Manifolds with wells of negative curvature

Invent. math. 103,471-495 (1991) l~verl tio~e$ mathematicae �9 Springer-Verlag 1991

Manifolds with wells of negative curvature* with an Appendix by Daniel Ruberman

K.D. Elworthy ~ and Steven Rosenberg 2 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 2 Mathematics Department, Boston University, Boston, MA 02215, USA

Oblatum 25-XI-1989 & 15-11-1990

Introduction

Bochner's Theorem for one forms states that a compact Riemannian m-manifold M with positive Ricci curvature has H 1 ( M ; R) = 0. Similarly HP(M; R) = 0 provided R p, the curvature term in the Weitzenb6ck formula for the Laplacian on p-forms, is positive. Myers improved Bochner's result by showing that positive Ricci curvature implies nl (M) is finite. This paper extends Bochner's Theorem, its analogues for p- forms, and Myers ' Theorem to metrics which have positive curvature on "most" of M. In addition, we find topological restrictions on the universal cover of M if R p is mostly positive. These obstructions are new even in the case R p strictly positive.

The results for Ricci curvature can be summarized as follows. Let M have a volume one metric of positive Ricci curvature except on a set of very small volume. If the well of negative curvature is allowed to be arbitrarily deep, there are no restrictions on M. If the well does not exceed a certain depth, then H ~(M; R) = 0. If the well is very shallow, g l ( M ) is finite.

In contrast, Wu [W] has recently shown by different techniques that if the well does not exceed a certain depth and the curvature is positive off a set of small diameter then r h (M) is finite. Thus Myers ' Theorem generalizes to deep wells of small diameter and to shallow wells of small volume.

In more detail, w 1 deals with Bochner's Theorem for one forms. Every compact manifold trivially admits a volume one metric with Ricci positive off a set of arbitrarily small diameter, but the curvature of the metric on this small set may tend to minus infinity as the diameter shrinks. If the negative Ricci curvature is bounded as the volume shrinks, the first cohomology group must vanish:

Theorem 1.12. Let JV = Jff(K, D, 1/, m) be the class of Riemannian m-manifolds with Ricci curvature greater than K, diameter less than D, and volume greater than V. For R o > 0 there exists a = a(Jff, Ro) > 0 such that if(M, g)ed/" has Ric(x) > R o

* Support was received from the EEC Stimulation Action Plan, NATO Collaborative Research Grants Programme 0232/87 and from NSF grant INT-8703344.

472 K.D. Elworthy and S. Rosenberg

. . . v o l ( X ) except on a set X wttn ~ less than a, then H i ( M ; R) = 0. In fact, H ~ ( M ' ; R)

= O for every finite cover M ' o f M.

Here Ric(x) > C means that the lowest eigenvalue of the Ricci tensor, considered as an endomorphism of the cotangent space at x ~ M, is at least C. Theorem 1.12 is proven by combining a strong form of semigroup domination (Kato's inequality) with estimates for Schr6dinger operators. Semigroup domination implies that H i ( M ; R) vanishes if the operator A + Ric' is positive, where A is the Laplacian on functions and Ric'(x) is the lowest eigenvalue of Ricci at x. Moreover, the positivity of A + Ric' on M is equivalent to the positivity of A + Ric' pulled back to a finite cover of M. Although the situation is apparently more complicated than usual "Laplacian plus potential" arguments since both the Laplacian and the potential are metric dependent, a general property of Schr6dinger operators finishes the proof:

Proposition 1.2. Let W: M ~ R be continuous and choose B > 0 and b < O. Then there exists a > 0 such that

(i) W > B except on a set X (ii) W >= b on X

(iii) vol(X) < a vol(M) =

implies d + I4 x > O. Moreover a = a(B, b, Jff).

Other approaches to Bochner theorems via semigroup domination are given in [Be], I-BB], [Ma].

w gives extensions of Myers' Theorem both to metrics with a small (i.e. small volume) deep well of negative Ricci curvature and to metrics which have a small shallow well of negative curvature. In contrast to Myers' and Wu's proofs via Jacobi fields, the techniques here depend on the Bochner theorems of Section 1 and deep results of Gromov. For example, using H I ( M ' ; R) = 0 for all finite covers of manifolds with metrics as in Theorem 1.12, we obtain a Myers' type theorem for metrics with small deep wells if the fundamental group is almost solvable:

Theorem 2.1. Assume M e Jff admits a metric g with Ric(x) > R o > 0 except on a set

X with vol(X) - - < a(JF, Ro), with a as in Theorem 1.12. l f n l ( M ) is almost solvable, vol(M)

then nx(M) is finite. In particular, i f nl ( M ) has polynomial growth, it is finite.

Myers' Theorem generalizes completely to manifolds with metrics with small shallow wells of negati,ve curvature. A combination of Gromov's Compactness Theorem, Gromov's theorem on groups of polynomial growth, and estimates of Milnor, Cheeger and Klingenberg shows that the fundamental group of such a manifold is almost nilpotent and hence finite by Theorem 2.1. More precisely, let ~r = ~'(S, D, V, m) be the class of Riemannian m-manifolds with sectional curvatures bounded in absolute value by S, diameter at most D and volume at least V.

Theorem 2.4. Let M e J r and choose R o and a as above. There exists 6 = 6(dr Ro) < 0 such that if (i) Ric(g) > R o except on a set A o f volume less than a

(ii) Ric(g) > t~ on A then ~1 ( M ) is finite.

Manifolds with wells of negative curvature 473

As suggested by M. Micallef, Theorems 2.1 and 2.4 could be strengthened by showing that under the Ricci flow a metric with a small deep well of negative curvature deforms to a metric of positive Ricci curvature, in which case n l ( M ) would be finite. J. Eisenberg and R. Hamilton have informed us of preliminary results of this type for wells of small diameter.

Vanishing theorems for p-forms (p > 1) are discussed in Section 3. Here the curvature term R p in the Weitzenb6ck formula does not control the qualitative geometry of the manifold the way the Ricci curvature does. Nevertheless, by semigroup domination HP(M; R) = 0 provided the operator A + R p' is positive, so Prop. 1.2 still give vanishing results (Theorem 3.1) analogous to Thm 1.12. This positivity is equivalent to a Brownian motion invariant v p being negative. Since Brownian motion on M covers Brownian motion on M, vP(M) = vP(~). By semigroup domination and Hodge theory on ,~/,

(0 .1) R P ( M ) > O = ~ v P ( f f l ) < O = , , I m [ H P ~ ( I V 1 ; R ) ~ H P ( f f l ; R ) ] = O

where H~ is cohomology with compact supports. In [ER I] it was shown that in fact v~< 0 implies H I ( M ; R ) = 0, but for higher forms the situation is more complicated:

Theorem 3.21. Let p be between 2 and m - 2. I f v p < O, and H~_ 1(itS; R) = 0, then Hp(M; R) = 0. In particular, v 2 < 0 implies H2(/~ , R) = 0.

Corollary 3.23. IrE(M ) of a compact manifold M with R 2 > 0 is a torsion group. Moreover, the orders of the elements of n2(M) is bounded.

For the case p = 1 in [ER I], showing H~(At; R) = 0 reduced to taking a line segment i n /~ , extending it to an infinite line leaving every compact subset, and verifying that every element of H I (/t~; R) integrates over the line to give zero. The analysis in the last step carries over to p-forms provided the integration is over an infinite p-cycle with bounded coefficients (Prop. 3.13). However, the topological problem of extending a finite p-chain to an infinite p-cycle with bounded coefficients is no longer trivial. The following partial answer to this problem, which relies on Dodziuk's work on L z homology and cohomology on covering spaces, is the main topological step in the proof of Theorem 3.21:

Theorem 3.12. Let p be between 1 and m - 1. l f v p < 0, then every simplicial p-cycle is the boundary of an infinite (p + 1)-chain with square summable coefficients.

In a different direction, by (0.1) no nonzero cohomology class in HP(/~; R) can have a representative closed form with compact support if v ~ < 0. Using basic properties of L 2 cohomology, it is possible to give a bound on how quickly~a representative form can decay as a function of distance from a fixed point in M:

Proposition 3.5. Let p be between 2 and m - 1. l f v p < 0 on M, then no nonzero class in Hp()~; R) has a representative form which decays faster than the growth of Ir 1 ( M ).

In the appendix, D. Ruberman gives examples which show that the main theorems of this paper and of [ER I] provide new topological obstructions to metrics with wells of negative curvature. As an example for Theorem 3.21, he constructs a compact manifold M with HP(M'; R) = 0 for all finite covers M' of M, Hp_ 1 (A~t; R) = 0, but Hp(AI; R) # 0. Thus M does not admit a metric with v p < 0; in particular, there is no metric with R p > 0. Moreover, this cannot be detected by passing to a finite cover of M and applying the usual Bochner argument.


