Manifolds Carrying Large Scalar Curvature

Stefan Bechtluft-Sachs

Email: stefan.bechtluft-sachs@mathematik.uni-regensburg.de

URL: http://www-nw.uni-regensburg.de/∼.bes08226.mathematik.uni-regensburg.de/

September 27, 2004

Abstract

Let W = S ⊗ E be a complex spinor bundle with vanishing first Chern class over a simply connected spin manifold M of dimension ≥ 5. Up to connected sums we prove that W admits a twisted Dirac operator with positive order-0-term in the Weitzenböck decomposition if and only if the characteristic numbers Â(T M)[M] and ch (E)Â(T M)[M] vanish. This is achieved by generalizing [2] to twisted Dirac operators.

1 Introduction

A key point in the Lichnerowicz argument, showing that the Â genus is an obstruction to the existence of a metric with positive scalar curvature, is the fact that the scalar curvature appears as the order-0-term in the Weitzenböck decomposition of the ordinary Dirac Laplacian D^{2}. It was shown in [2], [8] that positive scalar curvature can be preserved under surgeries in codimension ≥ 3. Within the class of simply connected spin manifolds of dimension ≥ 5 the cobordism relation is generated by surgeries of this type. Therefore all such manifolds admitting a metric of positive scalar curvature could be determined by computations in the spin cobordism ring (see [2], [8], [6], [7]).

Here we extend this to general Dirac operators (see [1], [4]). The role of scalar curvature is taken by the order-0-term in the Weitzenböck decomposition of a twisted Dirac operator. This term is positive if the scalar curvature is larger than a certain norm of the curvature endomorphism of the coefficient bundle. First we prove a surgery theorem for the order-0-term in the the Weitzenböck decomposition of twisted Dirac Laplacians D_{∇}^{2} (Theorem 1). Next we consider complex spinor bundles with trivial first Chern class over simply connected spin manifolds of dimension ≥ 5. Up to connected sums, we determine all spinor bundles within this class, which admit a Dirac operator with positive order-0-term in its Weitzenböck decomposition (Theorem 2). This is done by a computation in the cobordism ring ∑_{n,k} Ω_{n}^{spin}(BSU(k)) ⊗ ℚ.

2 Statement of Results

Let W be a complex spinor bundle over a spin manifold M. Then W is a twisted spinor bundle W = S ⊗ E, where S is the spinor bundle associated to the irreducible representation of the Clifford algebra and E is a complex vector bundle, see [1], [4]. To a Riemannian metric g on M and a Hermitian connection ∇ on E there is naturally associated the twisted Dirac operator D_{∇} acting on sections of W. The Weitzenböck decomposition of its Dirac laplacian D_{∇}^{2} reads ([1], [4])

D_{∇}^{2} = D^{∗}D + \frac{1}{4}s + ∑_{i,j} e_{i}e_{j} ⊗ R_{e_{i},e_{j}} ,

the sum being taken over an orthonormal basis {e_{i}} of the tangent space of M. Here D is the covariant derivative on W induced from the connection ∇ and the Levi-Civita connection on M. By s we denote the scalar curvature of M and by R the curvature tensor of ∇. We also define E(∇) := 4 ∑_{i,j} e_{i}e_{j} ⊗ R_{e_{i},e_{j}} and | ; |E(∇)| ; | (x) to be minus the smallest eigenvalue of the bundle endomorphism E(∇) at the point x ∈ M.

Assume additionally that M is simply connected, dim M ≥ 5 and c_{1}(E) = 0. We will show that rationally, i.e. after eventually passing to a suitable connected sum multiple of (M, E), the following are equivalent:

1. M admits a Riemannian metric g and E a Hermitian connection ∇, such that s(g) > | ; |E(∇)| ; | on M. We will then say that (M, E) admits large scalar curvature.

2. Both S and W admit an invertible Dirac operator.

3. The characteristic numbers Â(T M)[M] and ch (E)Â(T M)[M] vanish.

As | ; |E(∇)| ; | is always nonnegative, the implication (1) ⇒ (2) is immediate from the Weitzenböck decomposition. By the index theorem we have (2) ⇒ (3). So we are left with the implication (3) ⇒ (1).

Therefore we will first extend the surgery theorem for scalar curvature (cf. [2], [8]) to show that positivity of s + E(∇) can be preserved under surgeries of codimension at least 3.

