51. CMB 1999 (vol 42 pp. 190)
||Topological Quantum Field Theory and Strong Shift Equivalence |
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.
Keywords:knot, link, TQFT, symbolic dynamics, shift equivalence
Categories:57R99, 57M99, 54H20
52. CMB 1999 (vol 42 pp. 248)
||The Classification of $\Pin_4$-Bundles over a $4$-Complex |
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 $X$.
Categories:55N25, 55R10, 57S15
53. CMB 1999 (vol 42 pp. 149)
||A Note on Finite Dehn Fillings |
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
54. CMB 1999 (vol 42 pp. 46)
||Generic Partial Two-Point Sets Are Extendable |
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.
55. CMB 1999 (vol 42 pp. 52)
||Embedding Coverings in Bundles |
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 $d'=1$.
Categories:57M10, 55R25, 55S40, 57N35
56. CMB 1998 (vol 41 pp. 374)
||Normal invariants of lens spaces |
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
57. CMB 1998 (vol 41 pp. 252)
58. CMB 1998 (vol 41 pp. 140)
||Skein homology |
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.
59. CMB 1997 (vol 40 pp. 309)
||On the homology of finite abelian coverings of links |
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.
Keywords:Alexander ideal, branched covering, Dedekind domain,, knot, link.
60. CMB 1997 (vol 40 pp. 370)
||Which $3$-manifolds embed in $\Triod \times I \times I$? |
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.
Categories:57N10, 57N35, 57Q35
61. CMB 1997 (vol 40 pp. 204)
||The $\eta$-invariants of cusped hyperbolic $3$-manifolds |
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.
Categories:57M50, 53C30, 58G25