RATIONAL HOMOTOPY THEORY AND NONNEGATIVE CURVATURE JIANZHONG PAN Abstract. In this note , we answer positively a question by Bele- gradek and Kapovitch[2] about the relation between rational ho- motopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact nonnega- tive curved manifolds admit (complete) metrics with nonnegative curvature. 1. Introduction Given a Riemannian manifold M with metric <>: T M x T M ! T M an affine connection is a bilinear map r : V ec(M) x V ec(M) ! V ec(M) which satisfies the following o rfV W = frV W o rV (fW ) = (V f)W + frV W where f 2 C1 (M), V, W 2 V ec(M) An affine connection is called Levi-Civita connection if it satisfies also the following o X < V, W >=< rX V, W > + < V, rX W > o rV W - rW V - [V, W ] = 0 ____________ Date: Aug. 30, 2001. 1991 Mathematics Subject Classification. 53C20 53C40 55P10. Key words and phrases. curvature, derivation, homotopy equivalence. The first author is partially supported by the NSFC projects 10071087 , 1970* *1032 and ZD9603 of Chinese Academy of Science . 1 2 JIANZHONG PAN where [V, W ]f = (XY - Y X)f is the Lie bracket. A fundamental result in Riemannian geometry asserts that Theorem 1.1. For each Riemannian metric, there exists a unique Levi-Civita connection. Given a Riemannian manifold M with Levi-Civita connection , there is defined a curvature operator R : V ec(M) x V ec(M) x V ec(M) ! V ec(M) defined by R(X, Y )Z = rX rY Z - rY rX Z - r[X,YZ] From it one arrives at an important geometric invariant which is called Sectional curvature defined by < R(v, w)w, v > K(oe) = _________________ < v ^ w, v ^ w > where oe TpM is a tangent plane at p 2 M and v, w 2 oe span it. It is well known that K(oe) does not depend on the choice of spanning vectors. A well known question in Riemannian geometry is Question 1.2. Does the restriction on curvature imply the restriction on topology and vice versa? In particular, how does the positive(nonnegative) curvature restrict the topology of the underlining manifold? A Riemannian manifold is called positively (or nonnegatively) curved if, for any oe, K(oe) > 0 (or K(oe) 0). For compact manifold, we have the following classical Theorem 1.3. Let M be a compact Riemannian manifold with positive curvature. Then 8 >< finite group if dim M is odd ß1(M) = 0 if dim M is even and M is orientable >: Z2 if dim M is even and M is nonorientable The main concern of this note is on noncompact manifold. In this case there is the following RATIONAL HOMOTOPY THEORY AND NONNEGATIVE CURVATURE 3 Theorem 1.4. Let M be a complete noncompact Riemannian manifold with nonnegative curvature. Then M is diffeomorphic to the total space of the normal bundle of a compact totally geodesic submanifold which is called the soul. Another central question in Riemannian geometry is to what extent the converse is true, or in other words Question 1.5. Total spaces of which vector bundles over compact non- negatively curved manifolds admit (complete) metrics with nonnegative curvature? Previously, obstructions to the existence of nonnegatively curved metrics on vector bundles were only known for a flat soul [7]. No obstructions are known when the soul is simply-connected. In [2] an approach to the reduction of the problem to the vector bundle over simply connected manifold was initiated. The start point is another result of Cheeger and Gromoll [4] that a finite cover of any closed non- negatively curved manifold is diffeomorphic to a product of a torus and a simply-connected closed nonnegatively curved manifold. It turns out that a similar statement holds for open complete nonnegatively curved manifolds which is the basis of their analysis. Lemma 1.6. [2] Let (N, g) be a complete nonnegatively curved mani- fold. Then there exists a finite cover N0 of N diffeomorphic to a product M x T kwhere M is a complete open simply connected nonnegatively curved manifold. Moreover, if S0 is a soul of N0, then this diffeomor- phism can be chosen in