Theorem 1 Let E → M be a vectorbundle over the smooth manifold M. Assume that there is a Riemannian metric g on M and a unitary connection ∇ on E with s(g) > | ; |E(∇)| ; |. If the manifold M′ is produced from M by surgery in codimension more than 2 and such that the vector bundle E extends over the trace of the surgery giving a vector bundle E′ over M′ then there are a Riemannian metric g′ on M′ and a unitary connection ∇′ on E′ with s(g′) > ||E(∇′)||.

Now we look at simply connected spin manifolds M of dimension dim M ≥ 5 endowed with a complex vectorbundle E with vanishing first Chern class. Then E — and the spinor bundle W = S ⊗ E — are trivial over embedded 2-spheres. As in [2] we obtain that any cobordism can be replaced by a sequence of surgeries of codimension ≥ 3. Hence we can decide from the cobordism class of (M, E) in Ω_{n}^{spin}(BSU(k)), whether it admits large scalar curvature. We have

Theorem 2 Let E → M be a SU(r)-vectorbundle over the smooth simply connected spin manifold M of dimension ≥ 5. Then the following are equivalent:

1. For some q the q-fold connected sum (M, E)# … #(M, E) carries a metric g and a connection ∇ with s(g) > | ; |E(∇)| ; |.

2. Â(T M)[M] = 0 and ch (E)Â(T M)[M] = 0.

3 Proof of Theorem 1

Consider surgery on an embedded sphere S^{k} ≅ S ⊂ M^{k+l}, n = k + l, with trivial normal bundle and such that the restriction to S of the vector bundle E is trivial. M′ is then obtained by cutting out a tubular neighbourhood f : S^{k} × D^{l} ↪ M of S and glueing back D^{k+1} × S^{l−1} along the boundary S^{k} × S^{l−1}. In the end M′ will be described as a submanifold of Z := M × [0, δ] ∪_{f} D^{k+1} × D^{l}

Let S^{k} × D^{l} carry the metric and the connection induced via f from M. We can extend these data to all of D^{k+1}×D^{l}, such that in the vicinity of the boundary S^{k}×D^{l} they are compatible to a product structure of a collar neighbourhood. The metric and connection on Z are then obtained by glueing this handle D^{k+1} × D^{l} with the product metric and connection on M × [0, δ].

Let ϱ ≤ R be sufficiently small constants (e.g. less that the injectivity radius of Z) and denote by d(·, S) the distance from S. Define N_{r} := {x ∈ M | d(x, S) ≤ r} and Y_{ρ} = ∂N_{ρ}. If ρ ≤ R then the exponential map provides diffeomorphisms D^{n−k} × S^{k} ≅ ρDν(S, M) → N_{ρ} and S^{n−k−1} × S^{k} ≅ ρSν(S, M) → Y_{ρ}. Pick a decreasing real function ϕ(ρ) defined for ρ ≥ ϱ, vanishing for ρ ≥ R and such that all derivatives of its inverse function χ = ϕ^{−1} vanish at ϕ(ϱ). Let δ := ϕ(ϱ) and ψ(x) := ϕ(d(x, S)), x ∈ M. The result of the surgery is

M′ = {(m, t) | ϕ(d(m, S)) = t} ∪_{f} {x ∈ D^{k+1} × D^{n−k} | d(x, S^{k} × D^{n−k} = ϱ} .

We will show that one can find ϱ and ϕ such that on M′ we have s − E positive.

The calculations in 3.1 are much the same as in [2] and merely included for the reader’s convenience.

3.1 Scalar Curvature of M′

M′ is glued together from the graph X of ψ on M \ N_{ϱ} and a handle. We express the scalar curvature of X ⊂ M × ℝ at (m, t) in terms of the second fundamental form T of the submanifolds ψ^{−1}(t) ⊂ M at m. This is a straightforward calculation based on the Gauß equation.

Denote the derivation in direction of the ℝ-factor by ∂_{t} and the gradient of ψ by ∂ψ. Let r := −∂ψ/|∂ψ| = ∂ψ/ϕ′ and n̂ := (−∂ψ, ∂_{t})/\sqrt{1 + |∂ψ|^{2}} = (−ϕ′r, ∂_{t})/\sqrt{1 + ϕ′^{2}} be the normal unit vectors to ψ^{−1}(t) and X respectively. For a vector v ∈ T_{m}M define \overline{v} := (v, v(ψ) ∂_{t}) ∈ T_{(m,ψ(m))}X.

