Torus and Z/p actions on manifolds

Let G be either a finite cyclic group of prime order or S^1. We find new relations between cohomology of a manifold (or a Poincare duality space) M with a G-action on it and cohomology of the fixed point set, M^G. Our main tool is the notion of Poincare duality on the Leray spectral sequence of the map M_G -> BG. We apply our results to study group actions on 3-manifolds.Comment: To appear in Topology, 25 page

On ZpZp[u, v]-additive cyclic and constacyclic codes

Let $\mathbb{Z}_{p}$ be the ring of residue classes modulo a prime $p$. The $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes of length $(\alpha,\beta)$ is identify as $\mathbb{Z}_{p}[u,v][x]$-submodule of $\mathbb{Z}_{p}[x]/\langle x^{\alpha}-1\rangle \times \mathbb{Z}_{p}[u,v][x]/\langle x^{\beta}-1\rangle$ where $\mathbb{Z}_{p}[u,v]=\mathbb{Z}_{p}+u\mathbb{Z}_{p}+v\mathbb{Z}_{p}$ with $u^{2}=v^{2}=uv=vu=0$. In this article, we obtain the complete sets of generator polynomials, minimal generating sets for cyclic codes with length $\beta$ over $\mathbb{Z}_{p}[u,v]$ and $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic codes with length $(\alpha,\beta)$ respectively. We show that the Gray image of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive cyclic code with length $(\alpha,\beta)$ is either a QC code of length $4\alpha$ with index $4$ or a generalized QC code of length $(\alpha,3\beta)$ over $\mathbb{Z}_{p}$. Moreover, some structural properties like generating polynomials, minimal generating sets of $\mathbb{Z}_{p}\mathbb{Z}_{p}[u,v]$-additive constacyclic code with length $(\alpha,p-1)$ are determined.Comment: It is submitted to the journa

Completions of Z/(p)-Tate cohomology of periodic spectra

We construct splittings of some completions of the Z/(p)-Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n-1)'s as a spectrum, where t is shorthand for the fixed points of the Z/(p)-Tate cohomology spectrum (ie Mahowald's inverse limit of P_{-k} smash SE(n)). We also give a multiplicative splitting of tE(n) after a suitable base extension.Comment: 30 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper8.abs.htm

On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic pp-Groups

In this article we show that if ${\cal V}$ is the variety of polynilpotent groups of class row $(c_1,c_2,...,c_s),\ {\mathcal N}_{c_1,c_2,...,c_s}$, and $G\cong{\bf {Z}}_{p^{\alpha_1}}\stackrel{n}{*}{\bf {Z}}_{p^{\alpha_2}}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p^{\alpha_t} }$ is the $n$th nilpotent product of some cyclic $p$-groups, where $c_1\geq n$, $\alpha_1 \geq \alpha_2 \geq...\geq \alpha_t$ and $(q,p)=1$ for all primes $q$ less than or equal to $n$, then $|{\mathcal N}_{c_1,c_2,...,c_s}M(G)|=p^{d_m}$ if and only if $G\cong{\bf {Z}}_{p}\stackrel{n}{*}{\bf {Z}}_{p}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p }$ ($m$-copies), where $m=\sum _{i=1}^t \alpha_i$ and $d_m=\chi_{c_s+1}(...(\chi_{c_2+1}(\sum_{j=1}^n \chi_{c_1+j}(m)))...)$. Also, we extend the result to the multiple nilpotent product $G\cong{\bf {Z}}_{p^{\alpha_1}}\stackrel{n_1}{*}{\bf {Z}}_{p^{\alpha_2}}\stackrel{n_2}{*}...\stackrel{n_{t-1}}{*}{\bf{Z}}_{p^{\alpha_t} }$, where $c_1\geq n_1\geq...\geq n_{t-1}$. Finally a similar result is given for the $c$-nilpotent multiplier of $G\cong{\bf {Z}}_{p^{\alpha_1}}\stackrel{n}{*}{\bf {Z}}_{p^{\alpha_2}}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p^{\alpha_t}}$ with the different conditions $n \geq c$ and $(q,p)=1$ for all primes $q$ less than or equal to $n+c.$Comment: 10 page