VARIATIONS ON A CONJECTURE OF HALPERIN
Gregory Lupton
Abstract. Halperin has conjectured that the Serre spectral sequence of an*
*y fibra-
tion that has fibre space a certain kind of elliptic space should collaps*
*e at the E2-term.
In this paper we obtain an equivalent phrasing of this conjecture, in ter*
*ms of for-
mality relations between base and total spaces in such a fibration (Theor*
*em 3.4).
Also, we obtain results on relations between various numerical invariants*
* of the base,
total and fibre spaces in these fibrations. Some of our results give weak*
* versions of
Halperin's conjecture (Remark 4.4 and Corollary 4.5). We go on to establi*
*sh some of
these weakened forms of the conjecture (Theorem 4.7). In the last section*
*, we discuss
extensions of our results and suggest some possibilities for future work.
x1 _ Introduction
We begin with a description of the conjecture referred to in the title. In t*
*his pa-
per, all spaces are simply connected CW complexes and are of finite type over Q*
*, i.e.,
have finite-dimensional rational homology groups. A fibration F - j! E - p! B *
*is
said to be totally non-cohomologous to zero (abbreviated TNCZ) if the induced h*
*o-
momorphism j* : H*(E; Q) ! H*(F ; Q) is onto. This is a very strong condition to
place on a fibration. It is equivalent to requiring that the Serre spectral seq*
*uence (for
cohomology with rational coefficients) collapse at the E2-term (cf. [McC,Th.5.9*
*]).
In this case there is an isomorphism H*(E; Q) ~=H*(B; Q)H*(F ; Q) of H*(B; Q)-
modules. Thus a TNCZ fibration is somewhere between being trivial from the
rational homology point of view and being trivial from the rational cohomology
algebra point of view (cf. Example 1.2).
In the sequel we focus on certain fibre spaces F that satisfy the following *
*condi-
tions:
(1) H*(F ; Q) is finite-dimensional.
(2) ss*(F ) Q is finite-dimensional. P
(3) The Euler characteristic of F , i.e., i(-1)idim Hi(F ; Q) , is posit*
*ive.
A space that satisfies (1) and (2) is called (rationally) elliptic. See [Ha1], *
*[Fe,Ch.5]
or [Au] for a discussion of these spaces. It is known that elliptic spaces have*
* non-
negative Euler characteristic [Ha1]. So condition (3) further restricts F to b*
*eing
one of two types of elliptic space. We often refer to a space that satisfies co*
*nditions
(1)-(3) as a positively elliptic space. However, we also refer to such spaces i*
*n the
long-hand, as `elliptic with positive Euler characteristic', particularly when *
*stating
results.
______________
This paper was written whilst the author was a guest at the Max-Planck-Insti*
*tut f"ur Math-
ematik. The work was begun whilst the author was a visitor at the Universite C*
*atholique de
Louvain. Thanks to both institutions for their support.
Typeset by AM S-T*
*EX
1
2 GREGORY LUPTON
The conjecture of the title, with which we are concerned, is as follows:
1.1 Conjecture (Halperin). Let F be elliptic with positive Euler characteristi*
*c.
Then any fibration F ! E ! B is TNCZ.
This conjecture has been established in various cases, but in general it rem*
*ains
open. Some results that concern it are mentioned later in the introduction.
We point out that this paper does not resolve Conjecture 1.1, not even in sp*
*ecial
cases! Rather, as the title suggests, we are concerned here with variations on *
*the
theme provided by the conjecture. These variations come about by considering
consequences of Conjecture 1.1, assuming it to be true. The motivation is two-f*
*old:
First, it is hoped to open up new lines of approach to the conjecture itself. S*
*econd,
by looking at such consequences one can obtain weak versions of the conjecture.
On the one hand, these weak versions might prove more tractable than the origin*
*al.
On the other hand, they should lead to a fuller understanding of the conjecture.
Although we consider consequences of Conjecture 1.1, some of our results are
independent of the status of this conjecture and furthermore are interesting in*
* their
own right. For instance, Theorem 4.7 specializes to obtain the following result*
*: If
F ! E ! S2n+1 is a fibration with fibre a positively elliptic space and base an
odd sphere, then cat0(E) = cat0(F ) + 1. This result establishes a weak form of
Conjecture 1.1. But also, for instance, it can be viewed as a strong form of Ga*
*nea's
conjecture in the (very) restricted circumstances to which it applies.
Next, we outline the contents of the paper. This introductory section contin*
*ues
with a discussion of positively elliptic spaces and some of their properties. W*
*e go
on to discuss models of rational fibrations, the main technical tool that we us*
*e.