such a way that it takes S0 onto C x T kwhere C is a soul of M. By using this and characteristic classes technique, they proved that, in various case, the total spaces of rank k vector bundles over C x T admit no nonnegatively curved metric if they do not become the pullback of a bundle over C in a finite cover. The following is such an example Corollary 1.7. [1] Let B be a closed nonnegatively curved manifold. If ß1(B) contains a free abelian subgroup of rank four (two, respectively), 4 JIANZHONG PAN then for each k 2 (for k = 2, respectively) there exists a finite cover of B over which there exist infinitely many rank k vector bundles whose total spaces admit no nonnegatively curved metrics. Belegradek and Kapovitch [2] are thus lead to the following Definition 1.8. Given a closed smooth simply connected manifold C, a torus T , and a positive integer k, we say that a triple (C, T, k) is splitting rigid if any rank k vector bundle over C xT with nonnegatively curved total space splits, after passing to a finite cover, as the product of a rank k bundle over C and a rank zero bundle over T . Let H be the class of simply-connected CW-complexes whose rational cohomology algebra is finite dimensional, as a rational vector space, and has no nonzero derivations of negative degree(a homomorphism f : A ! A between graded algebras over rational is said to be derivations of negative degree k if f(uv) = f(u)v+(-1)kpuf(v) where u 2 A, v 2 Ap), see [2] for the reason to choose such a class H. For example, H contains any compact simply-connected Kähler manifold [6]. A natural question is Question 1.9. [2] Let C 2 H be a closed smooth manifold. Is (C, T, k) splitting rigid for any T and k? The main result in this note is a positive answer to this question Theorem 1.10. Let C 2 H be a closed smooth manifold. Then (C, T, k) is splitting rigid for any T and k. In this paper, all cohomology groups have rational coefficients, all manifolds and vector bundles are smooth; all topological spaces are homotopy equivalent to connected CW-complexes. [X, Y ] will be the based homotopy classes of based maps between them. map(X, Y ) is the space of maps from X to Y and map(X, Y )f is the connected component of map(X, Y ) which contains the map f : X ! Y . 2. A splitting criterion Given a finite cell complex C, define Char(k, C) to be the subspace of H*(C) which is the direct sum of [(k-1)=2]i=1H4i(C) and the subspace RATIONAL HOMOTOPY THEORY AND NONNEGATIVE CURVATURE 5 equal to Hk(C) if k is even, and to H4[k=2](C) if k is odd. Note that any rational characteristic class of a rank k vector bundle over C lies in the subalgebra of H*(C) generated by Char(k, C). Belegradek and Kapovitch transform the problem of a triple (C, T, k) being splitting rigid into a homotopy problem as follows Proposition 2.1. [2] Let C be a closed simply-connected manifold, T be a torus, k be a positive integer. If any self-homotopy equivalence of C x T maps Char(k, C) to itself, then the triple (C, T, k) is splitting rigid. We are thus led to compute the group of homotopy classes of self homotopy equivalences Aut(C x T ) of C x T . Before we can do this , let's recall a work by Booth and Heath [3] Given spaces with base point (X, x0) and (Y, y0), there is a natural map ' : map(X x Y, X x Y ) ! map(X, X) x map(Y, Y ) defined by '(f) = (g, h) where g(x) = ßX Of(x, y0) , h(y) = ßY Of(x0, y) and ßX : X x X, ßY : X x Y ! Y are projections to the factors X and Y respectively. Definition 2.2. Let X and Y be two spaces . We say X and Y have the induced equivalence property(IEP) if whenever f is a homotopy equivalence , then g, h defined above are homotopy equivalences. Remark 2.3. Let X and Y be such that for each i > 0, at least one of ßi(X) and ßi(Y ) is zero. Then they satisfy the IEP by Whitehead theorem. With the above notion, we can quote the following Theorem 2.4. Let X and Y be two spaces having IEP. Suppose further that [X, map(Y, Y )id] = 0, then there is a short exact sequence of groups and homomorphisms 1 ! [Y, map(X, X)id] !