At a point (m, t) ∈ X choose an orthonormal basis v_{1}, …, v_{n−1} of the orthogonal complement of the gradient ∂ψ in T_{m}M. We work in the orthonormal basis

(\overline{v}_{1}, …, \overline{v}_{n−1}, \frac{\overline{∂ψ}}{|\overline{∂ψ}|} = −\frac{(r, ϕ′ ∂_{t})}{\sqrt{1 + ϕ′^{2}}} )

of T_{(m,t)}X.

First we compare the second fundamental form T of the submanifolds ψ^{−1}(t) ⊂ M at m with the second fundamental form \overline{T} of X ⊂ M × ℝ at (m, t). For v, w perpendicular to ∂ψ we obtain

\overline{T} (\overline{v}, \overline{w}) = ⟨∇\overline{v}\overline{w} | n̂⟩ = ⟨∇_{v}w | (−ϕ′ r)⟩ /\sqrt{1 + ϕ′^{2}}
= T (v, w)\frac{−ϕ′}{\sqrt{1 + ϕ′^{2}}}
\overline{T} (\overline{v}, \overline{∂ψ}/|\overline{∂ψ}|) = − ⟨∇\overline{v}(r, ϕ′ ∂_{t}) | (−ϕ′r, ∂_{t})⟩ / (1 + ϕ′^{2})
= (ϕ′ ⟨∇_{v}r | r⟩ − v(ϕ′)) / (1 + ϕ′^{2})
= (ϕ′v(|r|^{2})/2 − v(ϕ′)) / (1 + ϕ′^{2}) = 0
\overline{T} (\overline{∂ψ}/|\overline{∂ψ}|, \overline{∂ψ}/|\overline{∂ψ}|) = ⟨∇_{(r,ϕ′ ∂_{t})}(r, ϕ′ ∂_{t}) | (−ϕ′r, ∂_{t})⟩ /(1 + ϕ′^{2})^{3/2}
= (⟨∇_{r}r | − ϕ′r⟩ + r(ϕ′)) /(1 + ϕ′^{2})^{3/2}
= \frac{ϕ′′}{(1 + ϕ′^{2})^{3/2}}

The Gauss formula then yields for the sectional curvature \overline{K} of the submanifold X:

\overline{K} (\overline{v}, \overline{w}) = K(v, w) + \frac{ϕ′^{2}}{1 + ϕ′^{2}} (T (v)T (w) − T (v, w)^{2})
\overline{K} (\overline{v}, \overline{∂ψ}/|\overline{∂ψ}|) = K^{M×ℝ} (v, \overline{∂ψ}/|\overline{∂ψ}|) − \frac{ϕ′ϕ′′}{(1 + ϕ′^{2})^{2}} T (v)

= \frac{1}{1 + ϕ′^{2}} K(v, r) − \frac{ϕ′ϕ′′}{(1 + ϕ′^{2})^{2}} T (v)

Taking sums over the basis above we end up with

\overline{s} = s + \frac{ϕ′^{2}}{1 + ϕ′^{2}} ∑_{i,j} (T (v_{i})T (v_{j}) − T (v_{i}, v_{j})^{2})

− \frac{ϕ′^{2}}{1 + ϕ′^{2}} 2 ∑_{i} K(v_{i}, r)  (3.1)

− \frac{ϕ′ϕ′′}{(1 + ϕ′^{2})^{2}} 2 ∑_{i} T (v_{i}) .

We will need the asymptotic behaviour when approaching S of the functions on N_{R} \ N_{ϱ} defined by the sums in (3.1):

Lemma 3.2 As ρ = d(x, S) → 0 the asymptotic behaviour of the functions A := ∑_{i,j}(T (v_{i})T (v_{j}) − T (v_{i}, v_{j})^{2}) = (Tr T)^{2} − Tr T^{2}, B := −2 ∑_{i} T (v_{i}) = 2 Tr T and C := 2 ∑_{i} K(v_{i}, r) = 2 Ric (r) is

A(x) = a_{2} ρ^{−2} + a_{1}(x) ρ^{−1} + a_{0}(x), B(x) = b_{1}ρ^{−1} + b_{0}(x) ,

with bounded functions a_{1}(x), a_{0}(x) and b_{0}(x) and positive constants a_{2} and b_{1}. C also extends to a bounded function on N_{R}.

