Formality of cochains on BG$BG$

Let G$G$ be a compact Lie group with maximal torus T$T$ . If |NG(T)/T|$|N_G(T)/T|$ is invertible in the field k$\mathsf {k}$ , then the algebra of cochains C∗(BG;k)$C^*(BG;\mathsf {k})$ is formal as an A∞$A_\infty$ algebra, or equivalently as a differential graded algebra.


INTRODUCTION
It is well known that over a field of characteristic zero, the algebra of cochains on the classifying space of a connected compact Lie group is formal as an ∞ algebra, or equivalently as a differential graded (DG) algebra. We prove that this is the case for a compact Lie group that is not necessarily connected, over any field in which the order of ( )∕ is invertible, where is a maximal torus in .
In contrast, even for a finite group with cyclic Sylow subgroups of order ⩾ 3 in characteristic , the algebra of cochains on the classifying space is not formal. The ∞ structure in this case is computed in [3].
Our notation is as follows. We write for a compact Lie group, for a maximal torus in and for the finite group ( )∕ . This a finite group acting on by conjugation, and the kernel of the action is ( )∕ . If is connected, then ( ) = , and is the Weyl group of . If is a prime, we write ∧ for the Bousfield-Kan -completion of a space , see [10].
Our strategy is quite different to the usual characteristic zero proof. After reducing to the case of a normal torus, we make an approximation to at the prime by a locally finite groupw ith̆∧ homotopy equivalent to ∧ , and then we put an internal grading on the group algebrȃ . This gives us a second, internal grading on cochains * (̆; ), and the ∞ structure maps provided by Kadeishvili's theorem on * (̆; ) have to preserve the internal grading. This then proves that they are all zero apart from the multiplication map 2 . Our main theorem is the following. Theorem 1.1. If is a compact Lie group with maximal torus , and does not divide | ( )∕ |, then * ( ; ) is a formal ∞ algebra.
In Section 2 we recall the proof of the characteristic zero theorem. The proof of Theorem 1.1 occupies Sections 3 and 4. In Section 5 we make some remarks that put our result in context.

CHARACTERISTIC ZERO
We include the well-known proof for the connected case in characteristic zero for the sake of convenience of comparison. It uses the fact that the algebra of rational cochains on a simply connected space has a commutative model as a DG algebra. Proof. In rational homotopy theory, if the Sullivan minimal model of a space has zero differential, then the rational homotopy type is formal. This is true for simply connected spaces whose rational cohomology is a polynomial ring on even degree generators tensored with an exterior algebra on odd degree generators. The reason is that we can choose arbitrary cocycles representing the generators of cohomology, and there are no relations to satisfy apart from (graded) commutativity, which is automatic because of the commutativity of the minimal model. In the case of a connected Lie group, the rational cohomology is isomorphic to the invariant ring * ( ; ℚ) , where is a maximal (compact) torus and is the Weyl group. This is a polynomial ring on even degree generators. So * ( ; ℚ) is formal, and hence so is * ( ; ) by extension of scalars. For more details and background on rational homotopy theory, we refer the reader to Proposition 15.5 of Félix, Halperin and Thomas [13], or Example 2.67 in the book of Félix, Oprea and Tanré [14]. □ Remark 2.2. In the theorem, for connected, Ω ∧ ℚ ≃ ∧ ℚ has homology an exterior algebra on odd degree generators, so it is again formal. In fact, the proof shows that ∧ ℚ and ∧ ℚ are both intrinsically formal.
The obstruction to extending the proof to other characteristics is that the algebra of mod cochains on a space usually does not have a commutative DG algebra model. However, we can give a modified version of this theorem using the method of internal gradings. We do this in several steps, in the rest of this paper.