The introduction finishes with a brief summary of some results on Conjecture 1.1
and some notational conventions. Section 2 is a short technical section, althou*
*gh in
Theorem 2.2 we obtain a very strong consequence of Conjecture 1.1. In Section 3
we relate the formality of E and B, for a class of fibrations F ! E ! B includi*
*ng
those to which Conjecture 1.1 applies. In Proposition 3.2, for example, we show
B formal implies E formal, under the hypothesis that Conjecture 1.1 is true. We
also obtain an equivalent phrasing of Conjecture 1.1, in Theorem 3.4. In Secti*
*on
4 we consider some numerical rational homotopy invariants. Under the hypothesis
that Conjecture 1.1 is true, we obtain inequalities that relate the values of t*
*hese
invariants on base, total and fibre spaces of a suitable fibration (Remark 4.4)*
*. These
inequalities can therefore be viewed as weak versions of Halperin's conjecture.*
* We
go on to establish these weakened forms of the conjecture in certain restricted
circumstances: For some of the invariants, we obtain complete results in case t*
*he
base space is a wedge of odd-dimensional spheres (Theorem 4.7). In the last sec*
*tion,
we discuss extensions of our earlier results. Here we suggest various direction*
*s for
future work, in part by offering specific questions on these topics.
The spaces F that feature in Conjecture 1.1 are clearly of a very restricted*
* kind.
We continue with a discussion of some of their properties. We have characterized
them by conditions (1)-(3) above. However, results of Halperin [Ha1] allow for
various characterizations. Halperin shows that, for an elliptic space F , the *
*three
conditions of positive Euler characteristic, Oss(F ) = 0 and Hodd(F ; Q) = 0 are
equivalent. Here, Oss(F ) denotes the so-called homotopy Euler characteristic *
*of
F . This is a number defined for any space that has finite-dimensional rational
VARIATIONS ON A CONJECTURE OF HALPERIN 3
P
homotopy by Oss(F ) := i(-1)idim ssi(F ) Q . The cohomology algebra of a
positively elliptic space F is zero in odd degrees and has a presentation of th*
*e form
H*(F ; Q) ~=Q[x1;_:_:;:xn]_(R
1; : :;:Rn)
with relations generated by a maximal regular sequence {R1; : :;:Rn} in the pol*
*y-
nomial algebra Q[x1; : :;:xn]. Here, the relations R1; : :;:Rn need not be homo*
*ge-
neous (length) polynomials. The minimal model of such a space is of a particula*
*rly
restricted form. Recall that the minimal model of a space X is a differential g*
*raded
(henceforth DG) algebra MX ; dX , that as a graded algebra is a free graded com-
mutative algebra (polynomial on even degree generators and exterior on odd degr*
*ee
generators). Also, its (degree +1) differential dX is decomposable, in the sen*
*se that
it induces the trivial differential after passing to the quotient module of ind*
*ecom-
posables, i.e., it has zero linear part. See [Gr-Mo], [Ha3] and [Ta] for the ba*
*sics of
minimal models and their use in rational homotopy theory. The book by Felix [Fe]
contains more recent material and references. Condition (1) above implies that
the minimal model has finite-dimensional cohomology, as the cohomology of the
minimal model is identified with that of the space. Condition (2) translates i*
*nto
the condition that the minimal model be finitely-generated as a free graded alg*
*e-
bra, since the algebra generators of the minimal model are identified, as a gra*
*ded
vector space, with the rational homotopy groups of the space. For an elliptic s*
*pace,
condition (3) greatly restricts the form of the minimal model further. It impl*
*ies,
for instance, that up to isomorphism the model is a pure model [Ha1]. This is to
say that it has the form
MF ; dF = (V even) (V odd); dF
with dF (V even) = 0 and dF (V odd) (V even). For an elliptic space, condition*
* (3)
further implies that the minimal model has the same number of even degree gen-
erators as odd degree generators. In symbols, this means dim (V even) = dim (V *
*odd)
and this fact corresponds to the condition that Oss(F ) = 0. We state one more
property of these remarkable spaces. We have said that the cohomology algebra
of a positively elliptic space is zero in odd degrees. In fact, more is true. S*
*uppose
V; d is any pure model, as above. We place a second grading on V by setting
V k= V even kV oddfor k 0. Since d(V even) = 0 and d(V odd) (V even),
the differential d decreases second degree by 1 and so V; d is a bigraded DG al-
gebra. This second grading passes to cohomology. Now, if V is the model of a
positively elliptic space, then we have H+ (V; d) = 0 (see [Ha1,Th.2]).