In fact since the codimension l of the submanifold S is ≥ 3 we have a_{2} = (l − 1)(l − 2)/2 > 0 and b_{1} = l − 1 > 0.

Proof: Consider the diffeomorphism S × ℝ^{l} = ν(S, M) → N_{R} given by the exponential map i.e. mapping (p, v) ↦ exp_{p} v. For unit speed curves p(t) in S and A_{t} in SO(l) define vectorfields h = \frac{d}{dt} exp_{p(t)} v, u = \frac{d}{dt} exp_{p} A_{t}v, r̃ = \frac{d}{dt} exp_{p} tv. Then for small ρ = |v| expand |u| = ρ + b_{u}ρ^{2} and |r̃| = ρ + b_{r}ρ^{2} with smooth functions b_{u}, b_{r}. We compute

T (\frac{u}{|u|}) = \frac{1}{|u|^{2}|r̃|} ⟨∇_{u}u | r̃⟩ = −\frac{1}{2|u|^{2}|r̃|}r̃(|u|^{2})

because u and r̃ commute and are mutually perpendicular. Since r = r̃/ \left|{ \atop r̃}\right| = \frac{∂}{∂ρ}, we infer from the asymptotics of |u|, that this is

\frac{−1}{2|u|^{2}} \frac{∂}{∂ρ} (ρ + b_{u}ρ^{2})^{2} = −\frac{1}{ρ} + O(1) .

A similiar computation shows that T (h/|h|) and T (u/|u|, h/|h|) are bounded. The Lemma then follows from polarisation. •

The scalar curvature of Y_{ρ} is also obtained from the Gauß formula (substitute ϕ′′ = 0 and ϕ′ = ∞ in (3.1)). Hence for small ρ we get:

s^{Y_{ρ}} = s + A − C = a_{2} ρ^{−2} + a_{1} ρ^{−1} + a_{0} − C .

3.2 The Curvature Endomorphism

The manifold X can also be viewed as obtained from M \ N_{ϱ} by blowing up the metric in direction of r. More precisely X is isometric to (M \ N_{ϱ}, \overline{g}) with

\overline{g}(v, w) := g(v, w) + g(v, ∂ψ)g(∂ψ, w) = g(v, w) + |∂ψ|^{2}g(v, r)g(r, w) .

Especially the length of r becomes \sqrt{1 + ϕ′^{2}}. The transition matrix between the metrics g and \overline{g} gives an isomorphism between the spinor bundles of (M, g) and of (M, \overline{g}). The pull back via this isomorphism of the curvature endomorphism of (M, \overline{g}) to the spinor bundle over (M, g) is:

\overline{E} = 4 ∑_{i,j} v_{i}v_{j} ⊗ R_{v_{i},v_{j}} + \frac{4}{\sqrt{1 + ϕ′^{2}}} ∑_{i} rv_{i} ⊗ R_{r,v_{i}}

= E − 4 (1 − \frac{1}{\sqrt{1 + ϕ′^{2}}} ) ∑_{i} rv_{i} ⊗ R_{r,v_{i}}  (3.3)

and its smallest eigenvalue is estimated by

| ; |\overline{E}| ; | ≤ | ; |E| ; | + 4 (1 − \frac{1}{\sqrt{1 + ϕ′^{2}}} ) | ; |∑_{i} rv_{i} ⊗ R_{r,v_{i}} | ; |

≤ | ; |E| ; | + 4 \frac{ϕ′^{2}}{1 + ϕ′^{2}} | ; |∑_{i} rv_{i} ⊗ R_{r,v_{i}} | ; |  (3.4)

Herein D := 4 ||∑_{i} rv_{i} ⊗ R_{r,v_{i}}|| extends to a bounded function on N_{R}.

