Contemporary Mathematics
The Rational Toomer invariant and Certain Elliptic Spaces
Gregory Lupton
Abstract.We give an explicit formula for the rational category of an ell*
*iptic
space whose minimal model has a homogeneous-length differential. We also
show that for such a space, there are no gaps in the sequence of integers
realized as the rational Toomer invariant of some cohomology class. With*
* an
additional hypothesis, we show a result from which we deduce the relation
dimH*(X; Q) 2 cat0(X).
1. Introduction
Since the fundamental paper of F'elix and Halperin [FH82 ], the so-called r*
*a-
tional Toomer invariant of a space has played a central role in the development
of results concerning rational category. The rational Toomer invariant of a spa*
*ce
X is a numerical rational homotopy invariant, denoted by e0(X), that provides a
lower bound for the rational category of a space. We recall its definition bel*
*ow.
In general, its value is (strictly) between the rational cup length and the rat*
*ional
category of the space. One can also consider the rational Toomer invariant of an
individual cohomology class. This is a finer invariant, whose supremum, taken o*
*ver
all cohomology classes of a space, retrieves the rational Toomer invariant of t*
*he
space. The rational Toomer invariant of a cohomology class can also be identifi*
*ed
with the strict, or essential, category weight of the cohomology class, in the *
*sense
of Rudyak and Strom (cf. [Rud99 , Str97]). Since we are only concerned with the
rational case, we will henceforth refer to the rational Toomer invariant simply*
* as
the Toomer invariant.
In this paper, we study the (rational) Toomer invariant for rationally ellip*
*tic
spaces. Recall that a simply connected space X is called rationally elliptic if*
* both
H*(X; Q) and ß*(X) Q are finite-dimensional vector spaces. For these spaces, the
Toomer invariant becomes all the more interesting, since it is known that cat0(*
*X) =
e0(X) for any elliptic space X, where cat0(X) denotes the rational category of *
*the
space X. Indeed, it is proved more generally in [FHL98 ] that cat0(X) = e0(X) *
*for
any space X that satisfies rational Poincar'e duality.
____________
2000 Mathematics Subject Classification. Primary 55P62, 55M30; Secondary 55*
*T10.
Key words and phrases. Rational category, elliptic space, Toomer invariant,*
* e0-invariant,
Moore length, minimal model, Wang sequence, Gysin sequence.
cO2002 American Mathematical Socie*
*ty
1
2 GREGORY LUPTON
We now outline our results and indicate the organization of the paper. The m*
*ain
result is Theorem 2.2. We assume that the minimal model of an elliptic space has
a homogeneous-length differential. Then part (A) of Theorem 2.2 gives a formula
for the Toomer invariant, and hence the rational category, of the elliptic spac*
*e.
The approach used to prove this part of the result leads to finer information a*
*bout
the Toomer invariant of individual cohomology classes. Paraphrasing the precise
statement, part (B) of Theorem 2.2 says that for an elliptic space whose minimal
model has a homogeneous-length differential, there are no gaps in the sequence *
*of
integers realized as the Toomer invariant of some cohomology class. Part (C) of
the result gives information about the location of cohomology classes that have
a given Toomer invariant. Our second result is Theorem 2.5. Here we show that
under a fairly mild additional hypothesis, there are at least two linearly inde*
*pendent
cohomology classes with Toomer invariant any integer strictly between zero and *
*the
Toomer invariant of the space. The inequality dim H*(X; Q) 2 cat0(X) results.
The paper ends with a brief section of examples and comments.
A Little Archaeology and Acknowledgements: The formula of part
(A) of Theorem 2.2 can be adapted into a lower bound for the rational category
of any elliptic space_cf. Remark 2.4. Indeed, an earlier version of this paper
included this as a separate result. During the refereeing process, however, it *
*was
brought to my attention that this lower bound appears as Corollary 3 in [GJ01 ].
Further, the methods used there to obtain the lower bound, amongst a number of
other interesting related results, are comparable to the methods of this paper.*
* A
forthcoming paper [CJ01 ] also contains a similar, but more general result. The
referee further pointed out to me that part (A) of Theorem 2.2 appears as a spe*
*cial
case of Theorem 1 (Theorem 7) of [LM01 ]. Their method of proof in that paper,
however, is substantially different from the one given here and does not yield *
*the
finer information given by parts (B) and (C) of Theorem 2.2. I thank for the re*
*feree
for bringing these articles to my attention.
