Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

Generalized Gottlieb and Whitehead center groups of space forms

Pages: 323 – 340

DOI: http://dx.doi.org/10.4310/HHA.2019.v21.n1.a15

Authors

Marek Golasiński (Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland)

Thiago de Melo (Institute of Geosciences and Exact Sciences, São Paulo State University, Rio Claro, SP, Brazil)

Abstract

We extend Oprea’s result that the Gottlieb group $G_1(\mathbb{S}^{2n+1}/H)$ is $\mathcal{Z}H$ (the center of $H$) and show that for a map $f : A \to \mathbb{S}^{2n+1}/H$, under some conditions on $A$, we have $G^f_1 (\mathbb{S}^{2n+1} / H)=\mathcal{Z}_H f_{*} (\pi_1(A))$, the centralizer of the image $f_{*} (\pi_1(A))$ in $H$. Then, we compute or estimate the groups $G^f_m (\mathbb{S}^{2n+1}/H)$ and $P^f_m (\mathbb{S}^{2n+1} / H)$ for certain $m \gt 1$.

Keywords

classifying space, Gottlieb group, homology group, homotopy group, Moore–Postnikov tower, $n$-equivalence, projective space, space form, Whitehead center group, Whitehead product.

2010 Mathematics Subject Classification

55Q15, 55Q52, 55R05, 57S17

Full Text (PDF format)

This work was supported by CAPES – Ciência sem Fronteiras, grant 88881.068125/2014-01.

Received 1 June 2018

Received revised 6 September 2018

Published 31 October 2018