Ross Geoghegan

Free abelian cohomology of groups and ends of universal covers

Abstract Let G be a group for which there exists a K(G, 1)-complex X having finite n-skeleton (for n = 1 or 2 this is equivalent to saying that G is finitely generated or finitely presented). If X n is the n-skeleton of the universal cover of X, X n is a locally finite complex whose proper homotopy invariants at ∞ (homology groups of the end, homotopy groups of the end etc.) in dimensions ≤ n are invariants of G which are closely related to H∗(G, Z G). This relationship is explained. When G is a suitable extension of an infinite group by an infinite group, we obtain new information about H∗(G, Z G) and about the proper homotopy type of X n .

One-parameter Fixed Point Theory

For a piecewise linear map/: X x / -> X, where Xis a compact PL H-manifold, n > 4, we analyze the problem of deforming /, with control, so s to remove some or all of the fixed point set = {(x, t) e X x / |/(jc, t) = x}. 1980 Mathematics Subject Classification (1985 Revision): 55M20, 57Q40.

The fundamental group at infinity

Abstract Let G be a finitely presented infinite group which is semistable at infinity, let X be a finite complex whose fundamental group is G, and let ω be a base ray in the universal covering space X . The fundamental group at ∞ of G is the topological group π 1 e ( X , ω) ≡ lim {π 1 ( X − L)∣L ⊂ X is compact } . We prove the following analogue of Hopfs theorem on ends: π 1 e ( X ,ω) is trivial, or is infinite cyclic, or is freely generated by a non-discrete pointed compact metric space; or else the natural representation of G in the outer automorphisms of π 1 e ( X ,ω) has torsion kernel. A related manifold result is: Let G be torsion free (not necessarily finitely presented) and act as covering transformations on a connected manifold M so that the quotient of M by any infinite cyclic subgroup is non-compact; if M is semistable at ∞ then the natural representation of G in the mapping class group of M is faithful. The latter theorem has applications in 3-manifold topology.

The Sigma invariants of Thompson's group F

Thompsons group F is the group of all increasing dyadic piecewise linear homeomorphisms of the closed unit interval. We compute Sigma^m(F) and Sigma^m(F;Z), the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F, and we show that Sigma^m(F) = Sigma^m(F;Z). As an application, we show that, for every m, F has subgroups of type F_{m-1} which are not of type F_{m}.

Nielsen Fixed Point Theory

Nielsen Fixed Point Theory combines the ideas of the Lefschetz Fixed Point Theorem with the fundamental group to produce a richer version of the Lefschetz Theory. Just as the Lefschetz theorem concerns the “Lefschetz number”, L ( f ) ∈ ℤ, associated with a suitable map f : X → X , Nielsen Theory produces an invariant called the “Reidemeister trace”, R ( f ), not a number but an element of a certain free Abelian group. In fact, R ( f ) depends on a base point v and a base path τ from v to f ( v ), so I will need notation later which acknowledges this; but not in this Introduction.

The stability problem in shape, and a Whitehead theorem in pro-homotopy

Theorem 3.1 is a Whitehead theorem in pro-homotopy for finite-dimensional pro-complexes. This is used to obtain necessary and sufficient algebraic conditions for a finite-dimensional tower of complexes to be pro-homotopy equivalent to a complex (§4) and for a finite-dimensional compact metric space to be pointed shape equivalent to an absolute neighborhood retract (§ 5).

Ascending Hnn Extensions of Finitely Generated Free Groups Are Hopfian

It is shown that every ascending HNN extension of a nitely generated free group is Hopan. An important ingredient in the proof is that under certain hypotheses on the group H ,i fG is an ascending HNN extension of H, then cd(G )= cd(H )+1 . 1. Statement of results Ag roupG is Hopan, provided that every surjective endomorphism of G is an automorphism. This notion originated in connection with Hopf’s question of whether a degree 1 map from a closed manifold to itself must be a homotopy equivalence. While it is easy to give examples of innitely generated groups which are not Hopan, for some time it was an open question as to whether every nitely generated group is indeed Hopan. The rst nitely generated example was given by B. Neumann in [13], and shortly thereafter the following nitely presented example was given by G. Higman in [8]: ha;s;tj a t = a 2 ;a s = a 2 i;

Threading Homotopies and DC Operating Points of Nonlinear Circuits

This paper studies continuation methods for finding isolated zeros of nonlinear functions. Given a nonlinear function

Sigma invariants of direct products of groups

F : {\Bbb R}^n \rightarrow {\Bbb R}^n

A note on the vanishing of Hn(G,ZG)

, a {\em threading homotopy} is a function