We finish this introductory section with a brief review of some ideas from
rational homotopy theory. All results of this paper are proved using standard t*
*ools
of the subject. We refer to [FHT01 ] for a general introduction to these techn*
*iques.
We recall some of the notation here. By a minimal algebra we mean a free graded
commutative algebra V , for some finite-type graded vector space V , together *
*with
a differential d of degree +1 that is decomposable, i.e., satisfies d: V ! 2V*
* . We
assume that the minimal algebra is simply connected, i.e., that the vector spac*
*e V
has no generators in degrees lower than 2. This assumption is necessary in order
to translate our algebraic results into topological ones, although it is not st*
*rictly
necessary for the algebraic results themselves. A minimal algebra V is called
elliptic if both V and the cohomology algebra H*( V ) are finite-dimensional ve*
*ctor
spaces. If {v1, . .,.vn} is a graded basis for V , then we write V as (v1, . *
*.,.vn).
A basis can always be chosen so that dv1 = 0 and for i 2, dvi2 (v1, . .,.vi-*
*1).
Every simply connected space with rational cohomology of finite-type has a
corresponding minimal model, which is a minimal algebra that encodes the ration*
*al
homotopy of the space. Although our results are stated and proved in purely
algebraic terms, they do admit topological interpretations via this corresponde*
*nce.
In particular, a simply connected space is rationally elliptic if its minimal m*
*odel
is an elliptic minimal algebra. Because of the correspondence between spaces and
THE RATIONAL TOOMER INVARIANT AND CERTAIN ELLIPTIC SPACES 3
their minimal models, this characterization of a rationally elliptic space coin*
*cides
with that given earlier.
We recall the definition, in minimal algebra terms, of the Toomer invariant *
*for
a cohomology class and for a space. Suppose that V is a minimal algebra. For
n 1, let pn denote projection onto the quotient differential graded (DG) alge*
*bra
obtained by factoring out the DG ideal generated by monomials of length at least
n + 1, thus
pn : V ! ___V____. n+1V
Define e0( V ) to be the smallest n such that pn induces an injection on cohomo*
*logy,
or set e0( V ) = 1 if there is no such smallest n (cf. [FHT01 , p.381]). This
can be extended to apply to individual cohomology classes as follows: Suppose
x 2 H*( V ) is some fixed, non-zero cohomology class. For obvious degree reason*
*s,
p*n(x) 6= 0 for large enough n. We define e0(x) to be the smallest n for which
p*n(x) 6= 0. If the set of integers {e0(x) | x 6= 0 2 H*( V )} has a maximum, t*
*hen
we see that e0( V ) is this maximum. Otherwise we have e0( V ) = 1.
Where convenient, we will use a standard observation about e0( V ) for a min-
imal algebra V whose cohomology satisfies Poincar'e duality. Namely, that if ~
denotes a fundamental class of H*( V ), then e0( V ) = e0(~) [FH82 , Lem.10.1].
It is well-known that an elliptic minimal algebra satisfies Poincar'e duality [*
*Hal77 ,
Th.3]. By the formal dimension of an elliptic minimal algebra V , we mean the
largest i for which Hi( V ) 6= 0. If x is an element of a graded vector space, *
*then
we denote the degree of x by |x|.
2. Main Results
We say that V has differential of homogeneous-length l if d: V ! lV .
A coformal minimal algebra has differential of homogeneous-length 2. In the
homogeneous-length differential case, the cohomology admits a second grading,
H+ ( V ) = k 1H*k( V ), given by length of representative cocycle. Thus x 6=
0 2 H*k( V ) if and only if e0(x) = k. We extend this second grading to include
the degree zero component by setting H*0( V ) equal to Q in degree zero, and ze*
*ro
elsewhere_see (1) below. Now introduce the following notation: If H*k( V ) 6= 0,
then set
nk = min{i | Hik( V ) 6= 0} and Nk = max{i | Hik( V ) 6= 0}.
We make some observations about the bigraded Poincar'e duality algebra H**( V ).
Denote e0( V ) by e, so that H*( V ) = ek=0H*k( V ). By definition, we have
(
(1) Hi0( V ) = Q i = 0 so n0 = N0 = 0.
0 i > 0
Now suppose that H*( V ) has formal dimension N. Since the second grading
comes from length of representative cocycle, we have
(
(2) Hie( V ) = Q i = N so ne = Ne = N.
0 i < N
We assemble our remaining remarks about H**( V ) into the following lemma.