3.3 Solution of The Differential Inequality

Finally we need to solve the differential estimate \overline{s} − | ; |\overline{E}| ; | > 0. From (3.1), (3.4) and Lemma 3.2 we infer that

\overline{s} − | ; |\overline{E}| ; | ≥ s − | ; |E| ; | + \frac{ϕ′^{2}}{1 + ϕ′^{2}} (A − D − C) + \frac{ϕ′ϕ′′}{(1 + ϕ′^{2})^{2}} B

= s − | ; |E| ; | + \frac{ϕ′^{2}}{1 + ϕ′^{2}} (a_{2}ρ^{−2} + a_{1}(x)ρ^{−1} + a_{0}(x) − D − C)

+ \frac{ϕ′ϕ′′}{(1 + ϕ′^{2})^{2}} (b_{1}ρ^{−1} + b_{0}(x))

So we have solved the problem on X if we can find a decreasing function ϕ on [ϱ, R] such that this expression is positive. Furthermore we need that ϕ vanishes identically near R and that all derivatives of its inverse function χ = ϕ^{−1} vanish at ϕ(ϱ) so that X will inherit a product metric and connection near its boundary.

Eventually after taking an even smaller value of R, we pick positive constants a, b such that on N_{R} the estimates aρ^{−2} ≤ a_{2}ρ^{−2} + a_{1}(x)ρ^{−1} + a_{0}(x) − D − C and b ≤ b_{1}ρ^{−1} + b_{0}(x) hold. Furtermore let ϵ := min(s − | ; |E| ; |) > 0. Then it suffices to solve

ϵ + ϕ′^{4}aρ^{−2} + ϕ′ϕ′′bρ^{−1} > 0  (3.5)

Consider the the differential equation ϕ′^{4}ρ^{−2} a/2+ϕ′ϕ′′ρ^{−1} b = 0 and its solutions

ϕ_{C}(ρ) = ∫_{ρ}^{R} \frac{1}{\sqrt{\frac{a}{b} log x + C}} dx

defined for ρ ≥ ϱ := e^{−Ca/b} for some C ∈ ℝ. For a sufficiently large value of C we can find a decreasing solution of ϵ + ϕ′ϕ′′bρ^{−1} > 0 in the intervall [R/2, R] which vanishes identically near R and extends ϕ_{C} smoothly from [ϱ, R/2] to [ϱ, R] to ensure the proper boundary condition at ρ = R. At the other boundary (3.5) for the inverse function χ reads ϵχ^{2}χ′^{4} + a − bχχ′′ ≥ 0. Let χ_{C} be the inverse function of ϕ_{C} on [0, ϕ(ϱ)] extended by the constant ϱ to all of ℝ^{+}. Then we have a/2 − bχ_{C}(y)χ′′_{C}(y) = 0 for all y ≠ ϕ(ϱ). But χ_{C} can clearly be smoothed keeping a − bχ(y)χ′′(y) ≥ 0.

3.4 The result of glueing

In the above we could make ϱ arbitrarily small. By the remark after Lemma 3.2 we thus may assume s − | ; |E| ; | positive on the handle H_{ϱ} := {x ∈ D^{k+1} × D^{l} | d(x, D^{k+1} × ϱS^{l−1}) = ϱ}. Since both X and D^{k+1} × D^{l} were produced to carry product metric and connection near their boundary, we can glue M′ = X ∪ H_{ϱ} metrically and obtain the desired metric and connection over M′. This proves theorem 1. •

4 Proof of Theorem 2

We will exhibit representatives (M, E) admitting large scalar curvature in every cobordism class in Ω_{n}^{spin}(BSU(r)) ⊗ ℚ with vanishing characteristic numbers Â(T M)[M] and ch (E)Â(T M)[M]. In the sequel cobordism classes will always be understood rationally, i.e. tensored with ℚ, but this will be supressed in the notation. We will produce suitable generators of Ω_{n}^{spin}(BSU(r)) first.

This vectorspace is trivial for n odd. The cobordism classes X = (M, E) ∈ Ω_{n}^{spin}(BSU(r)) are detected by the characteristic numbers

c_{J}p_{I}(X) = c_{J}(E)p_{I}(T M)[M] ,

where c_{J} = c_{r}^{j_{r}} ⋯ c_{2}^{j_{2}} and p_{I} = p_{r}^{i_{s}} ⋯ p_{1}^{i_{1}} for J = (j_{r}, …, j_{2}), I = (i_{s}, …, i_{1}) with 2(rj_{r} + ⋯ + 2j_{2}) + 4(si_{s} + ⋯ i_{1}) = n. First we will define X^{n}(J) ∈ Ω_{2n}^{spin}(BSU(r)) such that the matrix (c_{J′}(X^{n}(r, J)))_{J′,J} has full rank. We will construct appropriate bundles over products of the sphere S^{2} and the complex projective spaces ℂP^{2n+1}:

