Subject: quasifibrations Date: Wed, 16 May 2001 15:50:07 +0100 From: Tom Goodwillie To: Don Davis CC: dan@math.uiuc.edu, may@math.uchicago.edu, Charles Rezk , John Klein Sorry for spreading alarm and confusion. (My own confusion was not without reason, but I'm sorry I spread it around.) After receiving informative responses from a number of people, I can now give you the good news that the definition of quasifibration has remained constant over time for many many years and that it is in fact the one I first mentioned: Def 1: A map p:E->B of spaces is a quasifibration (QF) if for every point in B the canonical map from the fiber of p to the homotopy fiber of p is a weak homotopy equivalence. The definition one usually sees is logically equivalent to this, but reads a little differently. But that's not what confused me. Actually some authors add the condition that p should be surjective. But that's not what confused me either. What confused me was that in a 1956 Comptes Rendu Dold and Thom give a different definition. But it seems that that definition was short-lived; by 1958 Dold and Thom (and by 1959 Dold and Lashof) are defining QF as above. For purposes of the present discussion I am going to give the 1956 notion a new name, LQF. Def 2: A map p:E->B of spaces is a local quasifibration (LQF) if every point in B has arbitrarily small open neighborhoods U such that over U p is a QF. It's not obvious that LQF implies QF, but I think it does. Thus LQF is equivalent to saying that p restricts to be QF over every open subset of the base. Also, obviously if p restricts to be LQF on each member of an open cover then p itself is LQF. And of course LQF is preserved by restriction to open subsets of the base. Both of those last sentences are false if QF is substituted for LQF throughout. Thus LQF is strictly stronger than QF. The fact that LQF implies QF belongs to the same family of results that includes Quillen's "Theorem B". Another result in that family (which is often taken as the starting part for proving such results, I believe) is this: Lemma 1: p:E->B is a QF if B has an open cover, closed under finite intersections, such that over each member of the cover p is QF. Then there's the related notion I was proposing: Def 3: A map p:E->B of spaces is a universal quasifibration (UQF) if not only p but every map obtained from p by pullback (=base change) is QF. Clearly UQF implies LQF. This implication is strict, since LQF is not preserved by pullback. Also, UQF implies QF trivially (without going through LQF). Lemma 2: If p becomes QF under every base change to a closed disk, then p is UQF. [Remark: A map to a contractible space is QF iff the inclusion of each fiber into the total space is a weak equivalence.] Proof (easy, sketched): We have only to show that p is QF. This means showing that certain extension/lifting problems w.r.t p have solutions. Every such problem really "lives" over some disk that's mapped to B, so it's enough if we can solve it after pulling back to the disk. Lemma 3: p:E->B is a UQF if B has an open cover such that over each member of the cover p is UQF. Proof (sketched): By Lemma 2, we can assume B is a disk, and we have only to prove that p is QF. We may refine the cover so that it is a finite cover such that the union of the first k elements is contractible for all k>0. Inductively prove that for all k p is QF over that union. In the step from k-1 to k we are just going to use the easy fact (Mayer-Vietoris plus van Kampen) that if a space is the union of two open sets U and V and the inclusion of the intersection of U with V into U is a weak equivalence and likewise for V then the inclusion of U (or V) into the union is again a weak equivalence. Lemma 3 gives a possibly new route to Lemma 1. In fact, let's look at Lemma 1 in the case of an open cover by two sets U and V. By using path fibrations we can reduce to the case when E is the union of E_U and E_V and p(E_U) is contained in U and p(E_V) is contained in V and all three of the maps E_U -> U E_V -> V (E_U intersect E_V) -> (U intersect V) are fibrations. One sees easily that over U or V p is a UQF, and it follows that p is a a UQF I am pleased with the notion UQF because it is - closed under pullbacks - local in a rather strong sense, and - by Prop 1, not so hard to check sometimes. Tom Goodwillie