Ronnie Lee

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.

Finite group actions on P2(C)

Consider the question: which finite groups operate as symmetries of the complex projective plane P’(C)? Any finite subgroup of &X,(C) acts as a group of collineations and these give the linear models. The list of such groups is relatively short [MBD] but contains, for example, a groups of rank < 2, subgroups of U(Z), and the simple groups A,, A,, an PSL(F,). It turns out that these linear groups are the only ones which ca operate topologicaliy on P’(C) with reasonable ~~avior near the s~~~uIar set. An action is called ~“locally linear” if each singular point has an invariant neighborhood which is equivariantly homeomur~bic to a ~eigbborhood of 0 in a (real) representation space.

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

Computation of wall groups

However, most of these groups are still impossible to compute. Recent works of Browder [3], Lee [6], Shaneson [9] and Wall [l l] have been successful only in computing a very few special cases. All of their results rely more or less on the geometric interpretation of the Wall groups. In this paper we shall study the algebraic properties of the Wall groups and prove that, for each odd prime p, the Wall group L3(kp) is always zero (see (1.12)).

Cohomology of the satake compactification

ONE OF the interesting spaces in algebraic geometry is the quotient space of the Siegel upper half space E, under the action of the discrete group r, = Sp,(Z). In the language of algebraic geometry, this quotient space G,/F, is the coarse moduli space of principal polarized abelian varieties (for definition see [16]). More than thirty years ago, Satake discovered a natural compactification 6,*/r, of this space obtained by attaching lower dimensional Siegel spaces, so that as a set

Surgery on closed 4-manifolds with free fundamental group

Even though the disk embedding theorem is not available in dimension 4 for free fundamental groups, some surgery problems may be shown to have topological solutions. We prove that surgery problems may be solved if one considers closed 4-manifolds and the intersection pairing is extended from the integers, and prove a related splitting result.

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.

Smooth group actions on definite

In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4-manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this collection, further examples of smooth but generally nonalgebraic actions can be constructed by an equivariant connected sum along an orbit of isolated points. Given a smooth oriented 4-manifold X which is diffeomorphic to a connected sum of algebraic surfaces, we can ask: (i) which (finite) groups can act smoothly on X preserving the orientation, and (ii) how closely does a smooth action on X resemble some equivariant connected sum of algebraic actions on the algebraic surface factors of X? For the purposes of this paper we will restrict our attention to the simplest case, namely X p2(C) #...# p2(C), a connected sum of n copies of the complex projective plane (arranged so thatX is simply connected). Furthermore, ASSUMPTION. All actions will be assumed to induce the identity on H,(X, Z).

4

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.

-manifolds and moduli spaces

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