proceedings of the american mathematical society Volume 72, Number 2, November 1978 TWO RESULTS RELATING NILPOTENT SPACES AND COFIBRATIONS ROBERT H. LEWIS Abstract. We first prove a Blakers-Massey Theorem for nilpotent spaces: If (A", A) is an «-connected, n > 1, pair of nilpotent spaces, then under suitable conditions the map m (X, A) -» v^X/A is an isomorphism in dimension n + 1 and an epimorphism in dimension n + 2. Next, we dualize the well-known fact that if the total space of a fibration is nilpotent, so is the fiber. Our dual theorem can be used to construct new examples of finite nilpotent CW complexes. 0. Introduction. A topological space X is said to be nilpotent if its funda- mental group is nilpotent and if the homotopy groups mnX, n > 2, are nilpotent modules over the group ring ZttxX. Dror [1] motivated the study of nilpotent spaces by proving that such spaces satisfy the Whitehead Theorem. Since then, many theorems previously established for simply connected spaces have been generalized to include nilpotent spaces. The book by Hilton, Mislin, and Roitberg [4] contains some of these results. Because it is defined in terms of homotopy groups, nilpotency behaves well with respect to fibrations. For instance, a well-known theorem asserts that if the total space of a fibration is nilpotent, so is the fiber. The purpose of the present work is to establish two results connecting nilpotency to cofibrations. In § 1 we prove a Blakers-Massey Theorem for pairs of nilpotent spaces. In §2 we assume that the total space of a cofibration is nilpotent and show that under certain conditions the cofiber is nilpotent. As a corollary of this theorem we have an easy way of constructing many new examples of nilpotent spaces which are finite CW complexes. In discussing nilpotent modules we use the notation of Dror in [1]. Whenever convenient, "space" will mean path connected CW complex. We use K'(G, n) to denote a Moore space: a simply connected CW complex with the single nonvanishing homology group G in dimension n. All tensor products are taken over the ring Z of integers. 1. A Blakers-Massey Theorem for nilpotent spaces. Definition. We say that a space is nilpotent up to dimension n if mxX acts nilpotently on irkX for k = 1,2, ... ,n. Lemma 1. Let X be nilpotent up to dimension n and Y be nilpotent up to Presented to the Society, January 4, 1978; received by the editors October 17, 1977 and, in revised form, March 8, 1978. AMS (MOS) subject classifications (1970). Primary 55D30,55D05,55E35. Key words and phrases. Nilpotent space, cofibration, Blakers-Massey Theorem, Moore space. © American Mathematical Society 1978 403 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use