FINITE APPROXIMATIONS
In this section, we show how to approximate compact Lie groups by finite groups. This is closely related to the work of Dwyer and Wilkerson [12], but better suited to the internal grading method.
Let be a torus of rank acted on by a finite group , let ∈ 2 ( , ) be an element of order ⩾ 1 and let be the finite subgroup of consisting of elements whose th power is the identity. Then is in the image of 2 ( , ) → 2 ( , ).
Proof. We have a short exact sequence which gives an exact sequence If ∈ 2 ( , ) has order , then it is in the kernel of multiplication by on 2 ( , ), and hence, it is in the image of of 2 ( , ). □

Theorem 3.2. Let be a compact Lie group with normal maximal torus of rank and let = ∕ . Then the Bousfield-Kan -completion
∧ is homotopy equivalent tŏ∧ , wherĕis a locally finite group sitting in a short exact sequence with̆isomorphic to ∞ . If | | is coprime to , then this sequence splits.
Proof. The group sits in a short exact sequence This extension defines an element of 2 ( , ). Since | | annihilates 2 ( , ), the order of is a divisor of | |. So by the lemma, is in the image of 2 ( , ) → 2 ( , ). It follows that if we quotient out by , we have a split short exact sequence Choosing a splitting and taking the inverse image in , we obtain a subgroup̃of that sits in a diagram Let̃be the subgroup of generated bỹand ∞ , regarded as a discrete locally finite group, and let̃be the discrete locally finite subgroup of̃generated by and ∞ . Then we have a diagram The comparison map of spectral sequences is an isomorphism on the 2 page, and hence, the map̃→ induces an isomorphism * ( ; ) → * (̃; ). It therefore induces a homology equivalence * (̃; ) → * ( ; ) and a homotopy equivalence of Bousfield-Kan completions̃∧ → ∧ .
For the final statement about splitting, the group 2 ( , ) is annihilated by | | and by , and is hence zero.
We shall make use of the theorem by means of the following proposition. Proof. In Theorem 20.3 of Borel [8], it is proved that for connected, * ( ; ℚ) → * ( ; ℚ) is an isomorphism. Theorem I.5 of Feshbach [15] improves this to the statement that whether or not is connected, the inclusion → induces an isomorphism * ( ; Feshbach attributes this theorem to Borel without reference, but the argument is simple. The restriction followed by the Becker-Gottlieb transfer is equal to multiplication by the Euler characteristic ( ∕ ), which is equal to | |, and transfer followed by restriction is the sum of the conjugates under ( ), by the double coset formula. If ( , | |) = 1, then it follows using the five lemma on the long exact sequence in cohomology associated to the short exact sequence of coefficients that * ( ; ) → * ( ; ) is an isomorphism. This applies just as well to ( ), and so * ( ; ) → * ( ( ); ) is an isomorphism. Now complete at . □

FORMALITY
Let be a compact Lie group with maximal torus of rank , and let = ( )∕ . According to Theorem 3.2 and Proposition 3.3, there is a locally finite discrete group̆sitting in a short exact sequence with̆≅ ∞ , and a homotopy equivalencĕ∧ ≃ ∧ .
For the proof of formality, we shall need to make use of Kőnig's lemma from graph theory, so we begin by stating this. Proof. This is proved in Kőnig [17]. For a modern reference in English, see, for example, Diestel [11,Lemma 8.1.2]. An equivalent and possibly more familiar formulation is that an inverse limit of a sequence of non-empty finite sets is non-empty (see, for example, Bourbaki [9, III.7.4]). □ Proof. We firstly examine the casĕ=̆= ∞ . Regarding 1 as the unit circle in ℂ, let g = 2 ∕ be a generator for as a subgroup of 1 , so that g = g −1 . So ∞ is the union of these subgroups, regarded as an infinite discrete group. Set = g − 1, a generator for the radical ( ). Then Assuming that ⩾ 3, we have * ( ; ) = [ ] ⊗ Λ( ). This ring inherits an internal grading from the group algebra, and we have | | = (−2, −1) and | | = (−1, − 1 ). The restriction map from * ( +1 ; ) to * ( ; ) sends +1 to and +1 to zero. Therefore * ( ∞ ; ) = [ ] with | | = (−2, −1). Now the ∞ structure maps ∶ * ( ∞ ; ) ⊗ → * ( ∞ ; ) given by Kadeishvili's theorem [16] preserve the internal grading, and increase the homological grading by − 2. To prove that * ( ∞ ; ) is formal, we must show that we may take to be zero for ≠ 2. But for every non-zero element of * ( ∞ ; ), the homological grading is twice the internal grading. So for to be non-zero, we must have = 2. Thus * ( ∞ ; ) is a formal ∞ algebra. Similarly, for a larger rank torus,̆≅ ∞ , we put an internal ℤ[ 1 ]-grading on ∞ by adding the internal gradings on the factors ∞ , so that all degrees lie in the interval [0, ). This puts an internal grading on * ( ∞ ; ), so that it is a polynomial ring on indeterminates, all in degree (−2, −1). The same argument as in the case of ∞ now implies that * (̆; ) is formal.
Next, we come to the case wherĕ≅ ∞ is normal in̆and =̆∕̆is a finite ′ -group. In this case, the extension . So a splitting for + 1 gives a splitting for by taking th powers. Since there are only finitely many splittings at each stage, it follows using Kőnig's lemma that we may choose consistent splittings for all > 0. The graph to which the lemma is applied has as vertices the pairs consisting of a value of and a splitting for the above sequence. The edges go from the pairs with + 1 to the pairs with , by taking th powers. A ray in this graph consists of a consistent set of splittings for all > 0.
Let ,1 , … , , be bases for such a consistent set of splittings, so that +1, = , . We may now put a grading on ∞ in such a way that the degree of , ∈ is equal to 1∕ , and preserves the grading. Now we can put a grading on̆by choosing a copy of in̆complementary tŏ, and putting it in degree zero. Then inherits an internal grading, and the homological grading of all elements is again twice the internal grading. So as before, all with ≠ 2 in the ∞ structure on * (̆; ) are forced to be zero. It follows that * (̆; ) is a formal ∞ algebra. □ Proof of Theorem 1.1. By Theorem 3.2, the ∞ algebra * ( ; ) is quasi-isomorphic to * (̆; ), which by Theorem 4.2 is formal. □

