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51. CMB 2000 (vol 43 pp. 79)

König, Steffen
 Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings. Categories:20G05, 20C20, 16G30, 17B37, 57M25

52. CMB 1999 (vol 42 pp. 257)

Austin, David; Rolfsen, Dale
 Homotopy of Knots and the Alexander Polynomial Any knot in a 3-dimensional homology sphere is homotopic to a knot with trivial Alexander polynomial. Categories:57N10, 57M05, 57M25, 57N65

53. CMB 1999 (vol 42 pp. 190)

Gilmer, Patrick M.
 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 equivalenceCategories:57R99, 57M99, 54H20

54. CMB 1999 (vol 42 pp. 248)

Weber, Christian
 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

55. CMB 1999 (vol 42 pp. 149)

Boyer, S.; Zhang, X.
 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 $3$. Categories:57M25, 57R65

56. CMB 1999 (vol 42 pp. 46)

Dijkstra, Jan J.
 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. Category:57N05

57. CMB 1999 (vol 42 pp. 52)

Edmonds, Allan L.
 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

58. CMB 1998 (vol 41 pp. 374)

Young, Carmen M.
 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. Categories:57R65, 57S25

59. CMB 1998 (vol 41 pp. 252)

Yang, Qingjie
 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 $g>1$. Category:57H20

60. CMB 1998 (vol 41 pp. 140)

Bullock, Doug; Frohman, Charles; Kania-Bartoszyńska, Joanna
 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. Categories:57M25, 57M99

61. CMB 1997 (vol 40 pp. 309)

Hillman, J. A.; Sakuma, M.
 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.Category:57M25

62. CMB 1997 (vol 40 pp. 370)

Rolfsen, Dale; Zhongmou, Li
 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

63. CMB 1997 (vol 40 pp. 204)

Meyerhoff, Robert; Ouyang, Mingqing
 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
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