x ) × ^ Z ( Z G ( x )) ,where U ( x ) := { ˜ a ∈ ^ DZ G ( x ) | ∃ ˜ z ∈ ^ DZ G ( x ) : [˜ a, ˜ z ] = ξ ( x ) } . Similarly to (13) we then obtain q − ( x ) ∼ = (cid:16) U ( x ) / Ad ^ DZ G ( x ) (cid:17) × C Z ( Z G ( x )) . Finally, writing out the commutator shows that ∃ ˜ z ∈ ^ DZ G ( x ) : [˜ a, ˜ z ] = ξ ( x ) ⇐⇒ ξ ( x ) · (˜ a ) = (˜ a ) , hence U ( x ) / Ad ^ DZ G ( x ) ∼ = (cid:16) ^ DZ G ( x ) / Ad ^ DZ G ( x ) (cid:17) h ξ ( x ) i , and the lemma follows. (cid:3) Remark 5.13.
In the case p = 4 the defining condition for q − ( x ) reads π ( Z G ( x, y )) (2) ∼ = Z / ξ ( x ) to be any generator of π ( DZ G ( x )) (2) ∼ = Z /
4. The rest of the proofgoes through verbatim.Observe that the fibers q − ( x ) only depend on the face σ ⊆ A ( p ) such that x ∈ int( σ )and not on the specific point x chosen within int( σ ). That is, if x, y ∈ int( σ ), then q − ( x ) = q − ( y ) = q − ( b ( σ )), and likewise for pr . Together with the combinatorial structure of thealcove this allows for a more manageable description of R G ( p ) and Rep( Z , G ).To this end, recall that the faces of the alcove A are in one-to-one correspondence withthe proper subsets I ⊆ ˜∆ of the extended set of simple roots as follows: I ( ˜∆ ←→ σ I ∈ F r −| I | , where σ I := { x ∈ A | ∀ α ∈ I : x lies in the wall of A determined by α } . Since A is a simplex and σ I is an intersection of facets, σ I is a face of A .For example, if I = ∅ , then σ ∅ = A . If I = ˜∆ \{ α j } for some j ∈ { , . . . , r } , then σ I isthe vertex opposite the wall of A determined by α j . If x ∈ A and I = ˜∆( x ) (as defined inSection 5.1 eq. (2)), then σ I is the minimal face of A containing x . Also relevant to us is: Lemma 5.14.
Let m be an integer and let I m = { α j ∈ ˜∆ | m ∤ n ∨ j } . Then σ I m = A ( m ) . N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 27
Proof.
We have that x ∈ A ( m ) ⇐⇒ m | n ∨ ( x ) ⇐⇒ ∀ j ∈ ˜∆ \ ˜∆( x ) : m | n ∨ j ⇐⇒ I m ⊆ ˜∆( x ) ⇐⇒ ∀ α ∈ I m : x lies in the wall of A determined by α ⇐⇒ x ∈ σ I m , by definition of A ( m ), n ∨ ( x ), ˜∆( x ) and I m . (cid:3) It follows that A ( m ) is a face of A , hence a simplex.Let I denote the poset of proper subsets of ˜∆ partially ordered by inclusion. Note thatwhen I, J ∈ I and I ⊆ J , then σ J is a face of σ I . Therefore, the assignment I σ I definesa contravariant functor from I to Top sending the inclusion I ⊆ J to the opposite inclusion σ J ⊆ σ I . This functor encodes the face structure of the fundamental alcove. Abusingnotation, we denote it by A : I op → Top , I A ( I ) = σ I . Similarly, for an integer m we denote by A ( m ) : I ( m ) op → Top the restriction of A to the subposet I ( m ) ⊆ I consisting of those subsets I containing I m (seeLemma 5.14). It encodes the face structure of the simplex A ( m ).On the other hand, we can define the following covariant functors on I . Recall that if σ, τ are faces of A and τ ⊆ σ , then there is an inclusion Z G ( b ( σ )) Z G ( b ( τ )). This induces amap Z G ( b ( σ )) / Ad Z G ( b ( σ )) → Z G ( b ( τ )) / Ad Z G ( b ( τ )) (which is, in fact, a surjection rather thanan inclusion) and thus by (11) a mappr − ( b ( σ )) → pr − ( b ( τ )) . Now if σ, τ are faces of A ( p ) (where p is as in Lemma 5.12), then this restricts to a map q − ( b ( σ )) → q − ( b ( τ )) . This is perhaps most easily seen from the description of the fibers of q displayed in (15). Wehave that ( y ) ∈ q − ( b ( σ )) if and only if π ( Z G ( b ( σ ) , y )) ( p ) = 0. But then π ( Z G ( b ( τ ) , y )) ( p ) =0 by Lemma 5.3, hence ( y ) ∈ q − ( b ( τ )). Therefore, there are functors F : I → Top , I pr − ( b ( σ I ))and F ′ : I ( p ) → Top , I q − ( b ( σ I )) . We may now identify Rep( Z , G ) and R G ( p ) as coends of the functor pairs ( A, F ) and( A ( p ) , F ′ ), respectively. To this end, recall Definition 5.15.
Let C be a small category and M : C →
Top and L : C op → Top a pairof co- and contravariant functors. The coend L ⊗ C M is the topological space defined by L ⊗ C M = G c ∈ Ob( C ) L ( c ) × M ( c ) / ≈ , where the equivalence relation ≈ is given by( L ( i )( a ) , b ) ≈ ( a, M ( i )( b ))for all c, d ∈ C , i ∈ Mor C ( c, d ), a ∈ L ( d ) and b ∈ M ( c ). Notation.
For x ∈ L ( c ) and y ∈ M ( c ) we denote by x ⊗ y the equivalence class of ( x, y ) inthe coend.The following lemma explains the first one of the two homeomorphisms in (9). Lemma 5.16.
There is a commutative diagram R G ( p ) ∼ = / / (cid:15) (cid:15) A ( p ) ⊗ I ( p ) F ′ (cid:15) (cid:15) Rep( Z , G ) ∼ = / / A ⊗ I F in which the two vertical maps are inclusions of subspaces.Proof. We first construct the bottom map. Each (( x, y )) ∈ Rep( Z , G ) is represented by apair ( x, y ) with x ∈ A . Define h : Rep( Z , G ) → A ⊗ I F (( x, y )) x ⊗ ( y ) ( x ∈ σ ˜∆( x ) , ( y ) ∈ Z G ( x ) / Ad Z G ( x ) ) . To see that h is well defined suppose that (( x, y )) = (( x ′ , y ′ )) and both x, x ′ ∈ A . Then x = x ′ and there is g ∈ Z G ( x ) such that y ′ = y g . Hence, ( y ) = ( y ′ ) as elements of Z G ( x ) / Ad Z G ( x ) .To see that h is surjective observe that F ( ∅ ) = Z G ( b ( A )) / Ad Z G ( b ( A )) = T , since b ( A ) is aregular point of G . Hence, A × T appears as one of the disjoint summands in the definitionof the coend. Since for each σ ⊆ A the map T → Z G ( b ( σ )) / Ad Z G ( b ( σ )) is surjective, everyelement of A ⊗ I F has a representative in A × T and is therefore in the image of h .Finally, to see that h is injective suppose that x ⊗ ( y ) = x ′ ⊗ ( y ′ ). Since for each inclusion I ⊆ J the map σ J → σ I is injective, we must have x = x ′ . But then, ( y ) = ( y ′ ) as elementsof Z G ( x ) / Ad Z G ( x ) , so (( x, y )) = (( x ′ , y ′ )).Together this proves that h is a continuous bijection. Since Rep( Z , G ) is compact and A ⊗ I F is Hausdorff, h is a homeomorphism.To obtain the homeomorphism in the top row of the square, first observe that the inclusionmap G I ∈ I ( p ) σ I × F ′ ( I ) ֒ → G I ∈ I σ I × F ( I )induces a homeomorphism i of A ( p ) ⊗ I ( p ) F ′ onto its image in A ⊗ I F . The homeomorphism h from before restricts to a bijection of R G ( p ) with i ( A ( p ) ⊗ I ( p ) F ′ ). Since both R G ( p ) and i ( A ( p ) ⊗ I ( p ) F ) carry the subspace topology, this bijection is a homeomorphism. (cid:3) N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 29
We now turn our attention to the homotopy equivalence in (9). To this end, we introduceanother diagram on I and I ( p ). Given I ∈ I let t ( I ) t be the R -linear span of { α ∨ i | i ∈ I } ,and let T ( I ) T be the subtorus whose Lie algebra is t ( I ); in other words, T ( I ) is theimage of t ( I ) under exp : t → T . Note that T ( I ) = T ∩ DZ G ( b ( σ I )) can be identified withthe maximal torus of DZ G ( b ( σ I )). When I ⊆ J , then T ( I ) T ( J ), and there is a quotientmap T /T ( I ) → T /T ( J ). This gives rise to a diagram of tori¯ F : I → Top , I T /T ( I ) . We will not distinguish notationally between ¯ F and its restriction to I ( p ). Lemma 5.17.
There are natural equivalences of diagrams F ≃ −→ ¯ F and F ′ ≃ −→ ¯ F which fitinto a commutative diagram A ( p ) ⊗ I ( p ) F ′ ≃ / / (cid:15) (cid:15) A ( p ) ⊗ I ( p ) ¯ F (cid:15) (cid:15) A ⊗ I F ≃ / / A ⊗ I ¯ F where the two vertical maps are inclusions of subspaces. Suppose that K is a compact, simply–connected, simple Lie group. To prove the lemmawe need a fact concerning the action of the center Z ( K ) on K/ Ad K specified in (14). Let S K be a maximal torus, s its Lie algebra and assume that appropriate choices havebeen made that allow us to identify K/ Ad K with (the closure of) an alcove A K ⊆ s . Thenit is well known that the resulting action of Z ( K ) on A K is through affine isometries of s permuting the vertices of A K (see [11, Section 3.2]). Lemma 5.18.
Let K be a compact simply–connected semi-simple Lie group with center Z ( K ) . Then K/ Ad K is Z ( K ) -equivariantly contractible.Proof. It suffices to prove the lemma in the case where K is simple, because in the generalcase there is s > K ∼ = K × · · · × K s where each K i is simple. Hence, K/ Ad K ∼ = K / Ad K × · · · × K s / Ad K s , and the action of Z ( K ) ∼ = Z ( K ) × · · · × Z ( K s ) is through theaction of Z ( K i ) on K i / Ad K i for every i = 1 , . . . , s .Assuming that K is simple, we identify K/ Ad K with a Weyl alcove A K in the Lie algebra s of a maximal torus for K . Then A K is a simplex given in barycentric coordinates by A K = (X v ∈ V a v v | a v ∈ R > for all v ∈ V and X v ∈ V a v = 1 ) ⊆ s , where V ⊆ A K is the set of vertices.As Z ( K ) acts on A K through affine isometries of s permuting the vertices of A K , the actionis determined by this permutation action on the vertex set V . In particular, the barycenter b ( A K ) = (cid:0)P v ∈ V v (cid:1) / | V | is a global fixed point for the Z ( K )-action. Now let Γ Z ( K ) beany subgroup. Let V = V ⊔ · · · ⊔ V ℓ be the decomposition of V into orbits with respectto the permutation action of Γ on V . Then a point x = P v ∈ V a v v of A K is fixed by Γ ifand only if a v = a u for all v, u ∈ V j and all j = 1 . . . ℓ . Clearly, if x is fixed by Γ, thenso is ab ( A K ) + (1 − a ) x for every a ∈ [0 , A Γ K is a star-shaped domain in s with respect to the barycenter b ( A K ) and therefore contractible. Asthis holds for all subgroups Γ Z ( K ), A K is Z ( K )-equivariantly contractible. (cid:3) Proof of Lemma 5.17.
