Let $g\colon M^{2n}\rightarrow M^{2n}$ be a smooth map of period $m>2$ which preserves orientation. Suppose that the cyclic action defined by $g$ is regular and that the normal bundle of the fixed point set $F$ has a $g$-equivariant complex structure. Let $F\pitchfork F$ be the transverse self-intersection of $F$ with itself. If the $g$-signature $\Sign (g,M)$ is a rational integer and $n<\phi (m)$, then there exists a choice of orientations such that $\Sign(g,M)= \Sign F=\Sign(F\pitchfork F)$.

Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and $H^*$-harmonic spaces associated to a H\"ormander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

We give an algorithm for a surgery description of a $p$-fold cyclic branched cover of $B^3$ branched along a tangle. We generalize constructions of Montesinos and Akbulut-Kirby.

Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

Let $M$ be a compact, connected, orientable, irreducible 3-manifold with a torus boundary. It is known that if two Dehn fillings on $M$ along the boundary produce a reducible manifold and a manifold containing a Klein bottle, then the distance between the filling slopes is at most three. This paper gives a remarkably short proof of this result.

Define a group $G$ to be in the class $\mathcal{S}$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that $g^n \in K$ for all $g\in G$, $K$ has finite index in $G$. We show that a free product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of $C$ is finitely generated.

Erratum to {\it The Lehmer Polynomial and Pretzel Links}, Canad. J. Math. {\bf 44}(2001), 440--451.

We give a topological interpretation of the core group invariant of a surface embedded in $S^4$ \cite{F-R}, \cite{Ro}. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of $S^4$ with the surface as a branched set, and the infinite cyclic group. We present a generalization for unoriented surfaces, for other cyclic branched covers, and other codimension two embeddings of manifolds in spheres.

In this paper we find a formula for the Alexander polynomial $\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with $(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and $p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7} (x)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.

We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature.

In a study of the word problem for groups, R.~J.~Thompson considered a certain group $F$ of self-homeomorphisms of the Cantor set and showed, among other things, that $F$ is finitely presented. Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed that $F$ is the fundamental group of a finite two-complex $Z^2$ having Euler characteristic one and which is {\em Cockcroft}, in the sense that each map of the two-sphere into $Z^2$ is homologically trivial. We show that no proper covering complex of $Z^2$ is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group $F$ is Cockcroft.

In this paper, we show that the absolute value of the signature of the $2$-parallel version of a link is less than or equal to the nullity of it and show that the signature, nullity, and Minkowski units of the $2$-parallel version of a certain class of links are always equal to $0$, $2$, and $1$ respectively.

An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

Any knot in a 3-dimensional homology sphere is homotopic to a knot with trivial Alexander polynomial.

Let $M$ be a compact, connected, orientable 3-manifold whose boundary is a torus and whose interior admits a complete hyperbolic metric of finite volume. In this paper we show that if the minimal Culler-Shalen norm of a non-zero class in $H_1(\partial M)$ is larger than $8$, then the finite surgery conjecture holds for $M$. This means that there are at most $5$ Dehn fillings of $M$ which can yield manifolds having cyclic or finite fundamental groups and the distance between any slopes yielding such manifolds is at most $3$.

In this paper we show that the Lie-group $\Pin_4$ is isomorphic to the semidirect product $(\SU_2\times \SU_2)\timesv \Z/2$ where $\Z/2$ operates by flipping the factors. Using this structure theorem we prove a classification theorem for $\Pin_4$-bundles over a finite $4$-complex $X$.

Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of a closed $(d+1)$-dimensional manifold $M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R.~Williams' work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an $S^1$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a finite group $G$ which has been studied by Quinn. In this way, we recover a link invariant due to D.~Silver and S.~Williams. We also obtain a variation on the Silver-Williams invariant, by using the TQFT associated to $G$ in its unmodified form.

It is shown that under $\ZFC$ almost all planar compacta that meet every line in at most two points are subsets of sets that meet every line in exactly two points. This result was previously obtained by the author jointly with K.~Kunen and J.~van~Mill under the assumption that Martin's Axiom is valid.

If $V\to X$ is a vector bundle of fiber dimension $k$ and $Y\to X$ is a finite sheeted covering map of degree $d$, the implications for the Euler class $e(V)$ in $H^k(X)$ of $V$ implied by the existence of an embedding $Y\to V$ lifting the covering map are explored. In particular it is proved that $dd'e(V)=0$ where $d'$ is a certain divisor of $d-1$, and often $d'=1$.

We show that normal and stable normal invariants of polarized homotopy equivalences of lens spaces $M = L(2^m;\r)$ and $N = L(2^m;\s)$ are determined by certain $\ell$-polynomials evaluated on the elementary symmetric functions $\sigma_i(\rsquare)$ and $\sigma_i(\ssquare)$. Each polynomial $\ell_k$ appears as the homogeneous part of degree $k$ in the Hirzebruch multiplicative $L$-sequence. When $n = 8$, the elementary symmetric functions alone determine the relevant normal invariants.

In this note we determine which dihedral subgroups of $\GL_g(\D C)$ can be realized by group actions on Riemann surfaces of genus $g>1$.

A new class of homology groups associated to a 3-manifold is defined. The theories measure the syzygies between skein relations in a skein module. We investigate some of the properties of the homology theory associated to the Kauffman bracket.

Let $A$ be a finite abelian group and $M$ be a branched cover of an homology $3$-sphere, branched over a link $L$, with covering group $A$. We show that $H_1(M;Z[1/|A|])$ is determined as a $Z[1/|A|][A]$-module by the Alexander ideals of $L$ and certain ideal class invariants.

We classify the compact $3$-manifolds whose boundary is a union of $2$-spheres, and which embed in $T \times I \times I$, where $T$ is a triod and $I$ the unit interval. This class is described explicitly as the set of punctured handlebodies. We also show that any $3$-manifold in $T \times I \times I$ embeds in a punctured handlebody.

In this paper, we define the $\eta$-invariant for a cusped hyperbolic $3$-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.