Ian Hambleton

A non-extended hermitian form over ℤ[ℤ]

We describe a nonsingular hermitian form of rank 4 over the group ring Z[Z] which is not extended from the integers. Moreover, we show that under certain indefiniteness asumptions, every nonsingular hermitian form on a free Z[Z]module is extended from the integers. As a corollary, there exists a closed oriented 4-dimensional manifold with fundamental group Z which is not the connected sum of S × S with a simply-connected 4-manifold.

Nonorientable 4-manifolds with fundamental group of order 2

In this paper we classify nonorientable topological closed 4-manifolds with fundamental group Z/2 up to homeomorphism. Our results give a complete list of such manifolds, and show how they can be distinguished by explicit invariants including characteristic numbers and the ti-invariant associated to a normal PinC-structure by the spectral asymmetry of a certain Dirac operator. In contrast to the oriented case, there exist homotopy equivalent nonorientable topological 4-manifolds which are stably homeomorphic (after connected sum with S2 x S2) but not homeomorphic.

DETECTION THEOREMS FOR K-THEORY AND L-THEORY

Suppose G is a p-hyperelementary group and R is a commutative ring such that the order of G is a unit in R. Suppose J is either one of Quillen’s K-theory functors or one of Wall’s oriented L-theory functors. We show that J(RG) can be detected by applying J(R?) to the subquotients of G such that all normal abelian subgroups are cyclic. In 3.A.6 we show that such subquotients have a quite simple structure. We also show how to detect more general L-theory functors, in particular unoriented ones and those that arise in the study of codimension one submanifolds.

Bounded surgery and dihedral group actions on spheres

If a finite group G acts freely and simplicially on a complex homotopy equivalent to a sphere S, then G has periodic Tate cohomology: H (G; Z) ∼= H (G; Z) for all i > 0. Swan proved in [26] that this condition was also sufficient. For free topological actions on S itself, the first additional restriction is: Theorem. [19] A finite dihedral group does not act freely and topologically on S. Milnor’s argument used the compactness of S as well as the manifold structure. In fact, for dihedral groups with periodic cohomology, i. e. of order 2n where n is odd we have,

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.

TOPOLOGICAL 4-MANIFOLDS WITH GEOMETRICALLY TWO-DIMENSIONAL FUNDAMENTAL GROUPS

Closed oriented 4-manifolds with the same geometrically two-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w2-type, equivariant intersection form and the Kirby–Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed oriented 4-manifolds with solvable Baumslag–Solitar fundamental groups, including a precise realization result.

EQUIVARIANT CW-COMPLEXES AND THE ORBIT CATEGORY

We give a general framework for studying G-CW complexes via the orbit category. As an application we show that the symmetric group G = S5 admits a nite G-CW complex X homotopy equivalent to a sphere, with cyclic isotropy subgroups.

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

The computation of the projective surgery obstruction groups LP.(ZG), for G a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. For G a 2-hyperelementary group, the LP.(ZG) are detected by restriction to certain subquotients of G, and a complete set of invariants is given for oriented surgery obstructions.

-manifolds and moduli spaces

We express the signature modulo 4 of a closed, oriented, 4k‐dimensional PL manifold as a linear combination of its Euler characteristic and the new absolute torsion invariant defined by Korzeniewski [11]. Let F! E! B be a PL fibre bundle, where F, E and B are closed, connected, and compatibly oriented PL manifolds. We give a formula for the absolute torsion of the total space E in terms of the absolute torsion of the base and fibre, and then combine these two results to prove that the signature of E is congruent modulo 4 to the product of the signatures of F and B. 55R25