Subject: Question about geometric realization From: Peter May Date: Fri, 4 Feb 2005 14:51:31 -0600 (CST) To: dmd1@lehigh.edu Let f:X. -> Y. be a map of simplicial topological spaces. Under what circumstances can one deduce that |f|:|X| -> |Y| is a Serre fibration? That is a very interesting kind of question. To the best of my knowledge, the first result of roughly this kind was the hardest technical detail in ``The Geometry of Iterated Loop Spaces'', SLN 271 (1972), where I prove (12.7) that if f is a ``simplicial Hurewicz fibration'' (12.5) such that each Y_q is connected and Y is proper (a cofibration condition on degeneracies), then |f| is a quasi-fibration. Both the hypotheses and the conclusion are unsatisfactory. That both present problems is clear from the standard map p: E_*G -> B_*G for a grouplike topological monoid G (so the set of components of G is a group). In this case, p_q: G^{q+1} -> G^q is projection on the first q coordinates, which is about as nice as can be, but |p| is a quasi-fibration and not in general a Serre fibration. I vaguely recall that Waldhausen proved a result of as similar nature around the same time. Another interesting result in this direction is due to Don Anderson, ``Fibrations and geometric realization'', Bull. A.M.S. 84(1978). He proves (6.2) that if f:X -> Y is an ``epifibration'' of bisimplicial sets, then |f| is a (Serre) fibration. One might try to study this question in two steps. What is the most general condition on f such that |f| is a quasi-fibration? What is a reasonable criterion for a quasi-fibration to be a Serre fibration? I have no idea about the second. I would like some verifiable condition. The best result I know of the flavor that I would like is the theorem of Steinberger and West (with correction of the proof by Cauty) that a Serre fibration between CW complexes is necessarily a Hurewicz fibration. Thus I would like a theorem saying that a quasi-fibration such that (fill in the blank non-tautologously) is a Serre fibration.