It is worthwhile remarking that, despite the highly restricted form of F , t*
*here
are many examples of such spaces, some of which correspond to spaces familiar to
topologists and geometers: Even-dimensional spheres, complex projective spaces,
Grassmann manifolds and in general homogeneous spaces G=H with G a compact,
connected Lie group and H a closed subgroup of maximal rank are all examples of
positively elliptic spaces. Further, given any algebra presented as above, the*
*re is
some space F , necessarily positively elliptic, that realizes the algebra as it*
*s rational
cohomology algebra.
4 GREGORY LUPTON
Next, we survey some material on rational fibrations and their minimal model*
*s.
For a fuller discussion see [Ha2] or [Fe]. Consider a sequence of DG algebra ma*
*ps
of the form
B; dB -i!B V; D -q!V; d
in which i: B; dB ! B V; D is the inclusion and q :B V; D ! V; d is the
projection onto the quotient DG algebra of B V by the ideal generated by B+ .
This sequence is called a KS-extension of B; dB (for Koszul-Sullivan) if there*
* is a
well-ordered basis {vff}ff2Iof V such that, for each ff 2 I, D(1 vff) 2 B (V<*
*ff).
Here V V ~= C N with D(C) = 0, D : N ! (u) V injective and such that any
cocycle in the ideal I(N) of (u) V generated by N is exact. Clearly u 2 C,
and we show that V0 C, for any such decomposition. For suppose not, so that
D(V0) 6= 0. Choose a KS-basis V0 = . There is at least one of th*
*e xi
with non-zero differential, so let r be the largest subscript with D(xr) 6= 0. *
* Note
that xr 2 N in the decomposition. Now D(V0) is contained in the ideal of (u)V
12 GREGORY LUPTON
generated by elements {ux1; : :;:uxr-1 }, which we denote Ir-1 . For parity of *
*degree
reasons, (D - d)(V1) is contained in the ideal generated by u . V1. Also, we h*
*ave
d(V1) 2 V0. Let J denote the ideal of (u) V generated by u . V1 + 2 V0.
Then the image of D is contained in the ideal Ir-1 + J. Now D(xr) = uO for some
O and hence we obtain a D-cocycle uxr 2 I(N). This cannot be exact, as it is
not in the ideal Ir-1 + J. This contradicts the assumptions on the decomposition
C N. Therefore, we must have V0 C and so D(V0) = 0. From this it follows
easily that the fibration is TNCZ, because then q* :H((u) V ) ! H(V ) is
surjective onto the generators of H(V ).
Finally, we collect together the preceding results into the main result of t*
*he
section. As we see, we have obtained an equivalent formulation of Halperin's co*
*n-
jecture.
3.4 Theorem. Let F be elliptic with positive Euler characteristic. Then the f*
*ol-
lowing are equivalent:
(1) Any fibration with fibre F is TNCZ.
(2) For any fibration F ! E ! B in which B is formal, E is formal also.
(3) For any fibration F ! E ! S2n+1 , the total space E is formal.
Proof. The implication (1) ) (2) follows from Proposition 3.2. (2) ) (3) is obv*
*ious,
since spheres are formal spaces. Assume (3). Then Proposition 3.3 implies that
each fibration F ! E ! S2n+1 is TNCZ. Hence, from Meier's result (Theorem
1.5), any fibration F ! E ! B is TNCZ.
x4 _ Numerical Invariants
Here we consider some invariants related to the Lusternik-Schnirelmann cate-
gory. Recall that this is a numerical homotopy invariant of a space, defined as
one less than the smallest number of open sets required to cover the space, when
each is contractible in the space. As is usual in rational homotopy theory, we *
*have
`normalised' so that a sphere has category equal to 1. This invariant and its a*
*pprox-
imations have been much studied both in ordinary and rational homotopy theory.
See [Ja] for a recent survey with many references. Here we focus on four ratio-
nal homotopy invariants. We define these invariants and include some discussion,
before proceeding to the results:
(Rational) Cup-length: This is the nilpotency_as an algebra_of the rational
cohomology algebra of a space X. It is denoted here cup0(X). For example we
have cup0(CP n) = n for each n 1.
(Rational) Toomer's invariant: As in [Fe-Ha2,Rem.9.3] we describe this invar*
*iant
as follows: Let V; d be the minimal model of X. Consider the projection
pn :V ! ___V_____n+1: V
We obtain our rational invariant, denoted e0(X), by setting e0(X) n if (pn)* is
injective. In other words, e0(X) is the largest n for which some non-zero clas*
*s in
H(V ) is represented by a cocycle in n V .
(Rational) Category: This is the Lusternik-Schnirelmann category of the rati*
*o-
nalization of X. We denote it cat0(X) and, following [Fe-Ha2,Th.4.7], we descri*
*be
VARIATIONS ON A CONJECTURE OF HALPERIN 13
it in terms of the minimal model of X: Set cat0(X) n if the above projection pn
makes V into a retract of the quotient V=n+1 V .