Most of these results were announced in [ERI I ] ; the results in ~ j l -2 were stated there in terms of wells of narrow diameter and are therefore superseded by both Wu's work and the work in this paper.

Acknowledgements. We are very grateful to M. Gromov for suggesting that there should be extensions of Myers' Theorem, to D. Ruberman for providing the appendix, and to the referee for useful suggestions.

w Vanishing results for H~(M; R)

Let M be a compact connected Riemannian m-manifold. The Laplacian on one- forms has the Weitzenb6ck decomposition A 1 = 17" V + Ric with Vthe Levi-Civita connection and Ric the Ricci curvature considered as a section of End( T * M). We write Ric(x) > R o for x E M if the lowest eigenvalue of Ric(x) is at least R o. If Ric(x) > 0 for all x ~ M , the Weitzenb6ck formula shows that A 1 is a positive operator on L 2 one-forms, so by the Hodge theorem we conclude H i ( M ; R) = 0; this is Bochner's Theorem. If Ric(x) is negative on a set of arbitrarily small diameter or volume, Bochner's Theorem fails. In fact, on any compact manifold M we may make a coordinate chart U look metricly like a sphere minus a disk and shrink any metric on M - U until the diameter and volume are very small. Thus any compact manifold admits volume one metrics with Ricci curvature positive off an arbitrarily small set.

In this example the curvature becomes increasingly negative as the diameter of M - U shrinks. The main result of this section (Theorem 1.12) is that we obtain a vanishing theorem for H t ( M ' ; R) for all finite covers M' of M if we control the volume of the set where Ricci is negative in terms of bounds on Ricci. The region of negative curvature need not have small diameter and in fact need not be connected. Note that if M is two dimensional, the theorem follows immediately from the Gauss-Bonnet theorem, so we will always assume that M is at least three dimensional.

The proof is based on semigroup domination (also called Kato ' s inequality). As in [R], this states that for all t ~ R + and x ~ M

(1.1) l e - t~(co)(x)l ~ m" e - tt~ + Ric')lcol(x) .

Here A is the Laplacian on functions, Ric '(x) is the lowest eigenvalue of Ricci at x, and the norm of the one-form co is computed with respect to the metric. If A + Ric' is a positive operator on L2(M), the right hand side of (1.1) decays to zero as t - * 0o, forcing the left hand side to also decay. On the other hand, if co is a harmonic one-form the left hand side is constant in t. Thus A + Ric' > 0 implies H i ( M ; R ) = 0; a further argument (Prop. 1.8) shows H I ( M ' ; R ) = 0 for finite covers M'.

The following result is a general criterion for the positivity of a Schr/Sdinger operator with a potential having a deep well on a set of small volume.

Let ]B] denote the volume of a set B and let a = a ( x , y , z . . . . ) denote a constant which depends only on the quantities x, y, z , . . . .

Proposition 1.2. Let W: M --* R be continuous and choose B > 0 and b < O. Then there exists a ~ R + such that

(i) W > B except on a set X


(ii) W > b on X

(iii) IX[ < a

implies A + W > O. Moreover a = a(B, b, K, D, V, m), where K is a lower bound for the Ricci curvature of M, D is a upper bound for the diameter, V is a lower bound for the volume, and m is the dimension of M.

The proof depends on a simple modification of Li's L ~ estimate for eigenfunc- tions of the Laplacian I-L].

Lemma 1.3. Let W be a continuous nonconstant function on M, and let f be an eigenfunction of A + W with eigenvalue 2. Then there is a dimension constant 6(m) and a constant Q = Q(K, D, V, m) such that

m

i, f l l 2 < 6 ( m ) . ( ~ . - ~ m i . ) Z . e x p [ 6_(re)Q_ ] . l l f l l2

2]M1~(2- Wmin) Moreover, if 2 is the lowest eigenvalue of A + V, then

rn

1 :22 / t-2[Mli( Wma~- Wmln)

Proof o f 1.3. For ~ > �89 we have

(~. - Wmin)llfll~ -->-- j'(;~ -- w ) s f l 2~ = j l f l 2 ~ - 2 . f . ~ f

= ~ V(l f]2~-2f) �9 V f = (2~t - 1)~1f12~-21Vfl 2

2 ~ - 1 (1.5) - ~2 ~[Vlfl=l 2

:)" Let ~ = 2 ) ~ , where 90 is the Sobolev constant

~ = s u p { C r C infs~l~ [If-sl '"~'- '-< I[ Vf I I "~ 'V feC~(M)} "

For any function f i n the Sobolev space H~(M), 2

(1.6) I] Vf [[ 2 > 6(m)C~ [ ll f l[ 2~,~ - [Ml-~llf l l 2]

[L. Lemma 2]. The Sobolev constant is bounded below by the isoperimetric constant

(volta _ ~ N)m J ( M )

i~f ( m i n - { volta(Mr ), volta(M2)}) ~-1

where N ranges over the compact (m - 1)-dimensional submanifolds of M which divide M into open submanifolds M~, M 2 [Cha, p. 110]. By Croke's work [Cr], J (M) > Q(K, D, V, m). Thus we may replace g by Q in (1.6). Applying (1.6) to I f l ~ and using (1.5) yields

(1.7) 6(m)" Q-~(2 - W~,) ~ + IMI-~ tlft1221 > Ilflt _

476 K.D. E l w o r t h y a n d S. R o s e n b e r g

m Set fl - m - 2 and let ~ = flk for k = 0, 1, 2 . . . . . (1.7) becomes

1

6(m) Q - l I M I t ( 2 - Wmin) 2flk-----~-(+ 1 ]]fll2a~[MI-2-~

1

(1.8) > Ilfltzt~+llMI 2P ~+' 1

Multiplying (1.8) over k and using lim tlfrlplMl-~ = tlfll~, we get p ~ c O

1

,~(m) Q-~IMIZ()~- W., i . )Z/~k~ + 1 Ilfll~lM1-1 > IIf}l~ k = 0

F o r f l > l a n d c l > 0 1

k = O

[L, Appendix], from which the first part of the lemma follows. For the second statement in the lemma, we replace (1.5) by

2ct - 1 (W,.a~- W,.i.)llfll~, > ~2 ~lVl/ l~l 2

since the variational formula for the lowest eigenvalue shows that 2 < Wm,~- We now proceed as before. []

By I-L] and [Cr] 6(m) and Q can be explicitly estimated in terms of their parameters.

Proof of 1.2. L e t f b e a ground state of norm one for A + W; i.e. (A + W)f= 2f where 2 is the lowest eigenvalue of A + W. Since A + W + C > 0 for C >> 0, we may use the maximum principle for A + W + C to conclude t h a t f m a y be chosen positive, as is well known. Recall that A + W > 0 if and only if ~M Wf> O, since ~MWf=~M(A+ W)f=2~Mf Also, since W > P implies A + W > A + P , we may assume that W = B except on X.

Denote the right hand side of (1.4) by y2. Note that

1 = ~MT z < I l f J l ~ m f < Y~uf Pick e > 0, and set B, = {x~M:f< ~}. Then

Y-~ <- i f + ~ f<= e)B~l + Y(IM)- IBm)) B~ M - B,

so for fixed e < min{(YlM[) -1, Y}

YIMI- y-1 IB, l_-<

Y - e

We have 1 { f > e, W > 0} I > I MI - [B~ I - IX [, and writing ~ Wfas an integral over this set, the set X, and the rest of M we get

(1.9) ~ Wf> b" Y'lXl + (IMI - In~l - IXt)ne M


Thus the right hand side of (1.9) will be positive if

for some 5.

By 1

l -IB~IIMI -* > Z

IXI IMI

IXI 1 - IB~IIMI -~ - - < IMI b Y

1 - - - Be

Y Set e = k for some k to be determined. By the estimate for IB~I above,

y - 1 _ IMle k Y -1 1 1-IB~IIMI - l > Y [ M l _ l M l e ( k - 1 ) I M I Y k - 1

a standard volume comparison theorem, IMIy2 < Z ( B , b , K , D , V,m), so 1 1

k - 1 > ~ if k > 2 Z + l . In other words, whenever

1 - - <

2Z (1 b(2Z + 2) . ) k

the right hand side of (1.9) is positive. []

= a (B ,b ,K ,D , ~ m)

Remark. Let 2(q) be the highest eigenvalue of - (A + q. Ric') for q > 0. The graph of 2(q) as a function of q is convex [A] [ER I] and 2(0) = 0, so A + Ric' will

if d2(q) certainly be positive d2 q = 1 < 0. By standard perturbation techniques [Ka,

VII w eqns. 2.1, 2.21, 2.32], this condition is equivalent to ~ ~M Ric ' f 2 > 0, where {f~} is an orthonormal basis of the ground states. This means that the hypothesis in Proposition 1.2 actually forces A + q. Ric' > 0 for an open interval of q containing (0, 1].