4 GREGORY LUPTON
Lemma 2.1. Let H be a bigraded Poincar'e duality algebra of formal dimension
N that satisfies H = ek=0H*ktogether with (1) and (2) above. Suppose that H*k6*
*= 0
for k = 1, . .,.e - 1, and that p is some positive integer. Then with the above
notation, the following are equivalent:
(a) n1 = p and nk+1 nk + p for k = 1, . .,.e - 1.
(b) Nk+1 Nk + p for k = 0, . .,.e - 2 and N = Ne = Ne-1+ p.
Proof. Because H is a bigraded algebra, and also satisfies Poincar'e dualit*
*y,
it follows that there are non-degenerate pairings
Hik( V ) x HN-ie-k( V ) ! HNe( V ) ~=Q ,
for k = 1, . .,.e-1 and i = 1, . .,.N -1. Hence nk = N -Ne-k for k = 1, . .,.e-*
*1.
The equivalence of (a) and (b) follows.
To phrase the result, and to give its proof, we say that a graded vector spa*
*ce
is (i - 1)-connected if degree i is the first non-zero degree. Also, for a big*
*raded
cohomology algebra H**( V ) we shall refer to elements in Hi*( V ) as having up*
*per
degree i and to elements in H*k( V ) as having lower degree k. In our setting, *
*upper
degree corresponds to the usual topological degree, and lower degree correspond*
*s to
the Toomer invariant of a cohomology class. In this bigraded setting the notati*
*on
|x| still denotes the upper degree of the element x.
Theorem 2.2. Suppose V is an elliptic minimal algebra with homogeneous-
length l differential. Set
e = dimV odd+ (l - 2)dim V even.
Then we have
(A) cat0( V ) = e0( V ) = e;
(B) H*k( V ) 6= 0 for each k = 0, . .,.e;
(C) Suppose that V is (p - 1)-connected. Then the bigraded Poincar'e duality
algebra H( V ) satisfies the two equivalent conditions of Lemma 2.1.
Proof. We will prove that e0( V ) = e. We can include cat0( V ) in the
statement of part (A) due to the result cat0( V ) = e0( V ) for elliptic minimal
algebras [FHL98 ].
Write V = (x1, . .,.xn), with p = |x1| |x2| . . .|xn| and d(x1) = 0.
We argue by induction on the number of generators n. For n = 1 ellipticity requ*
*ires
that |x1| be odd. In this case e = 1 and all parts of the result are trivial.
Now assume inductively that the result holds for all elliptic minimal algebr*
*as
with homogeneous-length differential and fewer than n generators. Let ( W ; ~d)*
* =
(x2, . .,.xn; ~d) be the quotient obtained by factoring out the DG ideal gener*
*ated
by the generator x1. Then W is elliptic [Hal77 , Prop.1] and has n-1 generator*
*s.
Note that W also has differential of homogeneous-length l. Both V and W are
bigraded DG algebras, with the lower grading in each given by word-length. Thus
their cohomology algebras are bigraded in the sense discussed above. We will ma*
*ke
extensive use of this bigraded structure without further remark.
Set f = dimW odd+ (l - 2)dim W even, so that we have
(
(3) e = f + 1 if |x1| is odd
f + (l - 2)if |x1| is even.
THE RATIONAL TOOMER INVARIANT AND CERTAIN ELLIPTIC SPACES 5
Let M and N denote the formal dimensions of W and V respectively. Although
we do not use it here, we mention that there is a formula that describes these *
*formal
dimensions in terms of the degrees of the generators [Hal77 , Th.3].
Further, set mk = min{i | Hik( W ) 6= 0} and Mk = max{i | Hik( W ) 6= 0}, for
k = 0, . .,.f. From the induction hypothesis, we have e0( W ) = f, H*k( W ) 6= 0
for k = 0, . .,.f and HM*( W ) = H*f( W ) = HMf( W ) ~=Q. Also, m1 = |x2| and
mk+1 mk + |x2| for k = 1, . .,.f - 1. Equivalently, as per Lemma 2.1, we have
Mk+1 Mk + |x2| for k = 0, . .,.f - 2 and M = Mf = Mf-1 + |x2|. To prove the
induction step there are two cases, according as the parity of the degree of th*
*e first
generator x1.
Case I. |x1| = 2r+1 is odd: In this case e = f +1. Here we form the following
short exact sequence of DG vector spaces:
j p
0_____// W ____//_ V____// W ____//_0.
