Signature Operator Research Articles

Global Analysis on PL-Manifolds

The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with “constant volume density” implies smoothing. A geometric realization of ${\text {PL}}\left ( m \right ){\text {/O}}\left ( m \right )$ is given in terms of Riemannian metrics. A graded differential complex ${\Omega ^ {\ast } }( M )$ is constructed: it appears as a subcomplex of Sullivan’s complex of piecewise differentiable forms. In the complex ${\Omega ^{\ast }}( M )$ the operators $d$, $\ast$, $\delta$, $\Delta$ are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator.

Global analysis on PL-manifolds

The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with “constant volume density” implies smoothing. A geometric realization of PL ( m ) /O ( m ) {\text {PL}}\left ( m \right ){\text {/O}}\left ( m \right ) is given in terms of Riemannian metrics. A graded differential complex Ω ∗ ( M ) {\Omega ^ {\ast } }( M ) is constructed: it appears as a subcomplex of Sullivan’s complex of piecewise differentiable forms. In the complex Ω ∗ ( M ) {\Omega ^{\ast }}( M ) the operators d d , ∗ \ast , δ \delta , Δ \Delta are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator.

The signature theorem for V-manifolds

Here L(g, M) is the equivariant L-class defined by Atiyah_Singer[4]. By choosing suitable metrics and connections, we can express L(g, M) in differential forms. These characteristic forms are defined locally. So we can expect that the above formula still has a sense for a space which is locally homeomorphic to an orbit space of above. type. That is our theorem. The method of our proof is so-called “heat kernel-zeta function method”. We are greatly indebted to the papers, Seeley[9], Atiyah, Bott and Patodi[2] and Gilkey [5]. Our theorem is closely related to the Lefschetz fixed point formulae. Many mathematicians have worked in this field: The results in 82 are actually included in Giikey[61 and, as for the signature operator, our results are completely proved by Donnelly and Patodi[ lo]. But our computations are independent of their works and yield more explicit formula in general framework. For the precise informations of our computations, see Kawasaki [7].

On the heat equation and the index theorem

The main error occurs on page 306 where it is implicitly assumed that the coefficients of the two operators A*A and AA* (associated to the signature operator A) are polynomial functions in the gij, their derivatives and (det g)-l . As we shall show later this is not quite t r u e t h e coefficients also involve d ] f ~ and the inverses of the principal minors of the matrix gu" Thus the form m in (5.1) is not a regular invariant of the metric in the sense of w 2, and so the Gilkey Theorem as formulated on p. 284 does not apply. To correct this we shall widen the notion of regularity (so as to include, in particular, the form ~o above) and then check that our proof of Gitkey's Theorem still holds in this wider context. In w regularity was only defined for invariants of a Riemann structure g (i.e. satisfying the naturality or invariance property (2.3)). It will perhaps make for greater clarity if we introduce our new notion of regularity for any function of g, independently of the invariance property. We shall say that f(g) is a regular function of g if, in any coordinate system, we have