Let JV = ~ ( K , D, V, m) be the collection of Riemannian m-manifolds with Ricci curvature bounded below by K, diameter bounded above by D, and volume bounded below by V.

Theorem 1.10. Choose R o > O. There exists a = a(JV, Ro) such that a manifold

M ~ JV with I {x: Ric'(x) < R o } I < a has A + Ric' > 0. In particular, such a manifold IMI

has H i ( M ; R) = 0.

Proof This follows from Prop. 1.2 by setting W = Ric'. []

Remark. There is a shorter proof that Ha(M; R) = 0. Namely, let co be a non-zero harmonic one-form of norm one. Then

0 = (co, ACO> = ( V'co, V'co> + (Ric co, co>

> (Ric co, co) = ~ (Ric co, co)x dvol(x) M

Diagonalizing Ricci with respect to an orthonormal frame of one-forms { 0 i} at x and writing co = aiO ~, we see that (Ric co, co)~ > Ric'(x)lcol~, so we will get a contradiction if ~ Ric'(x)lco I~ > 0. Using Li's estimate for harmonic one-forms [L]


and [Cr] as before, we get m ro,o, l

D(m) ~ "exPLiM[~ g

Integrating Ric'log[ 2 over {Ric '< 0}, {1~ol 2 >= ~, Ric' > Ro}, and the rest of M leads as in (1.5) to SM Ric'l~ > 0.

The next results justify our effort to force A + Ric' > 0 as opposed to just proving Hi(M; R) = 0, since in general Hi(M; R) = 0 does not imply HI(M'; R) = 0 for a finite cover M' of M.

Proposition 1.11. Let (M, g) be a compact Riemannian manifold and let (M', g') be a finite cover of M with g' the pullback of g. Then A + Ric' > O for (M, g) if and only if A + R i c ' > O for (M', g').

Proofl. Let 2 and 2' denote the bottom of the spectrum for A + Ric' on M and M' with corresponding eigenfunct ionsfandf ' , respectively. Pul l ingfback to M' shows 2' < 2. As in the proof of Prop. 1.2f' may be chosen positive. We now average f ' over the automorphisms of each fiber of the cover to produce a nonzero eigenfunction of A + Ric' on M with eigenvalue 2'. Thus 2 < 2'.

ProoflI. Let E be expectation with respect to the Wiener measure for Brownian motion x~ starting at some fixed point x o ~ M, and let

v(M) = limt_.~ o t -1 In E[exp( - So Ric'(x~)ds)]. As in [A], - v(M) is the lowest eigenvalue of A + Ric' on M. Since Brownian motion on M' starting at some lift of Xo covers the Brownian motion on M, v(M) = v(M'). []

From Theorem 1.10 and the last Proposition we immediately obtain the main result:

Theorem 1.12. Choose R o > 0.

M~Jr with I{x: Ric'(x) < Ro} [ IMI

of M. []

There exists a = a(JV', Ro) such that a manifold

< a has HI(M'; R ) = Of or every finite cover M'

In contrast, the estimates in the remark above do not appear to behave well under passage to a finite cover.

w Results for nl(M),

In this section we give extensions of Myers' theorem that a compact manifold with positive Ricci curvature has finite fundamental group. The techniques differ for metrics with "deep wells" of negative curvature (i.e. metrics treated by Theorem 1.12) and metrics with "shallow wells" (see Theorem 2.4).

For metrics with deep wells of negative curvature, the crucial fact is that Theorem 1.12 gives the vanishing of H 1 for all finite covers. In contrast, Myers' proof and Wu's extension for metrics with deep wells of narrow diameter work directly with Jacobi fields on the manifold to prove nl finite and thus conclude H 1 = 0 .


Recall that a group is called almost solvable if it contains a solvable subgroup of finite index. In particular, a group of polynomial growth is almost solvable, since in fact a group has polynomial growth if and only if it is almost nilpotent [G 13.

Theorem 2.1. Let M be a compact manifold with HI (M' ; R ) = O f or every fn i te cover M' of M. I f nl ( M ) is almost solvable, then nl ( M ) is finite. In particular, if M

admits a metric with I{x: Ric'(x) < Ro} I < a( J/', Ro ), with a as in Theorem 1.12, and IM[

n l (M) is almost solvable, then nl( M ) is finite.

Proof. Let S be the solvable subgroup of finite index in nl. Associated to the descending central series for S, 1 = Sk+ l c S~ c Sk-~ c . . . c $1 ~ So = S ~ nl, where $1§ 1 = IS j, S j3, we have the tower of coverings l~--.~ Mk-,, Mk_ 1 - , , . . . - - * M o ~ M .

M o is a finite cover of M, so by hypothesis HI(Mo; R) = 0. By two applications of the universal coefficient theorem, it follows that H 1 (Mo; Z) = 0 and H 1 (Mo; Z) is a finitely generated torsion group; in particular, H~(Mo; Z) is finite. However,

g l ( M o ) So HI(Mo; Z) = [rq(Mo), rq(Mo)] = ST"

Thus the index o f S 1 in So = S is finite, so M 1 is a finite cover of Mo. Proceeding by induction, we conclude that the universal cover/~/of M is a finite cover of M. []

We now consider metrics with shallow wells of negative curvature. Recall that the growth function v(r) of n l is defined by choosing a finite set of generators for n~ and setting v(r) to be the number of distinct words in the generators and their inverses of length at most r. Standard estimates for the growth function show that if a manifold with infinite It~ admits a shallow well metric, the metric must be very distorted, in the sense that either the injectivity radius is very small at each point, or a generator o f g 1 has very long geodesic length. To state the result precisely, we fix a point x o of M.

Lemma 2.2. Suppose zq (M, Xo ) is infinite. For a set of generators G = {~1 . . . . . 7t} for nl(M, Xo) and for positive numbers, l, p and Ro, there exist 6 = 5(Ro, m, G, l, p, M) < 0 and a = a(sff, Ro) > 0 such that i f (M, g ) e s F satisfies

(i) some point of M has injectivity radius larger than p (ii) the shortest geodesic in Vi has length less than l for each i

(iii) Ric(g) > R o except on a set of volume less than a, then Ric(g) < 5 somewhere on M.

Proof. If no such 6 < 0 exists, choose a sequence of fii T 0 and metrics gi satisfying (i)-(iii) and Ric(gi) > 6i. Let xi be the point in M at which (i) holds for g... Since the growth function is unchanged if we relace G by ~, GV- ~ for a fixed path ), from x~ to xo, we may assume that xi = Xo. Let ~o be a lift of Xo to ~ , the universal cover of M. Let F~ be the Dirichlet fundamental domain for M in M-i.e. F~ is the closure of the set of points in M closer to ~o than to any translate of Xo under a deck transformation. By (i) the ball of radius p around ~o (with respect to g~) is contained in F~, and by (ii) the gcdistance from ~o to Vj. ~o is bounded above for all Vje G and for all i. By IMi] (cf. [G I, p. 723) the growth function of n~ with respect to G satisfies

(2.3) v(r) < C1' exp (CEx / - 6i r)


for some constants C 1, C 2 > 0 and all i. (In fact, we may take C 1 = 4ml"p -m and

C2 = 2mx/m - 1 l.) For fixed N e Z § the right hand side of(2.3) is bounded by C1 .r for r = 1, 2 . . . . . N for i >> 0. Taking N >> 0, we may conclude by [ G I , p. 71] that ~t~ contains a nilpotent subgroup of finite index.

Choose a > 0 as in Theorem 1.12. Then HI(M' ; R) = 0 for all finite covers M' of M. By the argument in Theorem 2.1, rtt must be finite. []

By another theorem of Gromov, Myers ' Theorem now generalizes to manifolds with shallow wells of negative curvature. Let ~ = J / (S , D, V, m ) c JV( - (m - 1)S, D, V, m) denote the set of Riemannian m-manifolds with sectional curvatures bounded in absolute value by S, volume at least V, and diameter at most D. Pick a positive number R o < (m - 1)S and choose a = a(R o, JV( - (m - 1)S, D, V, m)) such that Theorem 1.12 applies. There are only a finite number of diffeomorphism types in J r [Che], so we may assume that the generators for the possible fundamental groups have been chosen.

Theorem 2.4. Let (M, g) ~ ~[I and choose R o and a as above. There exists = ~ ( ~ , Ro) < 0 such that if

(i) Ric(g) > R o except on a set A of volume less than a (ii) Ric(g) > ~ on A then 7z 1 (M) is finite.