Here, p denotes the projection and j denotes the degree-(2r + 1) linear map def*
*ined
by
j(Ø) = (-1)|Ø|x1Ø
for Ø 2 W . On cohomology, it is clear that j* increases lower degree by 1, wh*
*ilst
p* preserves lower degree. The corresponding long exact sequence in cohomology
has connecting homomorphism described as follows: Suppose Ø 2 W is any ele-
ment. Regard Ø as an element of V and write dØ = ~dØ + x1`(Ø). This defines a
derivation ` : W ! W , of degree-(-2r), that satisfies ~d` = `d~. Hence we have
an induced derivation on cohomology `*: Hi( W ) ! Hi-2r( W ), also of degree-
(-2r). The derivation `* increases lower degree by (l -2). The resulting long e*
*xact
cohomology sequence (the Wang sequence) is therefore a long exact sequence of
bigraded cohomology groups as follows:
* j* p*
(4) . .H.i-1k-1-(l-2)(_W`)//_Hi-2r-1k-1(_W_)//_Hik( V_)__//Hik( W ) ... .
It follows immediately from this long exact sequence that Hi*( V ) = 0 for i
M + 2r + 2 and H*k( V ) = 0 for k f + 2 = e + 1. Furthermore, the long exact
sequence restricts to isomorphisms
*
(5) 0 ____//_HMk( W )j~=_//HM+2r+1k+1(_V_)_//0.
One sees from this that the formal dimensions of V and W are related by
N = M + 2r + 1 (cf. [FH82 , Th.10.4]). This relation is also evident from the
formula of Halperin referred to above. One also sees that e0( V ) = f +1 = e. T*
*his
proves part (A).
Now consider part (B). In the previous paragraph, we showed that HN*( V ) =
H*e( V ) = HNe( V ) ~=Q. It is automatic that H0*( V ) = H*0( V ) = H00( V ) ~=
Q. We consider the remaining values k = 1, . .,.e - 1. Denote the map
`*: H*k( W ) ! H*k+(l-2)( W )
by `*k. From the bigraded Wang sequence (4) we obtain a short exact sequence
* p*
(6) 0 _____//cokernel(`*k-1-(l-2))j//_H*k(_V_)//_kernel(`*k)_//0,
6 GREGORY LUPTON
for each k = 1, . .,.f. From the inductive hypothesis applied to W , we have
(7) mk+(l-2) mk + (l - 2)|x2| mk
and
(8) Mk+(l-2) Mk + (l - 2)|x2| Mk.
Since `*kis of negative degree, (7) and (8) imply respectively that both kernel*
*(`*k)
and cokernel(`*k-1-(l-2)) are non-zero for k = 1, . .,.f. It follows from (6) *
*that
dim H*k( V ) 2 for each k = 1, . .,.f = e - 1.
Finally, we establish (C) in the present case, by showing that H**( V ) sati*
*sfies
condition (a) of Lemma 2.1 with p = |x1|. We observe from the sequence (6) that
Hik( V ) = 0 for i < mk-1 + |x1|. For if i < mk-1 + |x1| mk-1 + |x2|, it foll*
*ows
that there can be no contribution to Hik( V ) from cokernel`*k-1-(l-2). On the
other hand, i < mk-1 + |x1| implies i < mk. Thus if i < mk-1 + |x1|, then neith*
*er
does kernel(`*k) contribute to Hik( V ). So we have the inequality
(9) nk mk-1 + |x1|
for k = l, . .,.f = e - 1. Next, recall that `*kis of negative degree. Hence ke*
*rnel(`*k)
begins in degree mk. From (6), therefore, we obtain
(10) nk mk
for k = l, . .,.f = e - 1. Combining the inequalities (9) and (10), with a shif*
*t of
subscript in the latter, gives
nk mk-1 + |x1| nk-1 + |x1|
for k = 2, . .,.f = e - 1. In addition, it is evident that n1 = |x1|. From (5),*
* we see
that nf+1 = mf + |x1| nf + |x1|, with the latter inequality obtained from (10*
*).
This establishes (C) and completes the induction step in Case I.
Case II. |x1| = 2r is even: In this case e = f + (l - 2). Here, consider the
short exact sequence (again of DG vector spaces) as follows:
j p
0 ____//_ V____// V_____// W_____//0,
where p denotes the projection and j the degree-(2r) map defined by j(Ø) = x1Ø
for Ø 2 V . The corresponding long exact sequence in cohomology has connecting
homomorphism as follows: Suppose ~dØ = 0 for Ø 2 W , so that dØ = x1Ø0= j(Ø0)
for some Ø0 2 V and then @*([Ø]) = [Ø0]. Here @* is of degree-(-2r + 1) and
if Ø 2 kW , then Ø0 2 k+(l-2)V . On passing to (bigraded) cohomology, @*
increases lower degree by (l - 2), p* preserves lower degree and j* increases l*
*ower
degree by one. In this case, therefore, the resulting long exact sequence of bi*
*graded
cohomology groups (the Gysin sequence) is as follows:
* p* @*
(11) . .H.i-2rk-1(_Vj)//_Hik( V_)__//Hik( W )___//_Hi-2r+1k+(l-2)(.V ) . . .