(Rational) Cone-length: This is the least number of steps required to build *
*the
rational homotopy type of a space X as a succession of cofibration sequences of
rational spaces. It is denoted cl0(X). Specifically, set cl0(X) = 0 if X 'Q *
* *
and cl0(X) = 1 if X has the rational homotopy type of a wedge of spheres. In
general, set cl0(X) n if there are spaces X1; A1; : :;:An-1 , each of which has
the homotopy type of a wedge of rational spheres, and n - 1 cofibration sequenc*
*es
Ai ! Xi ! Xi+1, for i = 1; : :;:n - 1, such that Xn 'Q X. In [Co] it is shown
that cl0(X) agrees with the `homotopical nilpotency' of the minimal model of X,
i.e., the least n for which the minimal model is quasi-isomorphic to a DG algeb*
*ra
that is of nilpotency n as an algebra. It is in this latter guise that we meet*
* this
invariant here.
We have been a little careless in phrasing the above definitions by implicit*
*ly
assuming these invariants finite. Although the case when one or other of these
is infinite does not seem so interesting in our context, it is allowed for, whe*
*re
appropriate, in the following results.
For these invariants, we always have inequalities as follows:
cup0(X) e0(X) cat0(X) cl0(X):
In the special case that X is a formal space, all these invariants agree. In t*
*his
case we will denote their common value nil0(X). This usage accords with Cornea's
homotopical nilpotency in this case. In particular, if F is a positively ellipt*
*ic space,
then it is formal and we have cup 0(F ) = e0(F ) = cat0(F ) = cl0(F ), which we
denote by nil0(F ).
We mention some examples to illustrate these invariants:
4.1 Examples If X = Sn , then X is formal, our four invariants agree and we
have nil0(X) = 1. If X = CP n, it is likewise formal and nil0(X) = n. Next,
suppose X = S2 _ S2 [ffe5, where ff = [1; [1; 2]], the triple Whitehead product
in ss4(S2 _ S2). Then we have cup 0(X) = 1 but e0(X) = cat0(X) = cl0(X) = 2.
A well-known example of Lemaire-Sigrist, developed by Felix-Halperin (cf. [Fe-
Ha2]), is X = (CP 2_ S2) [! e7 for a certain attaching map !. This space satisf*
*ies
cup 0(X) = e0(X) = 2 whilst cat0(X) = cl0(X) = 3. Furthermore, this example
has e0(Xn ) = 2n and cat0(Xn ) = 3n. This illustrates that e0(X) can be smaller
than cat0(X) by an arbitrarily large amount.
These invariants behave quite well for products of spaces. The product formu*
*la
cup 0(X x Y ) = cup0(X) + cup0(Y ) is well-known. It is easy to see that e0 lik*
*ewise
is additive for products [Fe-Ha2,Rem.9.3]. Recently, cat0 has been shown to be
additive for products [Fe-Ha-Le], and cl0 to be additive at least for products *
*of
rational Poincare duality spaces. Indeed, it has been shown that e0(X) = cat0(X*
*) =
cl0(X) whenever X is a rational Poincare duality space (see [Fe-Ha-Le] and [Co-
Fe-Le]). Now Halperin's conjecture asserts that in certain fibrations the total*
* space
is close to being a product of the base and fibre spaces. These remarks combine*
* to
suggest there should be good relations between these invariants for base, total*
* and
fibre spaces in such fibrations. We shall see that this is the case.
First we give some results that complement one of Jessup [Je,Prop.3.6]. Let
F ! E ! B be a TNCZ fibration with F formal. Then Jessup's result gives
14 GREGORY LUPTON
cat0(E) cat0(B) + nil0(F ). Actually, his result applies a little more genera*
*lly
and was proved for (what was then thought to be) a different numerical invarian*
*t,
Mcat 0. The conclusion for cat0 follows by a result of Hess [He], which identi*
*fied
cat0 with Mcat 0.
We specialize this result to the following:
4.2 Proposition. Let F ! E ! B be a fibration in which F is elliptic with
positive Euler characteristic and B is formal. If the fibration is TNCZ, then E*
* is
also formal and we have nil0(E) nil0(B) + nil0(F ).
Proof. The formality of E follows immediately from Proposition 3.2. Since E and
B are both formal, nil0= cat0 for these spaces and the conclusion follows from *
*the
result of Jessup mentioned above.
Next, consider the case in which B is not formal. Notice the following resu*
*lt
does not require that F be positively elliptic.
4.3 Proposition. Let F ! E ! B be a fibration with F formal. If the fibration
is TNCZ, then we have the following inequalities:
(1) cup0(E) cup0(B) + nil0(F ).
