The Rational Homotopy Lie Algebra of Classifying Spaces for Formal Two-Stage Spaces

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1 The Rational Homotopy Lie Algebra of Classifying Spaces for Formal Two-Stage Spaces Samuel Bruce Smith September 2, 1956 Abstract We describe the structure of the rational homotopy Lie algebra of the classifying space Baut 1 (X), when X is a formal space with a twostage Sullivan minimal model. Using known cases of a conjecture of S. Halperin, we compute the center and nilpotency of this graded Lie algebra for a large class of pure, formal spaces X. The latter calculation determines the rational homotopical nilpotency of the topological monoid aut 1 (X) for these X. Our results apply, in particular, when X is a complex or symplectic flag manifold. 1. Introduction. Given a CW complex X, let aut 1 (X) denote the identity component of the space of self-equivalences of X and Baut 1 (X) the classifying space for this topological monoid [5]. Recall that Baut 1 (X) classifies orientable fibrations with fibre X [19, 1] Mathematics Subject Classification. Primary 55P62, 55P15. Key words and phrases. Rational homotopical nilpotency; Self-equivalences; Halperin s conjecture 1

2 In this paper, we give an explicit description of the structure of the rational homotopy Lie algebra π (ΩBaut 1 (X)) IQ when X is a two-stage formal space. Our method is to compare the calculation of π (aut 1 (X)) IQ in [13] which gives the rational homotopy Lie algebra as vector space with Sullivan s differential graded Lie algebra model for Baut 1 (X) [20], which gives the Lie structure. Our most general result is the identification of cycle representatives in Sullivan s model for homotopy elements in π (aut 1 (X)) IQ. Using this result we show that a famous conjecture of Halperin concerning elliptic spaces with no odd rational cohomology actually implies the full structure of π (ΩBaut 1 (X)) IQ for a much larger class of spaces. Applying this result to confirmed cases of Halperin s conjecture [9, 12, 15], we compute the center and nilpotency of π (ΩBaut 1 (X)) IQ for many pure, formal spaces X. For the remainder of the paper, we assume all spaces X are simply connected complexes of finite type with dim(π (X) IQ) < +. This ensures that the rationalization of aut 1 (X) is a nilpotent H-group. A basic problem then is to compute the rational homotopical nilpotency, Hnil 0 (aut 1 (X)), which is defined to be the length of the longest rationally essential commutator in aut 1 (X). This problem has been studied integrally for general loop spaces [3, 2, 12] and, rationally, for the space of self-equivalences [13]. By [13, Theorem 3], Hnil 0 (aut 1 (X)) = nil(π (ΩBaut 1 (X)) IQ), where the right side is the nilpotency of the rational homotopy Lie algebra of the classifying space. Our results in 5 include the following new calculations of 2

3 the rational homotopical nilpotency of spaces of self-equivalences. 1.1 Examples. Given any graded vector space V, let min(v ) = min{n V n 0} and max(v ) = max{n V n 0}. Given any simply connected graded algebra A, let Q (A ) = A /A + A + denote the graded vector space of indecomposables of A. (1) Let S = S 2m 1 S 2m k S 2n 1 +1 S 2n l+1, a product of spheres, with 1 m 1 m k and 1 n 1 n l. If l > 0, let c = card{n i i = 1,..., l}. Then 1 if c = 0 Hnil 0 (aut 1 (S)) = c if 2m k n l + 1. c + 1 if 2m k > n l + 1 (2) More generally, suppose F is any space with H (F, IQ) a finite tensor product of polynomial, truncated polynomial and exterior algebras. Let K H (F, IQ) be a maximal free graded subalgebra and let c = card{n Q n (K) 0}. Then 1 c = 0 Hnil 0 (aut 1 (F )) = c if max(π (F ) IQ) max(q (K)) c + 1 if max(π (F ) IQ) > max(q (K)). (3) Let F = G/H, where G = U(n) (respectively, G = Sp(n)) and H = U(n 1 ) U(n k ) (respectively, H = Sp(n 1 ) Sp(n k )). Let m = n n k. Then Hnil 0 (aut 1 (F )) = max{n m, 1}. (4) Let F be any pure, elliptic, formal space (see 2, below, for definitions) with dim(q even (H (F, IQ))) 3. Let c = card{n Q 2n+1 (H (F, IQ)) 0}. Then 1 if c = 0 Hnil 0 (aut 1 (F )) = c if max(π (F ) Q) max(q odd (H (F, IQ))) c + 1 if max(π (F ) Q) > max(q odd (H (F, IQ))). 3

