Homotopy Theory and Models: Based on Lectures held at a DMV by Marc Aubry

By Marc Aubry

In retaining with the final goal of the "D.M.V.-Seminar" sequence, this ebook is princi­ pally a document on a bunch of lectures held at Blaubeuren by way of Professors H. J. Baues, S. Halperin and J.-M. Lemaire, from October 30 to November 7, 1988. those lec­ tures have been dedicated to offering an advent to the idea of versions in algebraic homotopy. the 3 teachers acted in live performance to supply a coherent exposition of the idea. beginning from a typical place to begin, every one of them then proceeded clearly to his personal topic of study. The reader who's already acquainted with their clinical paintings will surely provide the teachers their due. Having been requested via the audio system to tackle the accountability of writing down the notes, it looked as if it would me that the fabric elucidated within the brief span of fifteen hours was once too dense to seem, unedited, in e-book shape. a few amplification was once worthy. in fact I submitted to them the ultimate model of this publication, which acquired their approval. I thank them for this token of self belief. i'm additionally thankful to all 3 for his or her support and suggestion in scripting this booklet. i'm rather indebted to J.-M. Lemaire who was once certainly quite often consulted. For uncomplicated notions (in specific these pertaining to homotopy teams, CW­ complexes, (co)homology and homological algebra) the reader is suggested to consult the elemental books written via E. H. Spanier [47], R. M. Switzer [49] and G. Whitehead [52].

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Additional info for Homotopy Theory and Models: Based on Lectures held at a DMV Seminar in Blaubeuren by H.J. Baues, S. Halperin and J.-M. Lemaire

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O. Cg I lX 9 Let f: EA --+ C g be defined by f = -jH + G'TrF,. and let ~g = (g, H) defined by the push-out. g rv c ~g: CF,. g' that is, there is a homotopy Cf under X. Proof. 2. and follows from the diagram §2. o. o. o. 2 Lemma. Let us define the generalized homotopy groups ([47]): 7r~(X) = [~nA,XJ = 7r~(X,*) 7r~(X, Y) = [(~n-ICA, ~n-l A), (X, Y)J and consider the following diagram 7rf(CB,B) 1 (7rg )* f E 7rf(cg,*) -----+) 7rf(Cg,X) where j is induced by the inclusion * c X and B is I-connected.

1 Definition. Let C be a cofibration category with an initial object *. By (C 3 ) the canonical map * ~ X can be factored as * >---+ RMX~X. Q We call MX~X a cofibrant model of X. We now quote two basic facts on cofibrant models. 2 Lemma. If X is fibrant we can choose M X to be fibrant. Proof. Then a factors through RMX by a' (cf. 1). By axiom (Cd 0:' is a quasi-isomorphism. Then the factorization * >--+ RM X::.. X settles the question. 3 Proposition (uniqueness of cofibrant models). Let * >--+ MX~X p and * >--+ M'X~X p' be both cofibrant and fibrant models of the fibrant object X.

3 Lifting lemma. Consider the commutative diagram of solid arrows: g a) b) c) If X is fibrant there is a map h for which the upper triangle commutes. If X and Yare fibrant there is map h for which the upper triangle commutes and for which p 0 h is homotopic to g reI B. We call a map h with these properties a lifting for the diagram. If X and Yare fibrant, a lifting for the diagram is unique up to homotopy relB. 39 §3. Properties of colibration categories Proof. 2. Let h = rho This proves a). Now assume that X and Yare fibrant.

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