For J = (j_{r}, …, j_{2}) with ∑_{i=2}^{r} ij_{i} = n we define (r ×n)-matrices M_{J}^{r}. If r ≥ 4 let

M_{J}^{r} := ( 1 ⋯ 1 ; \underbrace{−1 ⋯ −1}_{j_{r}} 1 ⋯ 1  0 ; \underbrace{−1 ⋯ −1}_{j_{r}+j_{r−1}} ; · ; · ; · ; 1 ⋯ 1 ; 0  \underbrace{−1 ⋯ −1}_{j_{r}+j_{r−1}+...+j_{3}}  1 ⋯ 1 ; \underbrace{−1 ⋯ −1}_{j_{r}+j_{r−1}+...+j_{3}+2j_{2}} )

For r = 3 and J = (j_{3}, j_{2}) with j_{3} > 1 let

M_{J}^{3} = ( \begin{matrix}1 ⋯ 1 & 1 & 0 ⋯ 0 \\ −1 ⋯ −1 & 0 & 1 ⋯ 1 \\ \underbrace{0 ⋯ 0}_{j_{3}} & −1 & \underbrace{−1 ⋯ −1}_{j_{3}+2j_{2}−1} \end{matrix} )

Then let

X^{n}(r, J) = (\underbrace{S^{2} × ⋯ × S^{2}}_{n}, E(M_{J}^{r})),  (4.1)

with

E(M_{J}^{r}) = ⨁_{i=1}^{r} γ_{1}^{ϵ_{1,i}} ⊗ ⋯ ⊗ γ_{n}^{ϵ_{n,i}}  (4.2)

for M_{J}^{r} = (ϵ_{μ,ν})_{μ=1...n,ν=1...r}. Here γ_{q} is the canonical complex line bundle over the qth factor S^{2} in (4.1). Slightly abusing the system of notation above define

X^{2n+2}(2, (n + 1)) := (ℂP^{2n+1} × S^{2}, (η ⊗ γ) ⊕ (η^{−1} ⊗ γ^{−1}))  (4.3)

and

X^{2n+1}(3, (a, b, c)) := (ℂP^{2n+1}, η^{a} ⊕ η^{b} ⊕ η^{b})  (4.4)

for a, b, c ∈ ℤ, −n ≤ a, b, c ≤ n, a + b + c = 0, if n ≥ 2. If n = 1 we take a = 2, b = c = −1. In (4.3) and (4.4) η, γ denote the canonical bundles over ℂP^{2n+1} and S^{2}.

Lemma 4.5 The X^{n}(r, J) above admit large positive scalar curvature with exception of X^{2}(2, (2)) and X^{3}(3, ((2, −1, −1)))

Proof: In [3] Hitchin has proved that (ℂP^{q}, η^{s}) admits large scalar curvature if q ≥ 2s and that s−||E(η^{s})|| = 0 if q = |s| = 1. It is immediate from the definition

that | ; |E(E ⊕ F)| ; | = max(| ; |E(E)| ; | , | ; |E(F)| ; |) and | ; |E(E ⊗ F)| ; | ≤ | ; |E(E)| ; | + | ; |E(F)| ; |.

Thus we can estimate

||E(E(M_{J}^{r}))|| ≤ max_{i} (∑_{q=1}^{n} ϵ_{q,i} | ; |E(γ)| ; |) < n | ; |E(γ)| ; |

because, with the above exceptions, in every row of the matrices M_{J}^{r} at least one entry vanishes. Since the scalar curvature of the round S^{2} equals | ; |E(γ)| ; |, we thus get that the scalar curvature of S^{2} × … × S^{2} is larger than ||E(E(M_{J}^{r}))||. The cases involving ℂP^{2n+1} are similiar. •

Lemma 4.6 The matrix (c_{J′} (X^{n}(s, J)))_{J′,(s,J)}, s ≤ r, has full rank.

