Generalized Gottlieb and Whitehead center groups of space forms

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

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

Received 1 June 2018

Received revised 6 September 2018

Accepted 7 September 2018

Published 31 October 2018