26. CJM 2003 (vol 55 pp. 1080)
 Kellerhals, Ruth

Quaternions and Some Global Properties of Hyperbolic $5$Manifolds
We provide an explicit thick and thin decomposition for oriented
hyperbolic manifolds $M$ of dimension $5$. The result implies improved
universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$
compact, new estimates relating the injectivity radius and the diameter
of $M$ with $\rmvol_5(M)$. The quantification of the thin part is
based upon the identification of the isometry group of the universal
space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of
quaternionic $2\times 2$matrices with Dieudonn\'e determinant
$\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm
L} (2,\mathbb{H})$.
Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40 

27. CJM 2003 (vol 55 pp. 766)
 Kerler, Thomas

Homology TQFT's and the AlexanderReidemeister Invariant of 3Manifolds via Hopf Algebras and Skein Theory
We develop an explicit skeintheoretical algorithm to compute the
Alexander polynomial of a 3manifold from a surgery presentation
employing the methods used in the construction of quantum invariants
of 3manifolds. As a prerequisite we establish and prove a rather
unexpected equivalence between the topological quantum field theory
constructed by Frohman and Nicas using the homology of
$U(1)$representation varieties on the one side and the
combinatorially constructed Hennings TQFT based on the quasitriangular
Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^*
\mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL
(2,\mathbb{R})$equivariant functors and, as such, are isomorphic.
The $\SL (2,\mathbb{R})$action in the Hennings construction comes
from the natural action on $\mathcal{N}$ and in the case of the
FrohmanNicas theory from the HardLefschetz decomposition of the
$U(1)$moduli spaces given that they are naturally K\"ahler. The
irreducible components of this TQFT, corresponding to simple
representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus
yield a large family of homological TQFT's by taking sums and products.
We give several examples of TQFT's and invariants that appear to fit
into this family, such as Milnor and Reidemeister Torsion,
SeibergWitten theories, Casson type theories for homology circles
{\it \`a la} Donaldson, higher rank gauge theories following Frohman
and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of
ReshetikhinTuraev theories over the cyclotomic integers $\mathbb{Z}
[\zeta_p]$. We also conjecture that the Hennings TQFT for
quantum$\mathfrak{sl}_2$ is the product of the ReshetikhinTuraev
TQFT and such a homological TQFT.
Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27 

28. CJM 2003 (vol 55 pp. 636)
 Schwartzman, Sol

Higher Dimensional Asymptotic Cycles
Given a $p$dimensional oriented foliation of an $n$dimensional
compact manifold $M^n$ and a transversal invariant measure $\tau$,
Sullivan has defined an element of $H_p (M^n,R)$. This generalized
the notion of a $\mu$asymptotic cycle, which was originally defined
for actions of the real line on compact spaces preserving an invariant
measure $\mu$. In this onedimensional case there was a natural 11
correspondence between transversal invariant measures $\tau$ and
invariant measures $\mu$ when one had a smooth flow without stationary
points.
For what we call an oriented action of a connected Lie group on a
compact manifold we again get in this paper such a correspondence,
provided we have what we call a positive quantifier. (In the
onedimensional case such a quantifier is provided by the vector field
defining the flow.) Sufficient conditions for the existence of such a
quantifier are given, together with some applications.
Categories:57R30, 57S20 

29. CJM 2002 (vol 54 pp. 396)
 Lebel, AndrĂ©

Framed Stratified Sets in Morse Theory
In this paper, we present a smooth framework for some aspects of the
``geometry of CW complexes'', in the sense of Buoncristiano, Rourke
and Sanderson \cite{[BRS]}. We then apply these ideas to Morse
theory, in order to generalize results of Franks \cite{[F]} and
IriyeKono \cite{[IK]}.
More precisely, consider a Morse function $f$ on a closed manifold
$M$. We investigate the relations between the attaching maps in a CW
complex determined by $f$, and the moduli spaces of gradient flow
lines of $f$, with respect to some Riemannian metric on~$M$.
Categories:57R70, 57N80, 55N45 

