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Search: All articles in the CJM digital archive with keyword topology

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1. CJM Online first

Mihara, Tomoki
Cohomological Approach to Class Field Theory in Arithmetic Topology
We establish class field theory for $3$-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

Keywords:arithmetic topology, class field theory, branched covering, knots and prime numbers
Categories:11Z05, 18F15, 55N20, 57P05

2. CJM Online first

Kuribayashi, Katsuhiko; Menichi, Luc
The BV algebra in String Topology of classifying spaces
For almost any compact connected Lie group $G$ and any field $\mathbb{F}_p$, we compute the Batalin-Vilkovisky algebra $H^{*+\operatorname{dim }G}(LBG;\mathbb{F}_p)$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^*(H_*(G),H_*(G))$. Over $\mathbb{F}_2$, such isomorphism of Batalin-Vilkovisky algebras does not hold when $G=SO(3)$ or $G=G_2$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

Keywords:string topology, Batalin-Vilkovisky algebra, classifying space
Categories:55P50, 55R35, 81T40

3. CJM 2014 (vol 67 pp. 55)

Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina
On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ${\mathbb C}$, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ${\mathbb N}$-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of $\mathbb{N}$-graded Lie algebras $L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$ and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the structure of these algebras with the property $L=\langle L_1 \rangle$. In this paper we study those generated by the first and $q$-th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under some technical condition, there can only be one isomorphism type of such algebras. For $q=3$ we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification
Categories:17B70, 14F45

4. CJM 2010 (vol 63 pp. 436)

Mine, Kotaro; Sakai, Katsuro
Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces
Let $F$ be a non-separable LF-space homeomorphic to the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$, where $\aleph_0 < \tau_1 < \tau_2 < \cdots$. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $|K| \times F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$ with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$ (and $\operatorname{card} K^{(0)} \le \tau$), and, conversely, if $K$ is such a simplicial complex, then the product $|K| \times F$ can be embedded in $F$ as an open set, where $|K|$ is the polyhedron of $K$ with the metric topology.

Keywords:LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$
Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40

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