arXiv subjects

Marek Golasiński

Publications and source records attributed to Marek Golasiński.

6 recordsLinked to original sources

arXiv2018

Exponents of $[Ω(\mathbb S^{r+1}), Ω(Y)]$

We investigate the exponents of the total Cohen groups $[Ω(\mathbb S^{r+1}), Ω(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n+1})]$ and $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n})]$ coincide with the $p$-primary homotopy exponents of spheres $\mathbb S^{2n+1}$ and $\mathbb S^{2n}$, respectively. We further study the exponent problem when $Y$ is a space with the homotopy type of $Σ(n)/G$ for a homotopy $n$-sphere $Σ(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space $\mathbb{H}P^n$ for $1\le n\le \infty$.

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arXiv2017

Generalized Gottlieb and Whitehead center groups of space forms

We extend the Oprea's result $G_1(\mathbb{S}^{2n+1}/H)=\mathcal{Z}H$ to the 1st generalized Gottlieb group $G_1^f(\mathbb{S}^{2n+1}/H)$ for a map $f\colon A\to \mathbb{S}^{2n+1}/H$. Then, we compute or estimate the groups $G_m^f(\mathbb{S}^{2n+1}/H)$ and $P_m^f(\mathbb{S}^{2n+1}/H)$ for some $m>1$ and finite groups $H$.

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arXiv2017

Generalized Jiang and Gottlieb groups

Given a map $f\colon X \to Y$, we extend a Gottlieb's result to the generalized Gottlieb group $G^f(Y,f(x_0))$ and show that the canonical isomorphism $π_1(Y,f(x_0))\xrightarrow{\approx}\mathcal{D}(Y)$ restricts to an isomorphism $G^f(Y,f(x_0))\xrightarrow{\approx}\mathcal{D}^{\tilde{f}_0}(Y)$, where $\mathcal{D}^{\tilde{f}_0}(Y)$ is some subset of the group $\mathcal{D}(Y)$ of deck transformations of $Y$ for a fixed lifting $\tilde{f}_0$ of $f$ with respect to universal coverings of $X$ and $Y$, respectively.

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arXiv2016

On the higher Whitehead product

The Porter's approach (G.\ J.\ Porter, \textit{Higher order Whitehead products}, Topology \textbf{3} (1965), 123--135.) is used to derive some properties of higher order Whitehead products, similar to those ones for triple products obtained by Hardie in K.\ A.\ Hardie, \textit{On a construction of E.\ C.\ Zeeman}, Journal London Math.\ Soc.\ \textbf{35} (1960), 452--464.

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arXiv2015

On the higher order exterior and interior Whitehead products

We extend the notion of the exterior Whitehead product for maps $α_i:ΣA_i \to X_i$ for $i=1,\dots,n$, where $ΣA_i$ is the reduced suspension of $A_i$ and then, for the interior product with $X_i=J_{m_i}(X)$ as well. The main result stated in Theorem 3.10 generalizes Theorem 1.10 in K.\ A.\ Hardie, \textit{A generalization of the Hopf construction}, Quart.\ J.\ Math.\ Oxford Ser.\ (2) \textbf{12} (1961), 196--204. and concerns to the Hopf invariant of the generalized Hopf construction. We close the paper applying the Gray's construction $\circ$ (called the Theriault product) to a sequence $X_1,\dots,X_n$ of simply connected co-$H$-spaces to obtain a higher Gray--Whitehead product map \[w_n:Σ^{n-2}(X_1\circ\dots\circ X_n)\to T_1(X_1,\dots,X_n),\] where $T_1(X_1,\dots,X_n)$ is the fat wedge of $X_1,\dots,X_n$.

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arXiv2014

On the group structure of $[Ω\mathbb S^2, ΩY]$

Let $J(X)$ denote the James construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),ΩY]\cong \lim\limits_{\leftarrow} [J_n(X),ΩY]$ for any space $Y$. When $X=\mathbb S^1$, the circle, $J(\mathbb S^1)=ΩΣ\mathbb S^1=Ω\mathbb S^2$. Furthermore, there is a bijection between $[J(\mathbb S^1),ΩY]$ and the product $\prod_{i=2}^\infty π_i(Y)$, as sets. In this paper, we describe the group structure of $[J_n(\mathbb S^1),ΩY]$ by determining the co-multiplication structure on the suspension $ΣJ_n(\mathbb S^1)$.

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