FINAL REMARKS
It follows from Stasheff and Halperin [19,Theorem 9] that if is a space and is a commutative ring of coefficients such that * ( ; ) is a polynomial algebra, then * ( ; ) is formal. For example, * ( ( ); ℤ) and * ( ( ); ℤ) are polynomial rings on Chern classes, and * ( ( ); ℤ) is a polynomial ring on Pontryagin classes, so the algebras of cochains are formal for all commutative coefficients in these cases. Similarly, * ( ( ); 2 ) and * ( ( ); 2 ) are polynomial rings on Stiefel-Whitney classes, so the algebras of cochains are formal over any commutative ring in which 2 = 0.
In the case of a connected, simply connected compact Lie groups , Borel [4][5][6][7][8] and Steinberg [20] have investigated torsion in * ( ; ℤ). Borel comments at the end of Volume II of his OEuvres that the upshot of these papers is that the following are equivalent.
(2) * ( ; ) is an exterior algebra on odd degree classes. The primes for which this occurs are called the torsion primes for . They are a subset of the bad primes, which are those for which the multiplicity of some fundamental root in the dominant root is divisible by . But these are not the same set. For example, if = ( ), then there are no torsion primes, but 2 is a bad prime. If = 2 , then the only torsion prime is 2 but the bad primes are 2 and 3.
Putting these together, if is a connected, simply connected compact Lie group, is a field of characteristic and is not a torsion prime, then * ( ; ) is a polynomial ring on even classes, and * ( ; ) is formal.
Here is a If is a compact Lie group which is connected but not simply connected, then there exists a torus and a simply connected group , such that if is the torsion subgroup of 1 ( ), then sits in a short exact sequence So * ( ; ℤ) has -torsion if and only if * ( ; ℤ) has -torsion, which happens if and only if either divides | | or is a torsion prime for . Otherwise, * ( ; ) is a polynomial ring in even degree generators, and * ( ; ) is formal. The case where divides | | is studied in Borel [5], Mimura and Toda [18]. For non-connected compact Lie groups, the situation is quite different. For example, let be a semidirect product 2 ⋊ ℤ∕2, where ℤ∕2 acts on the 2-torus 2 by inverting every element. Then

A C K N O W L E D G E M E N T
The first author thanks the University of Warwick for its hospitality during January 2022, while this paper was in progress. The research of the second author and the visit of the first author were partly funded by EPSRC grant EP/P031080/1.

J O U R N A L I N F O R M AT I O N
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