(2) e0(E) e0(B) + nil0(F ).
Proof. Actually, for (1) the hypothesis of formality on F is redundant. Recall
from the introduction, that if a fibration F ! E ! B is TNCZ, then H*(E; Q) ~=
H*(B; Q)H*(F ; Q) as H*(B; Q)-modules. Under this isomorphism, two elements
of 1 H*(F ; Q) multiply as (1 x)(1 y) = 1 xy + O, for some O in the ideal
generated by H+ (B). It follows that cup0(E) cup0(B) + cup0(F ).
For (2), we work with a model of the fibration. Suppose that F is formal and
the fibration is TNCZ. Then it has a model W; dB ! W V; D ! V; d in
which D(V0) = 0. This follows from the argument in the first two paragraphs of
the proof of Proposition 3.1, replacing H(B) there by W . So let ff 2 n V be a
cocycle representative for some non-zero class in H(V ). Since F is formal, we *
*can
suppose that ff 2 n V0. In our model we have D(V0) = 0, so [ff] 2 H(W V ).
Furthermore, ff is not D-exact, since it is not d-exact. Let fi 2 m W be a coc*
*ycle
representative for some non-zero class of H(W ). Since we have H(W V ) ~=
H(W ) H(V ) as H(W )-modules, the product [fi][ff] = [fiff] is non-zero in
H(W V ). Now fffi 2 m+n (W V ) so e0(E) m + n.
Example 1.2 illustrates that inequality, as in Proposition 4.3, is the best *
*that
can be hoped for in general. But see below for stronger relations in special ca*
*ses.
If B is not formal in Proposition 4.3, there is no a priori reason why the i*
*nvariants
cup 0, e0 and cat0 for B or E should agree. Therefore, it can be thought of as
giving three distinct necessary conditions for Halperin's conjecture to be true*
*. We
summarise this in the following:
4.4 Remark. Let F ! E ! B be a fibration in which F is elliptic with positive
Euler characteristic. If B is formal and if the fibration is TNCZ, then E is a*
*lso
formal and nil0(E) nil0(B) + nil0(F ). For general B, if the fibration is TNCZ
VARIATIONS ON A CONJECTURE OF HALPERIN 15
then we have three inequalities
cup0(E) cup0(B) + nil0(F )
e0(E) e0(B) + nil0(F )
cat0(E) cat0(B) + nil0(F ):
Each of these inequalities gives a necessary condition for Halperin's conjectur*
*e to
be true.
A much stronger consequence follows if we restrict the base as follows:
4.5 Corollary. Let F ! E ! B be a fibration in which F is elliptic with positi*
*ve
Euler characteristic and B is rationally a wedge of odd spheres. If the fibrati*
*on is
TNCZ, then E is formal, the invariants cup 0(E), e0(E), cat0(E) and cl0(E) all
agree and their common value, nil0(E), satisfies nil0(E) = 1 + nil0(F ).
Proof. This follows from Theorem 2.2.
So far, we have collected together some consequences of Halperin's conjectur*
*e.
These consequences can be read as weak versions of Halperin's conjecture. We go
on to establish some of these weak versions of the conjecture. We can deal qui*
*te
well with the case in which the base B is a wedge of odd-dimensional spheres.
The next result generalizes part of [Fe-Ha2,Th.10.4(iv)].
4.6 Proposition. Let F ! E ! B be any fibration in which F is a rational
Poincare duality space and B is a wedge of odd-dimensional spheres. Then e0(E)
1 + e0(F ).
Proof. Suppose W; dB ! W V; D ! V; d is the minimal model of the
fibration. Observe that W V; D must actually be the minimal model of E, i.e.,
the differential D must be decomposable. Indeed, this is the case for any fibra*
*tion
in which F is a space with finite dimensional rational cohomology and B is a we*
*dge
of odd-dimensional spheres, as follows from [Ha2,Th.1.4(iii)]. Observe further,*
* that
since B is a wedge of spheres, it is both formal and coformal. Thus its minimal
model W; dB is a bigraded model in the sense discussed earlier, and the differe*
*ntial
is quadratic, i.e., dB :W ! 2W . We use these observations in the proof.
Suppose that e0(F ) = n. Since F is a Poincare duality space, the fundamental
class of F can be represented by a cocycle ff 2 n V (cf. [Fe,Lem.5.6.1]). Supp*
*ose
that u 2 W is a generator of lowest (odd) degree, so that dB (u) = 0. Consider *
*the
element uff 2 n+1 (W V ). There is no a priori reason why uff should be a
D-cocycle, but we will show the following:
Claim. There is an element j 2 n+1 (W V ), with j 2 (W )+ V , such that
D(uff + j) = 0.