For i M + 2, this sequence gives isomorphisms
j* i
0 _____//Hi-2r*( V_)~=//_H*( V_)___//0.
Since Hi*( V ) must be zero for sufficiently large i, it follows that N M - 2*
*r + 1
and that @*: HM*( W ) ! HM-2r+1*( V ) is an isomorphism. Thus we obtain
N = M -2r+1 (cf. [FH82 , Th.10.4]). Once again, this relation is also evident *
*from
THE RATIONAL TOOMER INVARIANT AND CERTAIN ELLIPTIC SPACES 7
the formula of Halperin referred to above. On the other hand, for k f +(l-2)+*
*2,
(11) gives isomorphisms
j* *
0_____//H*k-1( V_)~=_//Hk( V )___//_0.
Again, since H*k( V ) must also be zero for sufficiently large k, it follows th*
*at
e0( V ) f + (l - 2) = e. The isomorphism @* restricts to isomorphisms
p* M @* M-2r+1 j*
0 ____//_Hk ( W )_~=_//Hk+(l-2)( V_)___//0.
It follows that e0( V ) = f + (l - 2) = e, as desired for part (A).
For parts (B) and (C), we show H*k( V ) 6= 0 for k = 1, . .,.e and establish
condition (b) of Lemma 2.1, by using a (secondary) induction on k.
The secondary induction hypothesis is as follows:
(i)H*i( V ) 6= 0 for i = 1, . .,.k - 1,
(ii)Ni Ni-1+ 2r for i = 1, . .,.k - 1, and
(iii)Ni Mi-(l-2)- 2r + 1, for i = l - 1, . .,.k - 1.
This induction starts with k = l. Since the differential is of length l, there *
*are no
boundaries of length (l - 1). Therefore xi16= 0 2 H2rii( V ) for i = 1, . .,.*
*l - 1.
Furthermore, if ff 6= 0 2 HNii( V ) for i = 1, . .,.l-2, then x1ff 6= 0 2 HNi+2*
*ri+1( V ),
again because there are no boundaries of length (l - 1). Hence Ni Ni-1+ 2r
for i = 1, . .,.l - 1. This establishes (i) and (ii) with k = l. For (iii), we *
*must show
that Nl-1 M1- 2r + 1. So consider the following portion of the Gysin sequence:
_(j*)l-2_______________________________________________*
*______________________________________________________________@
____________________________________________________________*
*______________________________________________________________@
______________________&&_______________________________________*
*_______________________
(12) H*1( V )__p*_//H*1( W_)@*_//H*l-1( V_)j*//_H*l( V.)
The map (j*)l-2:H*1( V ) ! H*l-1( V ) is simply j* composed with itself (l - 2)-
times. In other words, this map is multiplication by the class of x1 (l - 2)-ti*
*mes.
Note that the diagram as displayed is not commutative. By our primary induction
hypothesis, we have some ff 6= 0 2 HM11( W ). If @*(ff) 6= 0 2 HM1-2r+1l-1( V )*
*, then
Nl-1 M1-2r +1, and (iii) holds for k = l. On the other hand, if @*(ff) = 0, th*
*en
exactness provides some fi 6= 0 2 HM11( V ) with p*(fi) = ff. Then (j*)l-2(fi) *
*6=
0 2 HM1+2r(l-2)l-1( V ), once more because there are no boundaries of length *
*l-1.
Since |(j*)l-2(fi)| = M1 + 2r(l - 2) > M1 - 2r + 1, we have shown (iii) for k =*
* l
under either of the two possible circumstances. This starts our induction.
