Transversality and the index theorem
The author presents a new proof of the index theorem based on transversality ideas of Thom. He achieves a proof for the Lipschitz and topological manifolds independent of the proof for smooth manifolds. Reading this paper assumes familiarity with work of Cheeger on the Hodge theory of Riemannian pseudomanifolds, of the author and Sullivan on the analytical proof of Novikov's theorem on rational Pontryagin classes, and of the author on the index of signature operators and the index theorem for topological manifolds. The formulation in the present paper allows a possible axiomatization of the index theory. The two main ideas used in the proofs of the Atiyah-Singer and Sullivan-Teleman index theorems involve the Chern character of the symbol of the operator and the L-character of the base manifold. However, the expression of the L-character in terms of Pontryagin classes is complicated. On the other hand, Thom's construction, which is based on transversality, gives a simple realization of the L-character for homology manifolds and this is the idea underlying the present proof. It is shown that the signature operator with coefficients in an arbitrary aspherical complex vector bundle splits in a product of two operators: a signature operator with constant coefficients over the support of the bundle and a signature operator over an even dimensional sphere. This then allows a proof by ''standard'' direct computations over the base manifold. These computations involve the Hirzebruch-Thom formula, the Bott isomorphism, Hodge theory, excision invariance of the analytical index, the multiplicativity of the index, and the index formula for spheres.
Hodge theory of Riemannian pseudomanifolds
scientific article; zbMATH DE number 3965051
scientific article