Proof. By [Che] and [Kli], the injectivity radius of M is bounded below by p = p(J / ) . The space of Riemannian manifolds with an upper bound on diameter, a lower bound on injectivity radius, and upper and lower bounds on sectional curvature is compact in Gromov ' s Lipschitz distance [G II, Thm. 8.28], [GW]. In particular, there exists C = C(dt ' ) such that any two metrics on M with these bounds are C-quasi-isometric. This immediately gives an upper bound for the length of the shortest geodesic for each generator of ~ (M). Choosing 6 < 0 greater than the 6 of Lemma 2.2, we see that 7t~(M) must be finite. []

w Results for p-forms

The curvature term R P e E n d ( A P T * M ) in the Weitzenb6ck formula A p = V*V + R p for the Laplacian on p-forms is more complicated than R 1 = Ric, but the

techniques of w 1 still give vanishing theorems for HP(M; R). Of course, HP(M; R) = 0 if R p is positive. More generally, if we define RP'(x) to be the lowest eigen-

value Of R p at x e M , then as in w H P ( M , R ) = 0 whenever v p = limt_.~t -1 l nE[exp ( - So RP'(xs)ds)] < 0. Moreover, v p < 0 if and only ifA + R p' > 0 and v ~ is the same on all finite covers of M.

In this section we will give topological obstructions to a manifold admitting a metric with v p < 0 and in particular to metrics with R p > 0. First of all, applying Proposition 1.2 to V = R p' gives the p-form analogue of Theorem 1.12:

Theorem 3.1. Let Jff = Y ( m , K, D, V) be as in w and choose Ro > O. There exists

a = a (Jq ' ,Ro)> 0 such that a manifold M e Y with I{x: RP' (x )< Ro}l< a has IMI

A + R p" > O. In particular, HP(M'; R) = O for everyfinite cover M ' of M. []

The theorem applies if, for example, the curvature operator ~ is negative on a set of small volume. Since R P ( x ) > p ( m - p ) ~ t ( x ) [ G M ] , there exists


a = a(Jff, R0) such that a manifold with I{x: 9~(x) < Ro}l < a is a real homology IMI

sphere (cf. Cor. 3.23). There are other obstructions to v ~ < 0 on the universal cover M of M. Since

these obstructions for M compact are immediate consequences of the invariance of v p under finite covers, from now on we will assume that ~zl(M) is infinite.

We briefly recall the known obstruction on M. As in the second proof of Proposition 1.11, vP(M) = vP(M) with the pullback metric. By semigroup domination applied to d P, v p < 0 implies exponential decay of the heat operator applied to a L 2 p-form on /~ , which in turn implies the vanishing of octe P()~), the space o f L 2 harmonic p-forms on /~ . An application of the L 2 Hodge Theorem of Kodaira-de Rham as in [Y], [ER I] gives

(3,2) augP(M) = 0 ~ Im[H~(M; R) ~ HP(M; R)] = 0

(where M can be replaced by any complete manifold). H e reH ~ denotes cohomology with compact support. Thus no nonzero class in H e ( M ) has a representative closed differential form of compact support.

An application of Poincar6 duality yields a little more information. Let HII denote locally finite homology. For any manifold N the Poincar6 duality isomorphism PD: H ' - P ( N ; R) ~ HiS(N; R) is characterized by the equation

(3.3) f 6b=S~ b ^ z PD[z] N

for all q~ with [~b]~H~(N;R). (The same equation characterizes the usual Poincar~ duality PD: H ~ - r ( N ; R ) ~ H p ( N ; R ) where q~ is now an arbitrary closed p-form.) The right hand side of (3.3) gives a nondegenerate pairing H~ x H m - P ~ R , which yields the isomorphism H~(N; R ) ~ H" -P (N; R). By Poincar6 duality the left hand side of (3.3) must also be a nondegenerate pairing f: HI (N; R) x HiI(N; R) --* R. To sum up, we have the isomorphisms

PD I PD (3.4) H " P(N; R) , HiS(N; R) , H f ( N ; R) , H,,_p(N; R) .

Of course, the first and last terms of (3.4) are also isomorphic by a universal coefficient theorem.

We now let N = / Q and assume v p < 0. Since RP. = *R "-p, where �9 is the Hodge star operator, v p = v m-p. Thus v p < 0 implies that the top map of the commutative diagram

H~- ~(~') --, H'~- ~(~)

,L PO l eo

is zero by~(3.2), so the bottom map is also zero. In other words , / fv p < 0 every p- cycle in M is the boundary of a Iocally finite (p + 1)-chain.

For p = l , v = v l < 0 in fact implies H ~ ( M ; R ) = 0 in (3.2) and hence //~(J~; Z) = 0 [ER I]. In particular, a compact 3-manifold with infinite ~z 1 and admitting a metric with v < 0 must be a K(r~, 1) (cf, Cor. 3.23). For p > 1 and manifolds of higher dimensions, we consider the following question:

Question. Ooe~ v p < 0 imply H~(A4; R) = 0?


By (3.4) and the equality v p = v m-p, an affirmative answer would also give H . ( M ; R) = H ~ ( M ; R) = 0, with �9 equal p or m - p.

The analysis that gave the affirmative answer in the p = 1 case carries over virtually unchanged to the case of general p, while the topology becomes much more complicated. Our results for p between 2 and m - 1 fall into two types:

(i) If v r < 0, any p-cycle in /~ is the boundary of an infinite chain with square summable coefficients (Theorem 3.12), and any non-zero class in HP(M; R) has no representative closed form which decays faster than the growth of 7tl(M ) (Prop�9 3.5).

(ii) If v p < 0 and H.,_p_ 1()0; R) = 0, then H~(M; R) = 0 (Thm. 3.21-3.22).

We begin by defining various L 2 homology and cohomology groups. Let N be an oriented connected manifold with a fixed smooth triangulation 2: = {ai} with the following property: there exists an integer K such that each p-simplex in the triangulationbelongs to at most K (p + 1)-simplices. This property can always be satisfied on M by lifting a triangulation of M.

Definitions and Notation. (i) Denote the set of p-simplices in Z by Sp. The set of l ~ p-chains (with respect to S,) is given by vpc {2)= {tr = ~ ~ 1 niai:aieSp, ni~R, o~pn~ < ~ }. (We assume ai + aj for i ~ej.) By the assumption on S, the boundary

erator satisfies ~: C~ 2) ~ C~2_ ) 1, so we may define the sets of 12 p-cycles Ztp 2) and 12 Z ( 2 )

p-boundaries B~ 2~. The pth l 2 homology group is/_/(2) = --p (~(2) is complete with - - p B (p2) �9 - - p

respect to the usual 12 inner product and t3 is a bounded operator with respect to t h e 12 norm I I./4t2) has the seminorm 7(2) (~(2). ,7(2) �9 ._p induced from ~p = _p . for ze~p , I[z]l = inf{ Iz']: z 'e [z]}.

(ii) The 12 p-cochains are defined to be C~'2)= {continuous homomorphisms 4~: C~ 2 )~ R}. Set 6: CI'2)---, CtS-~ x by (6~b)(a) = ~b(0a). The 12 p-cocycles and 12 p-

p p coboundaries are denoted by Z~2).~mp and Bt2~.~mp, respectively, and the p,h p

p Z ( 2 ) , s i m p simplicial 12 cohomoloyy 9roup i s H ( 2 ) , s i m p - p . B ( 2 ) , s i m p

p Let -p P - p Z (2), s imp B(2),simp denote the closure-of the se t B(2),simp. Set H(2),simp - p

B ( 2 ) , s i m p (iii) The compactly supported smooth forms on N have the Hodge inner

product and L 2 n o r m II �9 IIo once a complete metric is chosen. For nonnegative even integers k, the Sobolev space A p,k is the completion of the smooth L 2 p-forms with

k respect to the norm II~Ollk -- I[(I + A P)2-(-~ Ho �9 The closed and exact p-forms in A p'k are Z~fi~,dR = {~osAP'k: d~o = 0} and B~j~,dR = {dO: OeA p-Lk, dOeAp'k}, respect-

p,k Z~2~'dR When k = 0 we get the usual L 2 de Rham cohomol- ively. We s e t Ht2),dR -- B ~ , a R "

P ogy group, denoted Ht2),dR. Of course these homology and cohomology groups are of interest only if the

manifold N is noncompact.

It is known that v p < 0 implies that the infimum of the spectrum of A p on L 2 p-forms on N is positive (see e.g. [ER I, Prop. 4B]), and so 0 = o~r p - H~i),dg [Don~ w This implies that a de Rham representative of a nonzero class in HP(M; R) cannot decay too quickly:


Proposition 3.5. I f v p < 0, no nonzero class in H'(At ; R) has a representative form 0 with

(3.6) lim 7(kr)" sup IO[x = O. r ~ oo xCB, ( x o )

for all k ~ Z +.

Here 7(r) is the growth function of ~I(M) with respect to some finite set of generators, Br(Xo) is the ball of radius r around a fixed Xo ~ M, and [0[~ is measured via the Riemannian metric. Note that (3.6) is independent of the choice of metric on M. Moreover, if 7'(r) is the growth function of rh (M ) for some other set of generators, there exists k ~ Z + such that 7'(r) < ?(kr) [Mi] , so (3.6) is independent of choice of generators.

