指标定理（五） (9-8)

Years later，Ativah，Raoul Bott，and Vijay Patodi (1973) gave a new proof of the index theorem using the heat kernel. Due to (Teleman 1983)，(Teleman 1984)，the theorem is generalized to any abstract elliptic operator (Atiyah 1970) on a closed，oriented，topological manifold. Later，Connes-Donaldson-Sullivan-Teleman index theorem arises due to (Donaldson and Sullivan 1989)， (Connes, Sullivan and Teleman 1994). Also，we have equivariant index theorem，and index theorem for families of elliptic operators.²

The index of Dirac operators was used to formulate and then prove the Gromov-Lawson conjecture： A compact，spin，simply connected manifold of dimension less than or equal to five admits a metric of positive scalar curvature iff the index of the spin Dirac operator is zero. ³

Also，the theory has found applications in physics. In more recent years，there is a theory named nonabelian gauge field theory of C. N. Yang and R.L. Mills which has led to astonishing results in dimension four.

Yang-Mills theory can be plausibly considered a generalization of Dirac’s theory which encompasses three fundamental forces.The theory of connections， Dirac-type operators，and index theory all play an important role.

References

[1] 苏竞存，流形的拓扑学，武汉大学出版社，1992.

[2] Raoul Bott and Loring Tu. Differentiαl Forms in Algebrαic Topology. Springer-Verlag New York，1982.

[3] Otto Forster，Lectures on Riemαnn Surfαces. Springer-Verlag New York，1981.

[4] Allen Hatcher. Vector Bundles αnd K-Theory.

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