` Aut(X x Y ) ! Aut(X) x Aut(Y ) ! 1 which splits by a homomorphism oe : Aut(X) x Aut(Y ) ! Aut(X x Y ) given by oe(g, h) = g x h 6 JIANZHONG PAN Let X = C and Y = T where C, T be as in Theorem1.10. Then X and Y have IEP by the remark following the definition2.2. On the other hand, [C, map(T, T )id] = 0 since it is well known that map(T, T )id= T and C is 1-connected and thus first cohomology of C is trivial. Now given f 2 Aut(C x T ) , to prove that the induced homo- morphism in cohomology maps Char(k, C) to itself , it suffices to assume that f 2 Im([T, map(C, C)id]) by the exact sequence above. Recall that the map ` : [T, map(C, C)id] ! Aut(C x T ) is given by `(f)(x, y) = (f(y)(x), y). The above argument gives the following Corollary 2.5. Let C be a closed simply-connected manifold, T be a torus, k be a positive integer. Then the triple (C, T, k) is splitting rigid if, for any map f : T ! map(C, C)id , the adjoint f~ : T x C ! C induces a homomorphism in cohomology given by f~*(u) = 1 u for any u 2 H*(C) . 3. The proof of Theorem1.10 Proof of Theorem1.10. Let T = (S1)s. By Corollary 2.5, to prove Theorem1.10 , it suffices to prove that , for map f : T x C ! C such that f(y0, -) homotopic to id, it induces a homomorphism in cohomology given by f*(u) = 1 u. Given such f, for any u 2 H*(C), X X f*(u) = 1 u + ~i1...ik(u) 'i1. .'.ik k i1...ik where the first sum is taken over k from 1 to s and the second sum is taken over all (i1 . .i.k)0s such that 0 < i1 < . .<.ik < s + 1. Thus we get a sequence of maps ~i1...ik: Hn (C) ! Hn-k (C) where 0 < k < s+1 and 0 < i1 < . .<.ik < s + 1. To prove that f*(u) = 1 u , it suffices to prove that ~i1...ik= 0 where 0 < k < s + 1 and 0 < i1 < . .<.ik < s + 1 while for the proof of later we need to study the behaviour of these maps with respect to the cup product of cohomology. If u, v 2 H*(C) , then X X f*(uv) = 1 uv + ~i1...ik(uv) 'i1. .'.ik k i1...ik RATIONAL HOMOTOPY THEORY AND NONNEGATIVE CURVATURE 7 On the other hand f*(uv) = f*(u)f*(v). Using the formula for f*(uv), f*(u), f*(v) and comparing the terms associated with 'i1. .'.ik, we find the following equations X ~i1...ik(uv) = ~i1...ik(u)v+(-1)k|v|u~i1...ik(v) ~j1...jp(u)~l1...lq(v) 'i1* *. .'.ik where p + q = k with p > 0, q > 0 and the sum is taken over all partitions of i1, . .,.ik into j1 < . .<.jp and l1 < . .<.jq. Let k = 1. Then then above formula implies that ~i1is a derivation of degree -1 which is trivial by the condition of the Theorem1.10. The above formula in case k = 2 implies that ~i1i2is a derivation of degree -2 modulo products of derivations of degree -1 which are trivial. It follows that ~i1i2is a derivation of degree -2 and thus is trivial by the condition of the Theorem1.10. Inductively we can prove that all ~i1...ik are trivial which completes the proof of the Theorem1.10. _____________________- References [1]I. Belegradek and V. Kapovitch, Topological obstructions to nonnegative cur- vature, Math.DG/0001125. [2]I. Belegradek and V. Kapovitch, Obstructions to nonnegative curvature and rational homotopy theory, Math.DG/0007007. [3]P.I. Booth and P.R. Heath, On the groups E(X x Y ) and EBB(X xB Y ). (English) Groups of self-equivalences and related topics, Proc. Conf., Mon- treal/Can. 1988, Lect. Notes Math. 1425, 17-31 (1990). [4]J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonneg- ative curvature, Ann. of Math. 96 (1972), 413-443. [5]J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegati* *ve Ricci curvature, J. Differential Geom 6 (1971/72), 119-128. [6]W. Meier, Some topological properties of Kähler manifolds and homogeneous spaces, Math. Z. 183 (1983), no. 4, 473-481. [7]M. Özaydin and G. Walschap, Vector bundles with no soul, Proc. Amer. Math. Soc. 120 (1994), no. 2, 565-567. _____________________- Institute of Math.,Academia Sinica ,Beijing 100080, China E-mail address: pjz@math03.math.ac.cn