30. CJM 2001 (vol 53 pp. 1309)
31. CJM 2001 (vol 53 pp. 780)
 Nicolaescu, Liviu I.

SeibergWitten Invariants of Lens Spaces
We show that the SeibergWitten invariants of a lens space determine
and are determined by its CassonWalker invariant and its
ReidemeisterTuraev torsion.
Keywords:lens spaces, Seifert manifolds, SeibergWitten invariants, CassonWalker invariant, Reidemeister torsion, eta invariants, DedekindRademacher sums Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25 

32. CJM 2001 (vol 53 pp. 212)
 Puppe, V.

Group Actions and Codes
A $\mathbb{Z}_2$action with ``maximal number of isolated fixed
points'' ({\it i.e.}, with only isolated fixed points such that
$\dim_k (\oplus_i H^i(M;k)) =M^{\mathbb{Z}_2}, k = \mathbb{F}_2)$
on a $3$dimensional, closed manifold determines a binary selfdual
code of length $=M^{\mathbb{Z}_2}$. In turn this code determines
the cohomology algebra $H^*(M;k)$ and the equivariant cohomology
$H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary selfdual
codes one gets information about the cohomology type of $3$manifolds
which admit involutions with maximal number of isolated fixed points.
In particular, ``most'' cohomology types of closed $3$manifolds do
not admit such involutions. Generalizations of the above result are
possible in several directions, {\it e.g.}, one gets that ``most''
cohomology types (over $\mathbb{F}_2)$ of closed $3$manifolds do
not admit a nontrivial involution.
Keywords:Involutions, $3$manifolds, codes Categories:55M35, 57M60, 94B05, 05E20 

33. CJM 2000 (vol 52 pp. 293)
 Collin, Olivier

Floer Homology for Knots and $\SU(2)$Representations for Knot Complements and Cyclic Branched Covers
In this article, using 3orbifolds singular along a knot with
underlying space a homology sphere $Y^3$, the question of existence
of nontrivial and nonabelian $\SU(2)$representations of the
fundamental group of cyclic branched covers of $Y^3$ along a knot
is studied. We first use Floer Homology for knots to derive an
existence result of nonabelian $\SU(2)$representations of the
fundamental group of knot complements, for knots with a
nonvanishing equivariant signature. This provides information on
the existence of nontrivial and nonabelian
$\SU(2)$representations of the fundamental group of cyclic
branched covers. We illustrate the method with some examples of
knots in $S^3$.
Categories:57R57, 57M12, 57M25, 57M05 

34. CJM 1999 (vol 51 pp. 1035)
 Litherland, R. A.

The Homology of Abelian Covers of Knotted Graphs
Let $\tilde M$ be a regular branched cover of a homology 3sphere
$M$ with deck group $G\cong \zt^d$ and branch set a trivalent graph
$\Gamma$; such a cover is determined by a coloring of the edges of
$\Gamma$ with elements of $G$. For each index2 subgroup $H$ of
$G$, $M_H = \tilde M/H$ is a double branched cover of $M$. Sakuma
has proved that $H_1(\tilde M)$ is isomorphic, modulo 2torsion, to
$\bigoplus_H H_1(M_H)$, and has shown that $H_1(\tilde M)$ is
determined up to isomorphism by $\bigoplus_H H_1(M_H)$ in certain
cases; specifically, when $d=2$ and the coloring is such that the
branch set of each cover $M_H\to M$ is connected, and when $d=3$
and $\Gamma$ is the complete graph $K_4$. We prove this for a
larger class of coverings: when $d=2$, for any coloring of a
connected graph; when $d=3$ or $4$, for an infinite class of
colored graphs; and when $d=5$, for a single coloring of the
Petersen graph.
Categories:57M12, 57M25, 57M15 

35. CJM 1999 (vol 51 pp. 585)
 Mansfield, R.; MovahediLankarani, H.; Wells, R.

Smooth Finite Dimensional Embeddings
We give necessary and sufficient conditions for a normcompact subset
of a Hilbert space to admit a $C^1$ embedding into a finite dimensional
Euclidean space. Using quasibundles, we prove a structure theorem
saying that the stratum of $n$dimensional points is contained in an
$n$dimensional $C^1$ submanifold of the ambient Hilbert space. This
work sharpens and extends earlier results of G.~Glaeser on paratingents.
As byproducts we obtain smoothing theorems for compact subsets of
Hilbert space and disjunction theorems for locally compact subsets
of Euclidean space.
Keywords:tangent space, diffeomorphism, manifold, spherically compact, paratingent, quasibundle, embedding Categories:57R99, 58A20 

