59. Algorithmic aspects on homeomorphism problems (preprint, with Nabutovsky)
We show that homeomorphism of closed simply connected manifolds and for embeddings in codimension other than two is solvable by combining work of E. Brown, Sullivan on rational homotopy theory, and Grunewald-Segal on actions of arithmetic groups. On the other hand, we show that even for homotopy equivalent manifolds, one cannot determine homeomorphism in higher dimensions. This makes use of the results of [60].
60. Higher Rho Invariants (preprint)
Here I define invariants of "antisimple manifolds", i.e. manifolds with no middle dimensional handles, that are secondary to a solution of the Novikov conjecture. These are connected to the method of [27, 50] which drafts stratified spaces with good intersection homology properties into the service of analysis of the signature operator. The most novel aspect of the result is the use of acyclicity to help with eta invariants -- rather than for torsions.
61. Large Riemannian Manifolds which are flexible (preprint, with Dranishnikov and Ferry)
With one example, we disprove a coarse analogue of the Borel rigidity conjecture and of the Baum-Connes Conjecture, as well as a question of Gromov's regarding whether all uniformly contractible Riemannian n-manifolds have degree one Lipschitz maps to euclidean space. These analogues arise in work of Higson and Roe, and in our works on the Novikov conjecture [28,36,51]. The core of the construction is a refinement of Dranishnikov's celebrated infinite dimensional CE image of a finite complex.
62. Equivariant Periodicity for Abelian Group Actions (preprint, with Yan)
We give a periodicity theorem for isovariant structure sets for all abelian groups. For odd order group actions, Yan had accomplished this in his thesis, using a theory of periodicity manifolds. Here we introduce some stratified spaces to replace the nonexistent periodicity manifolds. These spaces give a way to systematically exploit some multiplicativity of the signature phenomena (i.e. cases where Atiyah's characteristic classes canonically vanish). This gives some new structure to equivariant topological surgery, and has implications for the equivariant nature of isovariant geometric problems, some of which appear in [47] and in [67].
63. Arithmetic manifolds of positive scalar curvature (preprint, with Block)
We begin an exploration of which noncompact manifolds have complete metrics of uniformly positive scalar curvature, a la the Gromov-Lawson-Rosenberg philosophy. For locally symmetric spaces we show that the existence of such a metric implies the fundamental group is arithmetic, and that in terms of such structure the necessary and sufficient condition is that the q-rank be at least three.
64. Neighborhoods in stratified spaces with two strata (preprint, with Hughes, Taylor, and Williams)
Here we prove the "teardrop neighborhood theorem" announced in my book, which gives a precise topological structure result for neighborhoods in stratified spaces in terms of approximate fibrations. (Neither fiber bundles nor block bundles exist in sufficient generality.) It implies local contractiiblity of certain homeomorphism groups, finishes off some points in Quinn's h-cobordism theorem for stratified spaces, and is useful for proving the stratifgied surgery exact sequence.
65. On smooth surgery II (preprint)
G/O does not have an H-space structure for which the surgery obstruction map is a homomorphism, unlike G/PL or G/Top.
66. Critical points of riemannian functionals and arithmetic groups (preprint, with Nabutovsky)
We show that there are an infinite number of (nonrecursively deep) local minima for diameter on the space of riemannian metrics of curvature between -1 and 1. The method uses ways of eliminating degenerations of riemannian metrics due to Gromov, compactness theory a la Cheeger and Gromov, and some logic to reduce the statement to a logico-geometric lemma: In every dimension at least five, there is a machine that produces homology n-spheres that either have nonzero simplicial norm or are the sphere in such a way that the inputs for which the sphere is produced is not a recursive set. The existence of a single homology sphere with nonzero simplicial norm does not seem to have been known beyond mension three. These are produced using some logic, work of Hausmann and Vogel on homology spheres, and results about the cohomology of airthmetic groups due to Borel-Wallach, and Clozel (extending the examples of Rappoport-Zink, where classes predicted by the Matsushima formula are shown not actually arise for arithmetic reasons).
67. Nonlinear averaging of embeddings and group actions (preprint)
We explore the connection between the gap hypothesis, usually assumed in the theory of compact transformation groups, and embedding theory, and show that one can often get classification theories up to torsion phenomena if one combines the already existent theories with analysis of the embeddings, even when the gap hypothesis fails. (This has applications to realization of linking numbers among components of fixed point sets on e sphere, general classification theory, and the equivariant Borel conjecture among others.)
As I think all careful readers would notice, I denoted by Emb the embeddings of F in M homotopic to a given map. (The carefree reader might imagine that I believed that if very high codimension all embeddings are isotopic.) The confusion that this could lead to does not effect any of the examples.
68. (with A. Nabutovsky) The fractal geometry of Riem/Diff I. (draft)
This paper begins an exploration of the geometry of the moduli space of isometry classes of Riemannian metrics (with two-sided curvature bounds) on a given compact smooth manifold M of dimension >4. We expand upon the methods introduced in #66, but here have a more geometric focus.
We view the moduli space as being divided up into "basins" each of which rises up from a local minimum of the diameter functional. The main result asserts that at infinitely many scales, the critical points that are deep from the point of view of that scale are dense at that scale. (These scales are getting larger and larger, so we are really having a "large scale fractal structure".) This means that there are infinitely many basins, and the basin have basins, which themselves have basins coming off them, and so on.
Essentially, the idea of scale that most naturally enters here is measured by the hierarchy of functions given by stopping times of Turing machines. Paths in the moduli space are essentially the same thing as computations. As a result, if one starts with a Turing machine whose degree is lower than that of another, the critical points produced will be shallower, but more dense.
We also get information about various filling functions (in the sense of Gromov). For instance, the homological filling function for S^n, n>4, is not bounded by a computable function, but for hyperbolic manifolds, (at least the low dimensional homology) function is, if one measures cycles that are based at some origin.
A future paper will deal with "antifractal properties", i.e. behaviors that distinguish basins of the same depth from one another.
Remark: Given the interest this paper seems to have generated in different quarters, it is being rewritten (summer 99), hopefully to be accessible to a wider audience. Hopefully, this will not take too long.
69. This is a survey.
70. The main result is that there are closed manifolds whose universal covers do not have 0 in the spectrum of laplacian on forms in any dimension.
71. We discuss the difference of the eta invariants of a manifold and its universal cover (taken in the L^2 sense). We show that it is a homotopy invariant for some groups with property T (a challenge suggested by work of Keswani). We also show that if M is of dimension 3 mod 4, and larger than 3, then this difference is never homotopy invariant; in particular there are infinitely many manifolds homotopy equivalent to such an M and not homeomorphic to it.
72. The main result extends work of Chichilnisky and Heal and shows that social choice is impossible for a finite connected noncontractible complex and any society with even a fixed number of agents.We discuss also infinite complexes and make remarks on a suggestion of Smales for avoiding the social choice paradoxes.