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