(3.6) IHkK^lHIo
OF NON-POSITIVE SECTIONAL CURVATURE
This chapter is based on the idea that a manifold which represents a non-trivial homology class in a compact K(TT, i)-manifold, should not carry positive scalar curva-ture. If the K(TT, i)-manifold is a torus, this is true (since any such submanifold is easily seen to be enlargeable). In this section we show that the statement is, in fact, true for any K(TC, i)-manifold of non-positive curvature. We shall present two arguments. The first argument uses the Relative Index Theorem for Families, proved in the thesis of Zhiyong Gao at Stony Brook. The second argument, given at the end of this section, uses only the Relative Index Theorem proved here. A comparison of the two arguments is illuminating.
The fundamental construction is as follows. Let K be a compact riemannian manifold of non-positive sectional curvature, and consider a compact oriented ^-mani-fold X which represents a non-zero class in H^(K$ QJ. That is, we assume there is a (smooth) map /: X -> K so that /*[X] 4= o in H^(K$ QJ. By taking a product of K with a torus of sufficiently high dimension, we can assume f: X c-^ K is an embedding.
We now pass to the universal covering TT : K -> K and set X = T^^X). Note that X is a properly embedded submanifold of K which is invariant under the deck group r ^ ^ ( K ) .
Since K carries non-positive curvature, the exponential map expy: T y K ~ > K is a diffeomorphism at each point y e K. Furthermore, exp.^ o y* = Y ° ^Py ^or a^ Y e r, since F acts by isometrics. We now construct a family of maps K -> S^ (where N = dim K) as follows. To begin, we shall assume that there exists a r-invariant framing a of TK. At each point y e K, the frame dy can be considered as an isomor-phism dy : TyK -^R^ We now fix a degree-one map
0: R^S^^
which is constant outside the unit disk and, say, 27r-bounded, and we fix s > o. Then for each y e K we consider the map
V , : K - ^ given by setting
(i3-i) 9yW = ^(^•exp,-1^)).
POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR 191
Observe that for any y ^ r we have
9w(T^) = ^(^(e-exp^Y^))) -^(^(Y.eexp,-1^)))
^((^exp^)))
==
9yW-Hence the family of maps 9 : K X fL -> S^ is invariant under the diagonal action of F.
We now restrict this family of maps to the invariant subset £ X 5CC K X JK. and divide by F. The resulting space
(13.2) Y = £ x r X is the total space of a bundle
(13.3) P : Y ^ K
over K/r ^ K, whose fibre is X. The above construction gives a map
(13.4) 9 : Y-^
which has compact support on each fibre. This map has the following basic property whose proof we postpone.
LEMMA 13.5. — Let (o == 9*(vol) eH^(Y, R) be the compactly supported cohomology class obtained by pulling back the fundamental cohomology class ofS^ via 9. Let
A : H^Ys^^H^^KsR)
denote " integration over the fibre ". Then P^ co =t= o.
NOTE. — This cohomology class p^ is essentially (a multiple of) the Poincar^ dual of [X]eH^(K;R).
i92 M I K H A E L G R O M O V A N D H . B L A I N E L A W S O N , J R .
We now suppose that X carries a metric of positive scalar curvature, and we lift this metric to a (r-invariant) metric on 5c. We observe now that there is a constant c> o (independent ofe) so that for each y e JK the restricted map <py : X -> S1^ is zc-contracting with respect to this new metric. Indeed, it was 2£7r-contracting with respect to the old (induced) metric; and the two metrics are finitely related since they are both lifted from metrics on X.
We now assume N even (as before) and fix a complex vector bundle Eo over SN
with Cy{Eo) =[= o. We fix also a unitary connection in Eg. We then lift Eg to the family Y via the map 9, i.e. we set E = 9^0 with the induced connection.
This gives us a family of vector bundles on the fibres of the family Y -» K. Each vector bundle has curvature whose norm is uniformly ^ c'e for some constant c\ Fur-thermore, since <p is constant at infinity, the bundle E is trivialized outside a compact subset.
We now consider two families of Dirac operators. Let D be the family of canoni-cal Dirac operators on spinors in the fibres (« 5c) of the family. Set
Dc^ == D ® C?
Eg E E D ® E
where k == dim^ E (cf. § 5).
For e sufficiently small, both operators will be pointwise strictly positive and the analytic indices ofD^ and Djg will be zero. However, by the Relative Index Theorem for Families ([Gao]), the relative topological index ind((DcA, D^) must be zero, and this index is given by the following formula:
(13.6) ind,(Dc,,DE) = A { c h E.A(Y/K)} e H*(K, %)
where A(Y/K) denotes the total A-class of the tangent bundle to the fibres of p : Y -> K, ch E == ch E — k denotes the reduced Chern character, and p^: H^(Y$ QJ -> H*(K; QJ denotes integration over the fibres.
Since E = cp*Eo and since ^(Ko) + o in H^S^, we see that ch E == <p*(ch Eo) = 9' [(N^-7), ^(Eo) ]
== ^*{k vol) = ku
for some non-zero number k. Since A(Y/K) = i + ..., we see that the non-zero term of lowest degree in formula (13.6) is kp^u), which is not zero by Lemma 13.5.