Proof of Claim. We argue by induction, using the second grading of the bigraded
model W . Let {bi;j}j2Ji be a (vector space) basis of (+ W )i, for each i 0. It
is convenient for our argument to have this basis be a monomial basis, so that *
*each
basis element has a certain length. Also, our element u is one of the basis ele*
*ments
b0;j, but we denote it u anyway so as to distinguish it.
16 GREGORY LUPTON
P
Write D(ff) = i0;j bi;ji;j; for suitable elements i;j 2 V . Each term
bi;ji;j2 n+1 (W V ), as D is decomposable. Then we have
X X
D(uff) = - u b0;j0;j - u bi;ji;j:
j i1;j
Now W; dB is the minimal model of a wedge of spheres, whose cohomology
has trivial products. It follows that each ub0;j 2 (2 W )0 is a dB -cocycle. *
* So
dB (j1;j) = ub0;jfor some j1;j2 (W )1. Furthermore, since W; dB is coformal,
each j1;jis of length one less than ub0;j. For each j, we have
D(j1;j0;j) = ub0;j0;j + (-1)|j1;j|j1;jD(0;j):
From the observation about the length of each j1;j, together with the fact that
D is decomposable, it follows that each j1;j0;j 2P n+1 (W V ) and each
j1;jD(0;j) 2 n+2 (W V ). Finally, write j(1) = jj1;j0;j, so that j(1) 2
n+1 (W V ) and j(1)2 (W )+ V . Then we have
X X X (1)
D(uff + j(1)) = - u bi;ji;j+ (-1)|j1;j|j1;jD(0;j) = bi;ji;j;
i1;j j i1;j
for suitable (1)i;j2 V , and each bi;j(1)i;j2 n+2 (W V ). This starts the ind*
*uc-
tion.
Now suppose inductively that we have an element j(r)2 n+1 (W V ) with
P (r)
j(r)2 (W )+ V and D(uff+j(r)) = ir;j bi;ji;j2 n+2 (W V ). Re-write
P (r)
jbr;jr;j, the part of D(uff + j(r)) that contains terms of lowest second deg*
*ree
in W , as follows: Let {ck}k2K be a (vector space) basis for V . Then write
X (r) X
br;jr;j = fir;kck
j k
for suitable terms fir;k2 (W )r. For i r+1, we have D(bi;j(r)i;j) 2 (W )r V .
So working modulo the ideal generated by (W )r in W V , we have
X
0 = D2(uff + j(r)) dB (fir;k) ck:
k
Hence dB (fir;k) = 0 for each k. Recall once again that W; dB is the bigraded
model, with H+ (W; dB ) = 0. Since r 1, it follows that each fir;k is dB -exa*
*ct.
So dB (jr+1;k) = fir;kfor some jr+1;k 2 (W )r+1.PAs W is coformal, jr+1;kck 2
n+1 (W V ) for each k. Now set j(r+1)= j(r)- kjr+1;kck. Note that j(r+1)2
n+1 (W V ) and j(r+1)2 (W )+ V . A straightforward check shows that
X (r) X
D(uff + j(r+1)) = bi;ji;j - (-1)|jr+1;k|jr+1;kD(ck)
ir+1;j k
X (r+1)
= bi;ji;j ;
ir+1;j
VARIATIONS ON A CONJECTURE OF HALPERIN 17
for suitable terms (r+1)i;j2 V with each bi;j(r+1)i;j2 n+2 (W V ). This
completes the inductive step.
Since B is simply connected, the lowest degree of a generator in Wr increases
strictly with r. Hence, by taking r sufficiently large, we obtain an element j(*
*r)as
in the claim, with D(uff + j(r)) = 0. End of Proof of Claim.
We now show that this cocycle is not D-exact. Suppose that D(i + O) = uff + j
for i 2 V and O 2 + W V . Then d(i) = 0. However, i has higher degree
than ff, which represents the fundamental class of F . Thus iP= d(a) for some
a 2 V so D(a) = i + O0 for some O0 2 + W V . Write O - O0 = i;jbi;jOi;j.
Working modulo the idealPin W V generated by 2 W0 + (W )+ , we have
D(O + i) = D(O - O0) = j(-1)|b0;j|b0;jd(O0;j) uff. This implies ff is d-exac*
*t,
which is a contradiction since ff represents the fundamental class of F . There*
*fore,
uff + j is a non-exact D-cocycle in n+1 (W V ). The result follows.
Next we give the main result of this section.
4.7 Theorem. Let F ! E ! B be a fibration in which F is elliptic with positive
Euler characteristic and B is a wedge of odd-dimensional spheres. Then for E we
have e0(E) = cat0(E) = cl0(E) and furthermore these equal nil0(F ) + 1.
