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Topology of Manifolds / Topologie des variétés
(Ronnie Lee and Ian Hambleton, Organizers)

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HANS BODEN, Department of Mathematics, Ohio State University Columbus, Ohio  43210-1174, USA
Integral and non-integral SU(3) generalizations of the Casson invariant, Part I

This talk will present a definition of a new generalized SU(3) Casson invariant for homology spheres. The new invariant is integer-valued, and one interesting problem is to compare it to the existing definition of the SU(3) Casson invariant, which is real-valued (and only conjecturally rational).

We will discuss recent computations (joint with C. Herald, P. Kirk, and E. Klassen) of both the real-valued and integer-valued invariant for families of homology spheres obtained by Dehn surgery on torus knots. This uses a careful analysis of Floer-type perturbations in the SU(3) context to express the difference between the two invariants as the sum total of the rho invariants, summed over the flat SU(2) moduli space. The rho invariants are then computed in terms of the C² spectral flow and Chern-Simons invariants for flat SU(2) connections.

LEO BUTLER, Queen's University, Kingston, Ontario  K7L 3N6
Topology and integrable geodesic flows

A standard technique in differential geometry is to impose a geometrical constraint and determine the topological implications of this. For example, it is well-known that if Mⁿ is compact and admits an everywhere negative curvature metric, then Mⁿ is covered by Rn.

What can one say if a compact riemannian manifold (M,g) has a Liouville-integrable geodesic flow? The answer depends on the smoothness of the first integrals of the geodesic flow. In the real-analytic category, it is known that p₁(M) must be almost abelian. It will be shown that there are real-analytic riemannian manifolds (Mⁿ,g) that have Liouville-integrable geodesic flows with C^¥ first integrals such that p₁(Mⁿ) is n-1-step torsion-free nilpotent. Bolsinov and Taimanov have provided an example where p₁(Mⁿ) is of exponential growth.

DIARMUID CROWLEY, Department of Mathematics, Indiana University, Indiana  47405, USA
Classification of 3-sphere bundles over the 4-sphere

I shall present a classification (join with Christine Escher) of the total spaces of 3-sphere bundles over the 4-sphere up to homotopy equivalence, homeomorphism, PL-homeomorphism and diffeomorphism. Also, I present the action of self homotopy equivalences on the structure set for all of these manifolds.

JAMES F. DAVIS, Department of Mathematics, Indiana University, Bloomington, Indiana  47405, USA
Algebraic K-theory of virtually cyclic groups

This is joint work with Bogdan Vajiac. We use controlled topology, à la Farrell-Jones, applied to the hyperbolic plane, to compute the Whitehead group and reduced projective class group of discrete groups which surject to the infinite dihedral group. An equivalent characterization of such a group is that it is an amalgamated product of two groups along a common index 2 subgroup. Our precise result is the following:

Theorem. Let G be a group surjecting to the infinite dihedral group D. Let H be inverse image of the infinite cyclic subgroup of index 2 of D. Then H is a semidirect product of a group with the infinite cyclic group, and, as such, has associated Nil groups, Nil⁺(H) and Nil^-(H). Since G is an amalgamated product, there is the associated Waldhausen Nil-group WNil(G). Then Nil⁺(H) is isomorphic to WNil(G).

Corollary. Let G be a group which is an amalgamated product of two groups along a common index 2 subgroup which is a finite group of square-free order. Then there is a Mayer-Vietoris exact sequence computing the Whitehead group and reduced projective class group of G.

CHRIS HERALD, Department of Mathematics, University of Nevada, Reno Nevada  89557, USA
Integral and non-integral SU(3) generalizations of the Casson invariant, Part II

This talk will present a definition of a new generalized SU(3) Casson invariant for homology spheres. The new invariant is integer-valued, and one interesting problem is to compare it to the existing definition of the SU(3) Casson invariant, which is real-valued (and only conjecturally rational).

We will discuss recent computations (joint with H. Boden, P. Kirk, and E. Klassen) of both the real-valued and integer-valued invariant for families of homology spheres obtained by Dehn surgery on torus knots. This uses a careful analysis of Floer-type perturbations in the SU(3) context to express the difference between the two invariants as the sum total of the rho invariants, summed over the flat SU(2) moduli space. The rho invariants are then computed in terms of the C² spectral flow and Chern-Simons invariants for flat SU(2) connections.

HEATHER JOHNSTON, Department of Mathematics, University of Massachusetts, Amherst, Massachusetts  01003
Homology manifold bordism: Everyone has an evil twin

Homology manifolds are spaces with the local homology, but not necessarily the local topology of manifolds. Bryant, Ferry, Mio and Weinberger have discovered a large class of examples of non-resolvable homology manifolds which have no manifold points whatsoever: For every simply connected topological manifold M of dimension > 5 and every integer congruent to 1 modulo 8, they constructed an ``evil twin'' homology manifold which is simple homotopy equivalent to M. Furthermore BFMW showed that there is a surgery exact sequence for homology manifolds.