4 2. Two-stage spaces and Halperin s conjecture. By a two-stage space X, we mean one whose rationalization X QI appears as the total space in a principal fibration of the form K 1 X QI q K 0, where K i = n K(Vi n, n) for some finite-dimensional graded rational vector spaces V i, i = 0, 1. The Sullivan minimal model (M X, d X ) for a two-stage space X is a two-stage differential graded algebra (dga). That is, M X = Λ(V 0 ) dx Λ(V 1 ), with d X (V 0 ) = 0 and d X (V 1 ) Λ(V 0 ). We may assume d X : V 1 Λ(V 0 ) is an injection. Fixing bases, we write V 0 = IQ(x 1,..., x m ) and V 1 = IQ(y 1,..., y n ) so that d X (x i ) = 0 and d X (y j ) = R j (x 1,..., x m ), a polynomial without linear term in the x i. By the main result of [6], every formal space X with finite-dimensional rational homotopy is a two-stage space. In fact, a two-stage space X, as above, is formal if and only if the sequence of polynomials R 1, R 2,..., R n forms a regular sequence in the free algebra Λ(V 0 ). An important class of examples are the F 0 -spaces, by which we mean spaces X with finite-dimensional cohomology and homotopy (elliptic spaces) such that H odd (X) = 0. The class includes the homogeneous spaces G/H of a connected Lie group by a closed subgroup of maximal rank [4]. Halperin proved F 0 -spaces are formal twostage in [8] where he also made the following conjecture: Halperin s conjecture. The rational Serre spectral sequence collapses at the E 2 term for every IQ-orientable fibration of the form X E B with X an F 0 -space. 4

5 Halperin s conjecture has been confirmed for the homogeneous spaces mentioned above [15] and in several other special cases [10, 9]. In [12, Theorem A], W. Meier showed that Halperin s conjecture is equivalent to the assertion that π even (aut 1 (X)) = 0 for all F 0 - spaces. His result implies Theorem 2.1 (Meier) Let X be an F 0 -space satisfying Halperin s conjecture. Then Baut 1 (X) is a rational H-space and Hnil 0 (aut 1 (X)) = 1. Proof. Of course, the second assertion is a consequence of the first. For the first, note that by Meier s result, π odd (Baut 1 (X)) = 0. Now observe that any space with only even rational homotopy is rationally an H-space. Theorem 2.1 shows that Halperin s conjecture implies the full structure of π (ΩBaut 1 (X)) for F 0 -spaces X. Our purpose is to extend this implication to a larger classes of spaces. For example, call a two-stage space F pure if its minimal model (Λ(Z), d F ) satisfies 1) Z = Z 0 Z 1 with Z 1 oddly graded and 2) d F (Z 1 ) Λ(Z even 0 ). Homogeneous spaces G/H of a connected Lie group by a closed, connected subgroup are pure by [4]. If F is pure, elliptic and formal then [6] implies F QI X Y where X is an F 0 -space and Y is a product of odd spheres. In 5, we give the complete structure of π (ΩBaut 1 (F )) when F factors this way for any Y with free cohomology as long as X satisfies the Halperin conjecture. 3. Rational homotopy of the space of self-equivalences. We describe the graded vector space π (ΩBaut 1 (X)) = π (aut 1 (X)) when 5

6 X is a two-stage space. With notation as in 2, define graded spaces L 0 (X) and L 1 (X) by setting, in degree n > 0, L n i (X) = H k (X) Vi n+k, k 0 i = 0, 1. Thus, for example, L 0 (X) is spanned by elements of the form α x i where α H (X) is homogeneous of degree strictly less than x i. The degree of α x i in L 0 (X) is then the difference x i α. Let L(X) = L 0 (X) L 1 (X). Define a linear map D : L 0 (X) L 1 (X) of degree 1 by D(α x i ) = { } Rj (1) α y j, j x i where we write {P } to denote the cohomology class in H (X) represented by an element P M X. Set L 0 (X) = ker{d : L 0 (X) L 1 (X)} and L 1 (X) = cok{d : L 0 (X) L 1 (X)}. We then have Theorem 3.1 Let X be any two-stage space. Then π (ΩBaut 1 (X)) = L 0 (X) L 1 (X). Proof. The principal fibration for X QI determines a fibration on function spaces map(x QI, K 1 ; 0) Φ q aut 1 (X QI ) map(x QI, K 0 ; q). By Thom s classical result [21], π (map(x QI, K i ; f)) = L i (X), i = 0, 1, f = q, 0. We thus have a long exact sequence L n+1 0 (X) 0 L n Φ 1 (X) π n (aut 1 (X)) L n 0 (X) 0 L n 1 1 (X). 6