This is a contradiction to the existence of positive scalar curvature on X.
In the argument above we assumed the existence of a r-invariant framing of TSL This amounts to assuming K is parallelizable, or just stably parallelizable since we can multiply K by S1. This assumption can be avoided by enhancing the family Y to include all oriented tangent frames to K. That is, at each point y e £, and for each oriented orthonormal tangent frame cr atj?, we define the map 9 = O^e-exp'"1^))) as in (13. i)
POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR 193 above. The group F acts naturally on the oriented frame bundle P(IC) and 9 is invariant under the diagonal action on P(IC) x SL Restricting to P(iC) X 5C and taking a quo-tient by r gives a family
(13.7) P : Z - ^ K
where Z = P(K) Xp 5^ and where the fibre at y e K is X X {o.o.n. frames atj/} ^ X X SO^.
The map 9 again descends to a map 9 : Z -> S^ and Lemma 13.5 continues to hold.
This gives us the following result.
THEOREM 13.8. — Let X be a compact spin manifold which represents a non-trivial rational homology class in some compact manifold of non-positive sectional curvature. Then X cannot carry a metric of positive scalar curvature.
Note that as a special case (when dim X = dim K) we retrieve the result that compact manifolds which admit maps of non-zero degree onto manifolds of non-positive curvature cannot carry K > o.
REMARK 13.9. — When TT^X (or, in fact, T^K.) is residually finite, the above theorem can be proved by using the usual Atiyah-Singer Index Theorem for families.
PROOF OF LEMMA 13.5. — By Poincar^ duality and the work of Thorn we know that there exists a compact oriented manifold X* of dimension N — n, and a map X* -> K so that the homology intersection
X ^ X ^ o.
We can assume the map is smooth and transversal to X.
Consider the covering space X* -> X* induced by the universal covering K -> K.
There is a natural lifting X* -> K, and we consider the F-equivariant map:
X* X X -> K X X. Dividing by this (diagonal) action of F gives us the commutative diagram of maps:
X - X r X —> K X p X -^ SN
(13.10) p
X* —————> K
The family X* Xp X -> X* is just the X-bundle over X* induced from the family Y = K Xr X -> K. The map 9 has compact support in each fibre.
Recall that by its construction, 9 is a map 9 : 'K X 1C -> RN/(RN — D^ == S^
and the preimage of o under 9 is the diagonal in £ X JK-. Furthermore, note that 9, 405
i94 M I K H A E L G R O M O V A N D H . B L A I N E L A W S O N , J R .
when restricted to X* X X, is a map between oriented N-dimensional manifolds, and the set ^"^(o) is just the set of intersection points of X* with X, i.e. it is the set Tr^X" n X) C K. Since these intersections are all transversal, o is a regular value of 9, and this remains true when we push 9 down to the quotient X* Xp X. On this quotient we see that 9-l(o) ^ TC'^X* n X)/F ^ X* n X, and we conclude that
deg(cp) = X * . X .
Furthermore, if <o = 9*^ e^^t(X* ^ ^) denotes the pull-back of the fundamental cohomology class of S^ then co = deg(9) coo where (x>o is the fundamental class of X* Xr X. It follows immediately that
A(co) =(X*.X)^
where co* is the fundamental cohomology class ofX*. In particular />,((o)[X*] =t= o, and so ^(co) + o in H^^K) as asserted. •
REMARK 3.11. — This Lemma also holds in the general case where K is replaced by the oriented frame bundle P(K). Essentially the same argument applies; one must integrate over the family of frames at the appropriate time.
Using the constructions in the proof of this Lemma, it is possible to give a proof of Theorem 13.8 without using the Index Theorem for Families. Again for simplicity we remain in the case where there exists an invariant framing.
SECOND PROOF OF THEOREM 13.8. — Let X* -> K be the dual manifold considered in the proof of Lemma 13.5, and let W s X* Xp X be the family constructed there.
Recall that
W-^>X*
is a fibre bundle with fibre X, and that for each e > o, 9 : W -> S
, C^J.J.^O. LJ.J.(AL JLV^J. ^/Uf^J.JL & -" V-»,
N
is a map which is constant at infinity and of non-zero degree.
Suppose now that X carries a metric with K ^ i. Choose an arbitrary metric g*
on X* and lift the product metric to X* X X. This lifted metric is r-invariant and descends to a metric on W.
Assume that s > o is given. Replace g* by eg* where c > i is a constant chosen sufficiently large, so that the product metric on X* x X has scalar curvature ^ —, and the map 9 is se^-contracting. For e sufficiently small we get a contradiction by applying the Relative Index Theorem as usual. •
POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR 195
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Institut des Hautes Etudes Scientifiques 35, route de Ghartres,
91440 Bures-sur-Yvette.
Department of Mathematics,
State University of New York at Stony Brook Stony Brook, N.Y. 11794
Manuscrit refu Ie 16 avril 1982.