Subject: quasifibrations Date: Mon, 14 May 2001 12:10:11 +0100 From: Tom Goodwillie This is basically a references-and-terminology question. I think the following is correct: Def: 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. Question 1: Is this what is usually meant by this term? I've noticed that the following notion, strictly stronger than QF and strictly weaker than Serre fibration, is often more useful for me: Def: A map p:E->B of spaces is a universal quasifibration (UQF) if for every map f:X->B of any space into B the resulting map X\times_B E -> X is a QF. Prop: p is a UQF iff for every map f:D->B of a compact disk (of any dimension) into B, for every point x in D, the resulting map {x}\times_B E -> D\times_B E is a weak homotopy equivalence. (The main thing to prove is that this last condition implies that p is a QF.) Besides being preserved by base change, UQF is also a local notion, in the sense that if p:E->B is such that each point in B has a neighborhood U such that p^{-1}U->U is a UQF then p itself must be a UQF. Question 2: I presume that this stuff is "well known", but where is it in the literature and what is the standard terminology? Tom Goodwillie