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Appendices Appendix HL: Homotopy Colimits and Fibrations We Give Here

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Appendices Appendix HL: Homotopy Colimits and Fibrations We Give Here Appendices Appendix HL: Homotopy colimits and fibrations We give here a briefreview of homotopy colimits and, more importantly, we list some of their crucial properties relating them to fibrations. These properties came to light after the appearance of [B-K]. The most important additional properties relate to the interaction between fibration and homotopy colimit and were first stated clearly by V. Puppe [Pu] in a paper dealing with Segal's characterization of loop spaces. In fact, much of what we need follows formally from the basic results of V. Puppe by induction on skeletons. Another issue we survey is the definition of homotopy colimits via free resolutions. The basic definition of homotopy colimits is given in [B-K] and [S], where the property of homotopy invariance and their associated mapping property is given; here however we mostly use a different approach. This different approach from a 'homotopical algebra' point of view is given in [DF-1] where an 'invariant' definition is given. We briefly recall that second (equivalent) definition via free resolution below. Compare also the brief review in the first three sections of [Dw-1]. Recently, two extensive expositions combining and relating these two approaches appeared: a shorter one by Chacholsky and a more detailed one by Hirschhorn. Basic property, examples Homotopy colimits in general are functors that assign a space to a strictly commutative diagram of spaces. One starts with an arbitrary diagram, namely a functor I ---+ (Spaces) where I is a small category that might be enriched over simplicial sets or topological spaces, namely in a simplicial category the morphism sets are equipped with the structure of a simplicial set and composition respects this structure. To such a diagram the functor hocolim (otherwise called homotopy direct limit and denoted holimj ) assigns a space hocolimX E(Spaces). A basic property ---+ I of this assignment is that it respects 'local weak equivalence of diagrams'. Appendix HL: Homotopy colimits 177 PROPOSITION: Let f: X --+ Y be a map of I-diagram of spaces. Thus f is a natural transformation between two functors X and Y. Assume that for each i E I the map --+ Y i is a weak equivalence. Then f induces a weak equivalence of spaces hocolim f: hocolim X ---- hocolim Y. I I I In some sense the functor hocolim is the 'closest one' to the actual direct limit or colimit that has this 'weak homotopy invariant' property. One can consider the functor hocolim in various categories {Spaces} of spaces that have an appropriate notion of weak equivalence or 'homotopy of maps' (for example, in chain complexes or in simplicial (abelian or not) groups). We will need to consider it only in two cases: pointed spaces S. and unpointed spaces S. It turns out that their values on these categories are different, as can be seen in the following examples: Examples: (1) The most elementary examples of homotopy colimit are disjoint unions of (unpointed) spaces or wedges of pointed spaces. These are homotopy colim• its over discrete categories, i.e. categories with only identity morphisms. A diagram over such a category is just a family of spaces. A pointed diagram is a family of pointed spaces. In this case the unpointed homotopy colimit is equal to the colimit itself. Already here the pointed and unpointed ho• motopy colimits are different. As always they are related by the cofibration BI --+ hocolimX --+ hocolim.X so that in the pointed case one gets the wedge I I VXi as the pointed homotopy colimit over categories with no non­identity maps.. (2) If the category I = G is a (discrete or continuous) group, then a diagram over G is a space with a group action and a pointed diagram is a space with a specified fixed point (fixed under the action of G). The homotopy colimit in the unpointed case is weakly equivalent to the Borel construction EG Xa X, and this is true whether X has fixed points or not. In fact X may well be a single point * and then hocolim­sf») ::: BG, the classifying space of G. If G here is a simplicial or a topological category, we get BG as a topological or simplicial space by the definition of homotopy colimit. If X is a pointed G­space, then hocolim.X, the pointed homotopy colimit, is the pointed Borel construction given as EG I><a X = (EG I>< X)jG where by (-jG) we mean here the quotient space with respect to the G­action. In contrast to the above the pointed Borel construction on a single point is contractible: EG I><a {*} ::: {*}. (See Chapter 2 for a discussion of the half­smash product (­) I>< (-).) 