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Classifying spaces and infinite symmetric products

1969; American Mathematical Society; Volume: 146; Linguagem: Inglês

10.1090/s0002-9947-1969-0251719-4

1088-6850

Michael C. McCord,

Advanced Topics in Algebra

l) 1. Introduction.The object of study in this paper is a construction B(G, X) which essentially includes the classifying space construction BG of Milgram [7] and Steenrod [12] and the infinite symmetric product construction SP(X) of Dold and Thorn [4] as special cases.§ §2, 3, and 4 are preliminary in nature.Let us just remark that §2 describes the category of spaces we are working in (a modification of Steenrod's [11]).We make the convention that in the rest of the paper all spaces, products, topological monoids, etc., are meant in the sense of this category.§5 deals with the algebraic and set-theoretic side of the construction B(•, ■).The definition is quite simple: If G is a monoid with unit e and X is a based set, then B(G, X) is the monoid of all functions u: X->G such that u(*) = e and such that u(x) = e for all but finitely many xe X.In §6 we topologize B(G, X) when G is an abelian topological monoid and X is a based space, in such a way that B(■, ■ ) becomes a bifunctor to the category of abelian topological monoids.If G is an abelian topological group, then so is B(G, X).This construction includes the following special cases (up to topological isomorphism): (1) (B(G, S°)xG.(2) B(G, Sl)xBa as in [7] or [12].(3) B(G, I)xEG as in [7] or [12].( 4) If G is a discrete abelian group, then B(G, Sn) is an Eilenberg-MacLane space K(G, n) (see §10). ( 5) If Z+ is the additive monoid of nonnegative integers, then B(Z+, X)xSP(X) as in [4].(6) If Z is the additive group of integers, then B(Z, X)xAG(X), defined in [4] as a certain quotient space of SP(X v X).(7) B(Z/mZ, X)xAG(X; m), defined in [4] as AG(X)/mAG(X).( 8) One could form topological monoids such as B(SP(X), Y), or more generally B(B(G, X), Y).In 6.13 it is seen that the latter is topologically isomorphic to B(G, X A Y).In §7 we generalize a result of Dold and Thorn [4] by showing that B(G, X) is a CW complex whenever G is a discrete abelian monoid and X is triangulable.For a fixed abelian topological monoid [group] G, the functor B(G, •) has a tendency to convert cofibrations A -> Z-> X/A to quasifibrations [fibrations] (1.1) B(G, A) -> B(G, X) -> B(G, X/A).