Proof I f v p < 0, any closed L 2 p-form 0 must equal d~ for s o m e L 2 (p - 1 )-form ~. Thus it suffices to show that a closed form 0 satisfying (3.6) lies in L 2.

Let r(x) = dist(x, Xo) and let D(r) = sup [0Ix. Then x C a . ( x o )

[0[~ < ~ OZ(r(x)) = ~ DZ(r)dv(r) ~1 M 0

with v(r) the volume of Br(xo) and the last integral a Stieljes integral. Integrating the last integral by parts gives

(3.7) lim D2(r)v(r) - ~ 2v(r )D(r )dD(r R~oo 0

Since v(r) < C ' 7 ( k r ) for some positive constants C and k [Mi] , the first limit in (3.7) is zero and for all r greater than some Ro, the integrand in the second term is less than a fixed e > 0. Thus (3.7) is bounded above by

Ro -- ~ 2v ( r )D(r )dD(r ) + lim [ - 2 e [ D ( R ) - D(Ro)]]

0 R~oo

and the limit is finite since D ( R ) --* 0 as R ~ oo. []

The L z de Rham theory is closely related to the I z simplicial theory when N = M: for all k > 0

( 3 . 8 ) ~ - p - - - - p , k ~--- H ( 2 ) , s i m p B(z),aR

[Dod I, Theorem 1 and Prop. 2.9]. When v p < 0, all the terms in (3.8) are zero, and a bootstrapping argument shows the middle term stays zero if the closure is omitted:

p Lemma 3.9. Ht2).aR = 0 implies Hfi~,aR = O.

Proof It suffices to show that z P ' k c B p'R. Take c o e z P ' k c Z P ' ~ B p'~ Then

~o = dO with O s A p - l ' ~ ~- j vgv - , @ d A V - 2 @ 6 A v by the L z Hodge Theorem

(A denoting compactly supported forms). We may assume that O e 6 A p, so d60 = O.

The norm of a form in A p'R is equivalent to the norm defined by summing the L 2 norms of the covariant derivatives of the form up to order k [Dod II, Thm. 1.3].


Since 6 can be written in terms of the covariant derivative, 6 is a bounded operator from A p'k to A p - l ' k -2 . Therefore 6dO = 6 ~ . A p - l "k - 2.

Thus (1 + A)O = (1 + d6 + 6d )O~A p - l '~ which implies O e A p-1"2. Continu- ing this process, we see that O~A p-Lk , which proves tomB p'k. []

Let S denote the de Rham map from p-forms to simplicial p-cochains given by S :~ ~ E~(~, , ~b).a*, where a* is the homomorphism dual to al. The Whitney map W satisfies ~ o W = I on cochains, so for k >> 0 the induced map

p ~ : Hl~i~dR Zl~2~'dR Z(2),simp p

-m- ~ -'~ p : H(z),simp (2),dR B(2),simp

is surjective [Dod I, Lemma 3.8]. (Dodziuk proves surjectivity for the closures of the denominators, but the proof is valid without the closures.)

We summarize the preceding discussion:

Summary 3.10. If v p < O, then

Z f ~ R ~ p p ~, ,~ H(2),dR ~ H ~ . a R ~- H(2),simp 0 = H ~ 2 ~ , ~ i m . - B~,~

The next step is the construction of a Poincar6 duality map in the 12 simplicial theory.

Proposition 3.11. For any oriented m-manifold N there is a natural map PD: H~2),slmp(N ) ~ H~_p( N ) such that the following diagram commutes:

H ~ ( N ) ~ Hf2),~imp(N )

PO ~ PD

H, ._p(N) ~ H~)_r,(N)

In addition, PD is continuous with respect to the L 2 seminorms.

Proof Recall the simplicial Poincar6 duality map on Hff: we take the top dimensional simplices Z m = { co i } (oriented so that the locally finite chain ~co~ is a cycle) and denote the front p-face and back (m - p)-face by r be%, respectively. For f i E [ f i l c H p, PD([ f l ] ) = [ ~,i fl( f(l)i)b(l)i']"

P We now use this definition to extend PD to H(E),simp. We need only show that fl ~ Z~E),~imp implies X f l ( f COl) boo i~ L~rn_p,,'~(2), as the usual algebraic calculation then

P shows that this sum is closed and passes to a map on Ht2),simp. Let Sm-p {a~}' Then 2;f12(a) = ][ fl II : < 00. By our assumption on the triangulation, each ai can be the back face of at most K m-simplices, so IPD(fl)I 2 < K,~flE(fcoi) ~ K 2 It/~ll 2.

Recall that v p < 0 implies that every p-cycle in 2~ is the boundary of a locally finite chain. We can now strengthen this result.

Theorem 3.12. Let p be between I and m - 1 inclusive. I f v p < O, then Im[Hp(M) H~2)()~)] = 0. Thus every p-cycle is the boundary o f an l 2 (p + 1)-chain.

Proof Since v p = v m-p, replacing p by m - p in (3.10) and (3.11) gives the diagram

--~ H t 2 ) , s i m p ( M ) = 0

PO ~ PD

Hv(~r ) ~ H~Z)(.~r) m--p Since PD is an isomorphism on H~ , the bottom line is the zero map. []


Another topological obstruction to v p being negative is obtained by gen- eralizing to p-forms an argument in [ER I] about integrating closed compactly supported one-forms in M over lines. (The proof in [ER I, Theorem 5a] can be considerably shortened [ER III], but the short proof does not seem to generalize to p-forms.)

To set the notation, we define the bounded p-chains C~ to be the infinite simplicial p-chains c = N n~ai with finite sup norm: II c II oo = sup I nil < oe. As before, the assumption on the triangulation implies that ~ takes C~ ~ to Cp ~_ 1, so we may define Z~ ~ B~, H~ ~ to be the bounded p-cycles, bounded p-boundaries, and bounded ffh-homolooy 9roup, respectively.

Proposition 3.13. Assume v p < O. Let [q~] be a class in HV( ffl) and let c be a bounded p-cycle. Then Sc (o = O. Equivalently, v p < 0 implies that every bounded p-cycle is the boundary of a locally finite chain.

Remark: In terms of bounded homology, the Proposition says that Im[HV(/~)--* CHom(H~~ = 0 and I m [ n ; ( / ~ ) - - * n l / ( M ) ] = 0, where CHom denotes continuous homomorphisms.

Proof We prove the proposition for any regular infinite cover M' of M with a smooth triangulation lifted from a triangulation of M. For the first statement, choose a closed compactly supported p-form ~b on M' and c = X n l a i e Z ~ ( M ' ) . Let F be the group of deck transformations of M'. Xp can be written as {g 'a*}oer; i= 1 . . . . . k where each as now denotes a fixed lift of one of the k p-

simplices in the triangulation of M. Thus c may be written c = ~ n~(9)(O" ~r,). g e F

i = 1 . . . . . k

Let Pt denote the heat operator on L z p-forms on M'. We will show that for all times t

(3.14) ~ P,~b = .~ ni(g) ~ PrO < oe C l , g ~ "OIi

First, the hypothesis v p < 0 implies there is a smooth function R < R p' with A + R > 0 on M (R p' need only be continuous). By the maximum principle the groundstate h for A + R is positive; we denote its eigenvalue by - v where v is negative. Denote the lift ofh to M' also by h. The h-transformed semioroup P~ for the heat flow on L 2 functions on M' is defined by

P ~ f ( x ) = ~ e v' po (h f ) (x)

where pO is the heat semigroup on functions. P~ has differential generator N given by

~ f = h A(hf) + v

4

Let {F~(x): t > 0} be a random flow of diffeomorphisms on M' such that each F~(x) has differential generator ~ and such that each F) (x) covers a corresponding random flow on M (see [E, Ch. VIII]).