Now we make the inductive step. Given the secondary inductive hypothesis
for some k l, we will show (i), (ii) and (iii) for i = k. Observe, that in a*
*ny
one Gysin sequence, some terms cannot appear due to the non-consecutive nature
of the indices. We take the following two versions of the Gysin sequence, each *
*of
8 GREGORY LUPTON
which features H*k( V ):
j*
________))_________________________*
*______________________________________________________________@
H*k-1-(l-2)( V ) H*k-1( V ) H*k( V )
_88____ r99
________|_ rrrr ||
__________ | * rrrr |
____ p*|| @rrrr p*|| _;;___
| rrrr | ________
| rrr | ______
fflffl|rr fflffl|_____
H*k-1-(l-2)( W ) H*k( W )
and
j*
________))___________________________*
*______________________________________________________________@
H*k-(l-2)( V ) H*k( V ) H*k+1( V )
__88____ tt:: |
_______| ttt |
___________| * tttt |
___ p*|| @tttt p*|| _::____
| ttt | ________
| ttt | _______
fflffl|tt fflffl|_____
H*k-(l-2)( W ) H*k+1( W )
We splice these together at the H*k( V ) term, then add the map (j*)l-2 = j* O
(j*)l-3:H*k-(l-2)( V ) ! H*k( V ), as we did in (12) when starting this inducti*
*on.
This gives the following diagram, that contains the maps to which we refer in o*
*ur
argument.
__j*_________________________________________________________*
*______________________________________________________________@
________________________________________________________________*
*______________________________________________________________@
_______________##_________________________________________________*
*______________________________________________________________@
. .H.*k-1-(l-2)( V ) H*k-(l-2)( V ) H*k-1( V ) H*( V ) . . .
kk55 _ mk66m___
kkkk ____ mmmm ___
p*|| p* ||kkkkk p*______mmmm p*_____
| kk|kk ___mmmm ____
| kkkkk mmmm___ ____
| @*kkkk | @*mmmm ____ ____
| kkkk | mmmm ____ ___
fflffl|kkk fflffl|mm fflffl______ fflffl______
H*k-1-(l-2)( W ) H*k-(l-2)( W )
We note again that this is not a commutative diagram.
Our primary inductive hypothesis gives some ff 6= 0 2 HMk-(l-2)k-(l-2)( W ).*
* If
@*(ff) 6= 0 2 HMk-(l-2)-2r+1k( V ), then Nk Mk-(l-2)- 2r + 1, and (iii) holds
for i = k. On the other hand, if @*(ff) = 0, then exactness provides some
fi 6= 0 2 HMk-(l-2)k-(l-2)( V ) with p*(fi) = ff. We claim that (j*)l-2(fi) 6=*
* 0 2
HMk-(l-2)+2r(l-2)k( V ). This claim follows from a combination of exactness and
degree considerations. For the case l = 2, we interpret (j*)0(fi) as fi. For l *
* 3,
Consider the sequence of elements {fi, j*(fi), . .,.(j*)t(fi), . .,.(j*)l-2(fi)*
*}. By ex-
actness, we have
* * *
ker(j ):Hk-(l-2)+(t-1)( V ) ! Hk-(l-2)+t( V )
* * *
= im (@ ): Hk-2(l-2)+(t-1)( W ) ! Hk-(l-2)+(t-1)( V )
THE RATIONAL TOOMER INVARIANT AND CERTAIN ELLIPTIC SPACES 9
for t = 1, . .,.l - 2. Now @*(H*k-2(l-2)+(t-1)( W )) is zero in upper degrees a*
*bove
degree Mk-2(l-2)+(t-1)- 2r + 1. Since |(j*)t-1(fi)| = Mk-(l-2)+ 2r(t - 1), and
since our primary induction hypothesis gives
Mk-(l-2) Mk-2(l-2)+(t-1)+ (l - 2) - (t - 1) |x2|,
it follows that (j*)t-1(fi) is not in ker(j*) for t = 1, . .,.l - 2. From the c*
*laim, we
have Nk Mk-(l-2)+ 2r(l - 2) > Mk-(l-2)- 2r + 1. Under either of the two
possible circumstances, therefore, we have shown (i) and (iii) for i = k.
We complete the inductive step by showing (ii) for i = k. Our secondary
induction hypothesis provides an element fl 6= 0 2 HNk-1k-1( V ). If j*(fl) 6=*
* 0 2
HNk-1+2rk( V ), then Nk Nk-1+ 2r, and (ii) holds for i = k. On the other hand,
if j*(fl) = 0, then exactness implies fl = @*(ffi), for some ffi 6= 0 2 HNk-1+2*
*r-1k-1-(l-2)( W ).
This implies Mk-1-(l-2) Nk-1+2r-1 which, combined with (iii) of the secondary
induction hypothesis gives Mk-1-(l-2)= Nk-1 + 2r - 1. Combining this equality
with (iii) for i = k, which we just established, and the primary induction hypo*
*thesis,
we have
Nk Mk-(l-2)- 2r + 1
Mk-(l-2)+ |x2| - 2r + 1
= Nk-1 + |x2| Nk-1 + |x1|.
