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
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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