36. CJM 1998 (vol 50 pp. 620)
37. CJM 1998 (vol 50 pp. 581)
 Kamiyama, Yasuhiko

The homology of singular polygon spaces
Let $M_n$ be the variety of spatial polygons $P= (a_1, a_2, \dots,
a_n)$ whose sides are vectors $a_i \in \text{\bf R}^3$ of length
$\vert a_i \vert=1 \; (1 \leq i \leq n),$ up to motion in
$\text{\bf R}^3.$ It is known that for odd $n$, $M_n$ is a
smooth manifold, while for even $n$, $M_n$ has conelike singular
points. For odd $n$, the rational homology of $M_n$ was determined
by Kirwan and Klyachko [6], [9]. The purpose of this paper is to
determine the rational homology of $M_n$ for even $n$. For even
$n$, let ${\tilde M}_n$ be the manifold obtained from $M_n$ by the
resolution of the singularities. Then we also determine the
integral homology of ${\tilde M}_n$.
Keywords:singular polygon space, homology Categories:14D20, 57N65 

38. CJM 1997 (vol 49 pp. 1323)
 Sankaran, Parameswaran; Zvengrowski, Peter

Stable parallelizability of partially oriented flag manifolds II
In the first paper with the same title the authors
were able to determine all partially oriented flag
manifolds that are stably parallelizable or
parallelizable, apart from four infinite families
that were undecided. Here, using more delicate
techniques (mainly Ktheory), we settle these
previously undecided families and show that none of
the manifolds in them is stably parallelizable,
apart from one 30dimensional manifold which still
remains undecided.
Categories:57R25, 55N15, 53C30 

39. CJM 1997 (vol 49 pp. 883)
40. CJM 1997 (vol 49 pp. 696)
41. CJM 1997 (vol 49 pp. 193)
 Casali, Maria Rita

Classifying PL $5$manifolds by regular genus: the boundary case
In the present paper, we face the problem of classifying classes of
orientable PL $5$manifolds $M^5$ with $h \geq 1$ boundary components,
by making use of a combinatorial invariant called {\it regular genus}
${\cal G}(M^5)$. In particular, a complete classification up to
regular genus five is obtained:
$${\cal G}(M^5) = \gG \leq 5 \Longrightarrow M^5 \cong \#_{\varrho
 \gbG}(\bdo) \# \smo_{\gbG},$$
where $\gbG = {\cal G}(\partial M^5)$ denotes the regular genus of
the boundary $\partial M^5$ and $\smo_{\gbG}$ denotes the connected
sum of $h\geq 1$ orientable $5$dimensional handlebodies
$\cmo_{\alpha_i}$ of genus $\alpha_i\geq 0$
($i=1,\ldots, h$), so that $\sum_{i=1}^h \alpha_i = \gbG.$
\par
Moreover, we give the following characterizations of orientable PL
$5$manifolds $M^5$ with boundary satisfying particular conditions
related to the ``gap'' between ${\cal G}(M^5)$ and either
${\cal G}(\partial M^5)$ or the rank of their fundamental group
$\rk\bigl(\pi_1(M^5)\bigr)$:
$$\displaylines{{\cal G}(\partial M^5)= {\cal G}(M^5)
= \varrho \Longleftrightarrow M^5 \cong \smo_{\gG}\cr
{\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)1 \Longleftrightarrow
M^5 \cong (\bdo) \# \smo_{\gbG}\cr
{\cal G}(\partial M^5)= \gbG = {\cal G}(M^5)2 \Longleftrightarrow
M^5 \cong \#_2 (\bdo) \# \smo_{\gbG}\cr
{\cal G}(M^5) = \rk\bigl(\pi_1(M^5)\bigr)= \varrho \Longleftrightarrow
M^5 \cong \#_{\gG  \gbG}(\bdo) \# \smo_{\gbG}.\cr}$$
\par
Further, the paper explains how the above results (together with
other known properties of regular genus of PL manifolds) may lead
to a combinatorial approach to $3$dimensional Poincar\'e Conjecture.
Categories:57N15, 57Q15, 05C10 