This establishes (ii) in the case i = k and hence completes our secondary induc*
*tion
step.
To complete the primary induction step in Case II, it only remains to show
Ne = Ne-1+ |x1|. To see this, observe that H**( V ) is a bigraded Poincar'e dua*
*lity
algebra that satisfies, in particular, n1 = N - Ne-1. Clearly n1 = |x1| and so
N = Ne-1+ |x1|. This completes the (primary) induction step in Case II.
In both Case I and Case II the induction has been completed and this proves
the result.
Remark 2.3. In the above, we referred to (4) as a Wang sequence and (11) as
a Gysin sequence. In Case I of the proof, we have a fibration sequence of minim*
*al
models (x1) ____//_ V____//_ W. Topologically, this corresponds to a fibration
sequence of spaces F ! E ! S2r+1, with base an odd-dimensional sphere. The
Wang sequence of this fibration, in the usual sense [Whi78 , p.319], correspon*
*ds to
(4). In Case II, our fibration sequence of minimal models corresponds topologic*
*ally
to a fibration sequence of spaces F ! E ! K(Q, 2r). Going one step back in the
fibre sequence gives a fibration sequence K(Q, 2r-1) ! F ! E. Since K(Q, 2r-1)
and S2r-1 have the same rational homotopy type, we end up rationally with a
fibration sequence S2r-1 ! F ! E, with fibre an odd-dimensional sphere. The
Gysin sequence of this fibration [Whi78 , p.357], corresponds to (11).
Remark 2.4. In the introduction, we mentioned that the formula in part (A)
of Theorem 2.2 can be adapted into a lower bound for the rational category of a*
*ny
elliptic space. Specifically, this goes as follows: We say V has differential *
*of length
at least l if d: V ! lV . Then with e as in the statement of Theorem 2.2, the
inequality cat0( V ) = e0( V ) e holds for any elliptic space. Once again, th*
*is is
Corollary 3 of [GJ01 ]. The proof of Theorem 2.2 is easily adapted to establish*
* this
inequality. One simply omits all reference to parts (B) and (C) of the proof, a*
*nd
uses the Wang and Gysin sequences in their ordinary, i.e., not bigraded, versio*
*ns
10 GREGORY LUPTON
in precisely the same way as they were used above. Likewise, the proof of part *
*(A)
of Theorem 2.2 can be given independently of the proof of parts (B) and (C). It
is interesting to note, however, that the proofs of these latter two parts cann*
*ot be
separated from each other.
For our last result, we add the hypothesis that ker(d: V odd! V ) is non-
zero to those of Theorem 2.2. This extra hypothesis admits a simple topological
translation. Namely, that the rational Hurewicz homomorphism h: ß*(X) Q !
H*(X; Q) is non-zero in some odd degree.
Theorem 2.5. Suppose V is an elliptic minimal algebra with homogeneous-
length l differential and ker(d: V odd! V ) non-zero. Then dim H*k( V ) 2
for each k = 1, . .,.e - 1, where e = cat0( V ) = e0( V ) is given by the formu*
*la of
Theorem 2.2.
Proof. Suppose u 2 V oddis a cocycle. Then we can write V = (x1, . .,.xn)
with x1 = u. Now argue exactly as in case I of the proof of Theorem 2.2. Observe
that our quotient W = (x2, . .,.xn; ~d) in the present result satisfies the h*
*ypothe-
ses of Theorem 2.2. Therefore, the part of the argument featuring the short exa*
*ct
sequence (6), with the inequalities (9) and (10), used there to establish the i*
*nductive
step, can be used here to conclude our result.
Corollary 2.6. Let X be an elliptic space whose minimal model has a homo-
geneous-length differential and whose rational Hurewicz homomorphism is non-zero
in some odd degree. Then dim H*(X; Q) 2 cat0(X) = 2 e0(X).
Proof. Let V denote the minimal model of X. From Theorem 2.5, we obtain
dim H*( V ) 2 cat0( V ) = 2 e0( V ). The statements about X follow from the
standard translation from minimal models to spaces.
3. Examples and Comments
There are a number of special cases of our main results, which either retrie*
*ve
well-known results or provide interesting examples.
Example 3.1. Consider the case in which l = 2. Part (A) of Theorem 2.2
specializes to retrieve part of [FH82 , Prop.10.6], where it is shown that e0(*
* V ) =
dimV oddfor V elliptic and coformal.