The surgery exact sequence of BFMW implies that there are topological manifolds, such as high dimensional tori, which are not simple homotopy equivalent to any other homology manifolds. Nonetheless, every high dimensional topological manifold M has a ``bordism evil twin'' N for each integer congruent to 1 modulo 8: there are maps f: M ® N and g: N® M so that both compositions are normally bordant to the appropriate identity map. Various bordism and transversality results for homology manifolds follow from this result and the BFMW surgery exact sequence for homology manifolds.

LOWELL JONES, Stoney Brook
Classifying chain complexes

Let C(*) denote a chain complex over a PIDR such that each module C(i) is finitely generated. Two such chain complexes are stably isomomorphic if they become isomorphic after direct summing with ``elementary'' chain complexes. (A chain complex is ``elementary'' if for some i the boundary map from C(i+1) to C(i) is an isomorphism and C(j) = 0 for all j not equal i+1 or i.) If all of the modules C(i) are free R-modules then the stable isomorphism type of C(*) is determined by its homology groups. However if some of the modules C(i) contain torsion elements then the classification problem up to stable isomorphism seems (to this non-expert) to be quite difficult.

YOUNG-HOON KIEM, Yale University, New Haven, Connecticut  06520-8283, USA
Intersection homology of representation spaces of surface groups

Let p be the fundamental group of a closed Riemann surface and consider the representation variety Hom(p,G)/G for a compact simply connected Lie group G. This is in general a singular pseudomanifold and hence intersection homology is an interesting invariant. We show that this intersection homology is naturally embedded into the G-equivariant cohomology of Hom(p,G), which can be computed by other means like equivariant Morse theory and nonabelian localization principle. The case where G = SU(2) will be discussed in detail.

SLAWOMIR KWASIK, Tulane University, New Orleans, Louisiana  70118, USA
3-dimensional surgery, UNil-groups and the Borel conjecture

In this talk we will describe a joint work with B. Jahren (Oslo University) relating 3-dimensional surgery to various problems in the topology of higher dimensional manifolds. Applications include vanishing of Cappell's exotic UNil-groups and relations between these groups and the Borel conjecture.

TIAN-JU LI, Department of Mathematics, Princeton Universtiy, Princeton, New Jersey  08544, USA and Department of Mathematics, Yale University, New Haven, Connecticut  06520, USA
Intersection forms of non-spin manifolds

We propose the `8/8' conjecture for smooth, non-spin four manifolds with even intersection forms and report the progress we have made towards this conjecture.

MICHAEL MCCOOEY, McMAster University, Hamilton, Ontario  L8S 4K1
Symmetry groups of four-manifolds

I shall discuss some recent work on the question of classifying the symmetry groups of four-manifolds:

If a compact (possibly finite) Lie group G admits a locally linear, homologically trivial action on a closed, simply connected four-manifold M with second Betti number at least three, then G must be isomorphic to a subgroup of S¹×S¹.

If G contains Z_p×Z_p, then M must be homeomorphic to a connected sum of copies of ±CP² and S²×S².

Time permitting, I will discuss the question of classifying the actions of Z_p×Z_p on these manifolds, in light of work of Orlik and Raymond which classifies actions of the full torus group.

ERIK PEDERSON, Binghamton
To be announced

RANJA ROY, Department of Mathematics, Union College, Schenectady, New York  12308-2311, USA
The trace conjecture-a counterexample

Following Baum and Connes well known Isomorphism conjecture is their Trace conjecture:

If G is a discrete group with torsion and C^*G the reduced C^*-algebra of G, the trace map tr:K₀(C^*G) ® \mathbb R maps K₀(C^*G) onto the additive subgroup of \mathbb Q generated by all rational numbers of the form [1/(n)], where n is the order of a finite subgroup of G.

We construct a counterexample to this conjecture using asphericalization techniques developed by Davis-Januszkiewicz.

DARIUS WILCZYNSKI, Utah State University, Logan, Utah  84321-3900, USA
Group actions on Hirzebruch surfaces

We prove that the complex automorphism groups of diffeomorphic Hirzebruch surfaces are isomorphic but pairwise non-conjugate subgroups of the full diffeomorphism group.

BRUCE WILLIAMS, University of Notre Dame, Notre Dame, Indiana  46556-5683, USA
Transfer maps for K-theory and L-theory

We discuss methods for computing transfers for fiber bundles when the structure group is a compact Lie group. A key tool is a sum formula which is also related to the problem of comparing parametrized analytic and ``combinatorial'' Reidemeister torsion.