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IJMMS 25:5 (2001) 311–313 PII. S0161171201004641 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON WHITEHEAD’S INEQUALITY, nil[X, G] ≤ cat X MARTIN ARKOWITZ (Received 16 February 2000) Abstract. A new proof of Whitehead’s inequality, nil [X, G] ≤ cat X, is given. 2000 Mathematics Subject Classiﬁcation. Primary 55M30, 55P45, 55Q05. One of the beautiful theorems of elementary homotopy theory is the result that nil[X, G] ≤ cat X. We begin by explaining the notation. Let X and G be based, connected topological spaces and let G be group-like. Thus there is a multiplication G×G → G on G which satisﬁes the group axioms up to homotopy [7, page 118]. Then the set [X, G] of based homotopy classes of maps from X to G inherits a group structure from G. For a nilpotent group π , nil π is the nilpotency class of π . In particular, nil π = 0 means that π is the trivial group and nil π ≤ 1 means that π is abelian. Finally, cat X denotes the Lusternik-Schnirelmann category of X, normalized so that contractible spaces have cat = 0. Theorem 1 [7, page 464]. With the above assumptions, nil[X, G] ≤ cat X. The proof given in [7, pages 462–464] uses the following deﬁnition of category [7, page 458]: cat X is the smallest nonnegative integer l such that the diagonal map X → X l+1 factors up to homotopy through the subspace of X l+1 with at least one coordinate equal to the base point. Recently, another equivalent deﬁnition of category given by the existence of cross-sections to certain ﬁbrations, called Ganea ﬁbrations, has been widely used. The purpose of this paper is to give a new proof of Whitehead’s theorem using this latter deﬁnition of category. For a space X, we deﬁne the Ganea ﬁbrations pn in Gn (X) → X Fn (X) → (1) inductively [3]: for n = 0 the ﬁbration is just ΩX → EX → X, the standard pathpn−1 in−1 space ﬁbration. Assume Fn−1 (X) → X is deﬁned and let Gn Gn−1 (X) → (X) = Gn−1 (X) ∪in−1 CFn−1 (X) be the mapping cone of in−1 . Deﬁne pn : Gn (X) → X as pn−1 on Gn−1 (X) and trivial on the cone CFn−1 (X). Replacing pn by an equivalent ﬁbre in pn X. The connection of the Gn (X) → map, we obtain the ﬁbre sequence Fn (X) → Ganea ﬁbrations to category is as follows (see [2, 4]): cat X ≤ n if and only if pn admits a cross-section. We now start the proof of the theorem. We begin in Lemma 2 with a general result which is probably known (see [5, page 22] and [6]). Let f : A → B be any map and 312 MARTIN ARKOWITZ consider the mapping cone sequence of f , f j q B → Cf → ΣA, A → (2) where Cf is the mapping cone of f and ΣA is the suspension of A. If G is any group-like space, we obtain a homomorphism q∗ : [ΣA, G] → [Cf , G]. Lemma 2. The image of q∗ is contained in the center of [Cf , G]. Proof. We sketch the proof which is based on the operation of [ΣA, G] on [Cf , G] [7, page 136]. We denote this operation by “·” and the group operation in [ΣA, G] and [Cf , G] by “+”. Then for a, b ∈ [ΣA, G] and x, y ∈ [Cf , G], it is easily seen (see [1] and also [5, page 5]) that (a + b) · (x + y) = (a · x) + (b · y). (3) Let e denote the homotopy class of the constant map. By taking b = e and x = e, we obtain a · y = q∗ (a) + y. (4) By taking a = e and y = e, we obtain b · x = x + q∗ (b) which we write as a · y = y + q∗ (a). (5) Thus Image q∗ is in the center of [Cf , G]. Lemma 3. For any space X and group-like space G, nil[Gk (X), G] ≤ k. Proof. This is proved by induction on k. Clearly, nil[G0 (X), G] = 0 since G0 (X) is contractible. Suppose the result is true for k−1. It suﬃces to show that nil[Gk (X), G] ≤ k. Consider the mapping cone sequence ik−1 jk−1 qk Fk−1 (X) → Gk−1 (X) → Gk (X) → ΣFk−1 (X), (6) where Gk (X) is the mapping cone of ik−1 , jk−1 is the inclusion, and qk is the projection. This gives an exact sequence of groups ∗ qk∗ jk−1 Gk (X), G → Gk−1 (X), G . ΣFk−1 (X), G → (7) By Lemma 2, Image qk∗ is contained in the center of [Gk (X), G]. By induction, nil[Gk−1 (X), G] ≤ k − 1. Therefore, nil[Gk (X), G] ≤ k. Now we complete the proof of the theorem. Suppose cat X = n. Thus there is a section s : X → Gn (X), that is, pn s is homotopic to the identity map. Hence s ∗ : [Gn (X), G] → [X, G] is onto. Since nil[Gn (X), G] ≤ n by Lemma 3, it follows that nil[X, G] ≤ n. Remark 4. By dualizing the Ganea ﬁbrations we obtain the Ganea coﬁbrations X → Cn (X) → Qn (X) [2, Section 4]. Then the cocategory of X is deﬁned to be the smallest integer n such that the coﬁbre map X → Cn (X) has a retraction. If C is a co-H-group, then an argument dual to the one above yields nil[C, Y ] ≤ cocat Y . ON WHITEHEAD’S INEQUALITY, nil [X, G] ≤ cat X 313 References [1] [2] [3] [4] [5] [6] [7] M. Arkowitz, Actions and coactions in a category, in preparation. , Equivalent deﬁnitions of the Ganea ﬁbrations and coﬁbrations, Manuscripta Math. 100 (1999), no. 2, 221–229. 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