Example 3.2. Consider the case in which V = (x1, . .,.xn) is a minimal
algebra with |xi| = 1 for each i. This type of example arises as the minimal mo*
*del
of a nilmanifold [Opr92 ]. Although we are primarily interested in the simply
connected case, the results proved here carry though verbatim for this nilmanif*
*old
case. Here e0(x) = i if and only if x 6= 0 2 Hi( V ). Thus the Toomer invariant*
* of
a cohomology class is identified with the degree and an integer i is realized a*
*s the
Toomer invariant of some class if and only bi 6= 0, where bi denotes the i'th B*
*etti
number of V , or of the nilmanifold of which V is the minimal model.
For degree reasons, the differential here must be homogeneous of length 2.
Therefore Theorem 2.5 and Corollary 2.6 specialize to yield the following well-
known result, which is essentially due to Dixmier [Dix55 ]:
Proposition 3.3. A nilmanifold X of dimension n has bi 2 for 1 i n-1
and hence dim H*(X; Q) 2 dim(X).
THE RATIONAL TOOMER INVARIANT AND CERTAIN ELLIPTIC SPACES 11
The conclusion of Theorem 2.5 obviously holds in many cases besides those
covered by the hypotheses. We cannot resist making the following conjecture:
Conjecture 3.4. Let V be elliptic with homogeneous-length differential. Ei-
ther dim H*k( V ) 2 for each k = 1, . .,.e - 1, where e = cat0( V ) = e0( V *
*) is
given by the formula of Theorem 2.2, or H*( V ) is a truncated polynomial algeb*
*ra
on a single generator.
In view of Theorem 2.5, one need only consider the case in which ker(d: V !
V ) is concentrated in even degrees. We believe that an argument as in case II*
* of
the proof of Theorem 2.2, together with a careful analysis of the exceptional c*
*ases,
will establish Conjecture 3.4 at least in the coformal case, l = 2. We have be*
*en
unable to prove the general case, however, using this approach.
It is also clear that in many cases to which Theorem 2.5 applies, and others
to which it does not, the inequalities dim H*k( V ) 2 for each k = 1, . .,.e*
* - 1
and dim H*(X; Q) 2 cat0(X) are by no means sharp. The following examples
show that, nonetheless, these inequalities are best possible for a general resu*
*lt of
this nature.
Example 3.5. Let Xl = CP l-1x S2r+1 for l 2. Then Xl is elliptic and
has minimal model with homogeneous-length l differential. As is well known,
cat0(Xl) = e0(Xl) = l + 1. On the other hand, we see that dim H*k(X; Q) = 2 for
k = 1, . .,.l. Thus we have dim H*(Xl; Q) = 2 cat0(Xl).
Example 3.6. Let V = (x1, x2, y1, y2, y3) with |x1| = |x2| = 2, |y1| =
|y2| = |y3| = 3 and differential d(x1) = 0 = d(x2), d(y1) = x21, d(y2) = x1x2
and d(y3) = x22. Then V is elliptic and coformal. As is easily checked, we have
cat0( V ) = e0( V ) = 3, and dim H*k( V ) = 2 for k = 1, 2. In particular, we
have dim H*( V ) = 2 cat0( V ).
It would be interesting to know whether there are other examples in which the
inequalities of Corollary 2.6 and Theorem 2.5 are sharp, or whether these examp*
*les
are essentially the only such. In particular, it would be interesting to find g*
*eneral
conditions under which the inequality of Corollary 2.6 could be (substantially)
strengthened.
Finally, we remark that the original motivation for this work came from a
question of Yves F'elix, as to whether there exists any space with "e0-gaps" in*
* its
cohomology. Precisely, we say that a space X has an "e0-gap" in its cohomology
if H*(X, Q) has an element whose Toomer invariant is k, but does not have any
element whose Toomer invariant is k - 1. A recent example due to Kahl and
Vandembroucq [KV01 ] shows that e0-gaps can occur in the cohomology of a finite
complex. Their example is actually a Poincar'e duality space, but it is hyperbo*
*lic
and not elliptic. This leaves the following question:
Question 3.7. Can an elliptic space have e0-gaps in its cohomology?
If it is not possible for an elliptic space to have e0-gaps in its cohomolog*
*y, then
it would seem reasonable to extend Conjecture 3.4 to the general elliptic space.
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Department of Mathematics, Cleveland State University, Cleveland OH 44115
E-mail address: Lupton@math.csuohio.edu