Proof: We compute the Chern class of the vectorbundle E(M_{J}^{r}): Denoting by x_{q} = c_{1}(γ_{q}) the generator of the second cohomology group of the qth factor S^{2} in (4.1) we obtain from (4.2) that:

c_{k}(E) = ∑_{μ_{1},...μ_{k},ν_{1},...ν_{k}} ϵ_{μ_{1},ν_{1}} ⋯ ϵ_{μ_{k},ν_{k}}x_{ν_{1}} ⋯ x_{ν_{k}},

where the μ_{s} respectively ν_{s} in this sum are pairwise distinct. Order the partitions I, J lexicographically. Observing that x_{s}^{2} = 0 we get for r ≥ 4 that

c_{I}(X^{n}(r, J)) = \{ \begin{matrix}0 & if I > J \\ ≠ 0 & if I = J \end{matrix}  (4.7)

Thus this part of the matrix is triangular. If r = 3, then a straightforward calculations gives that c_{j_{3},j_{2}}(X^{n}(3, (j_{3}, j_{2}))) = (−1)^{j_{3}+j_{2}−1}(j_{3} −1)(j_{3} +j_{2})j_{3}!(2j_{3} + 2j_{2} − 1)!. If n is even then j_{3} ≠ 1 and (4.7) still holds. For the remainder of the matrix we use the manifolds defined in (4.3) and (4.4). For r = 2 we clearly have c_{2}^{n+1}((η ⊗ γ) ⊕ (η^{−1} ⊗ γ^{−1})) = 2(−1)^{j_{2}}j_{2} ≠ 0. We are left with the case r = 3 and n odd. The Chernclasses of η^{a} ⊕ η^{b} ⊕ η^{b} are given by the elemenary symmetric polynomials σ_{3}, σ_{2} in a, b, c. Assume that the polynomial

P (a, b, c) := ∑_{j_{3},j_{2}} α_{j_{3},j_{2}}c_{j_{3},j_{2}}(X^{n}(3, (a, b, c))) = ∑_{j_{3},j_{2}} α_{j_{3},j_{2}}σ_{3}^{j_{3}}σ_{2}^{j_{2}}

of degree 2n + 1 vanishes for all a, b, c as after (4.4). Then the polynomial P (a, b, −a − b) vanishes for all a, b ∈ ℤ with −n ≤ a, b, a + b ≤ n. Since it is homogeneous it must be divisible by all (na + sb) and (sa + nb), s = 0 … n and if n ≥ 2 it must also contain (a − b) hence have degree at least 2n + 2. Therefore P vanishes on the entire plane a+b+c = 0. Since it does not contain σ_{1} and since there are no algebraic relations between the elementary symmetric polynomials, the coefficients α_{j_{3},j_{2}} are all 0.

•

Let K_{n,r} ⊂ Ω_{2n}^{spin}(BSU(r)) be the kernel of those c_{J}p_{I} with nontrivial I. We have shown that the span of the X^{n}(r, J) as above projects onto K_{n,r}. It is well known that ⨁_{n} Ω_{n}^{spin} is polynomially generated by the Kummer surface K and the quaternionic projective spaces ℍP^{n}, n ≥ 2. In view of the direct sum decomposition

Ω_{2n}^{spin}(BSU(r)) = ⨁_{p=0}^{n} K_{p,r} × Ω_{2n−2p}^{spin}  (4.8)

we infer from Lemma 4.6 that there is a basis of Ω_{n}^{spin}(BSU(r)) consisting of monomials in K, quaternionic projective spaces and one of the X^{n}(r, J). Among these only K, X^{2}(2, (2)) and X^{3}(3, ((2, −1, −1))) do not admit large scalar curvature. Therefore the only monomials not admitting large scalar curvature are of the form K^{d/4−1} × X^{2}(2, (2)) or K^{d/4} if the dimension d is divisible by 4 and K^{(d−2)/4−1} × X^{3}(3, ((2, −1, −1))) if the dimension is d = 2 mod 4. These monomials are also detected by the characteristic numbers Â and ch Â.

References

[1] N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Grundlehren der Mathematischen Wissenschaften, 298, Springer Verlag, Berlin and New York, 1991

[2] M. Gromov and H. B. Lawson, Jr., The classification of simply-connected manifolds of positive scalar curvature, Ann. of Math. 111, (1980), 423-434

[3] N. Hitchin, Harmonic spinors, Adv. Math. 14, 1974, 1–55

[4] H. B. Lawson and M.-L. Michelson, Spin Geometry, Princeton University Press, Princeton, 1989

[5] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris, Sér. A-B 257, 1963, 7–9

[6] T. Miyazaki, Simply connected spin manifolds with positive scalar curvature, Proc. Amer. Math. Soc. 93 (1985), no. 4, 730–734

[7] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no. 3, 511–540

[8] R. Schoen, S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159–183