7 We describe the linking homomorphism 0 for general maps f : X X in [16, Chapter 6]. When f is the identity our result shows 0 coincides with D. (See also the special case [17, Lemma 4.4].) Our goal in 4 is to determine the Lie structure on the spaces L i (X) i = 0, 1 when viewed as subspaces of the rational homotopy Lie algebra of the classifying space. We conclude this section by observing two consequences of Theorem 3.1 and Meier s work [12]: Theorem 3.2 Let X be an F 0 -space. Then X satisfies Halperin s conjecture if and only if π (ΩBaut 1 (X)) = L 1 (X). Proof. Note that L 0 (X) is evenly graded. Thus [12, Theorem A] implies X satisfies Halperin s conjecture if and only if L 0 (X) = 0. An extension to spaces with an F 0 -space as factor can be formulated as follows. Let Y be any two-stage space with Sullivan minimal model M Y = Λ(W 0 ) dy Λ(W 1 ). For i = 0, 1 let L i (X Y, Y ) L i (X Y ) denote the subspace given in degree n by L n i (X Y, Y ) = H k (X Y ) Wi k+n. k 0 We prove Theorem 3.3 Let X be an F 0 -space satisfying Halperin s conjecture and Y any two-stage space. Then L 0 (X Y ) = ker {D : L 0 (X Y, Y ) L 1 (X Y, Y )}. 7

8 Proof. Since X satisfies Halperin s conjecture, D(α x i ) 0 for all α H < xi (X) and all x i in our basis for V 0. By the rational Künneth Theorem and the definition of D it follows that D(α x i ) 0 when α H < xi (X Y ), as well. 4. Cycle representatives in Sullivan s model for Baut 1 (X). We next consider Sullivan s differential graded Lie algebra (dgla) model for Baut 1 (X) as described in [20, 11]. For n > 1, let Der n (M X ) denote the space of degree n derivations of the graded algebra M X. That is, Der n (M X ) consists of maps θ : M X M X lowering degrees by n and satisfying θ(xy) = θ(x)y + ( 1) n x xθ(y). For n = 1, we require, additionally, that θ commute with the differential d X. The Lie bracket of two derivations is the graded commutator: [θ 1, θ 2 ] = θ 1 θ 2 ( 1) θ 1 θ 2 θ 2 θ 1. Define a differential X by X (θ) = [d X, θ]. The pair (Der + (M X ), X ) is a dgla model for the rational homotopy of Baut 1 (X). (For a proof, use [7, Theorem 2] together with [14].) In particular, H(Der + (M X ), X ) = π (ΩBaut 1 (X)) IQ, as graded Lie algebras. A basis for Der + (M X ) as graded space can be obtained using elementary derivations. Suppose X is two-stage with notation as in 2. Then, given z k {x 1,..., x m, y 1,..., y n } and P M X homogeneous of degree strictly less than z k, let P z k denote the derivation which carries z k to P and vanishes on the other basis elements of V 0 V 1. Observe that X is given by (2) X (P z k ) = d X (P ) z k ( 1) z k P n j=1 P R j z k y k. 8

9 If X is formal there is a dga map ρ : (M X, d X ) (H (X), 0) inducing a homology isomorphism. The map of graded spaces p : Der + (M X ) L(X) defined by p(p z k ) = ρ(p ) z k is then a surjection. The map p induces an isomorphism on homology between (Der + (M X ), X ) and (L(X), D) as dg vector spaces. However, we are interested in the Lie structure on π (ΩBaut 1 (X)) and so look for the actual cycle representatives in Der + (M X ) for the subspaces L 0 (X) and L 1 (X),. Define subspaces D 0 (X), D 1 (X), and B(X) of Der + (M X ) by D 0 (X) = Span{P x i P Λ(V 0 ), P < x i, i {1,..., m}} D 1 (X) = Span{P y j P Λ(V 0 ), P < y j, j {1,..., n}} and B(X) = Span{P y k y j P Λ(V 0 ), P + y k < y j, j, k {1,..., n}}. Note D 0 (X) and D 1 (X) are sub Lie algebras and B(X) D 0 (X) D 1 (X) a sub-dgla of (Der + (M X ), X ). Since we have restricted to P Λ(V 0 ), we have X (D 1 (X)) = 0 and X (B(X) D 0 (X)) D 1 (X). Also p(b(x)) = 0. We prove Theorem 4.1 The diagram B(X) D 0 (X) X D 1 (X) p L 0 (X) D p L 1 (X) commutes up to sign and the vertical maps are surjections. Proof. Since X is hyperformal, Λ(V 0 ) contains all cocycle representatives for 9

