Edward Y. Miller

SELF-ADJOINT ELLIPTIC OPERATORS AND MANIFOLD DECOMPOSITIONS. PART I: LOW EIGENMODES AND STRETCHING

This paper is the first of a three-part investigation into the behavior of analytical invariants of manifolds that can be split into the union of two submanifolds. In this article, we will show how the low eigensolutions of a self-adjoint elliptic operator over such a manifold can be studied by a splicing construction. This construction yields an approximated solution of the operator whenever we have two L2-solutions on both sides and a common limiting value of two extended L2-solutions. In Part 11, the present analytic “Mayer-Vietoris” results on low eigensolutions and further analytic work will be used to obtain a decomposition theorem for spectral flows in terms of Maslov indices of Lagrangians. In Part I11 after comparing infinite- and finite-dimensional Lagrangians and determinant line bundles and then introducing “canonical perturbations” of Lagrangian subvarieties of symplectic varieties, we will study invariants of 3-manifolds, including Casson’s invariant. 0 1996 John Wiley & Sons, Inc.

Cohomology of harmonic forms on Riemannian manifolds with boundary

Abstract On a smooth compact manifold M, the cohomology of the complex of differential forms is isomorphic to the ordinary cohomology by the classical theorem of de Rham. When M has a Riemannian metric g, the harmonic forms constitute a subcomplex of the de Rham complex because the Laplacian commutes with exterior differentiation. When (M, g) has no boundary, all of its harmonic forms are closed, and hence the cohomology of this subcomplex is isomorphic to the ordinary cohomology by the classical theorem of Hodge. But when the boundary of (M, g) is non-empty, it is possible for a p-form to be harmonic without being closed, and some of these, which are exact, although not the exterior derivatives of harmonic p – 1-forms, represent an “echo” of the ordinary p – 1-dimensional cohomology within the p-dimensional harmonic cohomology.

Self‐adjoint elliptic operators and manifold decompositions Part II: Spectral flow and Maslov index

This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous “static” results in obtaining results on the decomposition of spectral flows. Some of these “dynamic” results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of “jumping Lagrangians.” In Part 111, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce “canonical perturbations” of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson’s invariant. 0 1996

The action of the Torelli group on the homology of representation spaces is nontrivial

Abstract The mapping class group and its subgroup, the Torelli group, of a Riemann surface M has a natural action on the space R SU(2) (M) of SU(2) -representations of the fundamental group of M and its subspace R SU(2) (M)) irred of irreducibles. In this paper we compute the cohomology H ∗ (R SU(2) (M)), H ∗ (R SU(2) (M) irred ) of both of these spaces and show that the induced action of the Torelli group is non-trivial.

Some invariants of spin manifolds

Abstract The Rochlin invariant of a compact 3-manifold with a fixed spin structure can be generalized to high dimensions. This paper explores these generalized Rochlin invariants and shows that they are spectral invariants.

Self‐adjoint elliptic operators and manifold decompositions Part III: Determinant line bundles and Lagrangian intersection

The theory of spectral flows developed in the series [10, 11, 12], and the present paper has a wide range of applications to important geometric operators on compact manifolds. To present our results on spectral flow and manifold decomposition, the present paper develops a theory of determinant line bundles and infinitedimensional Lagrangians associated to self-adjoint elliptic operators on compact manifolds. The trace-class properties of these infinite Lagrangians established here and the precise uniform estimates relating them to finite Lagrangians are crucial for such a determinant line bundle approach to analytical questions. As an application, we elucidate the Walker’s and other generalizations of Casson’s SU(2) representation theoretic invariant of 3-manifolds in terms of the -invariant of certain Dirac operators. This is carried out by introducing the technique of “canonical perturbations” of singular Lagrangian subvarieties in symplectic geometry. At the end of Part II of this series, we obtained a formulation of the spectral flow of a family of self-adjoint elliptic operators D(u) : L2(E)! L2(E) in terms

Eliminating depth cycles among triangles in three dimensions

Given

Splitting spaces with finite group actions

n

A sublattice of the Leech lattice

pairwise openly disjoint triangles in 3-space, their vertical depth relation may contain cycles. We show that, for any

On the maslov index

\varepsilon>0