A model structure via orbit spaces for equivariant homotopy

For a given group $G$ and a collection of subgroups $\mathcal F$ of $G$, we show that there exist a left induced model structure on the category of right $G$-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on the $H$-orbits for all $H$ in $\mathcal F$. This gives a model categorical criterion for maps that induce weak equivalences on $H$-orbits to be weak equivalences in the $\mathcal F$-model structure.Comment: 9 page

Obstructions for constructing equivariant fibrations

Let $G$ be a finite group and $\mathcal{H}$ be a family of subgroups of $G$ which is closed under conjugation and taking subgroups. Let $B$ be a $G$-$CW$-complex whose isotropy subgroups are in $\mathcal{H}$ and let $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ be a compatible family of $H$-spaces. A $G$-fibration over $B$ with fiber $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ is a $G$-equivariant fibration $p:E \rightarrow B$ where $p^{-1}(b)$ is $G_b$-homotopy equivalent to $F_{G_b}$ for each $b \in B$. In this paper, we develop an obstruction theory for constructing $G$-fibrations with fiber $\mathcal{F}$ over a given $G$-$CW$-complex $B$. Constructing $G$-fibrations with a prescribed fiber $\mathcal{F}$ is an important step in the construction of free $G$-actions on finite $CW$-complexes which are homotopy equivalent to a product of spheres

A note on small covers over cubes

In this paper, we obtain a bijection between the weakly Z(2)(n)-equivariant homeomorphism classes of small covers over an n-cube and the orbits of the action of Z(2) (sic) S-n on acyclic digraphs with n vertices given by local complementation and reordering of vertices. We obtain a similar formula for the number of orientable small covers over an n-cube. We also count the Z(2)(n)-equivariant homeomorphism classes of orientable small covers and estimate the ratio between this number and the number of Z(2)(n)-equivariant homeomorphism classes of small covers over an n-cube

A note on quantizations of Galois extensions

In Huru and Lychagin (2013), it is conjectured that the quantizations of splitting fields of products of quadratic polynomials, which are obtained by deforming the multiplication, are Clifford type algebras. In this paper, we prove this conjecture. (C) 2014 Elsevier B.V. All rights reserved

The number of orientable small covers over a product of simplices

In this paper, we give a formula for the number of orientable small covers over a product of simplices up to D-J equivalence. We also give an approximate value for the ratio between the number of small covers and the number of orientable small covers over a product of equidimensional simplices up to D-J equivalence