10 H (X). Commutativity up to sign follows from (1) and (2) above and the formality of X : the map ρ chooses cohomology classes multiplicatively. Set D 0 (X) = ker{ X : B(X) D 0 (X) D 1 (X)} and D 1 (X) = cok{ X : B(X) D 0 (X) D 1 (X)}. We may naturally view D 1 (X) as a subspace of Der + (M X ) by taking vector space complements in each degree. Our main result in this section is Theorem 4.2 If X is formal two-stage, the subspaces D 0 (X) and D 1 (X) contain all cycle representatives in Der + (M X ) for the subspaces L 0 (X) and L 1 (X) in π (ΩBaut 1 (X)), respectively. Proof. We first prove that p : Der + (M X ) L(X) induces surjections p : D i (X) L i (X), i = 0, 1. Lemma 4.3 The map p induces a bijection p : D 1 (X) L 1 (X). Proof. This is just a diagram chase. Lemma 4.4 The map p restricts to a surjection p : D 0 (X) L 0 (X). Proof. Suppose z = m i=1 α i x i L 0 (X) so that D(z) = 0. Choose cocycle representatives P i Λ(V 0 ) for the α i and let θ = m i=1 P i x i. By commutativity of the diagram, 0 = p( X (θ)) = m n i=1 j=1 10 P i R j x i y j.

11 The regularity of {R 1,..., R n } in Λ(V 0 ) implies m i=1 P i R j x i = for each j = 1,..., n where Q jk Λ(V 0 ). Set n Q jk R k k=1 n n m φ = Q jk y k y j B(X). j=1 k=1 i=1 Then X (θ + φ) = 0 and ρ(θ + φ) = z. We next show that the nontrivial boundaries in D 1 (X) are precisely X (B(X) D 0 (X)). Lemma 4.5 Given θ Der + (M X ) if X (θ) D 1 (X) then θ = φ + ξ where φ B(X) D 1 (X) and X (ξ) = 0. Proof. Decompose M X as graded vector space by setting F 0 = Λ(V 0 ) and, for each k > 0, F k = n j=1 y j F k 1. Note that d X (F k ) F k 1 where F 1 = {0}. Next, for each k 0, set D 0,k = Span{P x i P F k, P < x i, i {1,..., m}}, and D 1,k = Span{P y j P F k, P < y j, j {1,..., n}} so that D 0,0 = D 0 (X), D 1,0 = D 1 (X) and D 1,1 = B(X). Using (2), we obtain X (D 0,k ) D 0,k 1 D 1,k and X (D 1,k ) D 1,k 1. Write θ = k 0 θ 0,k + k 0 θ 1,k where θ i,k D i,k. Let φ = θ 0,0 + θ 1,1 and ξ = θ φ = θ 1,0 + i=0,1 k 0 θ i,k. Then X (θ) D 1,0 implies X (ξ) = 0. Finally, we show all boundaries in B(X) D 0 (X) vanish under p. 11

12 Lemma 4.6 Given θ Der + (M X ), if X (θ) B(X) D 0 (X) then p( X (θ)) = 0. Proof. Since p(b(x)) = 0, it suffices to assume X (θ) D 0 (X). But in this case, X (θ) = m i=1 d X (P i ) x i for some P i F 0 F 1. Clearly, p( X (θ)) = 0. Lemmas together imply Theorem Applications. Our most general result here is Theorem 5.1 Let X be any formal two-stage space. Then L 1 (X) is an abelian ideal of π (ΩBaut 1 (X)). Proof. We need only check that D 1 (X) is an abelian ideal of B(X) D 0 (X) D 1 (X). Note that, if P, Q Λ(V 0 ), then [P y j, Q y k ] = 0. Also, [P y j, Q x i ] = ±P Q x i y j and [P y j, Qy k y l ] = δ jk P Q y l, where δ jk is the Kronecker delta. We now focus on the case of a product X Y of two-stage spaces. Note that π (ΩBaut 1 (X)) is naturally a subspace of π (ΩBaut 1 (X Y )). We prove Theorem 5.2 Let X and Y be two-stage formal spaces with X an F 0 -space satisfying Halperin s conjecture. Then π (ΩBaut 1 (X)) center(π (Ω(Baut 1 (X Y ))). 12

