PROCEEDINGS OF THE AMERICANMATHEMATICAL SOCIETY Volume 113, Number 2, October 1991 VIETORIS-BEGLE THEOREM AND SPECTRA JERZY DYDAK AND GEORGE KOZLOWSKI (Communicated by James E. West) Abstract. The following generalization of the Vietoris-Begle Theorem is proved: Suppose {Ek}k>x is a CW spectrum and f:X'-*X is a closed surjective map of paracompact Hausdorff spaces such that Ind X = m < co . If /*: E (x) —► E {f~l(x)) is an isomorphism for all x € X and k = m0 , m0 + 1, ... , m0 + m , then /* : En(X) —► En(X ) is an isomorphism and /* : En+ (X) —> En+ (X1) is a monomorphism for n = m0 + m . Given a CW spectrum E = {Ek}k>x and a pointed CW complex K, one has cohomology groups En(K) for each integer « (see [Sw, Chapter 8]). They are defined as homotopy classes from the suspension spectrum of K to "LnE, where ifE is defined by 'ZnEk = Ek+n . In the case of an Q-spectrum (i.e., where the natural map Ek —► &Ek x is a homotopy equivalence for all k), En(K) is isomorphic to [K,En] (see [Sw, Theorem 8.42]). The groups En(X), X being any pointed topological space, are defined as dirlim{£'"(ArQ), p\ , A} , where {Xa, paß , A} is the Cech system of X (see [D-S, p. 21] for the definition of the Cech system of X). In this way one gets the Cech extension of the functor E" from pointed CW complexes to pointed spaces (see [D] for a general discussion of Cech extensions of functors). Again, if {Ek}k>1 is an Q-spectrum, then En(X) is isomorphic to [X,En]. A basic result is that every spectrum {Ek}k>x is isomorphic to an Q-spectrum (see [B, part 10 of Chapter II]). Essentially, the «th term of that spectrum is the telescope of En —> QEn^ ¡ —► clnEn+2 —»•••. In the case of an unpointed topological space X, we define the unreduced cohomology En(X) as En(X+), where X+ is X with a discrete base point added. Recall the classical Vietoris-Begle Theorem (see [S, p. 344]): Vietoris-Begle Theorem. Let f:X'-+X be a closed surjective map of paracom- pact Hausdorff'spaces. Assume that there is an « > 0 such that H (f~ (x)) = 0 (reduced Cech cohomology) for all x e X and for k < n . Then f* : H (X) —> H (X1) is an isomorphism for k < « and a monomorphism for k = « . Received by the editors February 18, 1990 and, in revised form, March 29, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 55N05; Secondary 55N20, 55P20. Key words and phrases. Vietoris-Begle theorem, spectra, cohomology. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 587 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use