51. 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.
52. 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
53. 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 invariants.
54. CMB 1998 (vol 41 pp. 252)
|Dihedral groups of automorphisms of compact Riemann surfaces |
In this note we determine which dihedral subgroups of $\GL_g(\D C)$ can be realized by group actions on Riemann surfaces of genus $g>1$.
55. 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.
56. 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.
57. 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
58. 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