26. 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 

27. 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 

28. CJM 1998 (vol 50 pp. 620)
29. 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 

30. 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 

31. CJM 1997 (vol 49 pp. 883)
32. CJM 1997 (vol 49 pp. 696)
33. 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 