Now for the proof of (3.14). Let J be an upper bound for the absolute values of the Jacobians of the embeddings as: Ap --* M' of the standard p-simplex Ap (with respect to some fixed coordinate charts on M lifted to M'.) Set K = supp ~b,


K t = (F~)- 1 (K), and let ZK be the characteristic function of K. Using the Girsanov- Cameron-Martin formula as in [ER I, Thms 3A, 5A], we get for fixed i

< ~ J I E evtXK(F)((g'a,)(s))) h(ff~(t((g.~))) ds A t ,

< C'e ' t~ I E[ZKt((g'a,)(s))]ds g zip

for some C > 0, since h is positive, bounded above, and bounded away from zero. Let C be the collection of all open sets of M' on which the projection to M is

injective, and let c(K) be the minimum number of elements of C needed to cover K. Since the diffeomorphisms F~ cover diffeomorphisms associated to the h-transform on M, c(KJ = c(K). A simple argument as in [ER, Theorem 5A] shows

I XK,((g'a,)(s)) < c(KJIAp] = c(K)lApl g Ap

Thus for some constant C

(3.15) ! Pt~) I <= C" c(K)" e" ,

proving (3.14). Moreover, from (3.15)

v p < 0 ~ !P,O ~ 0

as t ~ ~ . Thus it remains to show

IPt~b =I~b r c

just as in the usual Hodge theory for finite cycles. Let P~ denote the heat operator on (p - 1)-forms. Using the heat equation and

the equation P,A r = dP'td* dp, we-obtain

ItP,~b--~b) = - E n , ( g ) I di(P'*d*r C i , g g �9 a i 0

with the sum absolutely convergent. Let D be the maximum diameter of the ai, and set I, = {(i, g): dist(xo, g" tri) < Dn} for some fixed Xo e M'. For e > 0 there exists N such that for n > N

(3.16) 2 n,(g) I d I ( P ' ~ d * r 1 6 2 r <~ ( i , g ) a I n g �9 a i 0 c

Since O c = O,

" o ' a i 0 O ( g ' a D 0

t1~ ' (e',d*q~)d~ (3.17) < E IIc ~ I tg 'o,)* I,,+ 1 - 1 , , Ozh, 0


Now we reuse the analysis which gave (3.14) (without the h-transform) to the boundary integrals. Let C' be a lower bound for R p-1 on M' and let {Ft} be a random flow on M' analogous to {F~} but with A instead of ~ as generator. We have

~ [P',d*r ~ E[zr(F~((g.a,)(s)))]e-C"ds I. ~ A p I,~ OA r

=~', ~ E[zK((g 'ai) (s))]e-C"ds In O A p

< k.c(K).e-C'~.lOAel Thus

n = l pO = lira (g'cri)* (P'~d*~p)dr

n ~ cr~ 0 p

k ' c (K) < (1 - e-C")lDApl -- C '

so there exists N' with n > N' implying

E (o' o,)* I n + i - I n O A p 0

This equation combined with (3.16) and (3,17) shows that for any e > 0 we have I~cP,r - Scr < e.

Finally, since ~c r = 0, (3.4) shows that [c ] = 0 in H~Y(M'; R). []

The final theorem is a vanishing result for H~(,Q; R). In the next definition K ' denotes the interior of K.

Definitions. A triangulated m-manifold N is p-connected at infinity if for all finite subcomplexes of m-simplices K and all finite p-cycle c with support in N - K', I-c] = 0EHp(N; R) and [c ] = O~H~Y(N - K'; R) implies [c ] = OeHp(N - K'; R).

A manifold is boundedly p-connected at infinity if for all compact subsets K, [c ] = 0 ~ H p ( N ; R ) and [c ] = O ~ H ~ ( N - K ' ;R) implies [c] = O ~ H p ( N - K';R).

In other words, a manifold is p-connected at infinity if for every compact K and every locally finite (p + 1)-cycle z, the finite p-cycle c = 0(z - (z c~ K')) is the boundary of a finite (p + 1)-cycle in N - K ' . Thus z - ( z c ~ K ' ) may be reconnected in N - K'. Notice that 0-connectedness at infinity and boundedly 0-connectedness at infinity are the same as ordinary connectedness at infinity.

This homological definition of p-connectedness at infinity is due to Brown (unpublished); the analogous homotopy theoretic notion is in [Br] .

Lemma 3.18. (i) I f v p < O, M is boundedly (p - 1)-connected at infinity. (ii) I f v p < 0 and M is (p - 1)-connected at infinity, Hff(,~; R ) - - 0 . In fact, if

I m [ H P ( J ~ ; R ) ~ H P ( J ~ ; R ) ] = 0 and I~I is ( p - 1)-connected at infinity, then R ) = 0.

Proof (i) Pick a finite p-cycle c with c = De in h4 for some finite chain a and c = O~ m M - K ' for a locally finite chain cr = ~ niai. Replacing c by c N = 0% for % = ~ = 1 n~a~ for suitable N if necessary, we may assume that there is a compact neighborhood K a of K i n / ~ which does not intersect the supports of c and 0r If


c 4= 0 in H p ( 4 - K'), there is a closed p-form 0 on 4 - K with ~ 0 4= 0. Modify 0 inside K1 and extend over all of 4 to obtain 0 defined on ~ such that 0 = 0 outside of K~. Set r = d0"so supp ~b ~ K~. By Proposition 3.13

o= I 4,= I r a - a t ( a - a t ) O K 1 a c

Thus [c] vanishes in H p ( 4 - K'). (ii) Take ? E [ 7 ] z H ~ I ( 4 ) and a closed compactly supported p-form r By

hypothesis there exists a form 0 with ~b = dO. Let 7N be the N th partial sum in the chain ?. For N >> 0, Sr ~b = SrN ff = ~orN 0. By hypothesis again, d?N = ~ for some p-chain ~ with support in M - supp ~b. Then ~0rN 0 = ~, ~b = 0. By (3.4), this implies 0 = Ht / ' (4 ; R) ~- H y ( 4 ; R). []

Corollary 3.19. I f v p < 0 and

Ker[Hp(M - K'; R) ~ H~Y(M - K'; R)]

(3.20) K e r [ H p ( 4 - K'; R) ~ H~ ~ c~ (M - K' ; R)]

for all finite subsets K of Zm, then H ~ ( 4 ; R) = 0.

Proof Since v p < 0, by (i) of the Lemma 4 is boundedly ( p - 1)-connected at infinity. Under the hypothesis (3.20), M ( p - 1)-boundedly connected at infinity~ implies M is ( p - 1)-connected at infinity. By (ii) of the Lemma, H y ( M ; R ) = 0. []

This leads to the main vanishing result for M.

Theorem 3.21. Let p be between 2 and m - 2 inclusive. I f v p < 0 and H p _ l ( ~ ; R) = O, then H~'-V(M; R) "-~ Hp(M; R) = 0. In particular, v 2 < 0 implies H2(M; R) = 0 .

If p = l, v 1 < 0 alone implies H i = 0, and of course H~- 1 = 0 always.

Proof Replace p by m - p and then replace v " - p by the equal quantity v p. Proving the Theorem is then equivalent to showing

(3.22) vp < o , / 4 , . _ ~ _ 1 ( 4 ) = o ~ H ~ ( 4 ) = o

The exact sequences,

c~*(~) -~o o - , c ~ * ( ~ - K' ) -~ c ~ * ( ~ ) - . c ~ * ( ~ - K')

with* either ~ or I f induce long exact sequencesin the homoloKy of the pair ( M , M - K ' ) . For K a finite subset of Zm, H v ( M , M - K ' ) ~ - H ~ K') H~Y(M, M - K'), since

c ~ ( ~ ) c~*(~) c ~ ( ~ - K') - C~*(~ - K')


Consider the diagram

Hp+I(M, M - K') ,Hp(M-K') " ,Hp(M)

0oo

c~zl

Assume ~ e Hp(M - K') hasj3jl ~ = 0. By the hypothesis v p < 0 and Theorem 3.12, J2 = 0, SO t2 Jx Gt = 0. Choose f l e H ; § M -- K') with ~ f l =jl~c If we consider fl as an element of H ~ I + I ( M , M - K'), we have t3tI f l=j3j l~=O. By (3.22), O=H, ,_p_I (M)~-Hm-P-J (M)~-H~I+I (M ), so f l = 0 . Thus j l a 0. By Corol- lary 3.19, this shows H~(M) = O. []

Corollary 3.23. (i) Let M be a compact manifold with v 2 < O. Then n2(M ) is a torsion group. The orders of the elements of n2(M) are bounded.

(ii) Let M be a compact 4-manifold. I f the fundamental group of M is infinite and v 1 < 0, v 2 < 0, then M is a K(~, 1). I f the fundamental group is finite and v 2 < O, then M is covered by S 4.

(iii) Let JV(K,D, V,m) be as in w and pick R o > 0 . There exists a = a ( J V ' , R o ) > 0 such that if M e J f f has infinite fundamental group and

I { x e M : R P ' ( x ) < Ro}[ < a for p = 1 . . . . . m - 1, then ffl has the real homology of

IMI a point and every finite cover M' of M is a real homology sphere.

Vroof.~(i) By Thm. 3.21, H2(M; Z) | R = n z ( / ~ ; R) = 0. Since n2(M) -~ n2(/~) "~ H2(M; Z) by the Hurewicz Theorem, 1r2(M ) is a torsion group. Since n 2 is a finitely generated Z n 1-module, the order of any element divides the product of the order of the generators.

