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26. CJM 1999 (vol 51 pp. 585)

Mansfield, R.; Movahedi-Lankarani, H.; Wells, R.
Smooth Finite Dimensional Embeddings
We give necessary and sufficient conditions for a norm-compact 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

27. CJM 1998 (vol 50 pp. 620)

Sjerve, Denis; Yang, Qing Jie
The Eichler trace of $\bbd Z_p$ actions on Riemann surfaces
We study $\hbox{\Bbbvii Z}_p$ actions on compact connected Riemann surfaces via their associated Eichler traces. We determine the set of possible Eichler traces and determine the relationship between 2 actions if they have the same trace.

Categories:30F30, 57M60

28. 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 cone-like 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

29. 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 K-theory), we settle these previously undecided families and show that none of the manifolds in them is stably parallelizable, apart from one 30-dimensional manifold which still remains undecided.

Categories:57R25, 55N15, 53C30

30. CJM 1997 (vol 49 pp. 883)

Okounkov, Andrei
Proof of a conjecture of Goulden and Jackson
We prove an integration formula involving Jack polynomials conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with enumeration of maps in surfaces.

Categories:05E05, 43A85, 57M15

31. CJM 1997 (vol 49 pp. 696)

Charitos, Charalambos; Tsapogas, Georgios
Geodesic flow on ideal polyhedra
In this work we study the geodesic flow on $n$-dimensional ideal polyhedra and establish classical (for manifolds of negative curvature) results concerning the distribution of closed orbits of the flow.

Categories:57M20, 53C23

32. 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
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