Proof. In fact the proof will display a simple model for E of homotopical nilpo*
*tency
equal to 1 + nil0(F ). Let ae : V; d ! H(F ) be the bigraded model for F . By
construction we have ae(V1) = 0 and hence ae (V )+ = 0. Consider the ideal
ker ae V . This is a differential ideal since all boundaries are in kerae. Fur*
*ther, it
is an acyclic ideal since ae is a surjective quasi-isomorphism. Since B is form*
*al, the
fibration has a model H(B) ! H(B) V; D ! V; d. We claim that the ideal
H(B) kerae of H(B) V is also a differential acyclic ideal. To see this, we ar*
*gue
as follows:
Let {bi} be a basis for H+ (B). For any O 2 V , we can write
X
D(O) = dO + bii(O);
i
and a standard argument shows that this defines derivations i on V; d, each of
negative even degree according as the degree of the biP(cf. the result of Meier*
*, cited
as Theorem 1.5). Now for w 2 V1, we have D(w) = dw + ibii(w). For parity of
degree reasons we must have i(w) 2 (V )+ . It follows that i (V )+ (V )+
for each derivation i.
Next, if O 2 kerae V , write O = O0 + O+ where O0 2 V0 and O+ 2 (V )+ .
Since O 2 kerae, it follows that O0 2 kerae and hence O0 = dj for some j 2 (V )*
*1.
Then
X X
D(O) = d(O+ ) + bii(O0) + bii(O+ )
Xi Xi
= d(O+ ) + bii(dj) + bii(O+ )
Xi i
X
= d(O+ ) + bid i(j) + bii(O+ ):
i i
18 GREGORY LUPTON
This term is in H(B) kerae because kerae contains all boundaries in V and also,
as remarked earlier, i(O+ ) 2 i (V )+ (V )+ kerae. We have shown that
D(ker ae) H(B) kerae and since D = 0 on H(B), it follows that H(B) kerae is
D-stable. P
Further, suppose ff 2 H(B)ker ae is a D-cocycle. We can write ff = ff0+ ib*
*iffi
with ff0 and each ffi in kerae. Then D(ff) = 0 implies d(ff0) = 0 and hence ff0*
*= dj0
for some j0 2 kerae, as kerae is acyclic. WithoutPloss of generality, we can c*
*hoose
j0 2 (V )+ , as d(V0) = 0. Then D(j0) = d(j0) + i bii(j0). From an earlier
remark, each i(j0) 2 (V )+ kerae. So we have
X X
ff = D(j0) + bi ffi- i(j0) = D(j0) + biff0i
i i
for elements j0; ff0i2 kerae. As all products in H+ (B) are trivial, D(ff) = 0 *
*implies
each d(ff0i) = 0, so thatPff0i= dji for elementsPji 2 kerae. But D(biji) = -bi*
*d(ji)
and hence ff = D(j0 - i biji) with j0 - i biji 2 H(B) kerae. This shows
H(B) kerae is an acyclic ideal of H(B) V .
To finish, notice that the projection
q : H(B) V ! _H(B)__V_____H(B) kerae
is a quasi-isomorphism, as H(B) kerae is acyclic. If nil0(F ) = n, then >n V
>n V0 + (V )+ kerae and hence the quotient H(B) V=(H(B) kerae ) is a
nilpotent DG algebra of length n + 1, and is quasi-isomorphic to a KS-model for
E. Therefore, it follows from [Co] that cl0(E) n + 1. From Proposition 4.6, we
have n + 1 e0(E). But in general we have e0(E) cat0(E) cl0(E). Hence
these three invariants must agree and furthermore must equal n + 1.
x5 _ Discussion and Questions
In this last, somewhat informal, section we discuss the foregoing results an*
*d give
some additional ones. The intention is to indicate both limits on, and possibil*
*ities
for, development of the work in Sections 3 and 4. We also offer some specific
questions along these lines. All this is collected under two subheadings, accor*
*ding
as the topic relates to Section 3 or Section 4.
5.1 Formality.
One can attempt a quite general study of relations between the formality of
F , E and B in a fibration (cf. [Th2] and [Vi]). It is tempting to think that*
* the
very restrictive hypotheses of Proposition 3.2 might be relaxed, whilst keeping*
* the
conclusion. However, the following sort of example suggests these hypotheses a*
*re
actually sharp.
Example. There is a rational fibration S2 _ S2 _ S2 ! E ! S3 that is TNCZ
in which E is not formal. We omit details of this example, as it is similar to
[Th2,Ex.III.13] and other examples. We note that the fibre is formal with non-z*
*ero
cohomology only in even degrees. The fibration we have in mind is actually much
closer to being trivial than just TNCZ. It satisfies H*(E; Q) ~=H*(B; Q)H*(F ; *
*Q)
as algebras and admits a rational section.