(ii) Assume Inl(M)[ = o0. B y t h e theorem HI( /~ ; Z) = H2(M; R) = 0 since v 2 < 0. A torsion element in H2(M; Z) - H~(M; Z) comes from a torsion element in H2(M, M - K'; Z) for some compact set K. By a universal coefficient theorem, this element in turn comes from a tors ion element oeeH,(M, M - K'; Z). By Lemma 3.18 (i), v 1 < 0 implies M is connected at infinity, which forces Ho(M - K'; Z) = Z. The exact sequence of the pair (M, M - K') gives

0 = H~(M; Z) ~ Hi(h4, M - K' ; Z) ~ Ho(M - K' ; Z) --- ,Ho(M; Z)

showing that Hl ( /~ , ~ - K' ; Z) = 0. Thus H2(M; Z) = 0. Also, v 1 < 0 implies 0 = H ~ ( M ; Z) - Ha(M; Z). By the Hurewicz Theorem, all the homotopy groups of M vanish and M is a K(n~ 1). If zq is finite, v 2 < 0 implies H2(M; R ) = 0 and hence 0 = HZ(M; Z) = H2(M; Z). Since Ha(M; Z) ~- H~(M; Z) - HI(M; Z) = 0, M is a Z-homology sphere and hence a sphere by Feedman's solution of the Poincar6 conjecture.

(iii) By Theorem 3.1, M has v p < 0 for p = 1 . . . . . m - 1. The results follows from Theorems 3.1 and 3.21. []


Remark. Micallef and Moore prove in I-MM] that a compact m-manifold M with curvature operator positive on totally isotropic two planes has n k ( M ) = 0 for 2 < k < ~-. This curvature condition implies that the Weitzenbrck term R 2 is positive ifm is even [M. Micallef-personal communicat ion] , so v z < 0. (In fact, on a 4-manifold R a > 0 if and only if the curvature operator is positive on totally isotropic two planes I-MW-I. Whether this curvature condition on even dimensional manifolds implies R v > 0 for all p ~ 1, m - 1 is open.) Thus Cor. 3.23 (i) complements their work by giving the weaker result that n 2 is torsion under a weaker curvature condition.

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Arnold, L.: A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math. 44, 793-802 (1984) B&ard, P.: From vanishing theorems to estimating theorems: the Bochner tech- nique revisited. Bull Am. Math. Soc. 19, 371-406 (1988) Brrard, P., Besson, G.: Number of bound states and estimates on some geometric invariants. (Preprint) 1990 Brown, E.M.: Proper homotopy theory in simplicial complexes. In: Topology Conference, Virginia Polytechnic Institute and State University (Lect. Notes Math. Vol. 375, pp 41-46) Berlin Heidelberg New York: Springer,1974 Chavel, I.: Eigenvalues in Riemannian Geometry. Orlando, San Diego: Academic Press, 1984 Cheeger, J.: Finiteness theorems for Riemannian manifolds. Am J. Math. 92, 61-74 (1970) Croke, C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ec. Norm. Super. 13, 419-435 (1980) Dodziuk, J.: De Rham-Hodge theory for L2-cohomology of infinite coverings. Topology 16, 157-165 (1977) Dodziuk, J.: Sobolev spaces of differential forms and de Rham-Hodge isomorphisms. J. Differ. Geom. 16, 63-73 (1981) Donnelly, H.: The differential form spectrum of hyperbolic space. Manuscr. Math. 33, 365-385 (1981) Elworthy, K.D.: Stochastic Differential Equations on Manifolds. Cambridge Uni- versity Press, Cambridge: 1982 Elworthy, K.D., Rosenberg, S.: Generalized Bochner theorems and the spectrum of complete manifolds. Acta Appl. Math. 12, 1-33 (1988) Elworthy, K.D., Rosenberg, S.: Compact manifolds with a little negative curvature. Bull. Am. Maths Soc. 20, 41-44 (1989) Elworthy, K.D., Rosenberg, S.: Spectral bounds and the shape of manifolds near infinity. In: Simon B. et al. (eds.), IXth International Congress on Mathematical Physics Bristol: Adam Hilger, 1989, pp. 369-373 Gallot, S., Meyer, D.: Op&ateur de courbure et Laplacien des formes diflbr- entielles d'une varirt6 Riemannienne. J. Math. Pures Appl. 54, 259-284 (1975) Greene, R., Wu, H.: Lipschitz convergence of Riemannian manifolds. Pac. J. Math. 131, 119-141 (1988) Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. I.H.E.S. $3, 53-78 (1981) Gromov, M.: Structures M&riques pour les Vari&rs Riemanniennes. Paris: Cedic, 1981 Kato, T.: Perturbation Theory for Linear Operators. New York Berlin Heidelberg: Springer, 1966 Klingenberg, W.: Contributions of differential geometry in the large. Ann. Math. 69, 654-666 (1959) Li, P.: On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Super. 13, 451-469 (1980)


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Appendix: Homology and bounded homology of universal covers

Daniel Ruberman* Department of Mathematics, Brandeis University, Waltham, MA 02254, USA

The papers [ER 1, ER2] of D. Elworthy and S. Rosenberg give topological criteria which a manifold must satisfy in order for it to possess a Riemannian metric with certain positivity properties of its curvature. In this appendix, we construct a collection of manifolds which demonstrate the independence of their theorems from one another, and from the classical theorems of Bochner. To describe the examples, let W" denote a smooth manifold of dimension ~ 5, and let I~ be its universal cover. We will make statements concerning the collection of all finite covering spaces of W, and will let W' stand for an arbitrary finite covering of W. The notations v p, R p will be as in [ER2]. We find four different sorts of phe- nomena:

1. Manifolds W such that H 1 (W') = 0 (for all W') but H i (/~) 4= 0. This property depends only on the fundamental group, and works in all dimensions greater than or equal to 4. Such manifolds cannot admit metrics with v L < 0, by theorem 5A of [ER 1]. However the Bochner method will not show this, because of the fact about cohomology of the finite covers.

2. For n > 5 a n d [ n / 2 ] < p < n - 2, there are manifolds W n such that H P ( W ') = 0, but He(W) has not-trivial image in Hv(~') . By corollary 3D of [ER1], this

gives a family of manifolds which admit no metric with positive curvature operator on p-forms, but where this is not detected on finite covers.

3. For n > 5 and 2 =< p < [n/2] there are manifolds with HP(W') = Hp_l (W) =

0, but with Hp( I4 z) 4= 0. By theorem 3.21 of [ER2], such manifolds have no metric with v p < 0. Again, this is not detected until one passes to the universal cover.

* Author supported by an NSF Postdoctoral Fellowship.


4. For n >_- 5 and 2 < p < [n/2], there are manifolds with Hv(W') = 0, Hp(fie) H~f(fie) the zero map, but with the composition Hp(if/)--. Htp2~(fie) ~ H~O( fro having non-trivial image. For these manifolds, theorem 3.12 in [ER2] is used to show the nonexistence of metrics with v p < 0.

The control over the cohomology of all finite covers W' in examples 1, 2, and 4 is achieved by choosing the fundamental group so that Whas no finite covers at all, save the trivial one. Thus the verification of the statements about the cohomology of all the finite covering spaces involves checking the vanishing of a single cohomology group. To this end, we use the remarkable family of infinite, finitely presented, simple groups discovered by R.J. Thompson [T] , and generalized by G. Higman [H] . For the rest of this appendix, let G be any of the groups G(n, k) described in [H]. The only property of G we will use, besides its existence, is the fact that it contains elements of any finite order, as well as elements of infinite order.

Lemma A.0. The group G contains no (proper) subgroups of finite index. The same is true for G * G, the free product of G with itself.

Proof. If a group contains a subgroup of finite index, then it contains a normal subgroup of finite index. Since G is infinite, any subgroup of finite index is proper. So a subgroup of finite index would contradict the simplicity of G. If G * G has a finite index normal subgroup, then it has a surjection ~0 onto a non-trivial finite group F. But since G has no finite index subgroups, the composition of q~ with the inclusion of G into either free factor is trivial. Thus ~0 (and therefore F ) must also be trivial. []

This leads immediately to examples of the first kind.

Example A.1. Any finitely presented group is the fundamental group of a manifold in every dimension > 4. Let W be any manifold with fundamental group G �9 G. The vanishing of H ~(W) is equivalent to the vanishing of H i ( G . G ) = H l (G) ~ H 1 (G) = 0 since G is simple. The non-vanishing of H i (fie) is equival-

ent to the statement that fie has more than one end. This again is a statement which only involves the fundamental group [SW]. But the free product of infinite groups always has infinitely many ends, so W does as well. (It follows from the structure theory of Stallings [S] that a simple group has either 0 or 1 ends, which is why we take G �9 G.) []

The other examples are built using the following construction: Let n be a finitely presented group, and let X "§ be the regular neighborhood in R "+ 1 of a 2- complex corresponding to a finite presentation of n. Alternatively, X may be described as a manifold with a handle decomposition with a single 0-handle, and with 1-handles and 2-handles corresponding to the generators and relations in a finite presentation for ~. For n > 5, X is a manifold with boundary V" such that n~(V) ~ n l (X) = rc is an isomorphism, and such that H~(X; M) = 0 for i > 2 and for any coefficient module M, including twisted coefficients. It follows by Poincar6 duality that H~( V; M) vanishes for 2 < i < n - 2. Suppose that 2 < k < [n/2], and that A is an r x m matrix of elements in the group ring Z [n I (X ) ] . Then we construct a new manifold by adding m trivial k-handles to X, and then adding r (k + 1)-handles according to the attaching maps specified by the matrix A. (See Rourke and Sanderson [RS] for details on handle theory.) Let W be the boundary of the resulting manifold, i.e., W is the result of surgery on aX along the attaching spheres of the k and k + 1-handles.