We will only describe the equivalence F ′ ≃ −→ ¯ F as the equivalence F ≃ −→ ¯ F follows analogously. By Lemma 5.12, F ′ ( I ) = q − ( b ( σ I )) ∼ = (cid:16) ^ DZ G ( b ( σ I )) / Ad ^ DZ G ( b ( σ I )) (cid:17) h ξ ( b ( σ I )) i × C Z ( Z G ( b ( σ I ))) . Since both h ξ ( b ( σ I )) i and C act through the center of ^ DZ G ( b ( σ I )), Lemma 5.18 implies thatthe projection onto Z ( Z G ( b ( σ I ))) induces a homotopy equivalence F ′ ( I ) ≃ −→ Z ( Z G ( b ( σ I ))) /C , that is natural in I . The latter space can be identified naturally with Z G ( b ( σ I )) /DZ G ( b ( σ I ))as a look at the isomorphism (12) shows. On the other hand, as T ( I ) = T ∩ DZ G ( b ( σ I )) theinclusion T ֒ → Z G ( b ( σ I )) induces a natural isomorphism T /T ( I ) ∼ = Z G ( b ( σ I )) /DZ G ( b ( σ I )) . This proves the equivalence F ′ ≃ −→ ¯ F .To see that the natural equivalences F ′ ≃ −→ ¯ F and F ≃ −→ ¯ F induce homotopy equivalences A ( p ) ⊗ I ( p ) F ′ ≃ A ( p ) ⊗ I ( p ) ¯ F and A ⊗ I F ≃ A ⊗ I ¯ F we recognize the coends as homotopycolimits of the diagrams F ′ , F and ¯ F . For this recall that, if M : C →
Top is a smalldiagram of spaces, then a model for the homotopy colimit of M is the coendhocolim C M = B ( − / C ) ⊗ C M , where B ( − ) is the classifying space functor and c/ C denotes the category of objects under c ∈ C , see [17, XII. § A : I σ I and I B ( I/ I ) are naturallyisomorphic, we find thatRep( Z , G ) ∼ = A ⊗ I F ∼ = B ( − / I ) ⊗ I F = hocolim I F , and similarly, R G ( p ) ∼ = hocolim I ( p ) F ′ . The required homotopy equivalences are now implied by homotopy invariance of homotopycolimits. Finally, commutativity of the diagram follows by inspection. (cid:3)
The following lemma completes the proof of Proposition 5.10. Essentially, this is anidentification of the coend A ⊗ I ¯ F with the weighted projective space CP ( n ∨ ). This kind ofidentification is well known in toric topology (see e.g. [49, Section 5.3]), but we will give adirect proof here. Lemma 5.19.
There is a commutative diagram A ( p ) ⊗ I ( p ) ¯ F ∼ = / / (cid:15) (cid:15) CP ( n ∨ ( p )) ι p (cid:15) (cid:15) A ⊗ I ¯ F ∼ = / / CP ( n ∨ ) N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 31 in which the left hand vertical map is a subspace inclusion.Proof.
The top horizontal map is simply the restriction of the bottom map, so we firstconstruct the latter. To this end, we replace A ⊗ I ¯ F by the homeomorphic identificationspace ( A × T ) / ≈ where( x, t ) ≈ ( x ′ , t ′ ) ⇐⇒ x = x ′ and t − t ′ ∈ T ( ˜∆( x )) . The homeomorphism ( A × T ) / ≈ → A ⊗ I ¯ F is given by mapping ( x, t ) x ⊗ t .To define a homeomorphism of ( A × T ) / ≈ with CP ( n ∨ ) = S r +1 / S n ∨ we shall view S r +1 as the ( r + 1)-fold unreduced join S r +1 ∼ = S ∗ · · · ∗ S = ( S ) ∗ ( r +1) . It is convenient to write elements of ( S ) ∗ ( r +1) formally as tuples h a t , . . . , a r t r i with a i ∈ [0 , , t i ∈ S for i = 0 , . . . , r and r X i =0 a i = 1 , and subject to the identification 0 t = 0 t ′ for all t, t ′ ∈ S . The homeomorphism with theunit sphere S r +1 ⊆ C r +1 is then given by h a t , . . . , a r t r i 7→ ( a t , · · · , a r t r ) / k ( a t , · · · , a r t r ) k . Note that this map is S n ∨ -equivariant for the diagonal (weighted) action on ( S ) ∗ ( r +1) . Weidentify points of A with their barycentric coordinates a = ( a , . . . , a r ) ∈ ∆ r . Given t ∈ T ,we let ( t , . . . , t r ) ∈ ( S ) r denote the coordinates of t with respect to the decomposition S α ∨ × · · · × S α ∨ r ∼ = −→ S α ∨ · · · S α ∨ r = T , where S α ∨ i T is the one-parameter subgroup determined by the coroot α ∨ i . Define φ : ( A × T ) / ≈ → S ∗ ( r +1) / S n ∨ [ a , t ]
7→ h a e, a t , . . . , a r t r i S n ∨ , where e ∈ S is the identity element. We claim that φ is a continuous bijection of thecompact space ( A × T ) / ≈ with the Hausdorff space S ∗ ( r +1) / S n ∨ , thus a homeomorphism.To see that φ is well-defined it suffices to check that if a i = 0 for some i ∈ { , . . . , r } , then φ ([ a , ts ]) = φ ([( a , t )]) for any s ∈ S α ∨ i and t ∈ T . This is clear when 1 i r , because inthis case φ ([ a , t ]) is independent of t i by the identifications made in the join construction.Now suppose that a = 0 and let s ∈ S α ∨ T . Since − α ∨ = P ri =1 n ∨ i α ∨ i , we can parametrize S α ∨ by λ ( λ n ∨ , . . . , λ n ∨ r ) ∈ Q ri =1 S α ∨ i , thus s = ( λ n ∨ , . . . , λ n ∨ r ) for some λ ∈ S . Then φ ([( a , ts ]) = h e, a t λ n ∨ , . . . , a r t r λ n ∨ r i S n ∨ = h λ − n ∨ , a t , . . . , a r t r i S n ∨ = φ ([( a , t ]) . Here we used the weighted action of S n ∨ in the second equality. We conclude that φ is welldefined and continuous. Not to be confused with our notation S w for the circle group acting on S r +1 with weights w . It is clear that φ is surjective, that is, that every element of ( S ) ∗ ( r +1) is equivalent to oneof the form h a e, a t , . . . , a r t r i modulo S n ∨ .To see that φ is injective suppose that φ ([ a , t ]) = φ ([ a ′ , t ′ ]), that is, suppose that h a e, a t , . . . , a r t r i S n ∨ = h a ′ e, a ′ t ′ , . . . , a ′ r t ′ r i S n ∨ . Then there exists λ ∈ S such that(16) h a e, a t ′ , . . . , a r t ′ r i = h a λ, a t λ n ∨ , . . . , a r t r λ n ∨ r i . First, this implies that a = a ′ by the properties of the join construction. In addition, weclaim that t − t ′ ∈ T ( ˜∆( a )). From this it will follow that ( a , t ) ≈ ( a ′ , t ′ ), hence that φ isinjective. As a subset of { , . . . , r } the set ˜∆( a ) consists of those i such that a i = 0. Considerfirst the case 0 ˜∆( a ), that is, a = 0. From (16) we deduce that λ = e , and t i = t ′ i for all i ˜∆( a ). Therefore, t − t ′ ∈ T ( ˜∆( a )). In the case a = 0, we deduce that t − i t ′ i = λ n ∨ i for all i ˜∆( a ). Therefore, letting s = ( λ n ∨ , . . . , λ n ∨ r ) ∈ S α ∨ , we have that t − t ′ s − ∈ T ( ˜∆( a ) \{ } ),hence t − t ′ ∈ T ( ˜∆( a )). This completes the proof that φ is a homeomorphism.The top horizontal map in the diagram is obtained by restriction of φ . The subspace A ( p ) ⊗ I ( p ) ¯ F of A ⊗ I ¯ F consists of those [ a , t ] with a ∈ A ( p ). By definition, a ∈ A ( p ) isequivalent to a i = 0 for all i such that p ∤ n ∨ i . Thus φ maps A ( p ) ⊗ I ( p ) ¯ F homeomorphicallyonto the subspace (cid:8) h a t , . . . , a r t r i S n ∨ ∈ S ∗ ( r +1) / S n ∨ | a i = 0 if p ∤ n ∨ i (cid:9) ⊆ S ∗ ( r +1) / S n ∨ . This space is homeomorphic to the weighted projective space CP ( n ∨ ( p )). By inspection, oneidentifies the inclusion into S ∗ ( r +1) / S n ∨ with the map ι p : CP ( n ∨ ( p )) → CP ( n ∨ ) described inProposition 5.10. (cid:3) The cases E and E and the proofs of Theorems 5.21 and 5.22. Thus far wehave determined H ∗ ( X G ( p ); ( π ) ( p ) ) under the assumption that G
6∈ { E , E } or p >
2. Inthis subsection we calculate additional homology groups for G = E and G = E which areneeded to prove Theorem 5.21. Lemma 5.20.
Suppose that G = E or G = E . Then H Gk ( X G (2); ( π ) (2) ) ∼ = ( Z / , if k = 0 , , if k = 1 . Proof.