VARIATIONS ON A CONJECTURE OF HALPERIN 19
In case a weaker conclusion is acceptable, then of course there are more pos*
*si-
bilities. In Proposition 3.2, we can remove the hypothesis of ellipticity on th*
*e fibre
space, but so far only at the cost of a greatly weakened conclusion, thus:
Proposition. Let F ! E ! B be a fibration in which F and B are both formal.
If the fibration is TNCZ, then E has spherically-generated cohomology.
The proof of this proposition is omitted. It can be proved with an argument sim*
*ilar
to that of Proposition 3.2. The above example also suggests this might be a sha*
*rp
conclusion, without much more restrictive hypotheses.
It would be satisfying to have a converse of Proposition 3.2. In Proposition*
* 3.3,
we have such a result but only for a very special base. We offer the following *
*as a
specific question in this area:
Question 1. Let F ! E ! B be a fibration in which F is formal and elliptic and
B is formal. If E is formal, then is the fibration is TNCZ?
Of course, there are many variations on this type of question to investigate*
*. For
more results and examples along these lines, see [Th2] and [Vi].
Returning to Conjecture 1.1, it may be possible to apply Theorem 3.4, togeth*
*er
with an appropriate obstruction theory for the formality of E in such a fibrati*
*on.
Such an obstruction theory does exist, and I hope to develop this line of inves*
*tigation
in future work.
5.2 Numerical Invariants.
The results of Section 4 deal nicely with fibrations F ! E ! B, in which F is
positively elliptic and B is a wedge of odd-dimensional spheres, but only for t*
*he
invariants e0, cat0 and cl0.
Question 2. For a fibration F ! E ! B, in which F is elliptic with positive
Euler characteristic and B is a wedge of odd-dimensional spheres, is cup 0(E) =
1 + nil0(F )?
Of course, if Halperin's conjecture is true, it implies an affirmative answe*
*r to
Question 2 (Corollary 4.5). It is curious that Question 2 remains unresolved wh*
*ilst
its counterpart for the invariants e0, cat0 and cl0is established.
Another `asymmetry' in the results we have presented is the lack of an inequ*
*ality
for cone-length in Proposition 4.3. This gives our next specific question:
Question 3. Let F ! E ! B be a fibration with F formal. If the fibration is
TNCZ, then is cl0(E) cl0(B) + nil0(F )?
As regards extending the results presented in Section 4, we can show at leas*
*t the
following:
Proposition. Let F ! E ! B be a fibration in which F is elliptic with positive
Euler characteristic and B has e0(B) = 1. Then e0(E) 1 + nil0(F ).
Of course, if e0(B) = 1, then the base space B is rationally a wedge of sphe*
*res and
e0(B) = cat0(B) = cl0(B) = 1. Notice, however, that we allow even dimensional
spheres and also that we do not have Poincare duality in either the base or the
total spaces. The current proof of this Proposition uses a rather involved redu*
*ction
argument, similar to that of [Th1] or Proposition 4.6 here. This result sugges*
*ts
that the `next step' might be to investigate one of the following questions:
20 GREGORY LUPTON
Question 4 (a). Let F ! E ! B be a fibration in which F is elliptic with posit*
*ive
Euler characteristic and e0(B) = 2. Is e0(E) 2 + nil0(F )?
Question 4 (b). Let F ! E ! B be a fibration in which F is elliptic with posit*
*ive
Euler characteristic. Is e0(E) 1 + nil0(F )?
In the previous two paragraphs, we focussed on the e0-invariant. Choosing one
of the other three invariants gives similar questions to be investigated. Ther*
*e is
some overlap between all these questions, because of the results of [Co-Fe-Le] *
*and
[Fe-Ha-Le] mentioned earlier: e0(X) = cat0(X) = cl0(X) whenever X is a rational
Poincare duality space. Thus, if F ! E ! B is a fibration in which F and B
are rational Poincare duality spaces, then so too is E a rational Poincare dual*
*ity
space and hence the three invariants agree on each of F , E and B. Notice that
this observation obtains the first part of the conclusion to Theorem 4.7 in the
case B = S2n+1 . Finally, we mention that Jessup's result [Je,Prop.3.6] also sh*
*ows
that cat0(E) cat0(B) + 1 for any fibration in which F is positively elliptic _
without the hypothesis of TNCZ. These latter observations give some interesting
complements to the results presented here.
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Department of Mathematics, Cleveland State University, 2400 Euclid Avenue,
Cleveland OH 44115 U.S.A.
E-mail address: Lupton@math.csuohio.edu