13 Proof. By Theorem 3.2, π (ΩBaut 1 (X)) = L 1 (X). Since L 1 (X Y ) is abelian, it suffices to show [L 1 (X), L 0 (X Y )] = 0. Write (M Y, d Y ) = (Λ(W 0 ) Λ(W 1 ), d Y ), again for the Sullivan minimal model of Y. Choose bases {w 1,..., w s } and {v 1,..., v t } for W 0 and W 1, respectively. Let D 0 (X Y, Y ) = Span{P w i P Λ(V 0 W 0 ) P < w i i {1,..., s}} and B(X Y, Y ) = Span{P v k v j P Λ(V 0 W 0 ) P + v k < v j j, k {1,..., t}}. By Theorems 3.3 and 4.2, the elements of L 0 (X Y ) are represented by derivations in D 0 (X Y, Y ) B(X Y, Y ). Such derivations clearly commute with those in D 1 (X). Theorem 5.3 Let X and Y be as in Theorem 4.2. Then nil(π (ΩBaut 1 (X Y )) = nil(π (ΩBaut(Y )) + ɛ, where ɛ = 0 or 1. If max(π (X)) min(π (Y )) then ɛ = 0. Proof. We prove nil (π (ΩBaut 1 (X Y ))/L 1 (X Y )) nil(π (ΩBaut 1 (Y ))). The first statement then follows from the fact that L 1 (X Y ) is an abelian ideal. Consider the left Λ(V 0 )-action on M X Y induced by the isomorphism M X Y = MX M Y. It induces a left Λ(V 0 )-action on Der + (M X Y ) after truncating in positive degrees. Similarly, the left H (X)-action on H (X Y ) 13

14 induces one on L(X Y ). Observe that p : Der(M X Y ) L(X Y ) satisfies p(p θ) = ρ(p ) p(θ) where ρ : M X H (X) is the formalization map. The restriction of the Λ(V 0 )-action to D 0 (X Y, Y ) B(X Y, Y ) satisfies (i) X Y (P θ) = P X Y (θ) and (ii) [P θ 1, Q θ 2 ] = P Q [θ 1, θ 2 ], for P, Q Λ(V 0 ), θ 1, θ 2 D 0 (X Y, Y ) B(X Y, Y ). Similarly, the restriction of the H (X)-action to L 0 (X Y, Y ) satisfies (iii) D(α z) = α D(z) for α H (X) and z L 0 (X Y, Y ). Now suppose z 1, z 2 L 0 (X Y ) represent homogenous elements with nontrivial bracket. By Theorem 3.3, z 1, z 2 L 0 (X Y, Y ). For i = 1, 2, write z i = d j=1 α j z ij, where {α 1,..., α d } is a fixed additive basis for H (X) and z ij L 0 (Y ). By (iii), z ij L 0 (Y ). Use Theorem 3.3 to choose representatives θ ij D 0(Y ) for z ij. Then θ i = d j=1 P j θ ij D 0(X Y ) represents z i where the P ij Λ(V 0 ) are cocycle representatives for the α ij. By (i), θ ij D 0 (Y ). Moreover, by (ii), d d 0 p([θ 1, θ 2 ]) = ρ(p j P k )p([θ 1j, θ 2k ]). j=1 k=1 Thus there exist θ 1j, θ 2k D 0 (Y ) with p([θ 1j, θ 2k]) 0 in L 0 (Y ). For the second statement, note that our the degree hypothesis implies [D 1 (X), D 0 (X Y, Y ) B(X Y, Y )] = 0. 14