To start observe that Lemma 5.9 and Proposition 5.10 imply that(17) H Gk ( X G (4); ( π ) (2) ) ∼ = ( Z / , if k = 0 , , if k = 1 . In what follows we are going to prove that(18) H Gk ( X G (2) , X G (4); ( π ) (2) ) = 0 for k = 0 , . The result is then obtained using (17) and (18) in the long exact sequence in Bredon homologyassociated to the pair ( X G (2) , X G (4)) · · · → H Gk ( X G (4); ( π ) (2) ) → H Gk ( X G (2); ( π ) (2) ) → H Gk ( X G (2) , X G (4); ( π ) (2) ) → · · · N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 33
A standard argument using excision and the long exact sequence associated to the pair( X G (2) /X G (4) , X G (4) /X G (4)) shows that (18) follows from H Gk ( X G (2) /X G (4); ( π ) (2) ) = 0 for k = 0 , . We show this next. Let Q denote the coefficient system whose value is Z / G/G where we set Q ( G/G ) = 0, and for which each map Z / → Z / π ) (2) and Q agree when restricted to the quotientcomplex X G (2) /X G (4), hence H Gk ( X G (2) /X G (4); ( π ) (2) ) ∼ = H Gk ( X G (2) /X G (4); Q ) . Therefore, the proof of the lemma reduces to showing that H Gk ( X G (2) /X G (4); Q ) vanisheswhen k = 0 ,
1. To see this for k = 0 we observe that the coefficient system Q satisfies theconditions of Lemma 3.2. Moreover, X G (2) /X G (4) is path–connected as the same is true for X G (2), and the basepoint X G (4) /X G (4) is fixed by the G -action. Lemma 3.2 then impliesthat(19) H G ( X G (2) /X G (4); Q ) = 0 . To finish we handle the case k = 1. For this notice that Proposition 5.10 implies that, upto homotopy equivalence, both R G (2) and R G (4) can be identified with weighted projectivespaces. Therefore, for the constant coefficient Z / H Gk ( X G (2) /X G (4); Z / ∼ = H k ( R G (2) /R G (4); Z / ∼ = ( Z / , if k = 0 , , if k = 1 . Let K denote the coefficient system which is 0 everywhere, except for G/G where we set K ( G/G ) = Z /
2. Then we have a short exact sequence of coefficient systems0 → K → Z / → Q → . Using (19), (20) and the long exact sequence in Bredon homology associated to the aboveshort exact sequence we obtain the short exact sequence(21) 0 → H G ( X G (2) /X G (4); Q ) → H G ( X G (2) /X G (4); K ) → H G ( X G (2) /X G (4); Z / → . By definition of K , H G ∗ ( X G (2) /X G (4); K ) is the Z / G -fixed pointsof X G (2) /X G (4). The only point fixed is the basepoint, so H G ( X G (2) /X G (4); K ) ∼ = Z / H G ( X G (2) /X G (4); K ) → H G ( X G (2) /X G (4); Z / ∼ = Z / H G ( X G (2) /X G (4); Q ) = 0 , as we wanted to show. (cid:3) With the calculations of the previous lemma we have gathered all the necessary pieces toprove
Theorem 5.21.
Let G be a simply–connected and simple compact Lie group. Then π (Hom( Z , G )) ∼ = Z , and on this group the quotient map Hom( Z , G ) → Rep( Z , G ) induces multiplication by theDynkin index lcm { n ∨ , . . . , n ∨ r } where n ∨ , . . . , n ∨ r > are the coroot integers of G .Proof. To calculate π (Hom( Z , G )) our starting point is Lemma 3.7, where we showed that π (Hom( Z , G )) ∼ = Z ⊕ E , . By Corollary 5.7, the group E k, splits for each k > E k, = H Gk (Hom( Z , G ); π ) ∼ = M p ∈P H Gk ( X G ( p ); ( π ) ( p ) ) , where P is the set of primes dividing a coroot integer of G . For k = 1 each direct summandis trivial, either by Corollary 5.11 or by Lemma 5.20, hence E , = 0. This proves that π (Hom( Z , G )) ∼ = Z .Lemma 3.7 also showed that the degree of π ∗ : π (Hom( Z , G )) → π (Rep( Z , G )) equalsthe order of the finite group E , . When G is neither E nor E , then E , ∼ = L p ∈P Z /p by Corollary 5.11. On the other hand, when G = E or G = E , then Lemma 5.20 impliesthat E , ∼ = Z / ⊕ L p ∈P\{ } Z /p . By inspection (see Table 1), the order of E , equalslcm { n ∨ , . . . , n ∨ r } , and the proof of the theorem is complete. (cid:3) Combining Proposition 5.10 and Theorem 5.21 we derive the following more general result:
Theorem 5.22.
Suppose that G is a semisimple compact connected Lie group. Then thereis an extension → Z s → H (Hom( Z , G ) ; Z ) → H ( π ( G ) ; Z ) → , where s > is the number of simple factors in the Lie algebra of G .Proof. Consider the Serre spectral sequence for the universal covering sequenceHom( Z , ˜ G ) → Hom( Z , G ) → Bπ ( G ) . Inspection of the proof of [22, Lemma 2.2] shows that the action by deck translation of π ( G ) on Hom( Z , ˜ G ) is simply ( x, y ) ( ax, by ) where a, b ∈ π ( G ) are viewed as elements in thecenter of ˜ G . We claim that this action is trivial on H (Hom( Z , ˜ G ); Z ). Let us first treat thecase when G is simple. Then we can view H (Hom( Z , ˜ G ); Z ) ∼ = Z as a π ( G ) -submoduleof H (Rep( Z , ˜ G ); Z ) through Theorem 5.21. The action of Z ( ˜ G ) on Rep( Z , ˜ G ) given bytranslation of either coordinate is trivial on homology, because it extends to an action ofa maximal torus on S r +1 / S under the identification in (10). Since H (Hom( Z , ˜ G ); Z )is torsionfree, the differential d : H ( π ( G ) ; Z ) → H (Hom( Z , ˜ G ); Z ) is trivial, and theextension follows.The same argument applies when G is semisimple, with the only difference that nowHom( Z , ˜ G ) ∼ = Hom( Z , G ) × · · · × Hom( Z , G s ), hence H (Hom( Z , ˜ G ); Z ) ∼ = Z s , where G , . . . , G s are the simple factors in ˜ G . (cid:3) By degree of a map Z → Z we will always mean its absolute value. N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 35
Let G be simple, and suppose that the finite group π ( G ) is not cyclic; then neither is H ( π ( G ) ; Z ), and we deduce from Theorem 5.22 that H (Hom( Z , G ) ; Z ) necessarily hastorsion. This happens for G = P SO (4 n ) for any n >
1, since π ( P SO (4 n )) ∼ = Z / ⊕ Z / SO (3). One shows that H (Hom( Z , SO (3)) ; Z ) ∼ = Z (forexample, using the description Hom( Z , SO (3)) ∼ = ( S × Z / ( S ) ) / ( S × Z / ∗ ) provided in[46, Theorem 3.1]). In this case the extension is Z −→ Z → Z / n > p i,j : Hom( Z n , G ) → Hom( Z , G ) be the projection onto the i -th and j -thcomponent. Denote by p : Hom( Z n , G ) → Q i Let n > . Then the map p : Hom( Z n , G ) → Q ( n ) Hom( Z , G ) inducesan isomorphism p ∗ : π (Hom( Z n , G ) ) / torsion ∼ = −→ ( n ) M π (Hom( Z , G )) . Proof. For each 1 i < j n we have a natural inclusion of the i -th and j -th component I i,j : Hom( Z , G ) → Hom( Z n , G ) which inserts the identity in all the other components. Observe that, since π ( G ) = 0,for all i, j, k, l we have that ( p i,j ) ∗ ◦ ( I k,l ) ∗ = 0, unless i = k and j = l in which case( p i,j ) ∗ ◦ ( I i,j ) ∗ = id . This implies that p ∗ is surjective. By Corollary 2.2 and Theorem5.21, both the domain (modulo torsion) and codomain of p ∗ are free abelian groups of rank (cid:0) n (cid:1) . Since p ∗ is a surjective map of free abelian groups of the same rank it must be anisomorphism. (cid:3) Stability for commuting pairs in Spin groups In this section we study the stability behavior in the case of Spin groups.For m > Spin ( m ) is the universal covering group of SO ( m ). The standardinclusion SO ( m ) ֒ → SO ( m + 1), given by block sum with a 1 × Spin ( m ) → Spin ( m + 1). The purpose of this section is to prove Theorem 6.3 whichasserts that for m > Z , Spin ( m )) → Hom( Z , Spin ( m + 1))induces an isomorphism of second homotopy groups.We begin by proving a general homology stability result for representation spaces of spinorgroups which may be interesting on its own. Only a special case of it will be needed to proveTheorem 6.3. Let i ev ℓ − : Hom( Z , Spin (2 ℓ − → Hom( Z , Spin (2 ℓ ))denote the map obtained by iterating the stabilization map (22). Similarly, define i odd ℓ − : Hom( Z , Spin (2 ℓ − → Hom( Z , Spin (2 ℓ + 1)) . Let ¯ i ev ℓ − and ¯ i odd ℓ − denote the respective maps induced on representation spaces. Theorem 6.1. (i) When ℓ > , then the map (¯ i ev ℓ − ) ∗ : H k (Rep( Z , Spin (2 ℓ − Z ) → H k (Rep( Z , Spin (2 ℓ )); Z ) is an isomorphism for all k ℓ − and multiplication by for k = 2 ℓ − . (ii) When ℓ > , then the map (¯ i odd ℓ − ) ∗ : H k (Rep( Z , Spin (2 ℓ − Z ) → H k (Rep( Z , Spin (2 ℓ + 1)); Z ) is an isomorphism for all k ℓ − and multiplication by for k = 2 ℓ − .Proof. We begin by proving (i). For ℓ > Spin (2 ℓ ) is described by the rootsystem D ℓ (for ℓ = 3 there is an isomorphism D ∼ = A yielding the exceptional isomorphism Spin (6) ∼ = SU (4)). Let { α , . . . , α ℓ } be a set of simple roots. It may be chosen in such a waythat, when ℓ > 4, the subset { α , . . . , α ℓ } determines a subroot system of D ℓ of type D ℓ − and the image of Spin (2 ℓ − 2) in Spin (2 ℓ ) is the subgroup corresponding to this subrootsystem.Now let ℓ > 4. Let n ∨ ℓ − = (1 , , , . . . , , , ∈ Z ℓ be the tuple of coroot integers of Spin (2 ℓ − 2) (of which ℓ − i ev ℓ − isequivalent to a map f ℓ − : CP ( n ∨ ℓ − ) → CP ( n ∨ ℓ ), so that deg((¯ i ev ℓ − ) ∗ ) = deg(( f ℓ − ) ∗ ). Let( n ∨ ℓ − ) ′ ∈ Z ℓ − be the tuple consisting of the first ℓ − n ∨ ℓ − . We are going todescribe f ℓ − explicitly, and show that it fits into a commutative diagram(23) CP (( n ∨ ℓ − ) ′ ) (cid:15) (cid:15) & & ◆◆◆◆◆◆◆◆◆◆◆ CP ( n ∨ ℓ − ) f ℓ − / / CP ( n ∨ ℓ ) . Here the map CP (( n ∨ ℓ − ) ′ ) → CP ( n ∨ ℓ − ) is the “standard inclusion”, described in homoge-neous coordinates by [ z , . . . , z ℓ − ] [ z , . . . , z ℓ − , , CP (( n ∨ ℓ − ) ′ ) → CP ( n ∨ ℓ ) is the“standard inclusion” [ z , . . . , z ℓ − ] [ z , . . . , z ℓ − , , , w ∈ Z r +1 > , r > 1, and p w : CP r → CP ( w ) is the projection given by[ z , . . . , z r ] [ z w , . . . , z w r r ], then the degree of ( p w ) ∗ : H k ( CP r ; Z ) → H k ( CP ( w ); Z ) isgiven by(24) lcm (cid:26) w i · · · w i k gcd { w i , . . . , w i k } | i < · · · < i k r (cid:27) , see [28, p. 244]. This can be used to show that the inclusion CP (( n ∨ ℓ − ) ′ ) → CP ( n ∨ ℓ − )induces homology isomorphisms in degrees k ℓ − 6, whereas CP (( n ∨ ℓ − ) ′ ) → CP ( n ∨ ℓ )induces isomorphisms in degrees k ℓ − k = 2 ℓ − f ℓ − : CP ( n ∨ ℓ − ) → CP ( n ∨ ℓ ) induces isomorphisms in degrees k ℓ − k = 2 ℓ − f ℓ − . Fix a maximal torus T ℓ Spin (2 ℓ ), and let W ℓ be the correspond-ing Weyl group. The maximal torus T ℓ − Spin (2 ℓ − 2) is the subtorus of T ℓ generated bythe one-parameter subgroups corresponding to the coroots α ∨ , . . . , α ∨ ℓ . Let θ ℓ − : T ℓ − ֒ → T ℓ denote the inclusion. The map ¯ i ev ℓ − : Rep( Z , Spin (2 ℓ − → Rep( Z , Spin (2 ℓ )) may be N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 37 identified with the map ( T ℓ − × T ℓ − ) /W ℓ − → ( T ℓ × T ℓ ) /W ℓ induced by θ ℓ − . Let A ℓ de-note the fundamental alcove determined by { α , . . . , α ℓ } , and let A ℓ − be the alcove in t ℓ − determined by { α , . . . , α ℓ } . Our aim is to define a map f ℓ − making the following diagramcommute:(25) ( T ℓ − × T ℓ − ) /W ℓ − ≃ (cid:15) (cid:15) ¯ i ev ℓ − / / ( T ℓ × T ℓ ) /W ℓ ≃ (cid:15) (cid:15) ( A ℓ − × T ℓ − ) / ≈ f ℓ − / / ( A ℓ × T ℓ ) / ≈ The spaces in the bottom row are homeomorphic to weighted projective spaces, see Lemma5.19. The vertical maps are the equivalences established in the course of proving Proposition5.10. For example, the right hand map takes a W ℓ -orbit (( t , t )) to the equivalence classunder ≈ of any representative ( x, t ) of (( t , t )) with x ∈ A ℓ and t ∈ T ℓ .The inclusion θ ℓ − : T ℓ − ֒ → T ℓ descends to a map T ℓ − /W ℓ − → T ℓ /W ℓ , hence defines amap of alcoves ¯ θ ℓ − : A ℓ − → A ℓ . To define f ℓ − we begin by describing ¯ θ ℓ − in barycentriccoordinates. For this we will work with explicit formulae for the root system D ℓ which canbe found, for example, in [14, Plate IV]. Note that D ℓ is self-dual, which means that theexpressions for roots and weights shown in [14] also represent the coroots and coweights,respectively.We identify the Lie algebra t ℓ of T ℓ with R ℓ . With respect to the standard basis { e , . . . , e ℓ } of R ℓ , the simple coroots can be chosen to be(26) α ∨ i = e i − e i +1 for i = 1 , . . . , ℓ − , and α ∨ ℓ = e ℓ − + e ℓ . The Weyl group W ℓ acts on t ℓ by signed permutations e i 7→ ± e σ ( i ) ( σ ∈ Σ ℓ ) such that the totalnumber of negative signs is even. Note that t ℓ − = span( α ∨ , . . . , α ∨ ℓ ) = span( e , . . . , e ℓ ) ⊆ t ℓ .Recall from [14, VI § t ℓ , A ℓ is theconvex hull A ℓ = Conv(0 , ω ∨ /n , . . . , ω ∨ ℓ /n ℓ ) , where ω ∨ i is the i -th fundamental coweight (defined by α j ( ω ∨ i ) = δ ij ) and n i is the rootinteger associated to α i (which in the case of D ℓ equals the i -th coroot integer n ∨ i ). Let uswrite v i := ω ∨ i /n i , so that { , v , . . . , v ℓ } is the set of vertices of A ℓ . Let { , u , . . . , u ℓ − } denote the set of vertices of A ℓ − ⊆ t ℓ − ⊆ t ℓ = R ℓ . Using [14, Plate IV] we can write thevertices in terms of the standard basis of R ℓ . The result is displayed in Table 2.Let x ∈ A ℓ − . Then ¯ θ ℓ − ( x ) = wx , where w ∈ W ℓ is any element such that wx ∈ A ℓ . Let σ + ∈ W ℓ be the cyclic permutation mapping e i e i − for i = 2 , . . . , ℓ , and e e ℓ . Let σ − ∈ W ℓ be the element whose underlying permutation is σ + , but which sends e 7→ − e ℓ and e ℓ 7→ − e ℓ − . One can check that σ + u i = σ − u i = v i for all 1 i ℓ − 3, that σ + ( u ℓ − + u ℓ − ) = σ − ( u ℓ − + u ℓ − ) = 2 v ℓ − , and that σ + u ℓ − = σ − u ℓ − = ( v ℓ − + v ℓ ) / x = a u + · · · + a ℓ − u ℓ − ∈ A ℓ − with a i > a + · · · + a ℓ − a ℓ − a ℓ − , then, writing x = a u + · · · + a ℓ − u ℓ − + a ℓ − ( u ℓ − + u ℓ − ) + ( a ℓ − − a ℓ − ) u ℓ − , u = e v = e u = ( e + e ) v = ( e + e )... ... u ℓ − = ( e + · · · + e ℓ − ) v ℓ − = ( e + · · · + e ℓ − ) u ℓ − = ( e + · · · + e ℓ − − e ℓ ) v ℓ − = ( e + · · · + e ℓ − ) u ℓ − = ( e + · · · + e ℓ − + e ℓ ) v ℓ − = ( e + · · · + e ℓ − − e ℓ ) v ℓ = ( e + · · · + e ℓ − + e ℓ ) Table 2. Vertices of A ℓ − = Conv(0 , u , . . . , u ℓ − ) and A ℓ = Conv(0 , v , . . . , v ℓ ).we see that σ + x = a v + · · · + a ℓ − v ℓ − + 2 a ℓ − v ℓ − + a ℓ − − a ℓ − v ℓ − + v ℓ ) , which is a convex combination of 0 , v , . . . , v ℓ , hence a point in A ℓ . On the other hand, if a ℓ − > a ℓ − , then we find in a similar way that σ − x = a v + · · · + a ℓ − v ℓ − + 2 a ℓ − v ℓ − + a ℓ − − a ℓ − v ℓ − + v ℓ ) , which is again in A ℓ . From this we derive a description of the map ¯ θ ℓ − : A ℓ − → A ℓ inbarycentric coordinates:( a , . . . , a ℓ − ) ¯ θ ℓ − (cid:18) a , . . . , a ℓ − , { a ℓ − , a ℓ − } , | a ℓ − − a ℓ − | , | a ℓ − − a ℓ − | (cid:19) . Now let [ x, t ] ∈ ( A ℓ − × T ℓ − ) / ≈ , and let ( a , . . . , a ℓ − ) be the barycentric coordinates of x . Since we want the diagram (25) to commute, we are forced to set f ℓ − ([ x, t ]) := [¯ θ ℓ − ( x ) , wθ ℓ − ( t )] , where w = σ + if a ℓ − a ℓ − , and w = σ − if a ℓ − > a ℓ − . Let ( t , . . . , t ℓ − ) ∈ ( S ) ℓ − bethe coordinates of t ∈ T ℓ − with respect to α ∨ , . . . , α ∨ ℓ . Then it is easily verified, using theaction of W ℓ on the coroots α ∨ , . . . , α ∨ ℓ (see (26)), that σ + θ ℓ − ( t ) = ( t , . . . , t ℓ − , t ℓ − t ℓ − , t ℓ − , t ℓ − ) , σ − θ ℓ − ( t ) = ( t , . . . , t ℓ − , t ℓ − t ℓ − , t ℓ − , t ℓ − ) . Now it is not difficult to check that f ℓ − is well defined and continuous (for the definitionof the equivalence relation ≈ see the proof of Lemma 5.19). Moreover, the diagram (25)commutes by construction.Using the homeomorphism φ defined in the proof of Lemma 5.19 we could give a descriptionof f ℓ − : CP ( n ∨ ℓ − ) → CP ( n ∨ ℓ ) in homogeneous coordinates. Relevant to our proof, however, ismerely the observation that the restriction of f ℓ − to the first ℓ − CP ( n ∨ ℓ − ) (i.e., setting a ℓ − = a ℓ − = 0) equals the standard inclusion [ z , . . . , z ℓ − ] [ z , . . . , z ℓ − , , , θ ℓ − and wθ ℓ − given above, N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 39 as these coordinates depend only on a , . . . , a ℓ − and t , . . . , t ℓ − . This proves part (i) of thetheorem.Part (ii) for ℓ > Spin (2 ℓ + 1) is described bythe root system B ℓ . A basis { α , . . . , α ℓ } of simple roots can be chosen in such a way thatthe subgroup Spin (2 ℓ − 1) corresponds to the subroot system B ℓ − obtained by omitting thesimple root α . A calculation as before shows that the stabilization map ¯ i odd ℓ − is equivalentto the one induced by the map A ℓ − × T ℓ − → A ℓ × T ℓ sending(( a , . . . , a ℓ − ) , ( t , . . . , t ℓ − )) (( a , . . . , a ℓ − , a ℓ − , , ( t , . . . , t ℓ − , t ℓ − , t ℓ − )) . In particular, the restriction to the first ℓ − ℓ = 3 see Remark 6.4. (cid:3) Remark 6.2. Theorem 6.1 should be compared with the homology stability result of Ramrasand Stafa, [43, Theorem 1.1]. From their result one deduces that the rational homologygroups of Rep( Z , Spin (2 ℓ − Z , Spin (2 ℓ + 1)), and of Rep( Z , Spin (2 ℓ − Z , Spin (2 ℓ )), respectively, are abstractly isomorphic up to homological degree k ℓ − π for spaces of commuting pairs in spinor groups. As in theprevious theorem, it is natural to divide the analysis into the case of even and odd Spingroups; it will be enough, however, to treat the even case from which the general case maybe deduced. Theorem 6.3. For all m > the map Spin ( m ) → Spin ( m + 1) induces an isomorphism π (Hom( Z , Spin ( m ))) ∼ = −→ π (Hom( Z , Spin ( m + 1))) . Proof. We first focus on the range m > 6. The case m = 5 will be dealt with separately atthe end. Let ℓ > 4. By Theorem 5.21, all three groups in the sequence π (Hom( Z , Spin (2 ℓ − → π (Hom( Z , Spin (2 ℓ − → π (Hom( Z , Spin (2 ℓ )))are isomorphic to Z . Thus, if the composite map is an isomorphism, then so are the twocomponent maps. To prove the theorem in the range m > ℓ > i ev ℓ − : Hom( Z , Spin (2 ℓ − → Hom( Z , Spin (2 ℓ )) induces anisomorphism of second homotopy groups.Consider the commutative diagram π (Hom( Z , Spin (2 ℓ − ( i ev ℓ − ) ∗ / / ( π ℓ − ) ∗ (cid:15) (cid:15) π (Hom( Z , Spin (2 ℓ ))) ( π ℓ ) ∗ (cid:15) (cid:15) π (Rep( Z , Spin (2 ℓ − (¯ i ev ℓ − ) ∗ / / π (Rep( Z , Spin (2 ℓ )))in which the maps are the obvious ones. As all groups in the diagram are isomorphic to Z ,each map is determined by its degree. The degrees of the vertical maps are described by Theorem 5.21, from which we deduce thatdeg(( i ev ℓ − ) ∗ ) = deg((¯ i ev ℓ − ) ∗ ) deg(( π ℓ − ) ∗ )deg(( π ℓ ) ∗ ) = ( deg((¯ i ev ℓ − ) ∗ ) / ℓ = 4 , deg((¯ i ev ℓ − ) ∗ ) if ℓ > . Theorem 6.1 implies that deg(¯ i ev ℓ − ) ∗ = 1 if ℓ > i ev ℓ − ) ∗ = 2 if ℓ = 4, hencedeg(( i ev ℓ − ) ∗ ) = 1 for all ℓ > 4. This proves the theorem in the range m > Spin (5) → Spin (6) induces an isomorphism as stated. It isenough to show that the map π (Hom( Z , Spin (5))) → π (Hom( Z , Spin (6))) is surjectiveas both groups are isomorphic to Z . Consider the following sequence of matrix groups Spin (4) / / Spin (5) / / Spin (6) SU (2) × SU (2) ∼ = Sp (2) ∼ = SU (4) ∼ = Recall that Sp ( k ) = { A ∈ GL ( k, H ) | A † A = } . In particular, Sp (1) ∼ = SU (2) is the groupof quaternions of unit norm. It is wellknown, see [35, Theorem 5.20], that the first mapin the sequence is block sum Sp (1) → Sp (2), ( A, B ) A ⊕ B . The second map is theinclusion Sp (2) ֒ → SU (4) resulting from M (2 , H ) → M (4 , C ) , A + j B (cid:18) A − ¯ BB ¯ A (cid:19) , see [35, Theorem 5.21]. There is a permutation P ∈ Σ U (4) such that the compositionof SU (2) → Sp (2) → SU (4) is conjugate by P to block sum. As U (4) is path–connectedthe composition is homotopic to block sum. Then its restriction to the first SU (2)-factoris homotopic to the canonical inclusion of SU (2) into SU (4), which by Theorem 4.1 andRemark 4.2 induces an isomorphism π (Hom( Z , SU (2))) ∼ = π (Hom( Z , SU (4))). As aconsequence, the map π (Hom( Z , Spin (4))) → π (Hom( Z , Spin (6))) is surjective, hence sois π (Hom( Z , Spin (5))) → π (Hom( Z , Spin (6))). (cid:3) Remark 6.4. To prove the case ℓ = 3 of Theorem 6.1 (ii) we consider the sequence of mapsRep( Z , Spin (6)) / / , , Rep( Z , Spin (7)) / / Rep( Z , Spin (8)) / / Rep( Z , Spin (9))The two labels indicate the degree of the induced map of second homotopy groups as impliedby the case ℓ = 4 of Theorem 6.1. It follows that the first map in the sequence has degree2 on second homotopy groups. Now π (Rep( Z , Spin (5))) → π (Rep( Z , Spin (6))) is anisomorphism, by Theorem 6.3, hence π (Rep( Z , Spin (5))) → π (Rep( Z , Spin (7))) musthave degree 2. 7. The distinguished role of SU (2)In this section we continue to work with a fixed simply–connected simple compact Liegroup G . Let ν : SU (2) ֒ → G N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 41 be the embedding that corresponds to the highest root of G . Then the induced map ν ∗ : H ( SU (2); Z ) ∼ = −→ H ( G ; Z )is an isomorphism, see [18, III Proposition 10.2]. Theorem 7.1. The map ν : SU (2) ֒ → G induces an isomorphism π (Hom( Z , SU (2))) ∼ = π (Hom( Z , G )) . Proof. A standard argument with the semi-algebraic triangulation theorem [10, Theorem9.2.1] shows that we can give G = G × G a CW-structure in such a way that Hom( Z , G )is a subcomplex, hence the inclusion i : Hom( Z , G ) → G is a cofibration. Let us considerthe Puppe sequence of i ,Hom( Z , G ) i −→ G → G / Hom( Z , G ) ∂ −→ Σ Hom( Z , G ) − Σ i −−→ Σ G . By the K¨unneth theorem, and the fact that G is 2-connected, we have that H (Σ G ; Z ) ∼ = H ( G ; Z ) = 0 . Moreover, if j : G ∨ G → Hom( Z , G ) is the inclusion, then the composite map H ( G ∨ G ; Z ) j ∗ −→ H (Hom( Z , G ); Z ) i ∗ −→ H ( G ; Z )is an isomorphism. Hence, i ∗ is surjective. From the long exact homology sequence associatedto the Puppe sequence we derive the isomorphism ∂ ∗ : H ( G / Hom( Z , G ); Z ) ∼ = −→ H (Hom( Z , G ); Z ) . Let ν × ν : SU (2) / Hom( Z , SU (2)) → G / Hom( Z , G ) be the map induced by ν : SU (2) ֒ → G upon passage to quotients. By naturality of the connecting map ∂ and by the Hurewicztheorem, it suffices to show that there is an isomorphism( ν × ν ) ∗ : H ( SU (2) / Hom( Z , SU (2)); Z ) ∼ = −→ H ( G / Hom( Z , G ); Z ) . To see this we consider the commutator map G → G , ( x, y ) [ x, y ]. It is constant oncommuting pairs, hence induces a map γ : G / Hom( Z , G ) → G . We claim that when G = SU (2) the induced map γ ∗ : H ( SU (2) / Hom( Z , SU (2)); Z ) ∼ = −→ H ( SU (2); Z )is an isomorphism. Indeed, observe that SU (2) / Hom( Z , SU (2)) ∼ = Y + , where Y + is the one-point compactification of the space Y := SU (2) \ Hom( Z , SU (2)) ofnon-commuting pairs in SU (2). It is known that the commutator map restricted to Y isa locally trivial fiber bundle over SU (2) \{ } with compact fiber F := { ( x, y ) ∈ SU (2) | [ x, y ] = − } , see [7, VI 1 (a)]. In particular, as SU (2) \{ } is contractible (it is a spherewith one point removed), there is a homeomorphism Y ∼ = ( SU (2) \{ } ) × F under whichthe commutator map corresponds to the projection onto the first factor. Therefore, γ isequivalent to the map SU (2) ∧ F + → SU (2) induced by F + → S . Because this map has a section, and because H ( SU (2) ∧ F + ; Z ) ∼ = H (Hom( Z , SU (2)); Z ) ∼ = Z , the inducedmap H ( SU (2) ∧ F + ; Z ) → H ( SU (2); Z ) must be an isomorphism. Consequently, γ ∗ is anisomorphism in the case G = SU (2).Finally, we contemplate the commutative diagram H ( SU (2) / Hom( Z , SU (2)); Z ) γ ∗ ∼ = / / ( ν × ν ) ∗ (cid:15) (cid:15) H ( SU (2); Z ) ν ∗ ∼ = (cid:15) (cid:15) H ( G / Hom( Z , G ); Z ) γ ∗ / / H ( G ; Z )By Theorem 5.21 all groups in the diagram are isomorphic to Z , which forces both the lefthand vertical arrow as well as the bottom horizontal arrow to be isomorphisms. This finishesthe proof of the theorem. (cid:3) In the study of spaces of homomorphisms Hom(Γ , G ) it is natural to ask for the propertiesof the classifying space map B : Hom(Γ , G ) → map ∗ ( B Γ , BG ) , which takes a homomorphism φ : Γ → G to the classifying map Bφ : B Γ → BG of a flatprincipal G -bundle over B Γ with holonomy φ . On path–connected components B describesthe contribution of flat bundles to the set of isomorphism classes of principal G -bundles over B Γ; on higher homotopy groups there is a similar interpretation in terms of families of flatbundles (cf. [40, Section 7]). For Γ = Z we have the following corollary of Theorem 7.1. Corollary 7.2. The classifying space map B : Hom( Z , G ) → map ∗ ( S × S , BG ) induces an isomorphism on π k for all k . Here we identify B Z ≃ S × S up to homotopy. Proof. Since BG is 3-connected, the space map ∗ ( S × S , BG ) is path–connected. By ad-junction and the homotopy equivalence S k ∧ ( S × S ) ≃ S k +2 ∨ W S k +1 (for k > π k (map ∗ ( S × S , BG )) ∼ = [ S k ∧ ( S × S ) , BG ] ∼ = π k +1 ( G ) ⊕ π k ( G ) ⊕ π k ( G )for all k > 0. In particular, map ∗ ( S × S , BG ) is simply–connected. Since Hom( Z , G ) isalso path–connected and simply–connected, it only remains to prove the case k = 2 of thecorollary. We shall prove this case by reducing first to G = SU (2), and then further to thestable unitary group U = colim n →∞ U ( n ) for which the result is known.By Theorem 7.1, by naturality of the classifying space map, and because ν : SU (2) ֒ → G induces an isomorphism π ( SU (2)) ∼ = π ( G ), it suffices to show that π (Hom( Z , SU (2))) → π (map ∗ ( S × S , BSU (2))) ∼ = π ( SU (2))is an isomorphism. The isomorphism π ( SU (2)) ∼ = π ( U ) induced by the inclusion SU (2) → U is classical. On the other hand, we deduce from Theorem 4.1 and Remark 4.2 that N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 43 SU (2) → U also induces an isomorphism π (Hom( Z , SU (2))) ∼ = π (Hom( Z , U )). It is thenenough to show that the classifying space map induces an isomorphism π (Hom( Z , U )) ∼ = −→ π (map ∗ ( S × S , BU )) . This isomorphism is an application of [41, Theorem 3.4]. (cid:3) Loosely speaking, the corollary says that every principal G -bundle over S × ( S ) arisesfrom an S -family of flat bundles over ( S ) , and the associated family of holonomies S → Hom( Z , G ) is uniquely determined up to homotopy.Also observe that since π (Hom( Z , G )) → π (Rep( Z , G )) need not be an isomorphism,the map induced by B on second homotopy groups does not factor through Rep( Z , G ) ingeneral. This is in contrast to the fact that π (Hom(Γ , G )) → [ B Γ , BG ] factors through π (Rep(Γ , G )) so long as G is connected (cf. [1, Lemma 2.5]). Remark 7.3. Theorem 5.21 showed that the image of π (Hom( Z , G )) in π (Rep( Z , G )) hasindex lcm { n ∨ , . . . , n ∨ r } . With the ideas of the previous theorem we can give an alternativeexplanation of this fact. Without reference to Theorem 5.21, the proof of Theorem 7.1 andthe fact that π (Hom( Z , G )) has rank one as an abelian group (Corollary 2.2) show thatthe map ν : SU (2) → G induces an isomorphism π (Hom( Z , SU (2))) ∼ = π (Hom( Z , G )) / torsion . By Theorem 3.6, π ∗ : π (Hom( Z , SU (2))) → π (Rep( Z , SU (2))) is an isomorphism, hencethe degree of π ∗ : π (Hom( Z , G )) / torsion → π (Rep( Z , G )) equals the degree of the map π (Rep( Z , SU (2))) → π (Rep( Z , G )) induced by ν . To calculate the degree one identifiesthe map Rep( Z , SU (2)) → Rep( Z , G ) up to equivalence with a map CP (1 , → CP ( n ∨ )through a case-by-case argument with root systems similar to the proof of Theorem 6.3. Onefinds that, for each G , there exists j ∈ { , . . . , r } such that CP (1 , → CP ( n ∨ ) is homotopicto CP (1 , i j −→ CP (1 , . . . , p n ∨ −−→ CP ( n ∨ ) , where i j is the inclusion [ z , z ] [ z , , . . . , , z , , . . . , 0] with z in the j -th position, andthe second map is the projection [ z , . . . , z r ] [ z n ∨ , . . . , z n ∨ r r ]. By the formula displayed in(24), the degree of the composite map on second homology groups equals lcm { n ∨ , . . . , n ∨ r } ,independent of j . Remark 7.4. Consider a representation ρ : G → SU ( N ) and let D ρ be the Dynkin index of ρ , that is the degree of ρ ∗ : π ( G ) → π ( SU ( N )). Theorem 7.1 (or Corollary 7.2) shows thatthe degree of π (Hom( Z , G )) → π (Hom( Z , SU ( N ))) equals D ρ . On the other hand, as thedegree of π (Hom( Z , SU ( N ))) → π (Rep( Z , SU ( N ))) is one (Theorem 5.21), the degreeof π (Rep( Z , G )) → π (Rep( Z , SU ( N ))) equals D ρ /D , where D = gcd { D ρ | ρ : G → SU ( N ) , N ≥ } is the Dynkin index of G , which equals lcm { n ∨ , . . . , n ∨ r } as noted in theintroduction.8. Application: TC structures on trivial principal G -bundles In this section we provide a geometric interpretation for the calculations given in thisarticle. In particular, we show how examples of non-trivial transitionally commutative (TC) structures on trivial principal G -bundles can be constructed using non-trivial elements in π (Hom( Z , G )).8.1. TC structures on principal G -bundles. We start by reviewing the concept of a TCstructure on a principal G -bundle. Assume that G is a Lie group. We can associate to G asimplicial space, denoted [ B com G ] ∗ and defined by [ B com G ] n := Hom( Z n , G ) ⊂ G n . The faceand degeneracy maps in [ B com G ] ∗ are defined as the restriction of the face and degeneracymaps in the classical bar construction [ BG ] ∗ . The geometric realization of the simplicial space[ B com G ] ∗ is denoted by B com G and called the classifying space for commutativity in G [3, 4].The space B com G is naturally a subspace of BG and we denote by p com : E com G → B com G the restriction of the universal principal G -bundle p : EG → BG over B com G . The space B com G classifies principal G -bundles that come equipped with an additional structure thatwe will refer to as a transitionally commutative structure. To explain this further we needto recall some basic definitions from bundle theory.Suppose that q : E → X is a principal G -bundle with G acting on the right on E andthat X is a CW-complex. By local triviality we can find an open cover U = { U i } i ∈ I of X together with trivializations ϕ i : q − ( U i ) → U i × G for every i ∈ I . If U i and U j are such that U i ∩ U j = ∅ , then for every x ∈ U i ∩ U j and all g ∈ G we have ϕ i ϕ − j ( x, g ) = ( x, ρ i,j ( x ) g ).Here ρ i,j : U i ∩ U j → G is a continuous function called the transition function. The differenttransition functions satisfy the cocycle identity ρ i,k ( x ) = ρ i,j ( x ) ρ j,k ( x )for every x ∈ U i ∩ U j ∩ U k . Now, if for every i, j, k ∈ I and every x ∈ U i ∩ U j ∩ U k theelements ρ i,j ( x ) , ρ j,k ( x ) and ρ i,k ( x ) commute with each other, then we say that { ρ i,j } is acommutative cocycle.Following [4], we call a principal G -bundle q : E → X transitionally commutative if itadmits a trivialization in such a way that the corresponding transition functions define acommutative cocycle. Let f : X → BG be the classifying map of a principal G –bundle q : E → X over a finite CW–complex X . Then by [4, Theorem 2.2] it follows that f factorsthrough B com G , up to homotopy, if and only if q is transitionally commutative.Next we define the notion of a transitionally commutative structure on principal bundlesas in [25] and [44]. Definition 8.1. Suppose that q : E → X is a transitionally commutative principal G -bundle. A transitionally commutative structure (TC structure) on q : E → X is a choiceof map ˜ f : X → B com G , up to homotopy, such that i ˜ f : X → BG is a classifying map for q : E → X . Here i : B com G → BG denotes the natural inclusion.With the previous definition the set of TC structures on principal G -bundles over a space X is precisely the set [ X, B com G ] of homotopy classes of maps X → B com G . Example 8.2. Consider the trivial principal G -bundle pr : X × G → X . This bundleadmits a TC structure given by the homotopy class of a constant map X → B com G . Werefer to this TC structure as the trivial TC structure.Observe that the definition of TC structures implies that the same underlying principal G -bundle can admit TC structures in many different ways. N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 45 Examples of TC structures on the trivial bundle. Our next goal is to constructnon-trivial TC structures for the trivial principal G -bundle over S .Let E com G denote the homotopy fiber of the inclusion i : B com G → BG . As a directconsequence of [3, Theorem 6.3], the homotopy fiber sequence E com G → B com G → BG induces for every n > → π n ( E com G ) → π n ( B com G ) i ∗ −→ π n ( BG ) → . If G is connected, then both B com G and BG are simply–connected. In particular, for every n > S n , B com G ] ∼ = π n ( B com G ) so that π n ( B com G ) agrees with the set of TCstructures on principal G -bundles over S n . Let f : S n → B com G be a TC structure on thetrivial principal G -bundle over S n . Then [ f ] ∈ π n ( B com G ) belongs toKer( i ∗ : π n ( B com G ) → π n ( BG )) ∼ = π n ( E com G ) . In fact, elements in π n ( E com G ) correspond precisely to TC structures on the trivial principal G -bundle over S n .If G = SU ( m ) or G = Sp ( k ), then E com G is 3-connected by [4, Proposition 3.2]. Thisimplies that the lowest dimensional sphere for which a non-trivial TC structure on the trivialprincipal G -bundle can exist is S . For other simply–connected compact Lie groups G thismay not hold, as a result of the fact that Hom( Z n , G ) can be disconnected. However, thefollowing variation of B com G was considered in [4]. Let [ B com G ] ∗ denote the sub-simplicialspace of [ B com G ] ∗ defined by [ B com G ] n := Hom( Z n , G ) . Let E com G denote the homotopyfiber of the inclusion B com G → BG . The proof of [3, Theorem 6.3] shows that there is a splitexact sequence of the form (27) also when E com G and B com G are replaced by E com G and B com G , respectively. Therefore, [4, Proposition 3.2] implies that E com G is 3-connected if G is simply–connected. Moreover, π ( B com G ) ∼ = π ( E com G ) ⊕ Z if in addition G is assumedsimple. Corollary 8.3. Let G be a simply–connected simple compact Lie group. Then π ( E com G ) ∼ = Z and π ( B com G ) ∼ = Z ⊕ Z . Proof. As explained in [4] a model for E com G is the geometric realization of a simplicialspace [ E com G ] ∗ with n -simplices [ E com G ] n := Hom( Z n , G ) × G . To keep our proof shortwe refer to [4] for more details about the simplicial structure. The filtration of E com G arising from the simplicial structure leads to a spectral sequence (see [34, Theorem 11.14])which takes the form E p,q = H p ( H q ([ E com G ] ∗ ; Z )) = ⇒ H p + q ( E com G ; Z ) . The proof of [4, Proposition 3.3] shows that E p,q = 0 for all p, q > < p + q E p,q for p + q = 4. Since Hom( Z n , G ) × G is path–connected and simply–connected for all n > 0, we have that E p, = 0 for all p > E p, = 0 for all p > E , reads · · · → H (Hom( Z , G ) × G ; Z ) d −→ H ( G × G ; Z ) d −→ H ( G ) → , with differentials d k = P ki =0 ( − i ∂ i where ∂ i : [ E com G ] k → [ E com G ] k − is the i -th face map.Let i : G → G × G be the inclusion x (1 , x ). Then a short calculation shows that ker( d ) = Im(( i ) ∗ ). Moreover, if i ′ : G → Hom( Z , G ) × G denotes the inclusion x (1 , , x ), then d ( i ′ ) ∗ = ( i ) ∗ . Hence, E , = ker( d ) / Im( d ) = 0.The chain complex computing E , reads · · · → H (Hom( Z , G ) × G ; Z ) d −→ H (Hom( Z , G ) × G ; Z ) d −→ , because G is assumed simply–connected and therefore H ( G × G ; Z ) = 0 by the K¨unneththeorem. Now H (Hom( Z , G ) × G ; Z ) ∼ = H (Hom( Z , G ); Z ) ∼ = Z , by Theorem 5.21. As aconsequence, d factors through H (Hom( Z , G ) ; Z ) / torsion.We claim that d = 0, hence E , ∼ = Z . To see this let j : W Hom( Z , G ) → Hom( Z , G ) be the map induced by ( x, y ) ( x, y, x, y ) ( x, , y ), and ( x, y ) (1 , x, y ). ByCorollary 5.23, the map j ∗ induced by j on second homology groups is an isomorphismmodulo torsion. Therefore, to see that d = 0 it is enough to show that d j ∗ = 0, but this isan easy calculation.Finally, it is clear that the differential d : H ( G × G ; Z ) → H ( G ; Z ) in the chain complexcomputing E , is surjective, so E , = 0.It follows that the only non-trivial group in total degree 4 of the E -page is E , andthere are no non-trivial differentials originating from or arriving at E , . Hence, we concludethat H ( E com G ; Z ) ∼ = E , ∼ = Z . The isomorphism π ( E com G ) ∼ = Z is obtained from theHurewicz theorem and the fact that E com G is 3-connected. (cid:3) We show next how an element of π (Hom( Z , G )) ∼ = Z can be used to construct a TCstructure on the trivial principal G -bundle over S .Let β : S → Hom( Z , G ) be any map. If β and β are the components of β , then we havethat β ( x ) , β ( x ) ∈ G commute for all x ∈ S . We are going to