The effect of this operation on X is the same as doing the corresponding attachments of k and k + 1 cells. Since k is below the middle dimension, the effect on the homology of V = O(X) is the same. In fact, the k-dimensional homology of the universal cover l~(as a Z [ n 1 (X)]-module) is the homology of by the following (2-step) chain complex:

A

o - - , ( z [ ~ ( X ) l ) ~ , ( z [ r q ( x ) ] ) ~ - - , 0

The ordinary homology of W is described by the augmentation of this chain complex, in other words by substituting 1 for the elements of ~i in A and taking the resulting integral chain complex. The homology of other covers of W has a similar description. Moreover, for n > 5 and 2 < k < n - 2, the fundamental groups of X and its boundary are unaffected and will be denoted by ~.

To give examples 2, 3, and 4, we need to see what chain complexes are used to calculate cohomology with compact supports and bounded supports. We will stick to real coefficients in the rest of the discussion. We use the fact that cohomology with compact supports is Poincar6 dual to ordinary homology, and ordinary cohomology is Poincar6 dual to the Borel-Moore homology (H ~) based on locally finite chains. So in examples 2 and 4, k will be n - p. Let R denote the Z[n]. module consisting of formal real linear combinations of elements of re. Then the homology group HtkY(I~) is the cokernel of the matrix A, thought of as a module map from R" ~ R". Similarly, H~( I~ ) is the cokernel of A acting from R~ to R~, where R~ denotes the Z [ h i module of bounded formal real linear combinations of elements of n.

Example A.2. We will use the above construction to produce an example of the second type. i.e. where Hk(I~) goes non-trivially into H~I(W).

So let X be constructed as above, with fundamental group G, the Thompson- Higman group discussed above. Let m be any number divisible by two distinct primes, and let q~., be the m tn cyclotomic polynomial. According to [H], for any m, there is an element, say g, of G of order exactly m. Let W be obtained as above, using the 1 x 1 matrix ~km(g).

The key property of the polynomial ~b., is that ~bm(l) = 1 ILl. This means that Hk(W) = 0. By lemma A.0 there are no non-trivial finite covers of IV, so tha t part of the properties of Whas been established. To show that Hk(I~) ~ HtkI(W) is non- zero, it suffices to prove that the element 1 ~ R is not in the image of multiplication by@,,. But ~b,.(g) is a zero-divisor, since it is a factor of g" - 1, which vanishes because g has finite order. (The complementary factor to q~,, is non-zero in the group ring of Z[Z,] where Z. is the subgroup generated by O, so it is non-zero in Z [ G ].) If, say, 2-qS,.(g) = 0 (for 2 ~ Z [ G ]) then if 1 = ~b.,(g)-/~ for # ~ R, we would have

,~ = ~ ~ m ( ~ ) ' ~ = 0

which is a contradiction. []

Examples of the third type are easy to construct, and arise naturally in high dimension knot theory (see [R] for a different view of this construction.)

Example A.3. The manifolds constructed in this example may be taken to have fundamental group Z. In terms of the construction described above, the attaching matrix will again be a 1 x 1 matrix ~b(t). Here t is a generator of Z, and we regard ~b


as a polynomial. To get homology groups with the desired properties, choose r so that (a) 4~(1) = 1, and (b) no zero of ~b is a root of unity. The resulting manifold W has the homology of S t • S"-1 by (a). Since 7q(W) = Z, its finite covers are precisely the finite cyclic covers. It is well-known [F] (and may be derived from the chain complex for W) that (b) implies that all of these cyclic covers have the real homology of S 1 x S"- 1. Because the handles added to X are added along p and p - 1 dimensional spheres, and p < [n/2], we see that Hv_I(W) vanishes as required. Finally, H~( W; R) = R It, t - ~ ]/tk (t) which evidently has dimension de- gree (r - 1. []

The examples of manifolds W with Hp(I~) ~ H~I(if ') equal to the zero map, but Hp(W) ~ H~( i f ' ) non-trivial, are a little more subtle. They will be constructed as above, by adding a single p-handle and a single (p + 1)-handle. The attaching map will again be a I x 1 matrix dp(o)eZ[G], but 9 will be an element of infinite order in G. The discussion of the previous example shows that H~f(W) will vanish, if 4~ has the property that multiplicationby q~(9) is an isomorphism from R to itself. Similarly, the image of Hp(W) in H ~ ( W ) will be non-zero if there are finite linear combinations of elements of G in R~ which are not in the image of multiplication by q~(O).

Example A.4. The group G has an element, say y, of infinite order [HI. Let q~(9) = 92 + 29 + 1, and let W be the boundary of X w h ~ we h p+~ using ~ as attaching map, as in the previous examples. Note that the augmentation ~b( l ) , 0, so that the homology groups Hp(W) and Hp+ ~(W) vanish. Note that q~(g) = (O + 1) 2 has a formal inverse ~b- t = (1 - g + O: - �9 �9 �9 )2 which is a power series in O. Thus ~b(9 ) times the element 4)- ~ e R gives 1 e R, so the homology group HlpY(if') is zero.

The power series for ~b- ~ does not have bounded coefficients. This suggests that ~b does not act invertibly on R~, although, a priori, there could be a bounded inverse involving negative powers of g. To prove that q~ does not act invertibly, let H be the subgroup of G generated by 9, and note that R [ G ] splits up as a direct sum

RIO] = �9 R[H] H\G

The sum is taken over the right cosets of H, so that left multiplication by ~b preserves the summands. There are similar decompositions of R [ G ] and R| [ G ]. Hence the homology groups H~f(1~), H~~ and H~(I,V) are direct sums over H\G of the cokernels of (left) multiplication b'y ~b on R[H} , Roo [HI , and R[H ] , respectively.

It remains to show that the cokernel of r on Roo [ H ] is non-zero. To see this, suppose that ~b'7 = 1 for some 7 e R ~ [H] . Write 7 = ,~cjo j, then writing out 4~" 7 = 1 gives a recurrence relation for the coefficients cj. This relation is uniquely solved in terms of the initial data Co, cl :

((- 1)"+l(n(co + q ) - Co) ifn > 0;

c,= 1)"+l (n(co+q+l)+l -co) ifn < 0.

From these formulas, the coefficients are bounded as n ~ + 09 if and only if Co + cl = 0, and are bounded as n ~ - oo if and only if Co + cl = - 1. Hence 7 has unbounded coefficients in at least one direction, and the cokernel is non-zero.


Put more algebraically, the bounded chain complexes are induced from the

chain complexes R~ [ H ] -~ Ro~ [HI . Thus an element in the cokernel for H gives

one in the cokernel for G. Finally, we remark that the problem of when a polynomial q5 acts invertibly on Ro~[Z] can be determined in principle from the partial fraction expansion [ G K P ] of 1/~b. In particular, one can show that a polynomial has a bounded inverse if it has no repeated roots on the unit circle.

References

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[ER2]

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[GKP]

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[L] JR] [RS]

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[T]

Elworthy, K.D., Rosenberg, S.: Generalized Bochner theorems and the spectrum of complete manifolds. Acta. Appl. Math. 12, 1 33 (1988) Elworthy, K.D., Rosenberg, S.: Manifolds with wells of negative curvature Invent. Math. 103, 471-491 Fox, R.H.: A quick trip through knot theory. In: Fort, M.K. (ed.) Topology of 3- Manifolds. Englewood Cliffs, N J: Prentice-Hall 1962 Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. New York: Addison Wesley, 1989 Higman, G.: Finitely presented infinite simple groups (Notes Pure Math. vol. 8, Australia National University, Canberra 1974 Lang, S.: Algebra. London: Addison Wesley, 1971 Rolfsen, D.: Knots and Links. Berkeley: Publish or Perish, 1976 Rourke, C., Sanderson, B.: An Introduction to Piecewise-Linear Topology. Berlin Heidelberg New York: Springer 1972 Scott, P., Wall, C.T.C.: Topological methods in group theory. In: Homological Methods in Group Theory, LMS Lecture Note Series 36, Cambridge: Cambridge University Press, 1979 Stallings, J.: On torsion-free groups with infinitely many ends. Ann. Math. 88, 312-334 (1968) Thompson, R.J. unpublished

Manifolds with wells of negative curvature

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