15 When H (Y ) is free we get the complete answer. Namely, Theorem 5.4 Let X be an F 0 -space satisfying Halperin s conjecture and Y a nontrivial product of rational Eilenberg Maclane spaces. Let m = max(π (Y )) and c = card{n π n (Y ) 0}. Then and m center(π (ΩBaut 1 (X Y ))) = π (ΩBaut(X)) H m k (X) π m (Y ) k=1 nil(π (ΩBaut 1 (X Y ))) = { c + 1 if max(π (X)) > max(π (Y )) c if max(π (X)) max(π (Y )). Proof. Here W 1 = 0 and Λ(W 0 ) = H (Y ). Thus, as graded vector space, π (ΩBaut 1 (X Y )) = L 1 (X) H k (Y ) V1 n+k n 1 k 1 n 1 k 0 H k (X Y ) W n+k 0. For degree reasons, the subspace m k=1 H m k (X) π m (Y ) L 0 (X Y ) is central. We check that there are no other central elements. Let α w i H k (X Y ) W n+k 0 be nontrivial with w i = n + k < m. Choose w j W 0 with w j > w i. Let P Λ(V 0 W 0 ) represent α. Then [P w i, w i w j ] = P w j represents a nontrivial bracket involving α w i in π (ΩBaut 1 (X Y )). Next let α y j H k (Y ) V n+k 1 for k > 0, be nontrivial. Let P Λ(W 0 ) represent α and choose w i W 0 appearing in P. Then [1 w i, P y j ] = P w i y j represents a nontrivial bracket involving α w i. 15

16 Finally, if α H k (X Y ) is in the ideal generated by H + (Y ) then we may find w i appearing in its representative P. Again, [1 w i, P w j ] = P w i represents a nontrivial bracket involving α w i. w j For the nilpotence result, let w 1,..., w c denote elements of W 0 in strictly increasing order of degree. Then 1 w c = [1 w 1 [w 1 w 2 [ [w c 2 w c 1, w c 1 w c ] ]]] represents a maximal nontrivial iterated bracket in π (Ω(Baut 1 (Y ))) of length c. Thus nil 0 (aut 1 (Y )) = c. If max(π (X)) > max(π (Y )) choose y j V 1 with y j > w c. The elements 1 y j and w c y j in D 1 (X Y ) clearly cannot bound. Thus 1 y j = [1 w 1 [w 1 w 2 [ [w c 2 w c 1 [w c 1 w c, w c y j ]] ]]] represents an nontrivial iterated bracket of length c + 1. The first case follows from Theorem 5.3. Finally, note that, in the decomposition above, L 1 (X Y ) = L 1 (X) n 1 k 1 H k (Y ) V n+k 1. Since L 1 (X Y ) is an abelian ideal, any nontrivial iterated bracket in π (ΩBaut 1 (X Y )) can involve at most one element from this subspace and it must occur at the inner-most bracket. The nilpotence when max(π (X)) max(π (Y )) follows easily. 5.5 Remarks. Theorem 5.5 computes the center and nilpotence of the rational homotopy Lie algebra of the classifying space for all pure, elliptic, 16

17 formal spaces, modulo the Halperin conjecture. In particular, Examples 1.1 (1) and (2) follows from the proof of the Halperin conjecture for a truncated polynomial algebras [10]. Example 1.1 (3) follows from the main result of [15] the proof of the Halperin conjecture for homogeneous spaces G/H with rank G = rankh together with the work of Borel in [4] which implies U(n)/U(n 1 ) U(n k ) QI U(m)/U(n 1 ) U(n k ) S 2(m+1) 1 S 2n 1 Sp(n)/Sp(n 1 ) Sp(n k ) QI Sp(m)/Sp(n 1 ) Sp(n k ) S 4(m+1) 1 S 4n 1. Example 1.1 (4) follows from the main result of [9] together with the rational factorization of pure, formal, elliptic spaces implied by [6] as described in 2. As a final remark, we note that Theorem 5.2 implies that, when an F 0 -space X satisfies the Halperin conjecture, the homotopy Lie algebra of Baut 1 (X) is a factor in the homotopy Lie algebra of Baut 1 (X Y ) for any formal two-stage space Y. In [18], we investigate when this factorization occurs on the level of spaces. References [1] G. Allaud, On the classification of fibre spaces, Math. Z. 92 (1966), [2] M. Arkowitz and C. Curjel, Homotopy commutators of finite orders, (II), Quart. J. of Math. Oxford 15 (1964), [3] I. Berstein and T. Ganea, Homotopical nilpotency, Illinois J. Math. 5 (1961)

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