178 Appendix HL: Homotopy colimits A theorem of V. Puppe There are very useful relations between the concepts of 'fibre map', 'homotopy fibre' and that of homotopy colimits. These relations are useful since sometimes they allow one to commute the operation of taking the homotopy fibre of a map with that of taking homotopy colimits of a system of maps. In general one does not expect a 'left adjoint', such as a homotopy colimit, to commute with a 'right adjoint', such as taking a homotopy fibre of a map. In fact, in general, these operations do not commute. Example: The homotopy colimit of the two-arrow pushout diagram * f-- X -----> * is, of course, the suspension of X. Now given a map g: X --t Y with Y connected, taking its homotopy fibre does not commute with suspension f= But notice that taking homotopy groups commutes with linear homotopy colimits, namely in a tower of cofibrations of pointed spaces X: one has PROPOSITION: For a linear tower as above there is a natural isomorphism colim 7riX --t 7ri hocolimX. I Proof: This follows from the compactness of Si and the unit interval. I Now when we translate this to properties of homotopy fibres using long exact sequences and the fact that linear direct limits preserve exactness, we get PROPOSITION: Let Eo --t E 1 --t E 2 En --t E oo Po! ! ! Pn! Poo ! Bo --t B1 --t B2 Bn --t Boo be a tower of fibrations over connected pointed spaces Bi. There is a weak equivalence relating homotopy fibres as follows Appendix HL: Homotopy colimits 179 More surprising perhaps is that under certain conditions homotopy pushout also commutes with taking homotopy fibre [Pu]: THEOREM (V. PUPPE): Let [z B B B2- 0-t. 1 be a commutative diagram and suppose that each square is a homotopy pullback, i.e. both (iI, Jl) and (12, J2) induce a homotopy equivalence as the homotopy Fib(po) Fib(pI) Fib(P2) via the obvious maps. Then the maps of Fib(Pi) for i = 0,1,2 into the homotopy fibre of the maps of homotopy pushouts, namely are all weak equivalences, where 0 denotes the homotopy pnsbout, i.e. double mapping cylinder. General homotopy colimits of fibrations Since by induction one can build any homotopy colimit using direct towers, disjoint unions and pushouts, we can reach the following conclusion: PROPOSITION [PU]: Let -+ be any map ofI-diagrams with connected for all i E I. Assume that for each sp: i -+ j in mor I the square: E i 1 1 is a pullback square up to homotopy, so that the induced map on the homotopy fibres: is a weak equivalence. Suppose in addition, for simplicity, that I is connected (i.e. RI is a connected space). In this situation the homotopy fibre of the induced map 180 Appendix HL: Homotopy colimits on the homotopy colimits is equivalent to the common values of all the homotopy fibres, i.e. Fib(hocolimE --. hocolimB) I I is weakly equivalent to the 'common value' Fi . Diagrams over a fixed base space B The above theorem of V. Puppe has a very useful corollary, where instead of considering a diagram of fibrations with 'fixed homotopy type as a homotopy fibre' one considers diagrams of fibration sequences over a fixed base space B. See the examples following the next proposition. PROPOSITION: Let -!E ....... B be any map ofan I-diagram to a fixed connected space B. Thus all we assume is that all the squares strictly commute: Let !!: be the diagram of the homotopy fibres of the maps E, -- B i . In this situation the homotopy fibre of the map hocolim E -- hocolim B = B I I is equivalent to the homotopy colimit of the diagram of fibres: "hocolimF. I Proof: It is not hard to reduce this to the theorem of V. Puppe above by backing up the fibration to get a diagram of fibrations with a fixed homotopy type DB as the homotopy fibre. From the proposition above we get a fibration DB --+- hocolim F -- hocolim E I I that can be seen to be a principal one. Upon classifying this fibration sequence we get the desired fibration sequence over B. An easy direct proof can be obtained by decomposing each fibre map E; -- B into a commutative diagram of trivial fibrations Fi -- (Ei)u -- a over the diagram of simplices {u}=rB as explained below. This gives a 'double diagram' of fibrations Fi,u Ei,u --a, The original diagram is obtained by 'integrating', i.e. taking homotopy colimit over the index a that varies over the simplices of B. But since we are allowed to change the order of taking homotopy colimits we first take the homotopy colimit over the index i E I. Appendix HL: Homotopy colimits 181 Namely, we take first the hocolim over each simplex in B. This gives us a diagram of maps: (*) Notice the weak equivalence between fibre and total space that follows from the fact that for each individual i the map is an equivalence.
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