use the functions β and β to construct a commutative cocycle with values in G . The construction we give follows theidea of [44, Section 3] where the case G = O (2) was studied. Consider S = { ( x , x , x , x , x ) ∈ R | x + x + x + x + x = 1 } . We can cover S using the closed sets C , C and C given by C = { ( x , x , x , x , x ) ∈ S | x } ,C = { ( x , x , x , x , x ) ∈ S | x > , x > } ,C = { ( x , x , x , x , x ) ∈ S | x > , x } . Notice that C ∩ C ∩ C = { ( x , x , x , x , x ) ∈ S | x = 0 , x = 0 } ∼ = S . From now on we identify S with C ∩ C ∩ C . In addition, observe that C ∩ C ∼ = D andunder this identification the boundary S corresponds to C ∩ C ∩ C . The same is true for C ∩ C and C ∩ C .Recall that π ( G ) = 0, hence β : S → G is null-homotopic. Since C ∩ C ∼ = D , wecan find a continuous map ρ , : C ∩ C → G such that ρ , | C ∩ C ∩ C : C ∩ C ∩ C → G agrees with β . Similarly, the choice of a null-homotopy of β defines a continuous map ρ , : C ∩ C → G such that ρ , | C ∩ C ∩ C : C ∩ C ∩ C → G agrees with β . To define the N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 47 transition function ρ , : C ∩ C → G we consider the retraction r : C → C ∩ C definedby r ( x , x , x , x , x ) := (cid:18)q − x − x − x , x , x , x , (cid:19) . Then define ρ , : C ∩ C → G by ρ , (0 , x , x , x , x ) := ρ , (0 , x , x , x , − x ) ρ , ( r (0 , x , x , x , − x )) . For i > j we set ρ i,j := ρ − j,i . Notice that if x ∈ C ∩ C ∩ C , then x = (0 , x , x , x , 0) and r ( x ) = x . Therefore, for x ∈ C ∩ C ∩ C we have that ρ , ( x ) = ρ , ( x ) ρ , ( x ) by definitionof ρ , . Thus, { ρ i,j } satisfies the cocycle condition. Moreover, since ρ , ( x ) = β ( x ) and ρ , ( x ) = β ( x ) for all x ∈ C ∩ C ∩ C , we conclude that { ρ i,j } defines a commutativecocycle relative to the closed cover C := { C , C , C } of S .Let E β be the space defined by E β := ( C × G ⊔ C × G ⊔ C × G ) / ∼ , where ( j, x, g ) ∼ ( i, x, ρ ij ( x ) g ). The projection map E β → S induced by ( j, x, g ) x definesa principal G -bundle by [44, Lemma 3.1]. Lemma 8.4. The principal G -bundle E β → S is trivial.Proof. By [44, Lemma 3.2], the principal G -bundle E β is isomorphic to the bundle obtainedusing the clutching function ϕ : C ∩ ( C ∪ C ) ∼ = S → G given by ϕ ( x ) = ( ρ , ( x ) ρ , ( r ( x )) , if x ∈ C ∩ C ,ρ , ( x ) , if x ∈ C ∩ C . By construction this function satisfies ϕ (0 , x , x , x , x ) = ϕ (0 , x , x , x , − x ). This impliesthat ϕ factors through C ∩ C ∼ = D , hence ϕ is null-homotopic. Therefore, the principal G -bundle E β is trivial. (cid:3) Let N ∗ ( C ) denote the ˇCech nerve of the closed cover C of S . The commutative cocycle { ρ i,j } defines a simplicial map [ f β ] ∗ : N ∗ ( C ) → [ B com G ] ∗ sending x ∈ C i ∩ · · · ∩ C i n to( ρ i ,i ( x ) , . . . , ρ i n − ,i n ( x )) ∈ Hom( Z n , G ) . Upon geometric realization we obtain a map f β := | [ f β ] ∗ | : | N ∗ ( C ) | → B com G . We can choose open neighbourhoods U i ⊃ C i , i = 1 , , 3, such that for every choice ofindices the inclusion C i ∩ · · · ∩ C i n ֒ → U i ∩ · · · ∩ U i n is a homotopy equivalence. Let U = { U , U , U } be the resulting open cover of S . The map of ˇCech nerves N ∗ ( C ) → N ∗ ( U )is a levelwise homotopy equivalence of proper simplicial spaces, hence induces a homotopyequivalence | N ∗ ( C ) | ≃ | N ∗ ( U ) | . Since U is numerable, the natural map | N ∗ ( U ) | → S is ahomotopy equivalence by [45, Proposition 4.1]. We conclude that | N ∗ ( C ) | → S is a homotopyequivalence. Let λ : S → | N ∗ ( C ) | be a homotopy inverse. The argument of [44, Lemma 3.3]shows that f β λ : S → B com G is a TC structure on E β , i.e., that if β λ : S → BG classifies E β . Since E β is trivial, by Lemma 8.4, if β λ is null homotopic. Thus, f β λ factors, up tohomotopy, through the homotopy fiber E com G . Let us construct another commutative cocycle on S , this time representing a generatorof π ( BG ) ∼ = Z . To this end, we cover S by the contractible closed sets D = { ( x , x , x , x , x ) ∈ S | x } ,D = { ( x , x , x , x , x ) ∈ S | x > } . Observe that D ∩ D ∼ = S . Let τ , : D ∩ D → G represent a generator of π ( G ). To beconcrete, we choose τ , = ν , where ν : SU (2) → G is the map defined in Section 7. Then τ , defines trivially a commutative cocycle relative to the closed cover D := { D , D } . Asbefore, the cocycle { τ i,j } defines a simplicial map [ g ν ] ∗ : N ∗ ( D ) → [ B com G ] ∗ and thus amap g ν : | N ∗ ( D ) | → B com G . Upon choosing a homotopy equivalence λ ′ : S → | N ∗ ( D ) | oneobtains a classifying map ig ν λ ′ : S → BG for the bundle clutched by ν : S → G . Since ν represents a generator of π ( G ), the homotopy class of ig ν λ ′ generates π ( BG ).From now on we tacitly identify | N ∗ ( C ) | and | N ∗ ( D ) | with S and drop the homotopyequivalences λ and λ ′ from the notation. Theorem 8.5. Let [ β ] ∈ π (Hom( Z , G )) and [ ν ] ∈ π ( G ) be generators. Then the TCstructures [ f β ] and [ g ν ] generate π ( B com G ) ∼ = Z ⊕ Z . In particular, f β lifts to a generatorof π ( E com G ) ∼ = Z .Proof. The proof is by comparison of the spectral sequences associated to the simplicialspaces N ∗ ( C ), N ∗ ( D ) and [ B com G ] ∗ . Let us first consider the spectral sequence C E p,q = H p ( H q ( N ∗ ( C ); Z )) = ⇒ H p + q ( | N ∗ ( C ) | ; Z ) . In each degree N ∗ ( C ) is a disjoint union of contractible spaces and spaces homeomorphic to S . This readily shows that C E p,q = 0 unless q = 0 , 2. In the case q = 0 the chain complex H ( N ∗ ( C ); Z ) computes the homology of a 2-simplex, hence C E , = 0. In the case q = 2 weobserve that H ( N ( C ); Z ) ∼ = H ( C ∩ C ∩ C ; Z ) ∼ = Z , hence C E , is a quotient of Z . Infact, we must have C E , ∼ = Z , because H ( | N ∗ ( C ) | ; Z ) ∼ = Z . For the same reason, C E , isnot hit by any non-zero differential. Since C E , = 0, there is no non-zero differential leaving C E , either. We conclude that C E ∞ , ∼ = H ( C ∩ C ∩ C ; Z ) ∼ = Z is the only non-zero groupin total degree 4.The analysis of the spectral sequence { D E ∗ p,q } calculating H ∗ ( | N ∗ ( D ) | ; Z ) is very similar;one finds that D E ∞ , ∼ = H ( D ∩ D ; Z ) ∼ = Z is the only non-zero group in total degree 4.Finally, we consider the spectral sequence E p,q = H p ( H q ([ B com G ]; Z )) = ⇒ H p + q ( B com G ; Z ) . Since [ B com G ] is a one point space, we trivially have that E ,q = 0 for all q > 0. BecauseHom( Z n , G ) is path–connected and simply–connected for all n > 0, we further have that E , = E , = 0. On the other hand, we find that E , is a quotient of H ( G ; Z ) ∼ = Z , andlikewise E , is a quotient of H (Hom( Z , G ); Z ) ∼ = Z (Theorem 5.21). In Corollary 8.3 weshowed that H ( B com G ; Z ) ∼ = Z ⊕ Z ; but this is possible only if E , ∼ = Z and E , ∼ = Z andnone of these is hit by a non-zero differential. Furthermore, for degree reasons and because E , = 0, there are no non-zero differentials originating from either E , or E , . Hence, E ∞ , ∼ = H ( G ; Z ) ∼ = Z and E ∞ , ∼ = H (Hom( Z , G ); Z ) ∼ = Z . N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 49 The simplicial maps [ f β ] ∗ : N ∗ ( C ) → [ B com G ] ∗ and [ g ν ] ∗ : N ∗ ( D ) → [ B com G ] ∗ give riseto the following diagram of extensions:0 / / / / (cid:15) (cid:15) H ( | N ∗ ( C ) | ; Z ) / / ( f β ) ∗ (cid:15) (cid:15) C E ∞ , ∼ = Z / / ∼ = β ∗ (cid:15) (cid:15) / / E ∞ , ∼ = Z / / H ( B com G ; Z ) / / E ∞ , ∼ = Z / / / / D E ∞ , ∼ = Z / / ∼ = ν ∗ O O H ( | N ∗ ( D ) | ; Z ) / / ( g ν ) ∗ O O O O / / D E ∞ , → E ∞ , can be identified with the map( ν ) ∗ : H ( S ; Z ) → H ( G ; Z ), and C E ∞ , → E ∞ , may be identified with β ∗ : H ( S ; Z ) → H (Hom( Z , G ); Z ); both are isomorphisms by choice. The commutative diagram togetherwith the Hurewicz theorem imply that [ f β ] and [ g ν ] generate π ( B com G ). Since if β is nullhomotopic, [ f β ] ∈ Ker( i ∗ : π ( B com G ) → π ( BG )) ∼ = π ( E com G ) and it is clear that [ f β ]generates π ( E com G ). (cid:3) An explicit generator of π (Hom( Z , G )) . We finish our discussion by constructing amap β : S → Hom( Z , G ) whose homotopy class generates the group π (Hom( Z , G )) ∼ = Z .In theory, this enables us to write down an explicit commutative cocycle on S (relative to theclosed cover { C , C , C } described above) which represents the generator of π ( E com G ) ∼ = Z .By Theorem 7.1 it suffices to construct β in the case G = SU (2); composition with theembedding ν : SU (2) → G yields a generator for general G . Let T SU (2) be the maximaltorus consisting of all diagonal matrices and identify it with the unit circle S ⊆ C . Underthis identification the action of the Weyl group W ∼ = Z / S . Let I = [0 , 1] denote the unit interval and let ∆ = { ( s, t ) ∈ I | s t } . Our modelfor S will be the boundary of the prism P = ∆ × I . Consider the continuous map r : ∆ → ( S ) ⊂ Hom( Z , SU (2))( s, t ) ( e πis , e πit ) , whose image is the closure of a fundamental domain for the diagonal Z / S ) .The basic idea to construct the desired homotopy class is to choose a null homotopy of r | ∂ ∆ : ∂ ∆ → Hom( Z , SU (2)) , which exists because Hom( Z , SU (2)) is simply–connected. Up to homotopy, this induces amap ∆ /∂ ∆ ∼ = S → Hom( Z , SU (2)). However, some care must be taken as the choice ofnull homotopy will in general affect the resulting homotopy class.Let ρ : ( I, ∂I ) → ( S , 1) represent a generator of π ( S ) and let h : ( I, ∂I ) × I → ( SU (2) , any fixed null homotopy of iρ , where i : ( S , ֒ → ( SU (2) , 1) is the inclusion. Let i : SU (2) → Hom( Z , SU (2)) denote the inclusion of the first factor. Similarly, let i denote the inclusion of the second factor. Let d : SU (2) → Hom( Z , SU (2)) be the diagonal map. We now extend r to a continuous map β : ∂P → Hom( Z , SU (2))as follows: Define β | ∆ ×{ } := r and let β | ∆ ×{ } be the constant map with value (1 , ∂ ∆ is the union of the three intervals∆ = { s = 0 } , ∆ = { s = t } , ∆ = { t = 0 } , each of which is identified with I . Define β | ∆ × I := i h , β | ∆ × I := dh , and β | ∆ × I := i h .By inspection, these maps can be glued together and define a continuous map β : ∂P → Hom( Z , SU (2)). Proposition 8.6. The homotopy class of β generates π (Hom( Z , SU (2))) .Proof. By Theorem 5.21 it suffices to show that the composite map ∂P β −→ Hom( Z , SU (2)) π −→ Rep( Z , SU (2))represents a generator of π (Rep( Z , SU (2))). Since Rep( Z , SU (2)) ∼ = S , it is enough toverify that πβ has degree ± 1. Let int(∆) denote the relative interior of the bottom faceof ∂P , and let U ⊆ S be the image of int(∆) under πβ . Then U is open and πβ maps ∂P \ int(∆) into S \ U . Any x ∈ U has a unique preimage under πβ . By considering localdegrees it follows that πβ has degree ± (cid:3) Let ν : SU (2) → G be the embedding corresponding to the highest root of G . Corollary 8.7. The homotopy class of ( ν × ν ) β generates π (Hom( Z , G )) .Proof. This is immediate from Proposition 8.6 and Theorem 7.1. (cid:3) Remark 8.8. In [4, 6] a K-theory group was introduced, defined for a finite CW complex X by ˜ K com ( X ) := [ X, B com U ]. In [24, Proposition 5.2] it was shown that the inclusion SU (2) → U induces an isomorphism π ( B com SU (2)) ∼ = π ( B com U ), and ˜ K com ( S ) ∼ = Z ⊕ Z .Thus, the results of this section can be used to construct explicit commutative cocycles over S , relative to the closed covers C and D , representing the two generators of ˜ K com ( S ). References [1] A. Adem and F. R. Cohen. Commuting elements and spaces of homomorphisms. Math. Ann. 338, (2007),587–626.[2] A. Adem, F. Cohen and J.M. G´omez. Stable splittings, spaces of representations and almost commutingelements in Lie groups. Math. Proc. Camb. Phil. Soc. 149 (2010) 455-490.[3] A. Adem, F. R. Cohen E. Torres Giese. Commuting elements, simplicial spaces and filtrations of clas-sifying spaces. Math. Proc. Camb. Phil. Soc. 152, 2012, 1, 91–114.[4] A. Adem and J. M. G´omez. A classifying space for commutativity in Lie groups. Algebr. Geom. Topol.15, (2015), 1, 493–535[5] A. Adem and J. M. G´omez. Equivariant K-theory of compact Lie group actions with maximal rankisotropy. Journal of Topology 5, (2012), 431–457.[6] A. Adem, J. M. G´omez, J. Lind and U. Tillmann. Infinite loop spaces and nilpotent K-theory. Algebr.Geom. Topol. 17 (2017), no. 2, 869–893.[7] S. Akbulut and J. McCarthy. Cassons Invariant for Oriented Homology Spheres. Mathematical Notes36, Princeton University Press (1990) N THE SECOND HOMOTOPY GROUP OF SPACES OF COMMUTING ELEMENTS 51 [8] T. Baird. Cohomology of the space of commuting n -tuples in a compact Lie group. Algebr. Geom. Topol.7, (2007), 737–754.[9] J. Bernstein and O. Schwarzman. Chevalley’s theorem for complex crystallographic Coxeter groups. Funk. Anal. Prilozhen. (1978) 79–80 (Russian); Funct. Anal. Appl. (1978) 308–309 (1979) (Eng-lish]).[10] J. Bochnak, M. Coste, and M. Roy. Real algebraic geometry. Vol. 36. Springer Science & BusinessMedia, 2013.[11] A. Borel, R. Friedman J. W. Morgan. Almost commuting elements in compact Lie groups. Mem. Amer.Math. Soc., 157, (2002), 747, x+136,[12] I. Biswas, S. Lawton and D. Ramras. Fundamental groups of character varieties: surfaces and tori.Math. Z. 281 (2015), no. 1-2, 415425[13] A. Borel. Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes. Tohoku Mathe-matical Journal, Second Series 13.2 (1961): 216–240.[14] N. Bourbaki. Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, xii+300.[15] N. Bourbaki. Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005, xii+434.[16] T. Br¨ocker and T. tom Dieck. Representations of compact Lie groups. Graduate Texts in Mathematics,98, Springer-Verlag, New York, (1985), x+313.[17] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations. Lecture Notes inMathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972. v+348 pp.[18] R. Bott and H. Samelson. Applications of the Theory of Morse to Symmetric Spaces. American Journalof Mathematics 80, No. 4 (1958), 964-1029.[19] R. Friedman, J. W. Morgan, and E. Witten. Principal G -bundles over elliptic curves. MathematicalResearch Letters 5 (1998), 97–118.[20] W. Fulton. Introduction to toric varieties. No. 131. Princeton University Press, 1993.[21] W. Fulton and J. Harris. Representation theory. Graduate Texts in Mathematics, 129, (1991), xvi+551.[22] W. M. Goldman. Topological components of the space of representations. Invent Math. 93(3), 557-607(1988).[23] J. M. G´omez, A. Pettet and J. Souto. On the fundamental group of Hom( Z k , G ). Math. Z. 271, (2012),33–44.[24] S. P. Gritschacher. The spectrum for commutative complex K -theory. Algebr. Geom. Topol. 18, (2018),2, 1205–1249.[25] S. P. Gritschacher. Commutative K-theory. PhD thesis. University of Oxford, 2017.[26] S. Gritschacher and M. Hausmann. Commuting matrices and Atiyah’s Real K-theory. Journal of Topol-ogy 12 (2019) 832–853.[27] V. G. Kac and A. V. Smilga. Vacuum structure in supersymmetric Yang-Mills theories with any gaugegroup. 185–234, World Sci. Publ., River Edge, NJ, (2000).[28] T. Kawasaki. Cohomology of Twisted Projective Spaces and Lens Complexes. Math. Ann. 206 (1973),243–248.[29] S. Kumar and M. S. Narasimhan. Picard group of the moduli spaces of G -bundles. Math. Ann. 308(1997) 155–173.[30] Y. Laszlo. About G -bundles over elliptic curves. Annales de l’institut Fourier 48 (3) (1998), 413–424.[31] Y. Laszlo and C. Sorger. The line bundles on the moduli space of parabolic G –bundles over curves andtheir sections. Ann. Sci. ENS 30 (1997), 499–525.[32] S. Lawton and D. Ramras. Covering spaces of character varieties. New York J. Math. 21 (2015) 383–416.[33] E. Looijenga. Root systems and elliptic curves, Invent. Math. 38, (1976), 1732.[34] J. P. May. The geometry of iterated loop spaces. Vol. 271. Springer, 2006.[35] M. Mimura and H. Toda. Topology of Lie groups, I and II. Vol. 91. American Mathematical Soc., 1991.[36] M. S. Narasimhan and C. S. Seshadri. Stable and unitary vector bundles on a compact Riemann surface.Ann. of Math. (2) 82 (1965), 540–567. [37] D. H. Park and D. Y. Suh. Equivariant semi-algebraic triangulations of real algebraic G -varieties. KyushuJ. Math. 50 (1996), no. 1, 179–205.[38] A. Pettet and J. Souto. Commuting tuples in reductive groups and their maximal compact subgroups.Geom. Topol. 17, (2013), 2513–2593.[39] A. Ramanathan. Stable principal bundles on a compact Riemann surface. Math. Ann. 213 (1975),129–152.[40] D. A. Ramras. The homotopy groups of a homotopy group completion. Israel Journal of Mathematics234.1 (2019): 81-124.[41] D. A. Ramras. The stable moduli space of flat connections over a surface. Transactions of the AmericanMathematical Society. 363 no 2 (2011) 1061–1100.[42] D. A. Ramras and M. Stafa. Hibert-Poincare series for spaces of commuting elements in Lie groups.Math. Z. 292, 591610 (2019).[43] D. A. Ramras and M. Stafa. Homological Stability for Spaces of Commuting Elements in Lie Groups.International Mathematics Research Notices, (2020), 5.[44] D. A. Ramras and B. Villareal. Commutative cocycles and stable bundles over surfaces. Forum Mathe-maticum, 31(6), 1395-1415.[45] G. Segal, Classifying spaces and spectral sequences. Publications Mathmatiques de l’IHS 34 (1968):105-112.[46] D. Sjerve and E. Torres Giese. Fundamental groups of commuting elements in Lie groups. Bull. Lond.Math. Soc. 40 (2008), no. 1, 65–76.[47] N. E. Steenrod, Cohomology operations, and obstructions to extending continuous functions. Advancesin Math. 8, (1972), 371–416.[48] A. Tripathy. The symmetric power and etale realisation functors commute. arXiv:1502.01104.[49] V. Welker, G. Ziegler, and R ˇZivaljevi´c. Homotopy colimits – comparison lemmas for combinatorialapplications. J. Reine Angew. Math. 509 (1999), 117–149.[50] S. Willson. Equivariant homology theories on G -complexes. Transactions of the AMS, Vol. 212 (1975),pp. 155-171. Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2,Canada E-mail address : [email protected] Escuela de Matem´aticas, Universidad Nacional de Colombia sede Medell´ın, Medell´ın,Colombia E-mail address : [email protected] Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark E-mail address ::