




Topology of Manifolds / Topologie des variétés (Ronnie Lee and Ian Hambleton, Organizers)
 HANS BODEN, Department of Mathematics, Ohio State University
Columbus, Ohio 432101174, USA
Integral and nonintegral 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
integervalued, and one interesting problem is to compare
it to the existing definition of the SU(3) Casson invariant,
which is realvalued (and only conjecturally rational).
We will discuss recent computations (joint with C. Herald, P. Kirk, and
E. Klassen) of both the realvalued and integervalued invariant for
families of homology spheres obtained by Dehn surgery on torus knots.
This uses a careful analysis of Floertype 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 C^{2} spectral flow and ChernSimons 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 wellknown that if M^{n} is compact and admits
an everywhere negative curvature metric, then M^{n} is covered by
Rn.
What can one say if a compact riemannian manifold (M,g) has a
Liouvilleintegrable geodesic flow? The answer depends on the
smoothness of the first integrals of the geodesic flow. In the
realanalytic category, it is known that p_{1}(M) must be almost
abelian. It will be shown that there are realanalytic riemannian
manifolds (M^{n},g) that have Liouvilleintegrable geodesic flows with
C^{¥} first integrals such that p_{1}(M^{n}) is n1step
torsionfree nilpotent. Bolsinov and Taimanov have provided an example
where p_{1}(M^{n}) is of exponential growth.
 DIARMUID CROWLEY, Department of Mathematics, Indiana University, Indiana 47405, USA
Classification of 3sphere bundles over the 4sphere

I shall present a classification (join with Christine Escher) of the
total spaces of 3sphere bundles over the 4sphere up to homotopy
equivalence, homeomorphism, PLhomeomorphism 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 Ktheory of virtually cyclic groups

This is joint work with Bogdan Vajiac. We use controlled topology, à
la FarrellJones, 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 Nilgroup 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 squarefree order. Then there is a MayerVietoris 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 nonintegral 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
integervalued, and one interesting problem is to compare it to the
existing definition of the SU(3) Casson invariant, which is
realvalued (and only conjecturally rational).
We will discuss recent computations (joint with H. Boden, P. Kirk, and
E. Klassen) of both the realvalued and integervalued invariant for
families of homology spheres obtained by Dehn surgery on torus knots.
This uses a careful analysis of Floertype 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
C^{2} spectral flow and ChernSimons 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 nonresolvable
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 Rmodules 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 nonexpert) to be quite difficult.
 YOUNGHOON KIEM, Yale University, New Haven, Connecticut 065208283, 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 Gequivariant 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
3dimensional surgery, UNilgroups and the Borel conjecture

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

We propose the `8/8' conjecture for smooth, nonspin 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 fourmanifolds

I shall discuss some recent work on the question of classifying the
symmetry groups of fourmanifolds:
If a compact (possibly finite) Lie group G admits a locally linear,
homologically trivial action on a closed, simply connected
fourmanifold M with second Betti number at least three, then G
must be isomorphic to a subgroup of S^{1}×S^{1}.
If G contains Z_{p}×Z_{p}, then M must be
homeomorphic to a connected sum of copies of ±CP^{2} and
S^{2}×S^{2}.
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 123082311, USA
The trace conjecturea 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:K_{0}(C^{*}G) ® \mathbb R maps K_{0}(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 DavisJanuszkiewicz.
 DARIUS WILCZYNSKI, Utah State University, Logan, Utah 843213900, USA
Group actions on Hirzebruch surfaces

We prove that the complex automorphism groups of diffeomorphic
Hirzebruch surfaces are isomorphic but pairwise nonconjugate subgroups
of the full diffeomorphism group.
 BRUCE WILLIAMS, University of Notre Dame, Notre Dame, Indiana 465565683, USA
Transfer maps for Ktheory and Ltheory

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.

