Xiaofei ZHANG Yanlin LIU Hongzhi LIU

Abstract In this paper,the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.

Keywords Positive scalar curvature at infinity, K-theory of C∗-algebras,Higher index theory

1 Introduction

One of the most important applications of the Atiyah-Singer index theorem(see[1])in Riemannian geometry is on the study of manifolds of positive scalar curvature(see[8]).Under the assumption thatMis a compact spin manifold,in[9],Lichnerowicz established the formula index,it was shown that an enlargeable manifold cannot carry any positive scalar curvature metric(see[3]).

The index theoretic method can be generalized to study the positive scalar curvature metric outside a compact set of noncompact manifolds.In[4],Gromov and Lawson developed a relative index theorem to show the nonexistence of uniformly positive scalar curvature metric for a large class of noncompact manifolds.Roe studied the problem of the existence of positive scalar curvature metric outside a compact set on a complete manifold in terms of theK-theory of the Roe algebra and the coarse index(see[10-11]).

LetMbe ann-dimensional complete noncompact manifold with fundamental group Γ.In[2],Block and Weinberger proved a Bochner type vanishing theorem.More precisely,ifMcarries a uniformly positive scalar curvature metric offa compact setM1,andNis an(n−1)-dimensional submanifold ofMM1,then the higher index of the Dirac operator onN,viewed as aK-theory class invanishes.Note that there is no assumption on whetherNbears any positive scalar curvature metric.Block and Weinberger gave this Bochner type vanishing theorem to illustrate an obstruction to the existence of the uniformly positive scalar curvature metrics onMM1by the higher index of the Dirac operator onN.This in turn helped them to prove that the arithmetic manifold with-rank 1 or 2,carries no uniformly positive scalar curvature metric.

Block and Weinberger proved their Bochner type vanishing theorem by aKK-theory approach,which is a powerful tool but also a method that is famous for its great difficulty.The localization algebra approach arises for the reason that on the one hand usually it is as useful asKK-theory approach is,on the other hand it would greatly simplified theKK-theory method.In this paper,we take a localization algebra approach to provide a much more concise proof of their theorem.

Our proof is built on the definition of the equivariant coarse index introduced in[10]and theK-theory of the equivariant localization algebra introduced in[17].The advantage of Roe’s equivariant coarse index is that it can be represented by a geometrically defined operator.As shown in[17],theK-theory of the equivariant localization algebra is a homology theory,and satisfies a Mayer-Vietoris sequence.Our new proof is a direct computation of the equivariant coarse index of the Dirac operator via the Mayer-Vietoris sequence of theK-theory of the equivariant localization algebras.Our work is inspired by[11,15-16].Nevertheless,our method in this paper can also be adapted to simplify the proof of the main result of[16].

This paper is organized as follows.In Section 2,we recall the definition of theK-homology class and the equivariant coarse index of the Dirac operator in terms of theK-theory of equivariant localization algebra and equivariant Roe algebra.In Sections 3-4 we carry out some computations about theK-homology class and the equivariant coarse index of the Dirac operator respectively.In Section 5,we state and prove our main result.

2 Preliminarie s

In this section,we first recall the definition of the equivariant Roe algebra and the equivariant localization algebra.The Mayer-Vietoris sequence of theK-theory of the localization algebra is also briefly reviewed.We then recall the construction of the equivariant coarse index and theK-homology class of Dirac operator.

2.1 Equivariant Roe algebra and equivariant localization algebra

In this subsection,we recall the definitions of the equivariant Roe algebra and the equivariant localization algebra,and review the Mayer-Vietoris sequence of theK-theory of the equivariant localization algebra.We refer the readers to[10,14,17]for more details.

Suppose thatXis a proper metric space,i.e.,every closed ball inXis compact.AnXmodule is a separable Hilbert space equipped with a ∗-representation ofC0(X),the algebra of all continuous functions onXwhich vanish at infinity.AnX-module is called nondegenerate if the ∗-representation ofC0(X)is nondegenerate.AnX-module is said to be standard if no nonzero function inC0(X)acts as a compact operator.

IfX=Y∪ZwhereYandZareG-invariant closed subsets ofX,we have

2.2 K-homology class and equivariant coarse index of Dirac operator

In this subsection,we recall the construction of theK-homology class and the equivariant coarse index of the Dirac operator on a spin manifold.

WhenMis even dimensional,sinceDis an odd operator andχis an odd function,there exists the following decomposition

3 The K-Homology Class of Dirac Operator under the Connecting Map

Figure 1

Odd caseWhennis odd,

We need the following lemma.

Thusutis the desired representative element for[DM].

By the definition of the connecting map,we have

We have

we have

4 The Representative Class of the Coarse Index of Dirac Operator

where{ei}is an orthonormal basis ofTxM,we have

4.1 The dimension of M is odd

and it turns out that

and

Thanks to the triangle inequality,we have

we have

4.2 The dimension of M is even

Similarly we can show that

Thus we have

and

Thus we have

5 Main Result

In this section,we shall give a proof of the following Bochner type vanishing theorem which was first obtained by Block and Weinberger in[2].

Then we have

AcknowledgementsThe authors are grateful to Prof.Xiaoman Chen,Prof.Shengzhi Xu,Prof.Xiang Tang and Yi-Jun Yao for their guidance,and they also want to thank Prof.Zhizhang Xie and Prof.Guoliang Yu for their helpful comments.